Pore Size and Porosity in Packed Bed - Nahrain University

Pore Size and Porosity in Packed Bed Using RRSB Distribution

A Thesis Submitted to the College of Engineering

of Nahrain University in Partial Fulfillment of the Requirements for the Degree of

Master of Science in

Chemical Engineering

by

Suha Mahdi Saleh (BScin Chemical Engineering 2003)

Jamada Al-Awal 1427 June 2006

ABSTRACT

The aim of this research is to study pore size and porosity theoretically and experimentally in order to study the proprieties of packed bed There are different types of distribution

1 Log Normal distribution 2 RRSB-distribution 3 GGS-distribution 4 Rinkes- distribution 5 PM- distribution

RRSB distribution used in this research in order to get number percent used in theoritcal and experimental part

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H

In this research the mean pore diameter and the probability of pore diameter due to ( number length surface area and volume ) was found by using theoretical method suggested by Latif Experimentally packed bed was made by putting five types of spheres diameters (d1=6d2=8d3=10d4=20d5=26 mm) inside glass cylinder D=129 cm consists of sieve has pores of diameters 4 mm in bottom and then three kinds of impurities of diameters ( A=12-2 B=2-236 C=236-3 mm) passed through the packed bed the percent output of impurities and porosity was found in two three and four layers Its noticeable from experimental result that the percent output of impurities and porosity decreased with the increase of number of layers also the percent output of impurities decreased with the increase the size of impurities Experimental results are related with theoretical results in order to find the relation between mean pore diameter and percent output of impurities and porosity for each number percent of beds the increase in size of spheres lead to increase the mean pore diameter Also in this research porosity was found theoretically and experimentally Firstly porosity for mono system was calculated by using

I

Furnas equation and then determined experimentally in each experiment one kind of sphere used these spheres (d1=6d2=8d3=10d4=20d5=26 mm) putting known number of spheres inside cylinder glass D=129 cm and measure the height of the spheres in the cylinder to find the volume of bed in order to find the porosity The same procedure repeated for cylinder D=64 cm It is noticeable from theoretical and experimental results that the porosity increased with increase the diameter of spheres and decreased with increase the diameter of the bed contain the spheres Theoretical results compared with experimental results firstly in cylinder of D=129 cm and found the absolute error range 03 amp 259 Furnas equation modified under consideration the present work

⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε helliphellip (42)

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)

The absolute error in cylinder of D=64 cm lies between 186 amp 416 Furnas equation modified under consideration the present work

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε helliphellip(46)

Porosity for binary system was calculated theoretically by using Jeschar equation and Latif equation Then determined experimentally by putting known number of two kinds of spheres (d1=26d2=6mm) (d1=20d2=6mm)(d1=10d2=6mm)(d1=8d2=6mm)(d1=26d2=8mm)(d1=20d2=8mm)(d1=10d2=8mm)(d1=26d2=10mm)(d1=20d2=10mm)(d1=2620mm) in glass cylinder of D=129 cm and measure the height of the spheres in the cylinder to find the volume of bed in order to find the porosity Theoretical results compared with experimental results and found the absolute error in Jeschar is low equation range 485 to 17 but in Latif equation absolute error is high its range from 0985 to 683 Then Latif equation modified under consideration the present work absolute error found its range 004 to 138

( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε helliphelliphelliphelliphellip(47)

II

CONTENTS Abstract List of Contents Notations Abbreviations List of Tables List of Figures CHAPTER ONE Introduction CHAPTER TWO Literature Survey 21 Porous Media 211 Classification of Pores 211Pore Structure 22 Co ordination Number 23 Filtration 24 Sphere Packing Problem 241 Packing of Sphere 242 Packed Bed 25 The Methods of Determination of Pore Size Distribution 26 Review of Previous Work (pore size) 261 Theoretical Method to Calculate The Pore Size 27 Porosity 271 Review of Previous Work (porosity) 272 Theoretical Method to Calculate The Porosity CHAPTER THREE Experimental Work 31 Material and Tools(pore size) 32 Procedure of Experimental Work(pore size) 33 Material and Tools (porosity) 34 Procedure of Experimental Work(porosity) CHAPTER FOUR Results and Discussion 41 Presentation (pore size) 42 Relationships Between Mean Pore Diameter and Percent Output of Impurities

I III V VII VIII XV 1 3 4 5 5 6 7 7 9 10 11 12 16 16 18 19 21 22 22

24 31 39 45

52 54 56 56

43 Relationships Between Number of layers and Percent output of Impurities 44 Relationships between The size of impurities and Percent Output of Impurities 45 Relationships Between Mean Pore Diameter and Porosity 46 Relationships Between Number of Layers and Porosity 47 Presentations (porosity) 48 Derivation of Anew Empirical Equation for Mono System 49 Relation Between the Porosity and the Diameter of Spheres

III

410 Relation Between The Porosity and The Diameter of Cylinder which Contains The Sphere 411 Derivation of Anew Empirical Equation for Binary System CHAPTER SIX Conclusion and Recommendation 61 Conclusions 62 Recommendation References Appendix[A] A11 For Pore Size calculation A12 For probability calculation A13Calculation of the mean pore diameter A2 For Porosity Calculation Appendix[B] B1Theortical Resluts(Pore Size) B21Theortical Resluts(Porosity for Mono System ) B22Theortical Resluts (Porosity for Binary System) Appendix[C] C1 Experimental Resluts For Pore Size C21 Experimental Resluts (Porosity for mono system)

59 61 63 82 83 85 A-1 A-3 A-5 A-6 B-1 B-22 B-22 C-1 C-13 C-14 D-1 D-4 D-7

C22 Experimental Resluts Porosity for Binary System Appendix[D] Program number one Program Number Two Program Number Three

IV

NOTATIONS

ka Ratio between diameter of small and large particle

ma Ratio between diameter of medium and large particle

gA Area function of large spheres

kA Area function of small spheres

mA Area function of medium spheres A Area (cmsup2) d Particle mean diameter (mm) dc Mean value of pore diameter (mm)

Idc Pore diameter of three equal sizes of diameter (mm)

IIdc Pore diameter of two equal and one different sizes of diameter (mm)

IIIdc Pore diameter of three different sizes of diameter (mm) adcm Mean pore diameter due to area Idcm Mean pore diameter due to length ndcm Mean pore diameter due to number vdcm Mean pore diameter due to volume Large particle diameter (mm) gd

kd Small particle diameter (mm)

md Medium particle diameter (mm) D Diameter of the container (cylinder) (mm)

363d Diameter at =633 3H

3H Cumulative distribution K Single phase permeability (cmsup2)

21toKK Constants

gL Length function of large spheres

kL Length function of small spheres

mL Length function of medium spheres L Height of the sphere in the bed (mm) n Number of repetitions

gn Number of repetitions of large spheres

kn Number of repetitions of small spheres

mn Number of repetitions of medium spheres

gN Number function of large spheres

V

kN Number function of small spheres

mN Number function of medium spheres N Number of the sphere in the bed n P Pressureatm

iP Probability of finding the pores

rP Probability for diameters

raP Probability of diameters due to area

rlP Probability of diameters due to length

rnP Probability of diameters due to number

rvP Probability of diameters due to volume Q Surface tension dynecm r Type of distribution

gr Type of distribution for large particles

kr Type of distribution for small particles

mr Type of distribution for medium particles So Specific surface (cmsup2 cm 3 )

gV Volume function of large spheres (cm ) 3

kV Volume function of small spheres( cm ) 3

mV Volume function of medium spheres (cm 3 ) V Volume of the void (cm 3 )

TV Total volume(cm 3 )

PV Volume of the particle (cm 3 ) ε Porosity οε Average value of porosity between 1ε and 2ε

micro Viscositycp

VI

ABBREVIATIONS

RRSB RosinRammlerSperlingBennett GGS GatesGaudinSchuhmann PM PrerorskaMisek PBR Packed bed reactor PSM Pore size meter LC Liquid chromatography

VII

Tables

Table 31 Diameters Density of Spheres Table 32 Diameters and Weight of Impurities Table 33 Diameters and Number of Spheres Table 41 Calculated Values of Porosity Using Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 129 cm) Table 42 Calculated Values Porosity Using Modified Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 129 cm) Table 43 Calculated Values of Porosity Using Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 64 cm)

Table 44 Calculated Values Porosity Using Modified Furnas Equation for Mono System as Compared to The Porosity Calculated Experimentally with Absolute Error(in Cylinder of Diameter 64 cm)

Table 45 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=6) Table 46 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=6) Table 47 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=10d2=6) Table 48 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error

VIII

19 21 22 56 57 58 58 71 71 72 72

for Binary System(d1=8d2=6) Table 49 Theoretically Porosity as Compared to The for Binary System(d1=26d2=8) Table 410 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=8) Table 411 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=10d2=8) Table 412 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=10) Table 413 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=10) Table 414 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=20) Table 415 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=6) Table 416 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=6) Table 417 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=10d2=6) Table 418 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=8d2=6) Table 419 Porosity Calculated by Using Modified Latif Equation

IX

73 73 74 74 75 75 76 77 77 78 78

as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=8)

79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8

Table 420 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=8) Table 421 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=10d2=8) Table 422 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=10) Table 423 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=10) Table 424 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=20) Table B1 Arrangement of Particles to Determine the Diameter of the Pore Table B2 Results Obtained from Computer Program for

26201086 54321 ===== ddddd Table B3 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B4 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B5 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B6 Results Obtained from Computer Program for 26201086 54321 ===== ddddd

X

Table B7 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B8 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B9 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B10 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B11 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B12 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B13 Results Obtained from Computer Program for 26201086 54321 ===== ddddd Table B14 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 060540201010 54321 NNNNN ===== Table B15 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 06044020150150 54321 NNNNN ===== Table B16 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 0204301502020 54321 NNNNN ===== Table B17 Values of Mean Pore Diameter for

XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19

26201086 54321 ===== ddddd 0203480182020250 54321 NNNNN ===== Table B18 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 0202301502040 54321 NNNNN ===== Table B19 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 1070090060050 54321 NNNNN ===== Table B20 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 0202301180132050 54321 NNNNN ===== Table B21 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 090610140090070 54321 NNNNN ===== Table B22 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 15068009004500350 54321 NNNNN ===== Table B23 Values of Mean Pore Diameter for

26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22

010140150150550 54321 NNNNN ===== Table B24 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 20640090040030 54321 NNNNN ===== Table B25 Values of Mean Pore Diameter for 26201086 54321 ===== ddddd 01011500750150650 54321 NNNNN ===== Table B26 Porosity Calculated for Mono System by Using Furnas Equation Table B27 Calculated Values of Porosity for Binary System d1=26 d2=6 mm

XII

Table B28 Calculated Values of Porosity for Binary System d1=20 d2=6 mm Table B29 Calculated Values of Porosity for Binary System d1=10 d2=6 mm Table B30 Calculated Values of Porosity for Binary System

d1=8 d2=6 mm

B-23 B-23 B-24 B-24 B-25 B-25 B-26 B-26 B-27 C-1 C-2 C-3 C-4 C-5 C-6 C-7 C-8

Table B31 Calculated Values of Porosity for Binary System d1=26 d2=8 mm Table B32 Calculated Values of Porosity for Binary System d1=20 d2=8 mm Table B33 Calculated Values of Porosity for Binary System d1=10 d2=8 mm

Table B34 Calculated Values of Porosity for Binary System d1=26 d2=10 mm

Table B35 Calculated Values of Porosity for Binary System d1=20 d2=10 mm Table B36 Calculated Values of Porosity for Binary System d1=26 d2=20 mm Table C1 Results of Exp 1 Table C2 Results of Exp 2 Table C3 Results of Exp 3 Table C4 Results of Exp 4 Table C5 Results of Exp 5 Table C6 Results of Exp 6 Table C7 Results of Exp 7 Table C8 Results of Exp 8

XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18

Table C9 Results of Exp 9 Table C10 Results of Exp 10 Table C11 Results of Exp 11 Table C12 Results of Exp 12 Table C13 Height of Sphere in The Cylinders of Diameter 129 cm and 64 cm Table C14 Porosity in The Cylinders of Diameters 129 cm and 64cm Table C15 Experimentally Porosity for Binary System d1=26d2=6 mm Table C16 Experimentally Porosity for Binary System d1=20d2=6 mm Table C17 Experimentally Porosity for Binary System d1=10d2=6 mm Table C18 Experimentally Porosity for Binary System d1=8d2=6 mm Table C19 Experimentally Porosity for Binary System d1=26d2=8 mm Table C20 Experimentally Porosity for Binary System d1=20d2=8 mm Table C21 Experimentally Porosity for Binary System d1=10d2=8 mm Table C22 Experimentally Porosity for Binary System d1=26d2=10 mm Table C23 Experimentally Porosity for Binary System d1=20d2=10 mm Table C24 Experimentally Porosity for Binary System d1=26d2=20 mm

XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28

Figure 21 General Case Figure 22 A Pore by Two Different Sphere Diameters gt gg dd = kd Figure 22 B Pore by Two Different Sphere Diameters lt gg dd = kd Figure 23 Three Equal Sphere Diameters Figure 31 Main System of Experiment Work Figure 41 Pore Size Distribution Due to Number

Length Area and Volume for 060540201010 54321 NNNNN =====

Figure 42 Pore Size Distribution Due to Number












XV

Figure 48 Pore Size Distribution Due to Number Length Area and Volume

for 090610140090070 54321 NNNNN ===== Figure 49 Pore Size Distribution Due to Number








Figure 413 Mean Pore Diameter Due to Number Vs Percent Output of Impurities for Two Layers Figure 414 Mean Pore Diameter Due to Number Vs Percent Output of Impurities for Three Layers Figure 415 Mean Pore Diameter Due to Number Vs Percent Output of Impurities for Four Layers Figure 416 Mean Pore Diameter Due to Length Vs Percent Output of Impurities for Two Layers Figure 417 Mean Pore Diameter Due to Length Vs Percent Out of Impurities for Three Layers Figure 418 Mean Pore Diameter Due to Length Vs Percent Out of Impurities for Four Layers Figure 419 Mean Pore Diameter Due to Area Vs Percent

29 29

30 30 31 33 33 34 34 35 35

36

Output of Impurities for Two Layers

XVI

Figure 420 Mean Pore Diameter Due to Area Vs Percent Output of Impurities for Three Layers Figure 421 Mean Pore Diameter Due to Area Vs Percent Output of Impurities for Four Layers Figure 422 Mean Pore Diameter Due to Volume Vs Percent Output of Impurities for Two Layers Figure 423 Mean Pore Diameter Due to Volume Vs Percent Output of Impurities for Three Layers Figure 424 Mean Pore Diameter Due to Volume Vs Percent Output of Impurities for Four Layers Figure 425 Number of Layers Vs Percent Output of Impurities For 060540201010 54321 NNNNN ===== Figure 426 Number of Layers Vs Percent Output of Impurities For 06044020150150 54321 NNNNN ===== Figure 427 Number of Layers Vs Percent Output of Impurities For 0204301502020 54321 NNNNN ===== Figure 428 Number of Layers Vs Percent Output of Impurities For 0203480182020250 54321 NNNNN ===== Figure 429 Number of Layers Vs Percent Output of Impurities For 0202301502040 54321 NNNNN ===== Figure 430 Number of Layers Vs Percent Output of Impurities For 1070090060050 54321 NNNNN ===== Figure 431 Number of Layers Vs Percent Output of Impurities For 0202301180132050 54321 NNNNN ===== Figure 432 Number of Layers Vs Percent Output of Impurities For 090610140090070 54321 NNNNN ===== Figure 433 Number of Layers Vs Percent Output of Impurities For 15068009004500350 54321 NNNNN =====

36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII

Figure 434 Number of Layers Vs Percent Output of Impurities For 010140150150550 54321 NNNNN ===== Figure 435 Number of Layers Vs Percent Output of Impurities For 20640090040030 54321 NNNNN ===== Figure 436 Number of Layers Vs Percent Output of Impurities For 01011500750150650 54321 NNNNN ===== Figure 437 Size of Impurities Vs Percent Output of Impurities For 060540201010 54321 NNNNN ===== Figure 438 Size of Impurities Vs Percent Output of Impurities For 06044020150150 54321 NNNNN ===== Figure 439 Size of Impurities Vs Percent Output of Impurities For 0204301502020 54321 NNNNN ===== Figure 440 Size of Impurities Vs Percent Output of Impurities For 0203480182020250 54321 NNNNN ===== Figure 441 Size of Impurities Vs Percent Output of Impurities For 0202301502040 54321 NNNNN ===== Figure 442 Size of Impurities Vs Percent Output of Impurities For 1070090060050 54321 NNNNN ===== Figure 443 Size of Impurities Vs Percent Output of Impurities For 0202301180132050 54321 NNNNN ===== Figure 444 Size of Impurities Vs Percent Output of Impurities For 090610140090070 54321 NNNNN ===== Figure 445 Size of Impurities Vs Percent Output of Impurities For 15068009004500350 54321 NNNNN ===== Figure 446 Size of Impurities Vs Percent Output of Impurities

44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII

Figure 447 Size of Impurities Vs Percent Output of Impurities For 20640090040030 54321 NNNNN ===== Figure 448 Size of Impurities Vs Percent Output of Impurities For 01011500750150650 54321 NNNNN ===== Figure 449 Mean Pore Diameter Due to Number Vs Porosity Figure 450 Mean Pore Diameter Due to Length Vs Porosity Figure 451 Mean Pore diameter Due to Area Vs Porosity Figure 452 Mean Pore Diameter Due to Volume Vs Porosity Figure 453 Numbers of Layers Vs Porosity Figure 454 Ratio of dD vs Porosity in The Cylinder of Diameter 129 cm Figure 455 Ratio of dD vs Porosity in The Cylinder of Diameter 64 cm Figure 456 Diameter of Sphere Vs Porosity Determined Experimentally with Cylinder Diameter (64129) Respectively Figure 457 Diameter of Sphere Vs Porosity Calculated Theoretically with Cylinder Diameter (64129) Respectively Figure 458 Porosity Calculated in Jeschar Equation Vs Weight Percent of Spheres of d2 for Binary System Figure 459 Porosity Calculated in Latif Equation Vs Weight Percent of Spheres of d2 for Binary System Figure 460 Experimental Porosity for Binary System ( =26mm =6 mm) Vs Volume Fraction 1d 2d Figure 461 Experimental Porosity for Binary System

51 51 52 53 53 54 55 60

60 62 62

64 65 66 66

( =20mm =6 mm) Vs Volume Fraction 1d 2d

XIX

Figure 462 Experimental Porosity for Binary System ( =10mm =6 mm) Vs Volume Fraction 1d 2d Figure 463 Experimental Porosity for Binary System ( =8mm =6 mm) Vs Volume Fraction 1d 2d Figure 464 Experimental Porosity for Binary System ( =26mm =8 mm) Vs Volume Fraction 1d 2d Figure 465 Experimental Porosity for Binary System ( =20mm =8 mm) Vs Volume Fraction 1d 2d Figure 466 Experimental Porosity for Binary System ( =10mm =8 mm) Vs Volume Fraction 1d 2d Figure 467 Experimental Porosity for Binary System ( =26mm =10 mm) Vs Volume Fraction 1d 2d Figure 468 Experimental Porosity for Binary System ( =20mm =10 mm) Vs Volume Fraction 1d 2d Figure 469 Experimental Porosity for Binary System

67 67

68 68 69 69 70 70 ( =26mm =20 mm) Vs Volume Fraction 1d 2d

XX

CHAPTER ONE INTRODUCTION The study of pore size and porosity is necessary to study the packing The packing of particles is of great interest in a variety of scientific technological areas These includes for example catalytic reactors packed absorption distillation towers packed filter ceramics concrete powder metallurgy tabling and others(1234) The work with catalysts ceramics pharmaceuticals carbons building materials or with any other porous or divided solid needs quantitative information about pore size (5) Pore size parameters may be used to predict with confidence the in service performance of these construction materials Illustrations of progress in this direction have been presented for hydrated Portland cement concrete aggregates compacted fine grand soils and embankment shales(6) The unsaturated hydraulic conductivity of arouse sediment varies with capillary pressure which in turn depends on the complex arrangement of pore within the sediment Present theories indicate that un saturated conductivity is systematically related to the distribution of pore size(7) The need to improve the quality of mathematical simulations in such diverse fields as per collation diffusion and chemical reaction in catalysts requires the inclusion of statistic description of pore size distribution (8) Treating pores of various diameters as uniform in size eg average pore diameter (pore size distribution)tends to understanding the plugging effect of small pores by enzyme molecules during immobilization process especially for porous solids with broad or bimodal pore size distribution(9) Pore size plays dominate role in transport phenomena eg a number of seemingly comparable phenomena concepts are discussed drainage -imbibitions suction mercury porosimestry molecular diffusion viscous flow capillary potential(10)

Among the more promising uses of pore size distribution data is the explanation and ultimately the prediction of water permeability of compacted soils (11)

1

In addition to the pore size porosity is one of fundamental properties of abed of particulate materials Indication of porosity value have wide significance in a number of industrial applications such as 1-Powder metallurgy 2-Concrete production 3-Ceramics 4-Packing and transport The aim of this research is - 1-Description of pores using spherical particles 2-Calculate the probability of the pore diameter due to different particle size distribution (number length area and volume) in different number of packing layers 3-Weight percent of output impurities in two three and four layer 4-Determine the porosity in two three and four layer 5-Calculate porosity for mono system sphere by using Furnas equation and also determined experimentally then derive empirical equation for mono system 6-Calculate porosity for binary system by using Jeschar equation and Latif equation and also determined experimentally then derive empirical equation for binary system The above steps is necessary in many of the industrial process need to calculate the pore size and porosity in packing in order to purify materials from impurities Sedimentation process and separation of one material from another require the calculation of the particles diameters which pass through the pores that are formed in the packing particle

2

CHAPTER TWO

LITERATURE SURVEY

21 Porous Media A analysis of pore structure and pore radius distribution is necessary in order to construct on effective model for a porous medium (12) The following things are examples of porous media-glass spheres packed sand spheres crystalline calcium fluoride towers packed (13) with pebbles beryl saddles crashing rings granules load shot porous rocks such as limestone pumice dolomite etc fibrous aggregates such as cloth felt filter paper and finally catalytic particles(14) The models of a porous medium consist of lattice networks of cylindrical channels with distribution of radii and also of randomly packed rotund particles examples spheres as described by Iczkowski Mason and Haynes(15) Modern theory on laminar flow of fluids in porous media dates from the research of Darcy in 1958 he study the relation ship between parameters involved and designed the motion of permeability as structural parameter

AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)

Another land mark was Koseny-Carman equation 1937 which gives the

relative permeability ( ) 2

3

1 SohK

k εεminus

= (22) modern theory

on laminar flow of fluids (16) The significant properties of porous media is 1-Porosity which is a measure of the pore space and hence of the fluid capacity of the medium 2-Permeability which is a measure of ease with which fluids may traverse the medium under the influence of a driving pressure (17)

3

Relative permeability characteristics of a porous medium depend on the structural configuration of the void space in which flow take place There are different approaches have been developed for studying flow and displacement processes in disordered porous media these approaches are (Perrine 1961Heller1966Whitaker 1967Slattery 1967Peters 1984 Chang and Slatttery 1986)(18) 211 Classification of Pores Intuitively ldquoporerdquo are void spaces which must be distributed more less frequently through the material if the latter is to be called ldquoporousrdquo void is a space between particles extremely small voids in a solid are called ldquomolecular intersticesrdquo very large ones are called ldquocaverns porerdquo are void space intermediate between caverns and molecular interstices the limitation of their size is there fore intuitive and rather indefinite (19) A porous solid solid with cavities or channels which are deeper than they are wideconsists of solid matrix and pores The pore may be further classified as open pores closed end pores and isolated pores The classification is important for example only pores of the fist two types would permit the diffusion of a gas within the pores of the solid (20) Pore classified in to open pore cavity or channel with access to the surface and closed bed cavity not connected to the surface blind pore or dead end pore pore with a single connection to the surface and inter connected pore pore which communicates with other poresPore which are open at both sides of a membrane or porous plug are termed through pores The classification of pores according to size has been under discussion for many years but in the past the terms micro pore pore of internal width greater than 2 mmMacro pore pore of internal width greater than 50 mmhas been applied in different ways by physical and chemists and some other scientists in an attempt to clarify this situation Mesoporespore of internal width between 2mm and 50 mmare especially important in the context of adsorption Pore size is generally specified as the pore width has a precise meaning when geometrical shape is well defined nevertheless for most purposes the

4

limiting size is that of the smallest dimension and this is generally taken to represent the effective pore size(21) 212 Pore Structure Pore structure is modeled by using information about the pore size distribution that determines the capillary pressure The pore structure of un consolidated media as well as natural sandstones can in large be characterized in terms of pore size shapes distribution and topology (12) Pore structure implicitly presents in most of the proposed mechanisms of liquid movement in solid A summary of these mechanisms as a quoted by Peck and Wasan in 1974 is

1- Liquid diffusion due to difference in moisture concentrations 2- Liquid movement due to capillary pressure forces 3- Vapour diffusion in partly air ndashfilled pores due to differences in partial

pressure 4- Liquid or vapour flow due to differences in total pressure generated by

external pressure capillarity shrinkage or high temperature inside the moist materials

5- Liquid movement due gravity (22)

22 Co ordination Number The contacts between spheres are fundamental feature of packing which condition the mechanical stability and transport properties in the solid phase (23) Co ordination number is the number of contact points on a particle in multi component packed bed with size distribution had been obtained by Arakawa and Nishino and Powell from experiment or computer simulation but these data obtained for only or two particular types of size distribution and did not compare with the model or theoretical equations The computer simulation were developed for three components randomly Packed bed And studied the relation between co ordination numbers and size distribution these are log-normal log uniform rosin

5

Rammler and Andreasen The reason of using computer program simulation because it is difficult to measure the co ordination number in packed bed with size distribution experimentally (24) The average co ordination number is close to 6 whatever the size distribution because the stability criterion and precisely to the three contact position each new practice creates 6 contacts 3 for itself and 1 for each of the 3 supporting particle so expect for the top and bottom particles which are not analyzed(25) Co ordination number depends on the definition of a contact for which three cases are usually identified) (23) 1- Contact co ordination when sphere really touch one another 2- Close co ordination when spheres are with in a short distance of one

another say 10 of inters center distance 3- Neighbour hood co ordination when spheres are separated by about

2 (r1+r2)

Kissing number the number of equivalent hyper spheres in n dimensions which can touch an equivalent hyper sphere with out any intersections also sometimes called the Newton number contact number co ordination number Eric Weisstein correctly believed that the kissing number in three dimensions was 12 but the first proofs were not produced until the 19th century Conway and Sloane 1993 p21by Bender 1874 Hoppe 1874 and Gunther Leech 1956 after packing 12 spheres around the central one and the free space not enough to fit a 13th sphere(26) Many scientist have investigated the co ordination number of the packing of the spheres such as smith and Foote 1929Bennete and brown 1940 Gary 1959 Scott 1962 Clark 1965 Ridgeway and Tarbuk 1967 and Taradel and Bidean 1986 they used different techniques in their investigations to obtain the co ordination number(27)(28) 23 Filtration

Filtration is the physical or mechanical process of cleansing fluid where by particles in a fluid are captured retained or trapped during the passage of the fluid through porous media (29)

6

The size of the openings in a filter or pore size determines what particles get trapped and which ones slip through (30) Pore size rating is the pore size of the filter determined by the diameter of the particle that it can be expected to retain with a defined high degree of efficiency (31) Filtration efficiency and pressure drop are key properties depending on a pore size distribution (32) 24 Sphere Packing Problem The classical sphere packing problem is to find out how densely a large number of identical sphere can be packed together We know that the sphere do not fit together there is always some wasted space in between(33) Cracking Keplers solved sphere packing when he noticed that the familiars piles of neatly stacked oranges at supermarket represent a practical solution to this problem When super market personal stack oranges the bottom layer consists of rows that are staged by a half an orange placing oranges in the hollows formed by three adjacent oranges In the first produces the second layer and so on such as arrangement is known as face ndashcentered cubic packing(34) The assertion has long remained with rigorous proof but in august 1998 proof Thomas hales of the university of Michigan announced a computer based solution this proof is contained in over 250 manuscript pages and relies on over 3 gigabytes of computer files and so it will be some time before it has been rigorously by the scientific community to ensure that the Kepler conjecture is indeed proven (35) 241 Packing of Sphere The first study of spheres packing aimed to describe the structure of liquids and dealt with equal spheres Then the packing of spheres was used to analyze the properties of granular materials eg from the view point of percolation theory the present work was carried out to study the liquid ndashphase sintering processes by using computer method for random packing of spheres of unequal size this packing is built under gravity particle by particle (25)

7

An important property of multi particle systems is the packing density this is defined as the volume fraction of system occupied by solids (36) The important aspect is the structure of packing we distinguish between ordered structures Also called a periodic or lattice arrangement is one in which the centers of the sphere form very symmetric pattern called lattice In two dimensional Euclidean space German mathematician Carl Friedrich gauss proved that the regular arrangement of circle with the highest density is the hexagonal packing arrangement the density of this arrangement is

12143 =09069

Disordered structure (irregular or random packing of spheres) packing of differently sized Disordered structures are more important in practice but the order structure are more interesting for understanding of fundamental mechanisms because they allow theoretical calculation Arrangements in which the sphere are not arranged in lattice are called irregular or periodic arrangements Regular arrangement are easier to handle than irregular ones their high degree of symmetry makes it easier to classify them and to measure their densities(37) Random close packing of spheres in three dimensions gives packing densities in the rang 006 to 065(Jaeger and Nagel 1992Torquato etal2000)Hilbert and Cohn-Vossen(1999pp48-50) consider a tetrahedral lattice packing in which each sphere touch four neighbors and density is

163143 =03401 and cubic lattice

6143 =05236 and hexagonal lattice

33143 =06046(7)in three dimensional space gauss proved that regular

arrangements of sphere called cubic close packing (or face centered cubic ) and hexagonal close packing in both of this arrangement surrounded by 12 sphere and the packing density

18143 =074048 (71)For packing in three

dimensions CARogers (1958) showed that the maximum possible packing density lt 18(cos^-1

31 -

31 314)=77963557 (le Lionnais 1983) and this

result was subsequently improved to 77844 (Lindsey 1986) then 77836(Muder 1988) ( 38)

8

Natural materials involve particle may depart quite considerably from true spherical shape and more over are from uniform in size Non uniformity in size in general permits smaller particles to fill the pores between the larger particles thus resulting in a material reduction in porosity Angularity of the grains tends to produce bridging which results in random packing and higher porosities (17) It is only reasonable there fore to expect that spherical models as discussed before will not be entirely adequate to represent even the geometrical properties of porous media (not to speak of hydraulic ones they will help us to under stand some of the features better but actual correlation between grain sizes and pore sizes as have been proposed is based on experimental investigations rather than on theory atrue understanding of the effects of course can not be obtained in this manner (39) Theoretical and experimental studies of the effect of grain size on pore size have been conducted by Tickell Mechem and Mccurdy(1933)Hrubisek(1941) Rosen Feld (1949)Griffiths (1952) and Gaithe (1953) Kiesskatt and Matz (1951) and Wise (1954) discussed the dependence on the grain distribution (40) 242 Packed Bed When sphere are taken together as a packed bed there is no way of knowing from first principles such important properties as mechanical strength of packing its thermal conductivity and its permeability to fluids these properties depend on the way the particle fit together to form a continuous solid structure and corresponding void structure (24) Chemical engineers know this as a packed or porous bed ground water hydrologists call this flow through porous media They provide two methods the Idelchik and Darcyrsquos law The Idelchik method is valid for laminar or turbulent flow through the bed while Darcyrsquos law is valid only for laminar flow (Darcy Reynolds number lt10 )another difference is that only Darcyrsquos law requires entering the permeability (41) Liquid chromatography makes use of a packed bed of spherical particles in order to separate a mixture of un known compounds (42)It separate according to the pore size Only molecules that are smaller than the pore size can enter the pores In a packed bed adsorption very large pore size or macro porous adsorbents can provide high efficiency (43)

9

The effect of pore size in packed bed reactor (PBR) PBR having the largest pore size benefited from the recirculation (44) 25 The Methods of Determination of Pore Size Distribution The ratio of pore volume to specific surface area Vpa has been used for many years as a simple means of characterizing the pore size (21) The conventional method of determining pore size in the nitrogen adsorption method from plotting of nitrogen adsorption data versus the statistical film thickness from slopes the surface area calculated and from intercept obtain the volume of meso pores from surface area and volume the meso pore diameter is calculated Powder x-ray diffraction measurements determine pore size (45) Pore size meter series PSM enables to measure pore sizes of porous materials such as ceramical filter and membranes Permeability weighted pore size distribution calculated from wet flow curve (pressure drop vs volumetric flow rate of the wet sample )and dry flow curve (similar to wet flow but obtained on dry sample)(32) Pore size distribution in porous alumina bodies determined experimentally by the capsule free hot isostatic pressing technique (46) Capillary pressure curves give pore size distribution in the packing (16) Pore size of construction materials measured by mercury intrusion porosimetry(47) The pore size distribution of activated carbons in the range of openings below 13 mm has been determined by liquid chromatography (LC) (48)

10

26 Review of Previous Work (Pore Size) For spherical packing pore size is conveniently measured by pore diameter so that whenever we say pores size means diameter This pore size or pore diameter represents the diameter of the particle which may be able to pass through in the pore Pore size measurements have been made on a number of Indiana shales MKaneuji DNWinslow and WL Dolch 1980 and MSurendra 1980(4950) Filter aids which will represent the pore diameter and impurities have been represented as spherical particles by Latif (51) Fawaz calculated the pore size for three spheres of different size using theoretical method suggested by latif and experimentally he found the percentage output of impurities which passed through the packed bed He found the relation between mean pore diameter and percentage out of impurirties And the relation between mean pore diameter and packed bed composition in two methods triangle methods and in X-Y plane without triagle method (52)Omar pore size for binary system using theoretical method suggested by latif and experimentally he found the percentage output of impurities which passed through the packed bed The weight percent plotted vsimpurities diameter and the interceptions represent the mean pore diameter(53) AL-Obaidi calculated the pore size for binary spheres system using theoretical method suggested by Latif 1981 and compared it with experimental results obtained by Kreutz(54)

11

261 Theoretical Method to Calculate The Pore Size In this work we use the same procedure suggested by latif 1981 to find the pore size and the mean pore size of filter aid consisting of spheres and impurities as small spherical particles so the calculation procedure is as follows i The diameter of the ( ) which may be able to pass through the pore as shown in fig 23 which represents the general case

dcIII

( )

g

ciii dkkkd timesminusminus

=2

421

5244 helliphellip (23)

Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=

( ) 1324 kk2kk times+=

1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)

ii The diameter ( )as shown in fig 24a24b may be calculated by equation 22a and 22b for case (a)and(b)respectively

dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)

Equation (22a) is used if pore by two different sphere diameters dg=dggtdk

12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)

Equation (2-2b)is used if pore by two different sphere diameters dk=dk ltdg

Where dgdkak =

iiiThe diameter ( ) can be calculated by the equation written below when dc=dm=dg as in fig25

dCI

gci d1550d times= hellip(25) To calculate the probability that the diameter dc will occur is

nkdk

nmdm

ngdg

kmgi PrPrPr

nnn3P timestimes= helliphellip(26)

Where 3=++ kmg rrr Equation (26) is the general equation to calculate the probability due to number volume length and surface area so we must have the precent due to one of it and depend on it to find the probability due to other In this research we depend on the number percent to find the other as follows 1 To calculate the probability due to length We have the number percent of each type of diameters (d1 d2 d3 d4 and d5) and to find the percent of length as below

( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=

times+times+times+times+times= 5544332211

Where N number percent

13

dd mk and the choice diameters dgThen we sub and in eq 26 to find the probability due to length mk LL gL 2 To calculate the probability due to area

( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=

times+times+times+times+times=

Then we sub and in eq 26 to find the probability due to area mk AA gA

3 To find the probability due to volume

( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=


Then we sub and Vg in eq 26 to find the probability due to volume mk VV

sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity

The porosity is the proportion of the non-solid volume to the total volume of material and is defined by the ratio

TVV

=ε (28)

pT VVV minus=

Where is the non-solid volume (V pores and liquid) and is the total volume of material including the solid and non-solid parts (55)

TV

271 Review of Previous Work (Porosity) The packing of particulate solid materials enters into most aspects of applied sciences and has there fore been studied more or less continuously for a number of years The first study of the modes of packing of spheres and the porosity calculated therefore appears to have been under taken by sticker in 1899 science then the theory has been reviewed refined and extended by Smith Foote and Busang in 1929Gartor and Fraser in 1935Manegold in 1937Manegold and Solf in 1939Hrubisek in 1941and others(56575859) Several investigators (60 61 62 63 64 65 66) have measured the radial porosity profiles in abed packed with spheres and in abed packed with cylinders there results were found to be similar showing fluctuation of the porosity from the wall to the center Ouchiyama and Tanka (67) presented mathematical model for porosity estimation from particle size distribution Standish and YU (68) utilized the simplex centroid design proposed by Scheffe to get mathematical model for the porosity of ternary spheres system using regression methods

16

Latif(51) has farther more contributed to the prediction of porosity using experimental and statistical methods He developed empirical equation based on discontinuous size distribution Rasheed(69) experimented the porosity of ternary spheres system using discontinuous size distribution He suggested new predictive equation and compared the resulted porosity with theoretical values calculated according to empirical equations proposed by Ouchiyama(67) Using regression method AL-Azzawi(70) proposed mathematical model to find the porosity of quaternary spheres mixtures from knowledge of the weight proportions of component AL-Rammahi (71) produced a polynomial relation ship to calculate the porosity values from the diameter ratio particle diameter bed diameter using fitting equations for curves obtained experimentally for spheres mixture obeying RRSB size distribution

17

272 Theoretical Method to Calculate The Porosity The porosity of mono system may be calculated by Furnas equation (72) which takes care of the effect of the container wallFurnas used in his work concrete mixture

Dd 3403750 +=ε (29)

The porosity for binary system may be calculated by using two equation 1 Porosity for binary system may be calculated by using Jescher equation Jescher used in his work sphere particles(56)

helliphelliphelliphelliphelliphellip(210)

2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

2Also may be calculated by Latif equation Latif used in his work binary system of sphere particles the percentage weight of spheres from 0 to 25(51)

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18

CHAPTER THREE Experimental Work

31 Material and Tools(pore size) Spherical glass particles of five different sizes were used to achieve a packed bed The density of these spheres was calculated from weight and volume of spheres and the result was listed in the table 31

Table 31 Diameters Density of Spheres

No Diameter of spheres(mm)

density (gmcm ) 3

1 6 plusmn 2715 2 8 plusmn 2489 3 10 plusmn 2534 4 20 plusmn 2489 5 26 plusmn 2333

The tools used are listed below

1 Graduated glass cylinder (fig 31) with height of 365cm diameter of 129 cm and this cylinder is opened from the top and bottom

2 A sieve is used to retain the spheres that the packed bed consists of it is

connected to the cylinder by a resin material mixed with black cascade maker The parts are shown in figure 31

3 A digital balance (for accuracy of 01) was used to weight the mass of the

input and output 4 Venire with an accuracy of about 002 mm

5 Blower of velocity (13000 rmin) to introduce air after putting the sizes

of spheres and the impurities 6 A small sizes of spheres was used to represented the impurities and these

impurities are listed with their weight in table 32-

19

Figure 31 Main System of Experiment Work

20

Table 32 Diameters and Weight of Impurities

Sample of Impurities

Diameter of Impurities(mm)

Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457

32 Procedure of Experimental Work In each experiment the following steps were taken- 1The number percent of the five sizes of spheres was chosen to make the packed bed 2The number of layer was chosen and at the beginning make two layer were chosen 3Acertian number of each size that had the same number percent which was chosen in step 1 are taken to make the first layerand then put these inside the cylinder 4 The height of packed bed of two layers was measured and calculated the porosity from equation (28) 5 After that our configuration was completed and the system is ready for starting the experiment A certain weight is taken of each sample of impurities starting with sample A by entering the sample from the top of the cylinder The blower of (13000 rmin) was used to enter air from the top of cylinder for 3 min until the system reaches the steady state (here the steady state reaches when all the spheres all the spheres stop move)The resuts of the experiments listed in appendex [C] from table C1 to C12 6 Taking the out put of impurities (from the bottom of the cylinder) and weights it by the digital balance Notice all the above steps must be repeated for other size of impurities (B+C) Then all these steps must be repeated for three and four layers

21

Experimental Work (Porosity)

32 Material and Tools 1 Graduated cylinder with diameter of 129cm 2 Graduated cylinder with diameter of 64cm 3 Five groups of different size spheres the diameter and number of each group was listed in the table 33

Table 3 3 Diameters and Number of Spheres

No Diameter of sphere(mm)

Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70

45 Procedure of Experimental Work A-For Mono System 1The spheres of diameters 6mm were packed in the cylinder of diameter 129 cm and height (L)of these spheres in the cylinder was measured and listed in appendix [C] table C13 then the porosity was calculated from equation (28) and the value of porosity was listed in appendix[C] table C14 2 The same procedure above was repeated by using graduated cylinder of diameter 64 cm 3 All steps above must be repeated for spheres 8 10 20 and 26 mm diameter

22

B-For Binary System 1 Packed bed was made by choosing the weight percent of the two sizes of spheres Binary system of spheres of diameters (d1=26mmd2=6mm) were packed in the cylinder of diameter 129 cm and height (L)of these spheres in the cylinder was measured The porosity was calculated from equation (28) The value of the hight measured and the porosity was listed in appendix [C] table C15 2 The same procedure above was repeated by using another binary system of spheres(d1=20d2=6mm)(d1=10d2=6mm)(d1=8d2=6mm)(d1=26d2=8mm)(d1=20d2=8mm)(d1=10d2=8mm)(d1=26d2=10mm)(d1=20d2=10mm)(d1=2620mm) and the results were listed in appendix[C] tables C16 to C24

23

CHAPTER FOUR Results and Discussion

41 Presentation (pore size) In appendix [B] the principles of theoretical work for finding the pores sizes of the packed bed the probabilities of finding those pores and the mean pore diameters for each type distribution (due to number length area and volume)All these principles are given with detailed explanation for and with theoretical results are listed in tables B1 to B25 and figures shown from 41 to 412 In addition to theoretical work principles and their results In chapter three experiments were done for several packed beds with different compositions in each run and this difference due to variation in composition of small medium and large spheres of which the filter aid or packed is consisted of In each run the experiment was repeated with three sizes of impurities that were added in each experiment and the same experiments and size of impurities were repeated for two three and four layers and the results were given in tables C1 to C12 The results of experimental and theoretical works will discuss in this chapter These results are related to each other in away that we can study the properties of filter aid or packed bed Also reached suitable relations about the mean pore diameter and the weight percent of output impurities Also suitable relation between the number of layer and percent output of impurities must be found relation between the size of impurities and percent output of impurities Next relation between the number of layers and porosity was made Then relations between porosity and mean pore diameter would be down

24

Note All diameters above are in mm

Note that All the below figures are for

26d20d10d8dmm6d 54321 =====

000 100 200 300 400 500Diameter of the pore (dc) (mm)

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume

Figure 41 The Pore Size Distribution Due to Number Length Area and Volume for 060N540N20N10N10N 54321 =====

25

Figure 42 Pore Size Distribution Due to Number Length Area and Volume for 060N440N20N150N150N 54321 =====

000 100 200 300 400 500DIameter of the pore (dc) (mm)

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 44 Pore Size Distribution Due to Number Length Area and Volume for 20N250N 54321 = 020N3480N1820N= = ==

Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 45 Pore Size Distribution Due to Number Length Area and 020N230N150N20N40N 54321 =====

27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume

Figure 46 Pore Size Distribution Due to Number Length Area and


Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Fi e Size Distribution Due to Number Length Area and Vgure 47 Porolume for 020N230N1180N1320N50N 54321 =====

28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume

Figure 48 Pore Size Distribution Due to Number Length Area and Volume fo


r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume

000 100 200 300 400 500Diamet r of the pore (dc) (mm)

Figure 49 Pore Size Distribution Due to Number Length Area and or 150N680N090N0450N0350N 54321 =====

29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f

0 Pore Size Distribution Due to Number Length Area and or 010N140N150N150N550N 54321 =====

000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f

1 Pore Size Distribution Due to Number Length Area and

or 20N640N090N040N030N 54321 =====

30

100

Volume for Figure 412 Pore Size Distribution Due to Number Length Area and

010N1150N0750N150N650N 54321 =====

purities

e 6 8 10 20 and 26 mm respectively

42 Relationships Between Mean Pore Diameter and Percent Output of Im The constant parameters in this work are the diameters of small medium and large sphere that any packed bed made from five sizes of spheres these diameters ar Mean diameter of pores due to number is going to be related with the three types of impurities that will be passed from the packed bed and these types of impurities are 1 Spheres of 12-2 mm 2 Spheres of 2-236 mm 3 Spheres of 236-315 mm

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31

The above mixtures of spheres (impurities) are the same mixtures at are used in experimental work It is necessary to find out that these ixtures are used for the experiments of the length area and volume

istributions So they are constant even if the type of distribution is

1=6d2=8d3=10d4=20d5=26 mm

thmdchanged and the percent output of them is variable if the percents number length area or volumes for the bed are changed The curves in figures 413 to 415 show proportionality between the mean pore diameters due to number and the percent output It is clear from the figures they are proportional to each other(ie when the

umber of large spheres in any packed is increased so the weight of fine nspheres (impurities)that leave the packed bed will be increased)This is the principle of increasing the mean pore size for the bed The principle of reducing the mean pore diameter is done by increasing the number of small or medium spheres in the packed bed For the mean pore diameter due to length the relations are made by relating this mean pore diameter with the same values of percent output of impurities that are obtained from experimental part The difference etween the case of length percent and number percent is in the shape of b

the curve As shown in the figures 416 to 418 the results of the mean pore diameters(due to length ( ldcm ) that are related with percent output) are acceptable compared with rest type of distribution The same relations were made for mean pore diameter due to area ( adcm ) and volume ( vdcm )Figures 419 to 424 show the curves which relate the mean pore diameter due to area and volume with weight percent output for each type of impurities that are used in the experiment in this work The value of mean pore diameter are taken from appendix[B] from tables B14 to B24 Note that all the below figures are for diameters- d

32

100 150 200 250 300Mean pore diameter due to number (dcm) (mm)

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 413 Mean Pore Diameter Due to Number Vs Percent



000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm


Output of Impurities for Three Layers

33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


Output of Impurities for Four Layers

100 150 200 250 300 350Mean pore diameter due to length (dcm) (mm)

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 416 Mean Pore Diameter Due to Length Vs Percent Output of Impurities for Two Layers

34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 417 Mean Pore Diameter Due to Length Vs Percent Out

of Impurities for Three Layers


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 418 Mean Pore Diameter Due to Length Vs Percent

Out of Impurities for Four Layers

35

200 250 300 350Mean pore diameter due to area (dcm) (mm)

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 419 Mean Pore Diameter Due to Area Vs Percent



2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



250 300 350Mean pore diameter due to volume (dcm) (mm)

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 422 Mean Pore Diameter Due to Volume Vs Percent


37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38

43 Relationships Between Number of Layers and Percent Output of Impurities From the previous it is known that the same kinds of sizes of impurities were used in the experiments in case of two three and four layers The figures 425 to 436 show the proportionality between the number of layer and the percent output of impurities It is obvious from the three curves of 12-2 2-236 and 236-315 mm respectively that when this kind of impurities are passed from the packed bed in case of two layers a large weight of output of impurities that leave the packed bed appeared and this weight decreased in the case of three layers and a noticeable decrease was seen in the case of the four layers hence it is obvious from the figures that when increasing the number of layers a decrease in the weight of output of impurities (leaving the packed bed )occurs so the percent output of impurities will be decreased

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 4-25 Number of Layers Vs Percent Output of Impurities

For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 426 Number of Layers Vs Percent Output of Impurities

For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure431 Number of Layers Vs Percent Output of Impurities For 020N230N1180N1320N50N 54321 =====

42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

Figure 436 Number of Layers Vs Percent Output of Impurities For 010N1150N0750N150N650N 54321 =====

44 Relationships Between The Size of Impurities and Percent Output of Impurities In this section the size of impurities will be related with the weight percent output of impurities for each packed bed made from five sizes of spheres The curves of this relation are shown Figure 437 to 448These curves show the relation between the size of impurities and the percent output of impurities It is obvious from the figures that when impurities of large size are passed in the packed bed lead to decrease in the weight of output of impurities (leaving the packed bed) has occurred so this will lead to decrease in the weight of out put of impurities and this decrease is noticeable in the case of 12-22-236 and 236-315 mm respectively While the percent output of impurities will increase with the decrease in the size of impurities

45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer

160 200 240 280 320Size of impurities (mm)

Figure 437 Size of Impurities Vs Percent Output of Impurities For 060N540N20N10N10N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer

Figure 438 Size of Impurities Vs Percent Output of Impurities

For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51

45 Relationships Between Mean Pore Diameter and Porosity Porosity was calculated experimentally for each packed bed made from five sizes of spheres also calculated in the experiments in case of two three and four layers Mean pore diameter due to number will be related with porosity for each packed bed made from five sizes of spheres The curves of this relation are shown in figure 449 and these curves show the relation between mean pore diameter and porositythese figures show the proportionality between the mean pore diameter due to number and porosity for two three and four layer respectively The comparison between each curve for three types of impurities have been done in this section in each figure It is obvious that when the numbers of the large spheres increase there will be an increase in the porosity and these leads to the increase in the mean pore diameter The same relations were made for mean pore diameter due to length ( ) surface area ( ) and volume ( ) Figures show the curves of these relations are 450 to 452

ldcm adcm vdcm

080 120 160 200 240 280 320Mean pore diameter due number (dcm) (mm)

030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer

Figure 449 Mean Pore Diameter Due to Number Vs Porosity

52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer

Figure 450 Mean Pore Diameter Due to Length Vs Porosity

200 240 280 320 360Mean pore diameter due to area (dcm) (mm)

030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer

Figure 451 Mean Pore diameter Due to Area Vs Porosity

53

240 280 320 360Mean pore diameter due to volume (dcm) (mm)

030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer

Figure 452 Mean Pore Diameter Due to Volume Vs Porosity 46 Relationships Between Number of layers and Porosity In this sectionThe number of layers will be related with porosity calculated experimentally for each packed bed made from five sizes of spheres in case of two three and four layers The curves of this relation are shown in figure 453 This figure show the proportionality between the number of layers and porosity Its noticeable that when the number of layer increase a decrease in the value of porosity calculated occurs Therefore the largest value of porosity occurs in the case of two layers and porosity decreased in three layer and a noticeable decrease in the case of four layers

54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001

Figure 453 Numbers of Layers Vs Porosity

55

47 Presentations (Porosity) In appendix [B] the porosity for mono system of each group of spheres (6 8 10 20 and 26mm) was calculated by using Furnas equation and the result was listed in table B26 and also the porosity for binary system was calculated by using Jeschar equation and latif equation and listed the results in the tables B27 to B36The experimental work is given in chapter three which contains determining porosity for mono system experimentally by taking certain number of each group of sphere listed in table 31 and measure the height of spheres in two the cylinder (12964 cm) the height of sphere in the cylinder listed in table C13 and then determine the porosity to determine the porosity and the results are listed in the table C14 Now this chapter will discuss and explain the results of (theoretical and experimental) suitable relation between porosity of mono system and diameter of the cylinder which contains the sphere and the porosity will be related with the diameter of sphere 48 Derivation of Anew Empirical Equation for Mono System Furnas suggest the equation (29) for calculation the porosity of mono system which takes care of the effect of the container wall Porosity was calculated in appendix[B] and the result listed in table B26

Dd 3403750 +=ε helliphellip (29)

Table (41) presents the values calculated by equation (29) and the value of experimentally porosity The absolute error between 03 amp 259 Table 41 Calculated Values of Porosity Using Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 129 cm)

No Porosity determined

experimentally

Porosity calculated using Furnas

equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56

The modification of equation (29) under consideration of the porosity the present work leads to equation (41)

Ddf 3404180 +=ε helliphelliphellip(41)

Where (0418) the average value of experimental porosity of the present packing The absolute error lies now between 763 amp 982 Table 42 Calculated Values Porosity Using Modified Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 129 cm)


experimentally Porosity calculatedmodified equation

Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763

When the result of the present experimental table plotted as shown in Figure 454 The empirical equation (42) and (43)could be derived respectively

⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε helliphelliphellip(43)

Empirical equation which represent our research as linear equation (42) and (43) equation was in form of polynomial relation which can be used for further research under consideration of the effect of the surface area and the volume of the bed particle and the container

represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd

Table (43) presents the values calculated by equation (29) and porosity determined experimentally The absolute error between 186 amp 416

57

Table 43 Calculated Values of Porosity Using Furnas Equation for Mono System as Compared to The Porosity Determined Experimentally with Absolute Error(in Cylinder of Diameter 64 cm)


experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193


Ddf 3404610 +=ε helliphellip(44)

Where (0461) is the average value of experimental porosity of the present packing The absolute error lies now between 1888 amp 1451 Table 44 Calculated Values Porosity Using Modified Furnas Equation for Mono System as Compared to The Porosity Calculated Experimentally with Absolute Error(in Cylinder of Diameter 64 cm)

No Porosity

determined experimentally


equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451

When the result of the present experimental table plotted as shown in Figure 455 the empirical equation (45) and (46) could be derived respectively

58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd


Empirical equation which represent our research as linear equation(45) and equation (46) was in form of polynomial relation which can be used for further research under consideration of the effect of the surface area and the volume of the bed particle and the container represent by

2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd

The absolute error listed in the above table may be due to following reasons

1- The shape of the particles 2- The diameter ratio ( dD) 3- The packing method 4- The packing tools

59 Relation Between The Porosity and The Diameter of Spheres The porosity for mono system will be related with the diameter of spheres which made the packed bed The curves of this relation are shown in figures 454 and 455In the experimental part the group of spheres of diameter 6mm was packed in the cylinder and determined the porosity The same procedure repeated for 8 1020and 26It is noticeable that the porosity is larger in the case of 8mm than the porosity in the case of 6mm and the porosity is larger in the case of 10 mm than the porosity in the case of 8mm and so on therefore it is obvious that when the diameter of the spheres which made the packed bed increased the porosity will be increased In the theoretical part the value of porosity calculated by using Furnas equation will be increased when the diameter of the spheres which made the packed bed increased because the ratio of dD will be increased the porosity will be increased The porosity increased when the diameter of spheres which made the packed bed increased

59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically

Figure 454 Ratio of dD vs Porosity in The Cylinder of Diameter

129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically

Figure 455 Ratio of dD vs Porosity in The Cylinder of Diameter 64

cm

60

410 Relation Between The Porosity and The Diameter of Cylinder which Contains The Sphere The experimental work in chapter four was done first in the cylinder of diameter 129 and the same procedure was repeated with cylinder of 64 cm The curves of this relation are shown in figures 456 to 457The values of porosity calculated by Furnas are taken from the table B26 and the values of porosity determined experimentally are taken from the table C14 The figures 456 and 457 show the proportionality between the porosity and the diameter of the cylinder It is obvious in the experimental part the value of porosity in cylinder of diameter 129cm is smaller than the value of the porosity in the cylinder of diameter 64cm As a result the height of number of sphere in the cylinder of diameter 64 cm will be more than the height of the same number of sphere in the cylinder of 129 cm therefore the value of porosity will be larger In the theoretical part also the value of porosity in cylinder of diameter 129cm is smaller than the value of the porosity in the cylinder of diameter 64cmbecause the ratio of dD will be larger in the case of cylinder of diameter 64cmThis ratio will be effected in the value of porosity that calculated by Furnas equation the value of porosity will be increased when the diameter of the cylinder decreased

61

500 1000 1500 2000 2500 3000Diameter of sphere (mm)

036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm

Figure 456 Diameter of Sphere Vs Porosity Determined

Experimentally with Cylinder Diameter (64129) Respectively

500 1000 1500 2000 2500 3000

Diameter of sphere (mm)

036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm

Figure 457 Diameter of Sphere Vs Porosity Calculated

Theoretically with Cylinder Diameter (64129) Respectively

62

411 Derivation of Anew Empirical Equation for Binary System Porosity for binary system calculated theoretically by using Jeschar equation and Latif equation and the results was listed in appendix [B] in tables B27 to B36 and figures shown from 458 to 459 In appendix [C] porosity calculated experimentally and the results was listed in tables C14 to C24 figures shown form 460 to 469 Now theoretical results compared with experimental result and then absolute results was calculated The results was listed in tables from table 45 to 414 It is clear from the tables Jeschar equation give more acceptable result than Latif equation The absolute error in Jeschar equation is low its range 485 to 17 but in Latif equation absolute error is high its range from 0985 to 683 So Latif equation was modified under consideration of the porosity of the present work in order to get more acceptable results and the range of absolute error 004 to 138 The results of modified equation was listed in tables from 415 to 424

63

000 020 040 060 080 100Weight percent of spheres of d2

025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm

Figure 458 Porosity Calculated in Jeschar Equation Vs Weight

Percent of Spheres of d2 for Binary System

64

000 020 040 060 080 100Weight Percent of spheres of d2

000

010

020

030

040

050Po

rosi

ty

Figure 459 Porosity Calculated in Latif Equation Vs Weight Percent of Spheres of d2 for Binary System

65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048

Fig 460 Experimental Porosity for Binary System( =26mm =6

mm) Vs Volume Fraction 1d 2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity

Fig 464 Experimental Porosity for Binary System( =26

mm =8 mm) Vs Volume Fraction 1d

2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity

Fig 466 Experimental Porosity for Binary System( =10 mm 1d

2d =8 mm) Vs Volume Fraction

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70

Table 45 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=6)

V 2 Porosity


Porosity calculated by using Jeschar equation

Absolute error

Porosity calculated by using

Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02


V 2 Porosity Determined

Experimentally

Porosity Calculated by Using Jeschar

Equation

Absolute Error

Porosity Calculated by Using

Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 04034 01435 03677 0219 08936 04826 04009 0169 03729 0227 07887 0496 03945 0149 03776 0186 06853 0459 03858 016 038185 0169 05833 0455 03765 017 038549 0152 04828 04161 03687 0114 038855 0066 03836 04328 036403 0159 039097 00967 02857 0428 03644 0149 03927 00826 01892 04408 037095 0158 03937 0107 009396 04557 03842 0157 0394 0135 Table 48 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=8d2=6)


Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04262 0125 01958 0598 0894 04699 04208 0105 02195 0533 07894 04416 04074 00774 02287 0482 06862 04233 03898 0079 02463 04177 05843 03899 0371 00485 02646 0321 04838 04036 03542 0122 02836 0297 03845 0378 03428 0093 03033 0198 02866 03869 03409 0119 03235 0164 01898 04358 03529 01902 034409 02104 00943 04402 03816 0133 036508 01706 Table 410 Theoretically Porosity as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=20d2=8) V 2 Porosity

Determined Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75

The modification of Latif equation under consideration of the porosity of the present work leads to equation 47

( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝


Table 415 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=26d2=6)

V2 Porosity Determined

Experimentally

Porosity Calculated by Using Modified Latif

Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502

04783 03773 04219 01182 03794 0364 04198 01532 02821 03938 04152 00543 01865 04037 04086 00121 00925 0429 04021 00627 Table 417 Porosity Calculated by Using Modified Latif Equation as Compared to The Experimentally Porosity with Absolute Error for Binary System(d1=10d2=6)


Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104

Table 417 to 427 Demonstrates the differences in porosity and its cause will be as follows-

1- The effect of padding height measurement where as it is difficult to equalize it s surface

2- The effect of the vessels diameter concerning large diameters 3- Respective irregularity for the shapes of the applied bodes

81

CHAPTER FIVE

CONCLUSION AND RECOMMENDATION 51 Conclusions 1-The mean pore size will be the property of each packed bed and will be recognized from other different beds by mean pore size or diameter 2- The probability of finding pore sizes in any packed bed increased with increasing the diameter of the pore (as explained in theoretical part of research )depending on the equations are taken from Latif MN This proportionality is different for each type of distributions 3-The percent output of impurities which are passed through the packed bed decreases with the increasing the number of the layers and also decrease with the increase in the size of impurities and increasing the number of large spheres in the packed bed leads to an increase in the percent output and this in turn leads to an increase the mean pore diameter (due to number length surface area and volume)and an increase in the number of small or medium spheres leads to reduction in the output of impurities and that in turn leads to a reduction in the mean pore diameter (due to number length surface area and volume) 4-Porosity which is calculated in each kind of distribution decrease with the increasing the number of layers and also increased with the increasing of mean pore diameter (due to number length surface area and volume) 5-For mono size spherical particles system the porosity decreases as the bed diameter increases and the porosity increases as the diameter of spherical particles increases that leads to say the porosity of mono spherical particles system increases as the ratio (particle diameter bed diameter) increases Plot of diameter ratio against porosity was achieved and the advantages of this method are

i- It can be consider as a simple method ii- It has the ability of comparison the variation of porosity value with

particle size in certain bed iii- An equation can be obtained from it by found a relation ship that

describes the curve (so for any diameter ratio the value of porosity can be expected with out making any experiments)

82

6-For binary system mixing particles of different sizes is known to increase the over all porosity but if the small particles sufficiently small to penetrate the voids associated with the large particles the porosity will decrease therefore it can be say that the porosity values depends on the particle size and the diameter ratio 52 Recommendation 1- Developing Latif equation that gives us the relation between the probability and the diameter of the porous to give accurate results for all types of distribution 2-Further experiments for pore size should be made on other sizes of fine spheres or impurities to study the properties of the packed bed Also the number of layers should be increased 3-Further experiments for pore size should be made for another kind of packed bed which consists of more diameters of sphere and have another number percent 4-Further experimental work for pore size using various packing filtering liquids and suspended materials is required to find the usefulness of the theoretical method studying the effect of the particle shape the hydraulic gradient the surface forces and wall effects 5- Using video camera to observe the motion of the impurities when they exit from it and observe the wall effect and especially to study the decolmatage or remaining of the impurities 6-Improvement of the derived empirical equation that calculated porosity can be realized by addition experimental work Variable new sphere sizes may be attempted for this purpose 7-Although further investigation are needed to finding the effects of the surface of particles composing the bed and the surface of the container in the value of porosity 8-The effect of container wall in the porosity may further more be minimized using a smaller ratio particulate to container diameters

83

9-Another types of statistical function may be consider and relating porosity value with its variables finally comparison between the obtained results with the results of the present work were carried out 10-Using another packed bed with different diameter and increase the diameter of spheres that composing the bed to increase the porosity 11-Using another cylinder which contains the packed bed with smaller diameter to increase the porosity

84

References 1 AMarmur Powder Technology 44(1985)249 2 LHSRobleeRMBaird and JWTierneyAIChE4(1958)460 3 RFFedors Powder Technology 23(1979)225 4 NStandish and DEDorger Powder Technology 22(1979)122 5 Given on internet Particle characterization instruments httpwwwquantachromecommercury poromsimetrypore master GThtm 6 WJames Powder Technology 20(1978)115-120 7 Given on internet by ABenoit (predicting unsaturated hydraulic conductivity of coarse un consolidated sediment based on grain size )1992 at httpwwwostigovenergy citations productbibiojsposti id=5808958 8 Gp Androutsopoulos and ET Woodburn Powder Technology 33(1982)175-186 9 YJwang TCWU and CL chiangAiche Journal September 1989 vol 35 No 9 10 JVan Brakel Powder Technology 11(1975) 205 11 IGarcia ndashBengochea Dolch Jcolloid interface sci 38(1972)234-244 12 TS Ramakrishnan and DT Wasan Powder Technology 48(1986)99-124 13 LPaterson VHornof and GNeale Powder Technology 33(1982)265-268 14JD Bernol and JMason nature 1881960910 as cited in Ref 54 15 AKSinghal and PM Dranchuk Powder Technology 11(1975)45 16 IMahgoub CProst and JA Dodds Powder Technology 31(1982)143-151 17 MMuskat PHD (the flow of homogenous fluids through porous media) 1946 18 SHabib and SMuhammad Chemical Engineering Science vol 45 No1 pp163-182 1990 19E Adrian Scheidegger (The physics of flow through porous media) 1960 20 YNakano and JW Evans Powder Technology 35(1983)115-118 21 RFrancoise R Jean and SKenneth (the adsorption by powders and porous solids) 1999 22 GP Androutsopoulos and THJ Linardos Powder Technology 47(1986)9-15 23 PLegoff DLeclere and JDodds Powder Technology 42(1985)47-53 24 MSuzuki and TOshima Powder Technology 44(1985) 213-218 25 JRodriguez and CH Allibert and JMChaix Powder Technology 47(1986)25-33 26Given on internet (kissing number) http math world wolframcomkissing numberhtm 27 WAGray the packing of solid particles Chapman and Hall London 1968

85

28 KJIves The scientific basis of filtration London 1975 29 Given on internet (Filtration basis) http wwwtechno-filt combasic filtration htm 30 Given on internet (Water filtration guide) http wwwblank005tripodcomcompingequipmentguide filterhtm1 31 Given on internet (principles of filtration) http wwwpallcom17761-4740 32 Given on internet (pore size meter PSM 160165) http wwwtopas-gmbhdeporohtm 33 JBeck sums of distances between points on a sphere math 31(1984)33-41 34 Given on internet by I Peterson (Cracking Keplers sphere-packing problem) at August 15 1998 35 Given on internet by DrSJHeyes (Structures of simple inorganic solids) 2000at http wwwchemoxacukicheyesstructure of solidsecture 1lec 1htm 1 36 TStovall FDelarrard and MBuil Powder technology 48(1986)1-12 37 Given on internet (Wikipedia information about sphere packing) http wwwanswerscomtopicsphere packing 38 Given on internet by Eric WWeisstein (sphere packing) 1999 at httpmath worldwlframcomsphere packing htm 39 DScott Nature 188 1960908 40 KRidge way and KJ Tarbuck British Chemical Engineering 121967384 41 Given on internet by R Angle (packed bed calculator porous media permeameter) 2001at http wwwimnoengcomground waterpacked bed htm 42 Given on internet (Using computational fluid dynamics to simulate the flow through novel support structures for liquid chromatography) 2004 at httpwwwvubacbechisnicohtm 43 Given on internet by GondkarHManud AmritkarPaiaLali(Effect of adsorbent porosity on performance of expanded bed chromatography of proteins ) 2001 at httpwwwncbinlmnihgoventrezquery Fcgicmd=Retrieve amp db=pub medamplist uids=11386874amp dopt =abstract 44 Given on internet byA Jey (performance of anaerobic packed bed system with different media characterististics) 1996 at httpwwwiwapon inecomwst03405wst 034050453htm 45SJ Gregg and KSW Sing adsorption surface area and porosity 2nd Ed academic press New York 1982

86

46 Given on internet by YBP Kwan DJ Stephenson and JR Alcock (The dependence of pore size distribution on porosity in hot iso statically pressed porous alumina) 2001 at http wwwspring er linkcomapphomecontributionaspwasp=off 36f1 daqa 04 d7eae 5dc3ff 35fc 0003 amp referrer=parent amp back to 47 SDiamond and WLDolch JColloide interface sci 38(1972)234-244 48 BGerald and GDavid Chemical engineering science vol4 No11 p2079-2085 49 MSurendra phDThesis Purdue univ WLafayette Ind 1980 304pp 50 MKaneuji DNWinslow and WL DolchCem Concr Res (May 1980) 51 MNLatif PhDThesisBeitrag Zur ModellierungVonAuch Wemmschichten DresdemGermany 1981 52 NFawaz MSc Thesis Pore size in packed bed consisting of three sizes of spheres Saddam University 1997 53 HY AL-obaidi MScThesisPore size in packed of mixed spheres Saddam University 1994 54 MOmar MSc Thesis Pore size in binary mixture Saddam University 1999 55 Given on internet (Porosity) http wwwglossary oil fieldslbcomdisplaycfmterm=porosity 56 HJFraser and LCGarton JGeology 43(1935)785 57 JPHurbisckDMLecterePLegoffIndand EngChem62(1970) 58 WOSwith PhsRev 24(1929)1271 59 YLHE SZQin CJLim and JR Grace the Canadian Journal of Chemical Engineering 72(1994)561 60 LHSRoblee RMBaird and JWTierney AIChE 4(1958)460 61 RFbenenti and CBBrosilow AIChE 8(1962)359 62 DPHaughey and GSG Beverige ChemEngSci 21(1966)905 63 KRidgway and KJTarbuck ChemEngSci 23(1968)1147 65 MH VennetiGovindarao and GFFromentChemEngSci41(1986)533 64 DVortmeyer and JSchuster ChemEngSci 38(1983)1691 66 JKubie ChemEngSci 43(1988) 1403 67 NOuchiyama and TTankaIndEngChemFund 25(1986)125 68 NStandish and ABYu Powder Technology 49(1987)249 69 NSRasheed MScThesis Porosity determination of multi-component system using discontinuous size distribution Saddam University 1994 70 FRAlAzzawi MSCThesis Influence of the particle size distribution on the porosity of glass spheresSaddam University1996 71 AAAlRamahi MScThesis Porosity determination of multicomponent system Saddam University 1997 72 CCFurnas IndEngChem 23(1993)1052

87

APPENDIX [A]

SAMPLE OF CALCULATIONS A11 For pore size calculation

d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5

For the case of three different sizes of spheres- Example if we take d1 d d (d =6d =8d =10) 2 3 k m g

( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=

dgdkak = ( ) rarr a =06 le k

k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=

k 5 =927323 dc iii =119128 mm For the case of two large diameters and one small diameter- Example if we take d1 d d ( d k =6d =8d =8) 2 3 m g

( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphelliphellip (24a)

dc ii =1116843 mm For the case of two small diameters and one large diameter- Example if we take d1 d d ( d k =6d =6d =8) 2 3 m g

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphelliphellip(24b)

dc ii =1014832 mm

For the case of three equal diameters- Example if we take d1 d1 d1 (d k =6d =6d =6) m g

gci d1550d times= helliphellip (25)

dc i =0924 mm

A-2

A12 For probability calculation d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5

i- Calculate probability due to length

( ) ( ) ( ) ( ) ( )5544332211 NdNdNdNdNdL times+times+times+times+times=

L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr

nnn3P timestimes= helliphellip (26)

Where 3rrr kmg =++

P =000147 i

A-3

ii-Calculate the probability due to area

( ) ( ) ( ) ( ) ( )5254

243

232

221

21 NdNdNdNdNdA times+times+times+times+times=

A=28656

( ) ANdA kkk times= 2

A =001256 k

( ) ANdA mmm times= 2

A =002233 m

( ) ANdA ggg times= 2

A =0069793 g

According to eq 2-6 P =0000117446 i

iii-Calculate the probability due to volume

( ) ( ) ( ) ( ) ( )5354

343

332

321

31 NdNdNdNdNdV times+times+times+times+times=

V=564736

( ) VNdV kkk times= 3

V =000382 k

( ) VNdV mmm times= 3

V =0009066 m

A-4

( ) VNdV ggg times= 3 V =00354 g

P =7355E-06 i

A13 Calculation of the mean pore diameter

sum=

=n

iciicm dpd

1

helliphellip (27)

To determine the mean pore diameter due to number

sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm

To determine the mean pore diameter due to length

sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm

To determine the mean pore diameter due to area

sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm

To determine the mean pore diameter due to volume

sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5

A2 For Porosity calculation

1 For mono system

a) Furnas equation


Where

If we take sphere of diameter 6mm

In cylinder of diameter 129 cm

Dd =00465

ε=0375+034times00465=03908

in cylinder of diameter 64 cm

Dd =009375

ε=0375+034times009375=04069

Further values see table (B26)

b) Calculate the porosity experimentally from eqTV

V=ε

pT VVV minus=

For cylinder of D=129 cm

If we take spheres d=6mm number of sphere taken from table (33) N=1049 The height measured is 15cmFurther values see table (C13)

LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948

Further values see table (C14)

2 For binary system

AJeschar equation

If we take d1= 26d =6 mm 2

helliphelliphelliphellip (210)

2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235

Where ε from table (C14) To calculate the fractional volumes of sphere take basis 300 g

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7

Where presented in tables (B27) to (B36) 2W

21 ρρ amp From table (31) If we take 10 d 90 d1 2

0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=

Further values see from table (B27) to (B36)

( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]

B1Theortical Resluts(Pore Size) Equation of the theoretical work is applied using a suitable computer programs tables from B2 to B12 shows the results obtained by the computer programs that have been written for calculating the pore size distribution due to surface area number volume and calculating the mean pore diameter for different values of particle diameter Table B1 Arrangement of Particles to Determine the Diameter of The pore

No of Small

Spheres

No of Medium Spheres

No of Large

Spheres

Arrangement

dc(mm)

EquationUsed

3 0 0 d1 d1 d1 093 2-5 2 0 1 d1 d1 d 2 10148 2-4b 2 0 1 d1 d1 d 3 10779 2-4b 1 0 2 d1 d 2 d 2 1117 2-4a 1 1 1 d1 d 2 d 3 11913 2-3 0 3 0 d 2 d 2 d 2 124 2-5 2 0 1 d1 d1 d 4 1247 2-4b 1 0 2 d1 d 3 d 3 1277 2-4a 2 0 1 d1 d1 d 5 1291 2-4b 2 0 1 d 2 d 2 d 3 1328 2-4b 1 1 1 d1 d 2 d 4 1389 2-3 1 0 2 d 2 d 3 d 3 143 2-4a 1 1 1 d1 d 2 d 5 144 2-3 1 1 1 d1 d 3 d 4 1517 2-3 0 3 0 d 3 d 3 d 3 155 2-5 1 1 1 d1 d 3 d 5 157 2-3 2 0 1 d 2 d 2 d 4 1581 2-4b 2 0 2 d 2 d 2 d 5 16496 2-4b 1 1 1 d 2 d 3 d 4 1716 2-3 1 1 1 d 2 d 3 d 5 1789 2-3

B-2

1 0 2 d1 d 4 d 4 1861 2-4a 2 0 1 d 3 d 3 d 4 189 2-4b 1 1 1 d1 d 4 d 5 1916 2-3 2 0 1 d 3 d 3 d 5 19822 2-4b 1 0 2 d1 d 5 d 5 20715 2-4a 1 0 2 d 2 d 4 d 4 2155 2-4a 1 1 1 d 2 d 4 d 5 228 2-3 1 0 2 d 3 d 4 d 4 2393 2-4a 1 0 2 d 2 d 5 d 5 2424 2-4a 1 1 1 d 3 d 4 d 5 2532 2-3 1 0 2 d 3 d 5 d 5 2714 2-4a 0 3 0 d 4 d 4 d 4 31 2-5 2 0 1 d 4 d 4 d 5 3417 2-4b 1 0 2 d 4 d 5 d 5 3699 2-4a 0 0 3 d 5 d 5 d 5 403 2-5

Table B2 Results Obtained from Computer Program for 26d20d10d8d6d 54321 =====

N1 =01 N 2 =01 N 3 =02 N 4 =054 N 5 =006 L1 =0038 L 2 =005076 L 3 =01269 L 4 =06853 L 5 =00989 A1 =00126 A 2 =00223 A 3 =00698 A 4 =07538 A 5 =01415 V1 =000382 V 2 =000907 V 3 =00354 V 4 =07649 V 5 =01867 Diameter of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3

124 0026 0002724 19294E-4 11064E-5 1247 0042 0005704 54967E-4 44639E-5 1277 00542 0007543 73319E-4 59028E-5 1291 0056 0007973 80018E-4 67224E-5 1328 0062 0008954 9046E-4 75955E-5 1389 0094 0016899 21729E-3 2351E-4 143 0106 0019352 2499E-3 2692E-4 144 011 0020499 27374E-3 30806E-4

1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4

Table B3 Results Obtained from Computer Program for

26d20d10d8d6d 54321 =====

N1 =015 N 2 =015 N 3 =02 N 4 =044 N 5 =006 L1 =00622 L 2 =008299 L 3 =01383 L 4 =06086 L 5 =01079 A1 =002147 A 2 =003816 A 3 =00795 A 4 =06996 A 5 =01612 V1 =000663 V 2 =00157 V 3 =004095 V 4 =07208 V 5 =02159

Diameter of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05

124 00675 000894 71262E-4 422E-05 1247 00972 001602 16798E-3 137E-04 1277 0115 001957 20868E-3 171E-04 1291 0119 002083 37608E-3 199E-04 1328 0133 002368 4108E-3 23 E-04 1389 0192 00425 75467E-3 681 E-04 143 02101 004728 82703E-3 76 E-04 144 0218 005062 90627E-3 895 E-04

1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====

N1 =02 N 2 =02 N 3 =015 N 4 =043 N 5 =002 L1 =00894 L 2 =0119 L 3 =01118 L 4 =06408 L 5 =00387 A1 =00327 A 2 =005804 A 3 =006802 A 4 =07799 A 5 =00613 V1 =001057 V 2 =002505 V 3 =00367 V 4 =08417 V 5 =0086 Diameter

of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1


N1 = 025 N 2 = 02 N 3 = 0182 N 4 = 0348 N 5 = 002 L1 = 01209 L 2 = 0129 L 3 = 0147 L 4 = 0561 L 5 = 00419 A1 = 00467 A 2 = 00664 A 3 = 00944 A 4 = 07229 A 5 = 00702 V1 = 00155 V 2 = 00295 V 3 = 00524 V 4 =08014 V 5 =01012

B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1


N1 = 04 N 2 = 02 N 3 = 015 N 4 = 023 N 5 = 002 L1 = 0226 L 2 = 01506 L 3 = 0141 L 4 = 0433 L 5 =00489 A1 =00975 A 2 =00866 A 3 =01015 A 4 =06228 A 5 =00915 V1 =00341 V 2 =00405 V 3 =00593 V 4 =0727 V 5 =0139


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====

N1 = 005 N 2 = 006 N 3 = 009 N 4 = 07 N 5 = 01 L1 = 00164 L 2 = 00263 L 3 =00492 L 4 =0766 L 5 =0142 A1 =000497 A 2 =00106 A 3 = 00248 A 4 = 0773 A 5 = 01866 V1 = 000144 V 2 = 00041 V 3 = 0012 V 4 = 0748 V 5 = 0235 Diameter of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 125E-04 442E-06 123E-07 3E-09 10148 575E-04 256E-05 908E-07 286E-08 10779 125E-03 654E-05 275E-06 103E-07 1117 179E-03 994E-05 442E-06 176E-07

11913 341E-03 227E-04 123E-05 602E-07 124 3626E-03 245 E-04 135E-05 671E-07 1247 8876E-03 864 E-04 707E-05 533E-06 1277 00101 983 E-04 799E-05 596E-06

B-10

1291 00108 1098E-03 937E-05 742E-06 1328 00181 12 E-03 0000102 803E-06 1389 002441 318 E-03 0000346 346E-05 143 002587 3371 E-03 0000366 363E-05 144 002767 738 E-03 0000425 446E-05 1517 004657 7451 E-03 0000998 0000122 155 00473 7571 E-03 0001013 0000124 157 005 826 E-03 0001151 0000148 1581 00576 9844 E-03 0001412 0000186

16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1


N1 = 05 N 2 = 0132 N 3 = 0118 N 4 = 023 N 5 = 002 L1 =0289 L 2 =01019 L 3 =0114 L 4 =0444 L 5 =005 A1 =0125 A 2 =00588 A 3 =0082 A 4 =06399 A 5 =0094 V1 =00434 V 2 =00272 V 3 =00475 V 4 =074 V 5 =01415

B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1


N1 = 007 N 2 = 009 N 3 = 014 N 4 = 061 N 5 = 009 L1 = 00246 L 2 =0042 L 3 = 00819 L 4 = 0714 L 5 =0137 A1 = 00077 A 2 = 00176 A 3 =00428 A 4 = 0746 A 5 = 0186 V1 = 000227 V 2 = 00069 V 3 = 0021 V 4 = 07324 V 5 =0237


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 343E-04 149E-05 457E-07 117E-08 10148 1666E-03 913E-05 359E-06 118E-07 10779 3724 E-03 24E-04 112E-05 443E-07 1117 5425 E-03 371E-04 184E-05 768E-07 11913 10717 E-02 881E-04 531E-05 275E-06

124 11446 E-02 956E-04 586E-05 308E-06 1247 2041 E-02 2251E-03 191E-04 144E-05 1277 2453 E-02 2747 E-03 234 E-04 174E-05 1291 282 E-02 2995 E-03 267 E-04 211E-05 1328 292 E-02 3432 E-03 306 E-04 241E-05 1389 523 E-02 7874 E-03 913 E-04 93E-05 143 576 E-02 8724 E-03 101E-03 102 E-04 144 61 E-02 9576 E-03 1161 E-03 124 E-04

1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====

N1 = 0035 N 2 = 0045 N 3 = 009 N 4 = 068 N 5 = 015 L1 = 01107 L 2 = 00189 L 3 =00474 L 4 = 0717 L 5 = 02056 A1 = 000325 A 2 = 00074 A 3 = 00232 A 4 = 0701 A 5 = 0261 V1 = 000092 V 2 = 00028 V 3 = 00109 V 4 = 0664 V 5 = 0322


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 4287E-05 136E-06 343E-08 779E-10 10148 2083E-04 833E-06 269E-07 791E-09 10779 539E-04 258E-05 269E-07 358E-08 1117 7516E-04 377E-05 802E-07 576E-08 11913 1602E-03 975E-05 415E-06 228E-07

124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14

1291 5594E-03 518 E-04 403E-05 308E-06 1328 6141E-03 569 E-04 441E-05 334E-06 1389 12567E-02 1473E-03 145E-04 136E-05 143 001366 1601 E-03 157 E-04 147E-05 144 001508 186 E-03 195 E-04 196E-05

1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05

1581 00356 5648 E-03 758 E-04 964E-05 16496 003654 587 E-03 801 E-04 104E-04 1716 0053 9743 E-03 1524E-03 227 E-04 1789 00567 001085 1793 E-03 286 E-04 1861 01053 002792 6588 E-03 1502E-03 189 01218 003276 772 E-03 1742 E-03

1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1


N1 = 055 N 2 = 015 N 3 = 015 N 4 = 014 N 5 = 001 L1 = 0364 L 2 = 0132 L 3 = 0166 L 4 = 0309 L 5 =00287 A1 = 0185 A 2 = 00896 A 3 = 0139 A 4 = 05225 A 5 = 0063 V1 = 00724 V 2 = 00468 V 3 = 00914 V 4 = 0682 V 5 = 0107

B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1


N1 = 003 N 2 = 004 N 3 = 009 N 4 = 064 N 5 = 02 L1 = 00093 L 2 = 00165 L 3 = 0046 L 4 = 0659 L 5 = 0268 A1 = 000928 A 2 = 00063 A 3 = 0022 A 4 = 0634 A 5 = 0335 V1 = 000074 V 2 = 00023 V 3 = 00103 V 4 = 0585 V 5 = 0402 Diameter of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 27E-05 799E-07 191E-08 405E-10 10148 135E-04 506E-06 155E-07 425E-09 10779 378 E-04 17E-05 633E-07 211E-08 1117 522 E-04 246E-05 955E-07 333E-08 11913 117E-03 672E-05 322E-06 14E-07

124 1234 E-03 717E-05 348E-06 153E-07 1247 2962 E-03 242E-04 171E-05 111E-06 1277 3691 E-03 302 E-04 211E-05 135E-06 1291 4231 E-03 371 E-04 282E-05 201E-06 1328 4663 E-03 409 E-04 309E-05 218E-06 1389 9271 E-03 1015E-03 954E-05 825E-06 143 00102 1121 E-03 105E-04 9E-06 144 00117 1367 E-03 139 E-04 132E-05

1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05

1581 00291 4401 E-03 573 E-04 689E-05 16496 003005 462 E-03 613 E-04 755E-05 1716 004387 7648 E-03 115E-03 16 E-04 1789 004819 8878 E-03 1434 E-03 218 E-04 1861 0085 0020995 4658 E-03 978 E-04 189 01006 002526 5602 E-03 1163 E-03

B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====

N1 =065 N 2 =015 N 3 =0075 N 4 =0115 N 5 =001 L1 =0464 L 2 = 0143 L 3 = 00892 L 4 = 0273 L 5 = 00309 A1 = 02509 A 2 = 01029 A 3 = 00804 A 4 = 0493 A 5 = 00724 V1 = 01011 V 2 = 0055 V 3 = 0054 V 4 = 0663 V 5 = 0127


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19

Now to calculate the mean diameter (dc m )due to number length surface area and volume equation (27) must be applied so the results of the main diameter were (dc m ) also computed by computer program in appendix [D]

Table B14 Values of Mean Pore Diameter for 26d20d10d8d6d 54321 =====

060N540N20N10N10N 54321 =====

Table B15 Values of Mean Pore Diameter for

26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====

Mean pore diameter due to number 1978Mean pore diameter due to length 2613Mean pore diameter due to area 2915Mean pore diameter due to volume 3184


26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====


Mean pore diameter due to number 22045 Mean pore diameter due to length 2683 Mean pore diameter due to area 2999 Mean pore diameter due to volume 3204

B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22

B21Theortical Resluts(Porosity for Mono System ) At Packed bed composed of spheres of five diameters (6 8 10 20 and 26) porosity of each diameter could be calculated theoretically by using Furnas equation (29) for mono size system Table B26 Porosity Calculated for Mono System by Using Furnas Equation Ratio dD in the

Cylinder of Diameter 129cm

Porosity Ratio dD in the Cylinder of

Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131

B22Theortical Resluts (Porosity for Binary System) Table B27 Calculated Values of Porosity for Binary System d1=26 d2=6 mm Wight Percent of Sphere with d2=6

V2

Calculated Porosity

Using Jeschar Equation

Calculated Porosity

Using Latif Equation

100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23

Table B28 Calculated Values of Porosity for Binary System d1=20 d2=6 mm Wight Percent of Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644

Table B29 Calculated Values of Porosity for Binary system d1=10 d2=6 mm Wight Percent of Sphere with

d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417

Table B31 Calculated Values of Porosity for Binary System d1=26 d2=8 Wight Percent of Sphere with

d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25

Table B32 Calculated Values of Porosity for Binary System d1=20 d2=8

Wight Percent of Sphere with

d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504

APPENDIX[C] C1 Experimental Resluts For Pore Size The results of the experiments are given below in detail- Experiment 1

Table C1 Results of Exp 1

Diameter of Sphere

(mm)

Number of Sphere Cover

the Base of Cylinder

Number Percent of Sphere(N)

Number of Spheres in Each Layer

6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1

No of layers 2 3 4 Size of Impurities

(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17

Height of the Bed (cm) 32 45 58

No of Layers 2 3 4 Size of

Impurities (mm)

Percent Output (Wt )



12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57

236-315 457 38 30 16 Height of the bed (cm) 28 4 55


Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2

Experiment 3 Table C3 Results of Exp 3

Diameter of sphere(mm)





6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537

236-315 457 379 293 156 Height of the Bed (cm) 27 4 52


Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3


Diameter of Sphere(mm)





6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50


No of layers 2 3 4 Size of

Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of

Spheres in Each Layer

6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )

Percent output (Wt )

12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3

No of Layers 2 3 4 Size of Impurities

(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11




the Case of Cylinder



6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12

C21 Experimental Resluts (Porosity for mono system)

Table C13 Height of Sphere in The Cylinders of Diameter 129 cm and 64 cm

No Diameter of

spheres (mm) Height (L) (cm)

In the Cylinder of diameter 129 cm

Height (L) (cm) In the Cylinder of diameter 64 cm

1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63

Table C14 Porosity in The Cylinders of Diameters 129 cm and

64cm

No Diameter of spheres (mm)

Porosity In the Cylinder of diameter 129 cm


1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13

C22 Experimental Resluts (Porosity for Binary System) Table C15 Experimentally Porosity for Binary systemd1=26d2=6 mm

V 2 N1 N 2 L(cm) VT (cm ) 3 Porosity 1 0 978 16 20901 0471

08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443

Table C16 Experimentally Porosity for Binary System d1=20d2=6 mm


08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14

Table C17 Experimentally Porosity for binarySystemd1=10d2=6 mm


08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402


V 2 N1 N 2 L(cm) VT (cm 3 ) Porosity 1 0 450 18 235137 04872

09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325

Table C22 Experimentally Porosity for Binary Systemd1=26d2=10 mm


08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18

APPENDIX [D] COMPUTER PROGRAM

Program number one

REM REMPROGRAM USED TO CALCULATE THE PORE REMSIZEOF FIVE VALUES OF PARTICULATE DIAMETER REM CLS DIM N (5) F=J For I=1 to 5 READ N (I) NEXT I 5 PRINTINPUT THE DIAMETER VALUES DATA 06 08 1 2 26 DIM X (3) For I=1 to 3 Input X (I) NEXT I For I=1 to 2 For J=1 to 3-I If x (J) ltx (J+1) THEN 10 T=x (J) x (J) =x (J+1) x (J+1) =T 10 NEXT NEXT For I=1 to 3 PRINT x (I) Next I DK=X (1) DM=X (2) DG=X (3) IF X (1) =X (2) AND X (2) =X (3) THEN 20 IF DK=DM AND DK ltDG THEN 30 IF DG=DM AND DGgtDK THEN 40 IF DKltgtDM AND DMltgtDG THEN 50 20 PRINT THREE EQUAL DIAMETERS DP1=0155DG Z (F) =DP1 PRINTDP1= DP1 GOTO 60 30 PRINT TWO SMALL DIAMETER AND ONE LARGE A=DKDG

D-1

B= (((2A) + (A^2)) ^ (12)-A) C= (2+ (2B)) DP2= ((B^2)C)DG Z (F) =DP2 PRINT DP2= DP2 GOTO 60 40 PRINT TWO LARGE DIAMETERS AND ONE SMALL A=DKDG D= (((2a) + (A^-2)) ^ (12))-(A^-1) E= (2+ (2D)) DP3= ((D^2)E)DK Z (F) =DP3 PRINT DP3 GOTO 60 50 PRINT THREEDIFFERENT DIAMETER AM=DMDG A=DKDG K1= ((AM+1) ^2 (AM-1)) ^2 K2= (4AM) (AM+1) K3= ((A^2+AK2) ^ (12))-A K4= (K2+ (2K3))K1 K5= (K3^2)K1 DP4= (K4-((K4^2)-(4K5)) ^ (12))(DG2) Z (F) =DP4 PRINT DP4= DP4 60 PRINT IF YOU WANT TO CONTINUE PRESS 0 INPUT L IF Lltgt 0 THEN 80 F=F+1 IF Fltgt 36 THEN 5 FOR I=1 TO 34 FOR J=1 TO 35-I IF Z (J) ltZ (J+1) THEN 70 T=Z (J) Z (J) =Z (J+1) Z (J+1) =T 70 NEXT NEXT FOR I=1 to 35 PRINT Z ( I ) = Z (I) NEXT I 80 INPUT INTER 0 TO PRINT THE RESULTS H IF H=0 THEN 90 ELSE GOTO 100 90 FOR I=1 to 35

D-2

PRINT Z ( I ) = Z (I) NEXT I 100 FOR I=1 TO 5 PRINT ( I ) = N (I) NEXT I

D-3

Program Number Two

REM REM CALCULATING THE PROBABILITY OF PACKING REM BED FOR FIVE DIAMETERS BY USING THEIR REM NUMBER PERCENT REM R TYPE OF DISTRIBUTION REM RK TYPE OF DISTRIBUTION FOR SMALL PARTICALE REM RM TYPE OF DISTRIBUTION FOR MEDIUM PARTICALE REM CLS FOR I=1 TO 5 PRINT DIAMETER ( I ) = READ N (I) PRINT N (I) NEXT I DATA D1=N (1)D2=N (2)D3=N (3)D4=N (4)D5=N (5) PRINT ENTER THE NUMBER PERCENT OF EACH DIAMETER FOR I=1 TO 5 PRINT NUMBER PERCENT ( I ) = READ P (I) PRINT P (I) NEXT I DATA FOR I=1 TO 35 10 PRINT INPUT DIAMETER VALUE FOR I=1 TO 3 INPUT X (I) NEXT I FOR I=1 TO 2 FOR J=1 to 3-I IF X (J) ltX (J+1) THEN 15 T=X (J) X (J) X (J) =X (J+1) X (J+1) =T 15 NEXT NEXT DK=X (1) DM=X (2) DG=X (3) PRINT ENTER NUMBER PERCENT FOR EACH CHOICE DIAMETER FOR I=1 to 3 INPUT E (I) NEXT I IF X(1)ltgtX(2) AND X(2)ltgtX(3) AND X(1)ltgtX(3) THEN RG=1RM=1

D-4

RK=1 GOTO 20 IF X(1)=X(2) AND X(2)=X(3) AND X(1) =X(3) THEN RG=3 RM=0RK=0 GOTO 20 IF X(1)=X(2) AND X(1) ltX(3) AND X(2)ltX(3) THEN RG=1RM=0RK=2 GOTO 20 IF X (3) =X (2) AND X (3)gtX (1) THEN RG=2 RM=0 RK=1 GOTO 20 20 PRINT RG= RG PRINTRM= RM PRINTRK= RK REM CALCULATING THE FRACTIONAL OF RG RM RK B (1) =RM B (2) =RK B (3) =RG FOR I=1 TO 3 FACT =1 IF B (I) =0 THEN 30 FOR J=1 TO B (I) FACT=FACT J NEXT 30 F (I) =FACT PRINT FACT = FACT NEXT I PRN=6((E(1)^RK)(E(2)^RM)(E(3)^RG ))(F(1)F(2)F(3)) K (J) =PRN Z=Z+K (J) L=(N(1)P(1)+N(2)P(2)+N(3)P(3)+N(4)P(4)+N(5)P(5)) LK= (DKE (1))L LM= (DME (2))L LG= (DGE (3))L PRL=6((LK^RK)(LM^RM)(LG^RG)(F(1)F(2)F(3)) K(J)=PRL T=T+K(J) A=(P(1)(N(1)^2))+(P(2)(N(2)^2))+(P(3)(N(3)^2))+(P(4)(N(4)^2))+(P(5)(N(5)^2)) AK= (E (1)(DK^2))A AM= (E (2)(DM^2))A AG= (E (3)(DG^2))A PRA=6((AK^RK)(AM^RM)(AG^RG))(F(1)F(2)F(3)) K (J) =PRA Y=Y+K (J) V=(P(1)N(1)^3))+(P(2)(N(2)^3))+(P(3)(N(3)^3))+(P(4)(N(4)^3))+(P(5)(N(5)^3)) VK= (E (1)(DK^3))V VM= (E (2)(DM^3))V VG= (E (3)(DG^3))V

D-5

PRV=6((VK^RK)(VM^RM)(VG^RG)) (F (1)F (2)F (3)) K (J) =PRV W=W+PRV PRINT TAB (1) PRN PRN TAB (20)PRL= PRL TAB (40)PRA= PRA TAB (60)PRV= PRV PRINT TAB (1)Z Z TAB (20)T= T TAB (40)Y= Y TAB (60)W= W NEXT I INPUT U IF Ultgt 36 THEN 10 FOR I=1 TO 35 FOR J=1 TO 35-I IF K (J) ltK (J+1) THEN 40 T=K (J) K (J) =K (J+1) K (J+1) =T 40 NEXT NEXT

D-6

Program Number Three REMCalculating the mean pore diameter due to number REMLength Surface area and Volume REMdcn the mean pore diameter due to number REMdclthe mean pore diameter due to length REMdcathe mean pore diameter due to surface area REMdcv the mean pore diameter due to volume REMdcmn the mean pore diameter due to number REMdcml the mean pore diameter due to length REMdcma the mean pore diameter due to surface area REMdcmv the mean pore diameter due to volume CLS FOR I=1 TO 35 INPUT The diameter of the poresdc INPUT The Probability due to numberPrn INPUT The Probability due to lengthPrl INPUT The Probability due to surface area Pra INPUT The Probability due to volume Prv dcn=dcPrn Print dcn=dcn dcl=dcprl Print dcl=dcl dca=dcpra Print dca=dca dcv=dcprv Print dcv=dcv dcmn=dcmn+dcnPrint dcmn=dcmn dcml=dcml+dclPrint dcml=dcml dcma=dcma+dcaPrint dcma=dcma dcmvl=dcmv+dcvPrint dcmv=dcmv NEXT END

D-7

الخلاصة

ا في ك لأهميته ا وذل راج التي هدف البحث دراسة حجوم الفتحات والفراغية نظريا وعملي الاب تحتوي على حشوات

هناك أنواع مختلفة للتوزيع وهي

التوزيع اللوغارتمي الطبيعي 1 RRSB -توزيع 2 GGS -توزيع 3 Rinkes -توزيع 4 PM -توزيع 5

ة من اجل الحصول على النسبة RRSB - في هذا البحث استخدم توزيع التي استخدمت العددي

في الحسابات النظرية واستخدمت أيضا في العملي

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H

( Latif equation ة ة نظري تخدمت طريق وم الفتحات اس تعملت (ولحساب حج ث اس حيم وضعت داخل اسطوانة (d1=6d2=8d3=10d4=20d5=26) آرات مختلفة الأقطارخمس مل

ا سم l29 قطرها ه فتحات قطره ى منخل في وي عل م 4تحت اد توزيعات إحصائية ث حي مل م إيج تة والطولي تلأقطار الفتحا ل النسب العددي ة مث م ة بالاعتماد على نسب مختلف ة وت والمساحية و الحجمي

ا ح ابعملي ةس بة الوزني امية والنس وائب المس ن الش واع م ثلاث ان A=12-2 B=2-236 ) لC=236-3)

ة ملم دما تكون الحشوة مكون ة عن التي مررت خلال حشوة مكونة من آرات ذات خمس أقطار مختلفا من طبقتين ثم ثلاثة وأربعة ة للشوائب والمس ل ومن الملاحظ من النتائج العملية إن النسبة الوزني مية تق

بزيادة عدد الطبقات وبزيادة جحم الشوائب

ة بية الوزني وجدت علاقة بين النتائج العمليع والنظرية ادت الى ايجاد علاقة بين حجم الفتحة والنسم ادة حج م الفتحات وزي ادة حج د بزي امية تزي وائب والمس بة الش د ان نس امية حيث وج للشوائب والمس

ة يزيد من حجم الفتحاتالكرات المكونة للحشو

( Furnas equation طة ا بواس ادي نظري ام الأح بت بالنظ د حس امية فق بة للمس ا بالنس (أمين من الكرات في اسطوانة اولا ا بوضع عدد مع ا سم 129 وحسبت عملي اس 64ثاني سم وقي

ة حج ي وبمعرف م ارتفاع الكرات في الاسطوانة من اجل قياس الحجم الكل راغ ث م الكرات نجد حجم الف ومقارنة النتائج بالنتائج العملية ثم اشتقت معادلات جديدة نجد المسامية

9 المعادلات هي 259 amp 03 أ سم نسبة الخط 12 ذات قطرةبالاسطوان1

(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)

سم 416 amp 186 أ نسبة الخط المعادلات هي 64 ذات قطر ةبالاسطوان 2

⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)

(Jeschar and Latif equations ادلتين ( وحسبت فراغية النظام الثنائية نظريا بواسطة مع

رات ن الك وعين م ن ن ون م يط مك اع خل اس ارتف ا بقي (d1=20d2=6) (d1=26d2=6)وعملي(d1=10d2=6)(d1=8d2=6)(d1=26d2=8)(d1=20d2=8)(d1=10d2=8)

(d1=26d2=10)(d1=20d2=10)(d1=2620) سم لحساب الحجم 129 ملم في اسطوانة دة بحسب الظروف والكلي ثم حساب المسامية ة جدي تقاق معادل م اش مقارنة نتائجهما بالنتائج العملية وت

138 amp004 حيث تصبح هية المعادلالعملية حيث تقل باستخدامها نسبة الخطأ

( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير

لص شكري وتقديري وامتناني العميق للمشرف الدآتور محمد نصيف لما قدمه لي أود أن اعبر عن خا من توجيهات قيمة ونصائح سديدة طوال فترة أنجاز البحث

اعدة دائهم المس رين لإب ة النه ي جامع ة ف ة الكيمياوي م الهندس وظفي قس اتذة وم كر أس ا أن اش أود أيض

اللازمة أثناء هذا العمل

ى ولا أنسى أن أتقد رة البحث وخلال أصعب الظروف ال ي طوال فت م بالشكر والامتنان إلى من لازمن اعز من في الوجود إلى أمي وأبي وأخوتي وأخواتي فلهم جزيل الشكر

في الحشوة والمساميةلفتحاتقياس حجوم ا RRSB باستخدام توزيع

رسالة في جامعة النهرين مقدمة الى آلية الهندسة

وهي جزء من متطلبات نيل ماجستير في الهندسة الكيمياويةم علو

من قبل سهى مهدي صالح

)٢٠٠٣بكالوريوس علوم في الهندسة الكيمياوية(

١٤٢٧جمادى الأول ٢٠٠٦حزيران

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ABSTRACT

The aim of this research is to study pore size and porosity theoretically and experimentally in order to study the proprieties of packed bed There are different types of distribution

1 Log Normal distribution 2 RRSB-distribution 3 GGS-distribution 4 Rinkes- distribution 5 PM- distribution

RRSB distribution used in this research in order to get number percent used in theoritcal and experimental part

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H

In this research the mean pore diameter and the probability of pore diameter due to ( number length surface area and volume ) was found by using theoretical method suggested by Latif Experimentally packed bed was made by putting five types of spheres diameters (d1=6d2=8d3=10d4=20d5=26 mm) inside glass cylinder D=129 cm consists of sieve has pores of diameters 4 mm in bottom and then three kinds of impurities of diameters ( A=12-2 B=2-236 C=236-3 mm) passed through the packed bed the percent output of impurities and porosity was found in two three and four layers Its noticeable from experimental result that the percent output of impurities and porosity decreased with the increase of number of layers also the percent output of impurities decreased with the increase the size of impurities Experimental results are related with theoretical results in order to find the relation between mean pore diameter and percent output of impurities and porosity for each number percent of beds the increase in size of spheres lead to increase the mean pore diameter Also in this research porosity was found theoretically and experimentally Firstly porosity for mono system was calculated by using

I


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝


II


I III V VII VIII XV 1 3 4 5 5 6 7 7 9 10 11 12 16 16 18 19 21 22 22

24 31 39 45

52 54 56 56


III


59 61 63 82 83 85 A-1 A-3 A-5 A-6 B-1 B-22 B-22 C-1 C-13 C-14 D-1 D-4 D-7


IV

NOTATIONS













21toKK Constants








V

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝


II


I III V VII VIII XV 1 3 4 5 5 6 7 7 9 10 11 12 16 16 18 19 21 22 22

24 31 39 45

52 54 56 56


III


59 61 63 82 83 85 A-1 A-3 A-5 A-6 B-1 B-22 B-22 C-1 C-13 C-14 D-1 D-4 D-7


IV

NOTATIONS













21toKK Constants








V

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


I III V VII VIII XV 1 3 4 5 5 6 7 7 9 10 11 12 16 16 18 19 21 22 22

24 31 39 45

52 54 56 56


III


59 61 63 82 83 85 A-1 A-3 A-5 A-6 B-1 B-22 B-22 C-1 C-13 C-14 D-1 D-4 D-7


IV

NOTATIONS













21toKK Constants








V

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


59 61 63 82 83 85 A-1 A-3 A-5 A-6 B-1 B-22 B-22 C-1 C-13 C-14 D-1 D-4 D-7


IV

NOTATIONS













21toKK Constants








V

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc

NOTATIONS













21toKK Constants








V

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

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13doc

















micro Viscositycp

VI

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc

ABBREVIATIONS


VII

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc

Tables




VIII

19 21 22 56 57 58 58 71 71 72 72


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


IX

73 73 74 74 75 75 76 77 77 78 78


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


79 79 80 80 81 B-1 B-2 B-4 B-5 B-6 B-8



X


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











0doc

1doc

2doc

3doc

4doc

5doc

6doc

7doc

8doc

9doc

10doc

11doc

12doc

13doc


XI

B-9 B-10 B-12 B-13

B-14 B-16 B-17 B-19 B-19 B-19 B-19


26201086 54321 ===== ddddd

B-20 B-20 B-20

B-20 B-21 B-21 B-21 B-21 B-22

B-22


XII


d1=8 d2=6 mm





XIII

C-9 C-10 C-11 C-12

C-13 C-13 C-14 C-14 C-15 C-15 C-16 C-16 C-17 C-17 C-18 C-18


XIV

FIGURES 15 15 15 15 20 25 26

26 27 27 28 28















XV











29 29

30 30 31 33 33 34 34 35 35

36


XVI


36

37 37 38

38 39 40 40 41 41

42 42 43 43

XVII


44 44 45 46 46 47 47 48 48

49 49 50

50 For 010140150150550 54321 NNNNN =====

XVIII


51 51 52 53 53 54 55 60

60 62 62

64 65 66 66


XIX


67 67


XX



1


2

CHAPTER TWO

LITERATURE SURVEY


AL

ppMKQ ⎟

⎠⎞

⎜⎝⎛ minus

= 12 (21)



3

1 SohK

k εεminus



3


4







5



2 (r1+r2)



6


7


12143 =09069








31 -



8


9


10


11


dcIII

( )

g


=2

421


Where ( )( )

2

1 11⎥⎦

⎤⎢⎣

⎡minus+

=amamk

( )1amam4k2 +

=

( ) k21

2k2k3 akaak minus+=


1

235 kkk times=

dgdmam = (=1)

dgdkak = (le 1)


dcII

( )( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

= helliphellip(24a)


12

( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus

helliphellip(24b)


Where dgdkak =


dCI


nkdk

nmdm

ngdg

kmgi PrPrPr



( ) ( ) ( ) ( ) ( )( )( )( ) LNdL

LNdLLNdL

NdNdNdNdNdL

ggg

mmm

kkk

times=times=times=



13


( ) ( ) ( ) ( ) ( )( )( )( ) ANdA

ANdA

ANdA

NdNdNdNdNdA

g2gg

m2mm

k2kk

5254

243

232

221

21

times=

times=

times=




( ) ( ) ( ) ( ) ( )( )( )( ) VNdV

VNdV

VNdV

NdNdNdNdNdV

g3gg

m3mm

k3kk

5354

343

332

321

31

times=

times=

times=



sum=

=n

iciicm dpd

1

(27)

14

15

27 Porosity


TVV

=ε (28)

pT VVV minus=


TV


16


17


Dd 3403750 +=ε (29)



2

11

1

vx

ο

ο

εε

εminus

minus=

Where

( ) 260

1

2

2

211

1vsin

dd

vv

Xπ

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=


( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

18





density (gmcm ) 3











19


20




Weight (gm)

A 12-2 312 B 2-236 144 C 236-3 457


21





Number of sphere

1 6 1049 2 8 766 3 10 459 4 20 109 5 26 70


22


23



24



26d20d10d8dmm6d 54321 =====


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


25



000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


26

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



27

100

000

020

040

060

080

Prob

abili

ty

Number

Length

Surface area

Volume



Volume for 10N70N090N060N050N 54321 =====

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume



28

100

000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume



r 090N610N140N090N070N 54321 =====

Volume f

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area Volume



29

000

020

040

060

080

100

Prob

abili

ty

Number

Length

Surface area

Volume


Figure 41Volume f


000

020

040

060

080

00

Prob

abili

ty

1Number

Length

Surface area

Volume


Figure 41Volume f


or 20N640N090N040N030N 54321 =====

30

100


010N1150N0750N150N650N 54321 =====

purities



000

020

040

060

080Pr

obab

ility

Number

Length

Surface area

Volume


31



1=6d2=8d3=10d4=20d5=26 mm




32


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

ut p

ut (w

t)

12-2 mm

2-236 mm

236-315 mm



33


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


34


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



35


6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



36


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



37


2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



38


200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N540N20N10N10N 54321 =====

39

200 250 300 350 400Number of layer

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 060N440N20N150N150N 54321 =====


200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm

For 020N430N150N20N20N 54321 =====

40

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N3480N1820N20N250N 54321 =====

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 020N230N150N20N40N 54321 =====

41

200 250 300 350 400

Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 10N70N090N060N050N 54321 =====

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


42

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


43

200 250 300 350 400Number of layer

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 010N140N150N150N550N 54321 =====

200 250 300 350 400Number of layer

6000

8000

10000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm


For 20N640N090N040N030N 54321 =====

44

200 250 300 350 400Number of layer

000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

12-2 mm

2-236 mm

236-315 mm



45

000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer




000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 060N440N20N150N150N 54321 =====

46


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N430N150N20N20N 54321 =====


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N3480N1820N20N250N 54321 =====

47


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 020N230N150N20N40N 54321 =====

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



48


000

2000

4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



4000

6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 090N610N140N090N070N 54321 =====

49


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 150N680N090N0450N0350N 54321 =====


000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


For 010N140N150N150N550N 54321 =====

50


6000

8000

10000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer



000

2000

4000

6000

8000

Perc

ent o

utpu

t (w

t)

two layer

three layer

four layer


51


ldcm adcm vdcm


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


52


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer



030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


53


030

035

040

045

050

055

Poro

sity

two layer

three layer

four layer


54

200 250 300 350 400Number of layer

030

035

040

045

050Po

rosi

ty

055

N1=01N2=01N3=02N4=054N5=006

N1=015N2=015N3=02N4=044N5=006

N1=02N2=02N3=015N4=043N5=002

N1=025N2=02N3=0182N4=0348N5=002

N1=04N2=02N3=015N4=023N5=002

N1=005N2=006N3=009N4=07N5=01

N1=05N2=0132N3=0118N4=023N5=002

N1=007N2=009N3=014N4=061N5=009

N1=0035N2=0045N3=009N4=068N5=015

N1=055N2=015N3=015N4=014N5=001

N1=003N2=004N3=009N4=064N5=02

N1=065N2=015N3=0075N4=0115N5=001


55





experimentally


equation

Absolute error

1 0395 03908 00106 2 04 039608 00098 3 0412 04013 00259 4 0429 04277 0003 5 0452 04435 00187

56






Absolute error

1 0395 04338 00982 2 04 043908 00977 3 0412 04443 007839 4 0429 04707 00972 5 0452 04865 00763


⎟⎠⎞

⎜⎝⎛+=


32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



represent by 2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd


57



experimentally


equation

Absolute error

1 04146 04069 00186 2 043 04175 002907 3 04466 0428 00416 4 04905 04812 001896 5 05232 05131 00193




No Porosity



equation

Absolute error

1 04146 04929 01888 2 043 05035 01709 3 04466 05141 01511 4 04905 056725 01565 5 05232 05991 01451


58

⎟⎠⎞

⎜⎝⎛+=


32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd



2

⎟⎠⎞

⎜⎝⎛

Dd and

3

⎟⎠⎞

⎜⎝⎛

Dd




59

000 010 020 030

Ratio of dD

038

040

042

044

046

Poro

sity

experimentally

theoretically


129 cm

000 010 020 030 040 050Ratio of dD

040

042

044

046

048

050

052

054

Poro

sity

experimentally

theoretically


cm

60


61


036

040

044

048

052

056

Poro

sity

D=64 cm

D=129 cm



500 1000 1500 2000 2500 3000


036

040

044

048

052

Poro

sity

D=64 cm

D=129 cm



62


63


025

030

035

040

045

Poro

sity

d1=26d2=6 mm

d1=20d2=6 mm

d1=10d2=6 mm

d1=8d2=6 mm

d1=26d2=8 mm

d1=20d2=8 mm

d1=10d2=8 mm

d1=26d2=10 mm

d1=20d2=10 mm

d1=26d2=20 mm



64


000

010

020

030

040

050Po

rosi

ty


65

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

Poro

sity

048



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

036

038

040

042

044

046

048

Poro

sity



66

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

040

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

037

038

039

040

041

042

043

044

045

046

047

048

049

Poro

sity



67

010 020 030 040 050 060 070 080 090 100Volume Fraction

038

040

042

044

046

048

050

Poro

sity



2d

010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

68

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

69

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

042

044

046

048

050

Poro

sity



2d

000 010 020 030 040 050 060 070 080 090 100Volume Fraction

045

046

047

048

049

050

Poro

sity



2d

70


V 2 Porosity



Absolute error


Latif equation

Absolute error

1 0471 04236 01 01492 06832 08855 04454 0416 0066 01675 0624 07746 0408 03985 00233 01869 05419 06672 04176 03767 00979 02078 05023 05631 03797 03545 0066 02297 0395

05 03893 03354 01384 02439 03735 03642 03989 03231 019 0277 03055 02691 03616 0305 0156 03023 01639 01768 04099 03379 01756 03285 0198 0087 04443 03722 01622 0355 02



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 041196 0125 01912 0594 08919 04233 04063 004 02154 04911 07858 04338 03925 0095 0225 0481 06815 038 03745 00145 02431 036 0579 04083 03557 01288 02619 0358 04783 03773 0339 0102 02813 02544 03794 0364 03279 0099 03014 0172 02821 03938 03266 01706 0322 01823 01865 04037 0339 016 0343 01503 00925 0429 03678 0125 03644 02566

71



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0471 03974 0156 045386 00364 08919 0483 03959 0179 045035 0067 07858 0462 039212 015 0446 00339 06815 0457 03869 0153 04418 00332 0579 0434 03815 0121 04367 00064

04783 0447 03769 0157 04312 00354 03738 0442 037441 0154 042488 00396 028209 0438 03751 0143 04187 00438 01865 0449 03791 0157 041169 00849 00925 0462 03871 0162 040417 0125

72



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error




Equation

Absolute Error


Latif Equation

Absolute Error

1 04872 04146 0149 02508 04852 09 04809 04107 0146 02727 0432 08 0494 04008 0189 0277 0439 07 0458 03872 0154 02907 03652 06 0451 03722 0175 03046 0325 05 04129 03582 0132 03187 0228 04 04057 03486 01407 03329 0179 03 03984 03465 01302 03472 01285 02 04133 03555 0139 03614 0126 01 0442 03783 0144 03754 01507

73



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 04965 04322 0129 02417 0513 08923 04787 04276 01067 02656 0445 07864 04492 04156 00748 02708 03972 06824 04611 04007 00655 02857 06902 058 04316 03845 01091 03007 03033 0479 04387 03705 01554 03157 02804 03803 0421 03615 01413 03308 0214 02829 04259 036104 01522 03457 01883 01871 04377 03721 0149 03604 01766 0093 04411 03962 0102 03748 01503

74



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error



Experimentally


Equation

Absolute Error


Latif Equation

Absolute Error

1 0488 04407 00969 04648 00475 0894 0479 04394 00827 04608 00182 07894 0479 04359 00899 045505 005 06862 0491 0431 01222 04494 00847 05843 0491 04259 01326 04433 00971 04838 0498 04216 01534 04368 01229 03845 0486 04191 0138 04297 01158 02865 0458 04195 0084 04221 00784 01898 0469 04232 00976 041406 01171 0094 0497 04305 01338 04055 0184

75


( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝




Experimentally


Equation

Absolute Error

1 0471 04481 00486 08855 04454 0439 00144 07746 0408 04314 00573 06672 04176 04236 00144 05631 03797 04161 00958

05 03893 04116 00573 03642 03989 04019 00075 02691 03616 03953 00932 01768 04099 03889 00512 0087 04443 03828 0138

76



Experimentally


Equation

Absolute Error

1 0471 04585 00265 08919 04233 04508 00649 07858 04338 04433 00219 06815 038 04359 01471 0579 04083 04288 00502



Experimentally


Equation

Absolute Error

1 0471 04872 00344 08936 04826 04806 00041 07887 0496 0474 00443 06853 0459 04677 00189 05833 0455 04614 0014 04828 04161 04553 00942 03836 04328 04492 00379 02857 0428 04441 00376 01892 04408 04374 00077 009396 04557 04317 00527

77



Experimentally


Equation

Absolute Error

1 0471 04968 00548 08919 0483 04905 00155 07858 0462 04842 0048 06815 0457 04781 00462 0579 0434 04721 00877



Experimentally


Equation

Absolute Error

1 04872 04595 00568 0894 04699 0452 00381

07894 04416 04446 00068 06862 04233 04374 00333 05843 03899 04304 01038 04838 04036 04483 0111 03845 0378 04167 01024 02866 03869 04101 00599 01898 04358 04036 00739 00943 04402 03972 00977

78



Experimentally


Equation

Absolute Error

1 04872 04702 00349 09 04809 04634 00364 08 0494 04567 00755 07 0458 04517 00137 06 0451 04434 00168 05 04129 04368 00579 04 04057 04302 00604 03 03984 04237 00635 02 04133 04172 00094 01 0442 04108 00352



Experimentally


Equation

Absolute Error

1 04872 04997 00256 09016 04881 04939 00119 08028 04592 04882 00631 07037 046 04825 00489 06043 0461 04767 0034 05044 0428 04709 01137 04043 04295 04652 00831 03037 04305 04594 00671 02029 04315 04536 0051 01016 04325 04478 00353

79



Experimentally


Equation

Absolute Error

1 04965 04686 00561 08923 04787 04612 00366 07864 04492 04541 00108 06824 04611 04471 00304

058 04316 04402 00199 0479 04387 04335 00118

03803 0421 0427 00143 02829 04259 04207 00122 01871 04377 04144 00534 0093 04411 04083 00742



Experimentally


Equation

Absolute Error

1 0496 04795 0033 0898 0466 04729 0015

07971 0465 04664 0003 06963 0464 04599 0009 05957 0429 04535 0056 04956 0424 04472 0056 03958 0428 04408 00309 02963 0427 04346 00185 01972 0426 04384 00063 0098 0425 04222 00058


80


Experimentally


Equation

Absolute Error

1 0488 04979 00245 0894 0479 04917 00272

07894 0479 04856 00121 06862 0491 04796 00244 05843 0491 04736 00362 04838 0498 04678 00605 03845 0486 0462 00486 02865 0458 0456 00044 01898 0469 04507 0039 0094 0497 04452 0104




81

CHAPTER FIVE





82


83


84


85


86


87

APPENDIX [A]


d1 =6 d =8 d =10 d =20 d 5 =26 2 3 4

Example N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5


( )

g


=2

421

5244 (23)

Where ( )( )

2

11 ⎤

⎢⎣

⎡ +=

amamk

1 ⎥⎦minus

dgdmam = rarr am=08

k1 =26244

( )1amam4k2 +

=

k =17777 2

( ) k

212k

2k3 akaak minus+=


k 3 =05944

A-1

( ) 1324 kk2kk times+= k =778544 4

1235 kkk times=


( )

( ) g

k212

kk

2

k212

kk

ciii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+



( )( ) k

1k

212kk

21

k212

kk

cii daaa222

aaa2d times

⎥⎦⎤

⎢⎣⎡ minus++

⎥⎦⎤

⎢⎣⎡ minus+

=minusminus

minusminus


dc ii =1014832 mm



dc i =0924 mm

A-2


N1 =01 N =01 N 3 =02 N =054 N =006 2 4 5



L=1576 For d k =6d =8d =10 m g

( ) LNdL kkk times=

L =003807 k

( ) LNdL mmm times=

L =005076 m

( ) LNdL ggg times=

L =01269 g


nkdk

nmdm

ngdg

kmgi PrPrPr


Where 3rrr kmg =++

P =000147 i

A-3


( ) ( ) ( ) ( ) ( )5254

243

232

221


A=28656


A =001256 k


A =002233 m


A =0069793 g



( ) ( ) ( ) ( ) ( )5354

343

332

321


V=564736


V =000382 k


V =0009066 m

A-4


P =7355E-06 i


sum=

=n

iciicm dpd

1

helliphellip (27)


sum=

=n

icirncm dpnd

1

rarr =ndcm 22045 mm


sum=

=n

icirlcm dpld

1

rarr 2683 mm =ldcm


sum=

=n

iciracm dpad

1

rarr =adcm 2999 mm


sum=

=n

icirvcm dpvd

1

rarr =vdcm 3204 mm

A-5


1 For mono system

a) Furnas equation


Where



Dd =00465

ε=0375+034times00465=03908


Dd =009375

ε=0375+034times009375=04069



V=ε

pT VVV minus=



LdVT2

4π

=

A-6

VT =19594777cmsup3

NdVP3

6π

=

V=1185786 cmsup3

Vp=1959477-1185786=77368 cmsup3

ε =03948


2 For binary system

AJeschar equation



2v1x1

1

ο

ο

εε

εminus

minus=

221 datdat εε

εο+

=

οε =04235


⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

1

1

2

2

2

2

2

ρρ

ρWW

W

v helliphellip (A1)

A-7



0871550

3327852270

71479230

71479230

2

v =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=


( ) 2vsin60

1

2

2

2

dd

vv11

1X π

⎟⎟⎠

⎞⎜⎜⎝

⎛minus+

=

X=010798

ε=03722


B Latif equation


08715502 v =

( )29152002830

1

22196039940

v

ddv

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (211)

( ) 35502660871550196039940

08715509152002830

=⎟⎠⎞

⎜⎝⎛times+=ε

times+

A-8

B-1

APPENDIX [B]


No of Small

Spheres


No of Large

Spheres

Arrangement

dc(mm)

EquationUsed


B-2




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0001 552E-05 19814E-6 5595E-8 10148 0004 0000276 12549E-5 4539E-7 10779 001 0000828 45578E-5 2008E-6 1117 0013 0001122 6437E-5 2951E-6 11913 0025 0002593 1818E-4 10319E-5

B-3


1517 0175 0040363 67017E-3 92966E-4 155 0183 0042406 70416E-3 97406E-4 157 019 0045276 77858E-3 11258E-3

1581 0206 0050573 89134E-3 13144E-3 16496 0208 0051338 9125E-3 13605E-3 1716 0273 0077822 16173E-2 28338E-3 1789 028 0081648 17496E-2 31935E-3 1861 0367 013528 38904E-2 99077E-3 189 0432 0168386 49918E-2 12785E-2

1916 0452 018388 57956E-2 16063E-2 19822 0458 0188661 60023E-2 1677E-2 20715 046 018978 60778E-2 00172 2155 0547 026129 98838E-2 3308E-2 228 0567 0281948 01131 40848E-2

2393 0742 0460723 0232 0103 2424 0743 0462215 02334 010395 2532 0782 0513859 02781 01343 2714 0784 0517589 02823 0138

31 0941 0839388 07105 05855 3417 0994 0978829 09517 091327 3699 09998 099897 099699 09933 403 1 1 1 1

B-4


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 3375E-04 241E-04 9891E-6 292E-07 10148 00135 0001205 62642E-5 237E-06 10779 0027 0002809 17254E-4 777E-06 1117 00371 0004095 2663E-4 127E-05 11913 00641 000837 65705E-4 383E-05


1517 0297 008198 16226E-2 000207 155 0305 00846 16729E-2 0002138 157 0316 009017 1838E-2 000249

1581 0346 01027 19084E-2 0003025 16496 035 010497 1979E-2 0003185 1716 0429 01468 3252E-2 000597 1789 044 015419 35458E-2 0006804 1861 0527 02233 6698E-2 001714 189 0579 02581 80246E-2 002077

1916 0604 02826 94774E-2 002696

B-5

19822 0611 02888 9783E-2 002805 20715 0612 029097 9951E-2 002898 2155 0699 03832 0156 00535 228 0723 041585 01814 00682

2393 0839 05692 02981 013199 2424 0841 0572 03011 013418 2532 0873 06264 03549 01724 2714 0875 06313 036109 01781

31 09601 085665 07036 05526 3417 0995 09765 09403 0889 3699 09997 09978 09949 09898 403 1 1 1 1


26d20d10d8d6d 54321 =====


of the pores (dc)

Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0008 715E-04 34806E-5 11806E-6 10148 0032 0003573 22043E-4 9575E-6 10779 005 0006253 43797E-4 2187E-5 1117 0074 0010063 76797E-4 4177E-5 11913 011 001721 1541E-3 10007E-4

124 0118 0018903 1737E-3 1158E-4 1247 0169 0034268 4231E-3 3978E-4 1277 0183 0037618 4684E-3 44054E-4 1291 0185 0038548 48804E-3 4694E-4 1328 0203 0043312 52671E-3 49851E-4 1389 0307 0084284 14135E-2 18355E-3

B-6

143 032 0088751 14941E-2 19367E-3 144 0325 0091229 15638E-2 2073E-3

1517 0402 0129647 2603E-2 40321E-3 155 0406 0131043 26345E-2 4082E-3 157 0409 0133366 27162E-2 42817E-3

1581 0461 0160681 35045E-2 58661E-3 16496 0463 0162333 35664E-2 6028E-3 1716 0541 0213557 541393E-2 1067E-2 1789 0544 0216655 55592E-2 11145E-2 1861 0655 0326784 01152 33606E-2 189 0684 03508 012599 37007E-2

1916 0695 0364119 013536 41597E-2 19822 0696 0365571 01362 4194E-2 20715 0696 0365974 013658 42179E-2 2155 0807 0512814 02425 9541E-2 228 0817 0530573 025915 01063

2393 09007 0668259 03833 01843 2424 0901 0668796 03839 01848 2532 0909 0685448 040343 020078 2714 0909 0685952 04042 020159

31 0988 094908 08786 079782 3417 09994 0996815 09904 09806 3699 0999992 0999702 09993 099926 403 1 1 1 1



B-7


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 00156 000177 102E-04 376E-06 10148 00531 00074 536 E-04 251E-05 10779 00872 00139 000115 631E-05 1117 01172 00199 000177 104E-04 11913 01718 00337 000353 248 E-04

124 01798 00358 000382 273 E-04 1247 02451 006045 000855 854 E-04 1277 02699 006827 0009795 982 E-04 1291 02737 00701 001025 000106 1328 02955 00774 00115 000119 1389 03999 013 00249 000339 143 04198 01383 00267 000364 144 04258 014227 002802 00039

1517 05208 02021 00471 00078 155 05268 02052 004797 000798 157 05323 02097 004983 000847

1581 05741 02377 005938 00106 16496 05764 023981 00603 00108 1716 06524 03036 00875 00182 1789 06568 03084 00901 001918 1861 07476 042268 01632 00491 189 07822 045896 01825 00557

1916 07927 0476 01967 006329 19822 07947 04787 01986 00641 20715 07949 04794 01993 00646 2155 08676 06013 03032 01214 228 08759 06195 03234 01357

2393 09421 075825 04712 02367 2424 09423 07589 04721 02376 2532 09499 07796 050086 0263 2714 09502 07804 050225 02647

31 09923 095724 0879 07794 3417 09996 0997 098896 09743

B-8

3699 0999992 09998 09996 09989 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0064 00115 926E-04 398E-05 10148 016 00346 0003396 181E-04 10779 0232 00563 000629 389 E-04 1117 028 007165 000849 556 E-04 11913 0352 01005 001363 000105

124 036 01039 00143 00011 1247 04704 01703 0032 000366 1277 04974 01838 003505 000402 1291 0507 01913 003766 00045 1328 0525 02009 003995 000479 1389 06354 028936 007151 00108 143 06489 02984 007419 001125 144 06585 03083 007883 00124

1517 07413 03913 011582 00212 155 07447 039407 011687 002144 157 07519 040338 01223 002313

1581 07795 043288 013633 00267 16496 07819 043619 013839 00274 1716 08233 04915 01713 003785 1789 08269 04977 01761 003985

B-9

1861 08903 062489 02895 009402 189 09059 06508 030879 01017

1916 09169 06794 03421 01224 19822 09183 06823 034496 01238 20715 09187 06839 03474 012582 2155 09505 076866 044823 019002 228 095601 078769 047786 02145

2393 097982 08672 0596 03086 2424 098005 08683 05982 03109 2532 098419 088609 06329 03469 2714 09844 088709 06355 03503

31 09965 09684 0877 073481 3417 099971 09957 09835 09552 3699 0999992 099878 099915 09973 403 1 1 1 1


26d20d10d8d6d 54321 =====


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



B-10


16496 00586 001014 0001475 0000198 1716 00813 00161 0002696 0000419 1789 00846 001718 0002991 0000488 1861 01581 00461 0011897 0002906 189 01751 00516 0013329 0003229 1916 01961 00624 0017629 0004748

19822 01985 00634 0017974 0004849 20715 02 00644 0018493 0005087 2155 02882 01106 0037493 0011964 228 03134 01278 0046666 001628 2393 04457 021438 0091198 0036406 2424 04475 021598 0092306 0037084 2532 04853 024816 0113807 0049717 2714 0488 025114 0116402 00517

31 0831 070035 05782 0469706 3417 0978 095062 091268 08633 3699 0999 09971 09934 098684 403 1 1 1 1



B-11


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0125 00243 196 E-03 821E-05 10148 0224 004998 473 E-03 236E-04 10779 03125 007866 859 E-03 505 E-04 1117 03386 008769 988 E-03 602 E-04

11913 03854 01079 00135 938 E-04 124 038766 01089 001371 958 E-04 1247 056016 02208 004379 5153E-03 1277 058105 0232 00463 5447 E-03 1291 059605 02447 00507 6249 E-03 1328 06022 02482 00516 6354 E-03 1389 069329 0327 00798 00116 143 06988 03309 008103 00118 144 070673 03398 00852 00128 1517 078815 04278 01246 00219 155 07898 04292 01252 00221 157 07969 04392 01309 00238 1581 08089 0453 01376 00255

16496 08099 04546 01386 002577 1716 08314 04856 01571 00315 1789 08333 0489 01598 00326 1861 09126 06605 03136 010407 189 09223 06778 03265 01091 1916 0936 07166 03717 01364

19822 09369 07185 03736 01373 20715 09375 07207 03769 013995 2155 09584 07811 04491 01847 228 09621 07947 04703 020176 2393 098082 08621 05712 02798 2424 09809 08629 05727 028147 2532 09842 08781 06023 03113 2714 09844 08789 06045 03142

31 09965 09666 08665 072 3417 09997 09964 098199 09527

B-12

3699 099999 09997 09989 09971 403 1 1 1 1




Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 969 E-02 1821E-02 2635 E-03 334 E-04 155 0996 1876 E-02 2714 E-03 343 E-04 157 01049 2042 E-02 3081 E-03 411 E-04

1581 01197 2423 E-02 3775 E-03 516 E-04 16496 01219 2496E-02 3947 E-03 55 E-04 1716 0168 3976 E-02 7318 E-03 1188 E-03 1789 01748 426 E-02 8158 E-03 1395 E-03

B-13

1861 025298 8024 E-02 2101E-02 5046 E-03 189 02888 94634 E-02 25108 E-02 6015 E-03

1916 03119 010907 3152 E-02 8378 E-03 19822 03172 01118 32538 E-02 8691 E-03 20715 03189 01132 3334 E-02 9074 E-03 2155 04194 01777 6271 E-02 20202 E-02 228 0449 02025 7736 E-02 27405 E-02

2393 06053 03279 01488 61198 E-02 2424 06075 033 01506 6236 E-02 2532 06536 03784 01862 8423 E-02 2714 0657 0383 01907 8777 E-02

31 08839 07475 06057 04806 3417 09844 09572 09161 0862 3699 099927 09974 09935 09854 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )


124 1693E-03 104E-04 455E-06 25E-07 1247 4192E-03 368 E-04 268E-05 194E-06 1277 50427E-03 443 E-04 32E-05 227E-06

B-14


1517 002793 4119 E-03 512 E-04 599E-05 155 002866 4226 E-03 525 E-04 612E-05 157 003149 4874 E-03 643 E-04 807E-05


1916 01432 004255 00113 292 E-03 19822 01468 004394 00117 3037 E-03 20715 01492 004534 00124 322 E-03 2155 02116 00746 00233 7035 E-03 228 02392 009138 00315 001063

2393 0364 01645 00657 002514 2424 0367 016693 00672 002601 2532 04221 020888 00927 004007 2714 04282 021489 00975 004348

31 07426 058335 04425 033578 3417 09507 090032 08284 076076 3699 09966 09912 097228 09667 403 1 1 1 1



B-15


Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 01664 00483 6308E-03 379E-04 10148 03025 010103 001548 1115E-03 10779 04386 01669 002982 2551 E-03 1117 04757 018609 003427 3026 E-03

11913 055 023402 004817 4883 E-03 124 05534 02363 004889 4986 E-03 1247 06804 03593 01024 00157 1277 07176 03893 011327 001752 1291 07266 04007 011973 00192 1328 07368 04094 01231 001981 1389 0806 04988 017499 003367 143 08162 05097 018026 003484 144 08211 0518 01865 003702 1517 08904 06298 02676 00641 155 08938 06344 02704 006486 157 08988 06447 028016 00691 1581 09082 0661 02927 00736

16496 09089 06625 029425 00743 1716 09278 07032 033357 00918 1789 09291 07069 03383 009455 1861 09615 08113 048969 019566 189 09709 08367 05204 02128 1916 09755 085607 05569 024448

19822 09762 08584 05607 024717 20715 09764 08593 056287 024966 2155 098519 089727 06363 0315 228 09865 09043 06539 03355 2393 09952 09517 07687 046318 2424 09953 09521 07697 04648 2532 09966 09609 07974 05049 2714 099662 09613 079908 050799

31 09994 09908 09418 08257 3417 099996 0999 099347 09753 3699 099999 09998 09997 099876

B-16

403 1 1 1 1



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )



1517 0022 3071 E-03 366 E-04 399E-05 155 00228 3171 E-03 377 E-04 41E-05 157 0026 3863 E-03 496 E-04 593E-05


B-17

1916 01237 00351 9007 E-03 2206 E-03 19822 01285 003683 9506 E-03 2334 E-03 20715 01321 003883 0010405 2692 E-03 2155 01813 006036 001805 5094 E-03 228 021198 007786 00261 8392 E-03

2393 03226 013844 005298 0018946 2424 0327 0141995 00551 002019 2532 03965 01912 008349 003457 2714 04073 02012 009099 00395

31 06694 04884 03457 023974 3417 09152 08384 074928 065205 3699 0992 09806 09624 093510 403 1 1 1 1


26d20d10d8d6d 54321 =====



Probability Due to

Number( rnP )

Probability Due to

Length( rlP )

Probability Due to

Area ( raP )

Probability Due to

Volume( rvP )

093 0275 00997 1579 E-02 1035E-03 10148 0465 019178 352 E-02 273 E-03 10779 05598 02493 5042 E-02 439 E-03 1117 06037 02776 58399 E-02 532 E-03 11913 06476 03130 7086 E-02 713 E-03

124 065 03159 7195 E-02 73 E-03 1247 07967 049237 01651 276E-02 1277 08077 05034 01699 285 E-02 1291 08203 052338 01837 324 E-02 1328 08254 05288 01862 329 E-02

B-18

1389 08927 06374 02626 5517 E-02 143 08952 06408 02646 5566 E-02 144 09011 065308 02759 599 E-02

1517 09347 07209 03356 8163 E-02 155 09351 07216 03361 8179 E-02 157 0938 07293 03449 859 E-02

1581 09458 074602 03605 9203 E-02 16496 09465 07479 03628 9319 E-02 1716 09542 076879 03873 01051 1789 09549 07711 03909 01073 1861 09807 08752 057403 024065 189 09826 08817 058359 024645

1916 09871 090525 06374 029738 19822 09873 09059 06388 029849 20715 098749 09073 064277 030335 2155 09935 09393 071789 037627 228 09945 09466 073997 040413

2393 099746 09666 079864 04753 2424 09975 096699 08003 047795 2532 099802 09715 08175 050514 2714 099805 097177 081877 05077

31 09996 09922 09387 07989 3417 099996 099916 09916 09658 3699 099999 09999 09994 09976 403 1 1 1 1

B-19



060N540N20N10N10N 54321 =====


26d20d10d8d6d 54321 ===== 060N440N20N150N150N 54321 =====



26d20d10d8d6d 54321 ===== 020N430N150N20N20N 54321 =====



26d20d10d8d6d 54321 ===== 020N3480N1820N20N250N 54321 =====



B-20


020N230N150N20N40N 54321 =====



26d20d10d8d6d 54321 ===== 10N70N090N060N050N 54321 =====



26d20d10d8d6d 54321 ===== 020N230N1180N1320N50N 54321 =====



26d20d10d8d6d 54321 ===== 090N610N140N090N070N 54321 =====


B-21


150N680N090N0450N0350N 54321 =====



26d20d10d8d6d 54321 ===== 010N140N150N150N550N 54321 =====



26d20d10d8d6d 54321 ===== 20N640N090N040N030N 54321 =====



26d20d10d8d6d 54321 ===== 010N1150N0750N150N650N 54321 =====


B-22




Diameter 64 cm

Porosity

00465 03908 009375 04069 0062 039608 0125 04175 00775 04013 015625 0428 015503 04277 03125 04812 020155 04435 040625 05131


V2

Calculated Porosity


Calculated Porosity


100 1 04236 01492 90 08855 0416 01675 80 07746 03985 01869 70 06672 03767 02078 60 05631 03545 02297 50 05 03354 02439 40 03642 03231 0277 30 02691 0305 03023 20 01768 03379 03285 10 0087 03722 0355

B-23


V2

Calculated Porosity


Calculated Porosity


100 1 041196 01912 90 08919 04063 02154 80 07858 03925 0225 70 06815 03745 02431 60 0579 03557 02619 50 04783 0339 02813 40 03794 03279 03014 30 02821 03266 0322 20 01865 0339 0343 10 00925 03678 03644


d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 04034 03677 90 08936 04009 03729 80 07887 03945 03776 70 06853 03858 038185 60 05833 03765 038549 50 04828 03687 038855 40 03836 036403 039097 30 02857 03644 03927 20 01892 037095 03937 10 009396 03842 0394

B-24


Wight Percent of

Sphere with d2=6

V2

Calculated Porosity


Calculated Porosity


100 1 03974 045386 90 08919 03959 045035 80 07858 039212 0446 70 06815 03869 04418 60 0579 03815 04367 50 04783 03769 04312 40 03738 037441 042488 30 028209 03751 04187 20 01865 03791 041169 10 00925 03871 040417


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04262 01958 90 0894 04208 02195 80 07894 04074 02287 70 06862 03898 02463 60 05843 0371 02646 50 04838 03542 02836 40 03845 03428 030329 30 02866 03409 032346 20 01898 03529 034409 10 00943 03816 036508

B-25



d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 04146 02508 90 09 04107 027275 80 08 04008 027708 70 07 03872 02907 60 06 03722 030458 50 05 03582 031867 40 04 03486 033289 30 03 03465 034716 20 02 03555 036138 10 01 03783 03754


d2=8

V2

Calculated Porosity


Calculated Porosity


100 1 0406 048236 90 09016 0405 047621 80 08028 04024 0469585 70 07037 03986 046247 60 06043 03944 04548 50 05044 039064 044666 40 04043 03883 043792 30 03037 03881 04286 20 02029 03909 0418686 10 01016 03969 040818

B-26


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04322 02417 90 08923 04276 026557 80 07864 04156 0270856 70 06824 04007 028572 60 058 03845 030072 50 0479 03705 031573 40 03803 03615 033078 30 02829 036104 034568 20 01871 03721 0360376 10 0093 03962 037476


d2=10

V2

Calculated Porosity


Calculated Porosity


100 1 04206 030959 90 0898 04175 031919 80 07971 04096 032859 70 06963 03985 03378 60 05957 03864 0346846 50 04956 03755 03555 40 03958 03683 036386 30 02963 03675 037172 20 01972 03764 0379 10 0098 03932 03857

B-27


d2=20

V2

Calculated Porosity


Calculated Porosity


100 1 04407 046483 90 0894 04394 04608 80 07894 04359 045505 70 06862 0431 044944 60 05843 04259 04433 50 04838 04216 043676 40 03845 04191 042969 30 02865 04195 0422125 20 01898 04232 0414064 10 0094 04305 0405504



Diameter of Sphere

(mm)





6 387 01 39 8 193 01 19 10 125 02 25 20 29 054 16 26 19 006 1


(mm)

Weight of

Spheres Input(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 309 26 245

2-236 144 1389 1175 60 236-315 457 40 32 17



Impurities (mm)




12-2 99 8333 7853 2-236 9646 81597 4167

236-315 8753 70 37199 Porosity 0497 047 0451

C-1

Experiment 2


Diameter of sphere

(mm)





6 387 015 58 8 193 015 29 10 125 02 25 20 29 044 13 26 19 006 1


Impurities (mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 307 26 245 2-236 144 133 1152 57



Impurities (mm)




12-2 9839 8173 641 2-236 9236 80 39833

236-315 8315 6565 35 Porosity 0479 0467 0449

C-2







6 387 02 77 8 193 02 39 10 125 015 19 20 29 043 13 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 305 25 19 2-236 144 126 110 537



Impurities (mm)




12-2 9775 80128 6089 2-236 875 7639 3729

236-315 8293 6411 3414 Porosity 045 0444 0429

C-3







6 387 025 97 8 193 02 39 10 125 0182 23 20 29 0348 10 26 19 002 1


(mm)

Weight of spheres

Input (gm)

Weight of spheres

out put (gm)

Weight of spheres

out put (gm)

Weight of

spheres out put

(gm) 12-2 312 301 235 188 2-236 144 1258 100 50



Impurities (mm)




12-2 9647 7532 6025 2-236 8736 6944 3472

236-315 7789 4814 3042 Porosity 044 0424 0416

C-4







6 387 04 154 8 193 02 39

10 125 015 19 20 29 023 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 29 21 16 2-236 144 115 80 45



Impurities (mm)




12-2 9295 6731 5224 2-236 7986 5556 3125

236-315 7287 4376 2626 Porosity 0427 0418 0399

C-5



Number of Sphere

Cover the Base of

Cylinder



6 387 005 19 8 193 006 12 10 125 009 11 20 29 07 20 26 19 01 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 27 26 2-236 144 142 122 1172



Impurities (mm)




12-2 100 8654 8333 2-236 9861 8422 8139

236-315 9256 7834 6258 Porosity 0518 049 0484

C-6






Number of


6 387 05 193 8 193 0132 26 10 125 0118 15 20 29 032 7 26 19 002 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm)

Weight Of

Spheres out put

(gm) 12-2 312 26 19 119 2-236 144 110 62 34



impurities (mm)

Percent output

(Wt )

Percent output

(Wt )


12-2 8333 609 3814 2-236 7639 43 2361

236-315 70 3939 2188 Porosity 0408 0387 0376

C-7







6 387 007 27 8 193 009 17

10 125 014 18 20 29 061 18 26 19 009 2


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 265 25 2-236 144 140 118 100



Impurities (mm)




12-2 9936 849 8013 2-236 9722 8194 6944

236-315 8972 7221 547 Porosity 0504 0485 0475

C-8







6 387 0035 14 8 193 0045 9 10 125 009 11 20 29 068 20 26 19 015 3


(mm)

Weight of

Spheres Input (gm)

Weight of Spheres

Out put (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 31 28 265 2-236 144 143 126 120



Impurities (mm)




12-2 100 8974 8494 2-236 993 875 8333

236-315 941 7877 6346 Porosity 0535 0512 0499

C-9

Experiment 10



Number of Sphere

Cover the Base of

Cylinder



6 387 055 212 8 193 015 29 10 125 015 19 20 29 014 4 26 19 001 1


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 236 16 84 2-236 144 105 50 276



Impurities (mm)




12-2 7564 5128 2692 2-236 7292 3472 1917

236-315 6565 3282 1575 Porosity 0385 0374 035

C-10

Experiment 11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm) 12-2 312 312 297 287 2-236 144 144 140 125



Impurities (mm)




12-2 100 9519 9199 2-236 100 9722 868

236-315 9628 8534 6783 Porosity 0535 0512 0499

C-11







6 387 003 12 8 193 004 8 10 125 009 11 20 29 064 19 26 19 02 4


(mm)

Weight of

Spheres Input (gm)

Weight of

Spheres Out put

(gm)

Weight of

Spheres Out put

(gm)

Weight Of

Spheres Out put

(gm) 12-2 312 22 155 8 2-236 144 100 45 26



Impurities (mm)




12-2 7051 4968 2564 2-236 6944 3125 1806

236-315 6127 2626 1422 Porosity 0346 0331 0323

C-12



No Diameter of




1 6 15 42 2 8 262 2996 3 10 3128 135 4 20 659 112 5 26 9 63


64cm




1 6 0395 0415 2 8 04 043 3 10 0412 0447 4 20 0429 049 5 26 0452 0523

C-13



08855 1 880 15 195948 04454 07746 3 782 15 195948 0408 06672 4 684 15 195948 04176 05631 6 587 15 195948 03797

05 7 489 15 195948 03893 03642 8 391 15 195948 03989 02691 10 293 15 195948 03616 01768 11 196 16 20901 04099 0087 13 98 18 235137 04443



08919 3 880 15 195948 04233 07858 5 782 15 195948 04338 06815 8 684 14 18288 038 0579 11 587 15 195948 04083

04783 13 489 14 18288 03773 03794 16 391 14 18288 0364 02821 19 293 15 195948 03938 01865 21 196 15 195948 04037 00925 24 98 16 20901 0429

C-14



08936 23 880 165 21554 04826 07887 45 782 17 22207 0496 06853 68 684 16 20901 0459 05833 91 587 16 20901 0455 04828 113 489 15 195948 04161 03836 135 391 155 20248 04328 02857 158 293 155 20248 0428 01892 181 196 16 20901 04408 009396 203 98 165 21554 04557



08919 45 880 165 21554 048307858 90 782 16 20901 046206815 135 684 16 20901 04570579 180 587 155 20248 0434

04783 225 489 16 20901 044703738 270 391 16 20901 0442028209 315 293 16 20901 043801865 360 196 165 21554 044900925 405 98 17 22207 0462

C-15



0894 1 405 17 22207 0469907894 3 360 17 22207 0441606862 4 315 16 20901 0423305843 6 270 16 20901 0389904838 7 225 16 20901 0403603845 8 180 15 195948 037802866 10 135 16 20901 0386901898 11 90 17 22207 0435800943 13 45 18 235137 04402



09 3 405 18 235137 0480908 5 360 18 235137 049407 8 315 17 22207 045806 11 270 17 22207 045105 13 225 155 20248 0412904 16 180 155 20248 0405703 19 135 155 20248 0398402 21 90 155 20248 0413301 24 45 155 20248 0442

C-16



09016 23 405 18 235137 0488108028 45 360 17 22207 0459207037 68 315 17 22207 04606043 91 270 17 22207 046105044 113 225 16 20901 042804043 135 180 16 20901 0429503037 158 135 16 20901 0430502029 181 90 16 20901 0431501016 203 45 16 20901

04325



08923 1 204 17 22207 0478707864 3 181 17 22207 0449206824 4 158 17 22207 04611

058 6 135 17 22207 043160479 7 113 17 22207 04387

03803 8 91 16 20901 042102829 10 68 17 22207 0425901871 11 45 17 22207 043770093 13 23 18 235137 04411

C-17



0898 3 204 17 22207 046607971 5 181 17 22207 046506963 8 158 17 22207 046405957 11 135 16 20901 042904956 13 113 16 20901 042403958 16 91 16 20901 042802963 19 68 16 20901 042701972 21 45 16 20901 04260098 24 23 16 20901 0425



0894 1 24 17 22207 047907894 3 21 18 235137 047906862 4 19 18 235137 049105843 6 16 19 2482 049104838 7 13 19 2482 049803845 8 11 18 23514 048602865 10 8 18 23514 045801898 11 5 18 23514 04690094 13 3 2 2612 0497

C-18


Program number one


D-1


D-2


D-3

Program Number Two


D-4


D-5


D-6


D-7

الخلاصة






⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus=

n

3633 d

dexp1H











(42)⎟⎠⎞

⎜⎝⎛+=

Ddf 34580380170ε

32

1127869406134750 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (43)


⎟⎠⎞

⎜⎝⎛+=

Ddf 33190388290ε (45)

32

11278694061333180 ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛+=

Dd

Dd

Ddfε (46)





( )204068012830

1

22054660454880

v

f ddv

minus

⎟⎟⎠

⎞⎜⎜⎝

⎛+=ε (47)

شكر وتقدير











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Pore Size and Porosity in Packed Bed - Nahrain University

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