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Brazilian Journal of Chemical EngineeringPrint versionISSN 0104-6632

Braz. J. Chem. Eng. vol. 15 no. 3 So Paulo Sept. 1998

doi: 10.1590/S0104-66321998000300002

CALCULATING CAPACITY TRENDS IN ROTARYDRYERS

C.R.F. PACHECO1

and S.S. STELLA2

1Escola Politcnica da USP - Departamento de EngenhariaQumica,

Caixa Postal 61548 - CEP 05424-970Phone (011) 818-5765 - Fax (011) 211-3020 - S.Paulo - SP-

Brazil2Hercules Inc., Av. Roberto Simonsen, 500 CEP 15140-000 -

Paulnea - SP - Brazil

(Received: August 15, 1997; Accepted: April 19, 1998)

Abstract - This paper provides a methodology developed for the calculation of the feed rateand of the exit air conditions in an adiabatic rotary dryer, which operates with granular, non-porous solids having only unbound surface moisture. Some aspects related to the algorithmare also discussed in greater detail, such as the behavior of the wet-bulb temperatures alongthe dryer and the selection of initial values for the iterative loops. The results have beencompared with published data from commercial rotary dryers, and predictions compare within10% of the available data. The methodology can be used to evaluate trends in the behaviorof a rotary dryer where the operating parameters vary, and it is useful for the practicalengineer, who has to manage several problems commonly encountered in the operation of arotary dryer installed in a chemical plant.Keywords:Rotary dryers, drying systems, drying analysis.

INTRODUCTION

Ongoing market globalization has been pushing companies for a reduction in prices, alongwith an increase in quality. For any industrial operation this scenario implies a reduction ofproduction costs and a tightening of the specification ranges.

The rotary dryer is a piece of equipment which is of relatively common use in the chemicalprocess industries, due to its simplicity and versatility in handling different types of solids.The ability to estimate its operating characteristics is of major importance either in theproduction planning of an existing plant or in the design of a new one.

The purpose of this paper is to develop an algorithm for estimating the production capacity ofexisting rotary dryers.

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For a practical engineer involved with either operation or design of chemical plants, thisalgorithm may be helpful in several situations:

If one or more process conditions of an existing rotary dryer change, what is the newcapacity for the same product specifications?

a.

During the process of purchasing a new rotary dryer several proposals are normallyreceived. Can the offered systems do the specific job?

b.

There is a possibility of buying a second hand rotary dryer from a given supplier. In thiscase, how much product in specific conditions could be processed?c.

Literature on rotary dryers is primarily focused on design methodology and on fundamentalparameters needed to understand the physical phenomena involved in this equipment. In thispaper we consider an alternative point of view which represents a contribution to the analysisof the performance of existing rotary dryers.

We developed a new algorithm using an integral analysis methodology and assuming anadiabatic dryer operating with granular, non-porous solids. A comparison between the resultsobtained using our procedure and the available data from commercial rotary dryers wasshown to be quite satisfactory for the purposes above.

This paper is organized as follows: firstly, we present the equations which describe thebehavior of a rotary dryer; secondly, a careful discussion of the temperature profiles in thedrying region is performed. Then, we develop the structure of the algorithm and an analysisof the physical conditions which establish the restrictions needed to assure convergence. Acomparison to commercial rotary dryers is performed, and finally, an example of utilization isgiven. A copy of the executable program is available to the reader upon request.

THEORETICAL BACKGROUND

Tsao and Wheelock (1967) presented a set of general equations which describe the behaviorof a rotary dryer.

Here, we use those equations to calculate the production capacity of an existing rotary dryerof known diameter and length in different scenarios.

A rotary dryer operating with granular, non-porous solids with unbound surface moisture maybe divided, in a simple scheme, in three zones:

I - a first one where the solids are heated to the wet-bulb temperature of the drying airwithout losing any moisture,

II - a second one where the solids lose all the desired moisture while remaining at the wet-bulb temperature of the air and

III - a third one where again the temperature of the solids rises without any further moisture

loss.Figure 1 sketches the temperature profiles in the three before mentioned zones of suchsimplified model, for both counter flow and parallel flow. In the equations below, thedistinction between the two stream arrangements is made labeling the variable Sg whichassumes the value +1 for the counter flow arrangement and -1 for the parallel flow.




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Figure1:

Temperature profiles in the three zones of the rotary dryer as proposed by the model.

In both cases the following parameters are assumed to be known:

dryer geometry: length (Z) and internal diameter (D);for the solids: specific heat (CS) , inlet and outlet moisture content (X1and X2forcounter flow or X2and X1for parallel flow) and temperature (TS1and TS2for counter flowor TS2and TS1for parallel flow);

for the drying air: mass flow rate (G), inlet temperature (T2), ambient temperature (T3)and relative humidity (WR3).

The algorithm calculates the mass flow rate of solids (L) that can be processed in the dryerand the air temperature (T1) and humidity (W1) at the outlet.

For the situations sketched in Figure 1the following equations apply:

Overall water balance:

(1)

Overall enthalpy balance:

(2)

where: HS= enthalpy of the solids

H = enthalpy of the air, calculated from its temperature and humidity

Enthalpy balance between points 2 and 4:




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(3)

and between 4 and 5:

(4)

The kinetics of the process is expressed in terms of the Number of Heat Transfer Units (NT)and the Length of a Transfer Unit (LT).

The Number of Heat Transfer Units (NT) is related to the fraction of the initial heat transferdriving force existing at the air outlet. This is defined as:

(5)

In order to integrate equation (5), Tsao and Wheelock (1967) have assumed that the heatcapacities of both streams have little variation along the dryer. With this assumption, thetemperature profiles become linear and integration for each zone of the dryer gives,respectively:

(6)

(7)

(8)

The Length of a Transfer Unit (LT) is defined by:

(9)

where: CG= specific heat of the air, kJ kg-1oC-1

GS= air mass velocity, kg m-2 s-1,

Ua= overall heat transfer coefficient, W m-3oC-1.

Several methods for estimating the overall heat transfer coefficients have been described andwere summarized by Baker (1983). According to him, none of the correlations reviewed in hisarticle can be recommended with reasonable degree of confidence. However, the correlationproposed by Friedmann and Marshall (1949) is considered the most reliable, since it is basedon extensive and careful experimental data. Their correlation has the following form:

(10)

where K=244.7 for Ua in W m-3oC-1, GSin kg m

-2 s-1 and D in m.

The above correlation is valid for peripheral shell speeds between 0.2 and 0.5 m s -1 andholdups between 2 and 8 %.

The constant (K) takes into account factors that could influence the available heat transferarea, namely: particle size distribution, shell rotation speed, material holdup in the dryer andshape and number of flights.

The dryer length is related to NTand LTby:

, (11)

where




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(12)

With this set of equations, it is possible to fully describe the behavior of a rotary dryer.

Analysis Of The Temperature Profiles

One point that needs to be discussed more profoundly is the behavior of the wet-bulbtemperature along zone II of the dryer.

The temperature profiles shown in Figure 1suggest that the wet-bulb temperature decreasesin the direction of the air flow along zone II. In fact, the wet-bulb temperature can rise or falldepending on the operating conditions. This behavior may be understood in the followingway:

The effect of drying on a material for a counter flow arrangement implies that:

X5 > X4(13)

Developing this inequality using the enthalpy balance between points 4 and 5, one obtains:

(14)

where: Cw= water specific heat, kJ kgoC-1

TW= air wet-bulb temperature,oC

Rearranging the above inequality and taking into account that, for a given pressure, H can becalculated as a function of TWonly, the results are:

(15)

where:

(16)

The left-hand side of inequality (15) is a function of TW4and TW5, where r is a parameter.Therefore, we can write:

f(TW4,TW5) >0 (17)

As stated above, drying can only occur if condition (17) is met.

Figure 2shows a plot of the above inequality for three different values of r, in the full linesrepresents the region where f(Tw4, Tw5) >0 and in the dotted lines the negative one. The

behavior of f(TW4,TW5) and the resulting wet-bulb temperature profile are summarized asbelow.

The same analysis would show a similar behavior for a dryer operating in parallel flowarrangement.

value of r f(TW4,TW5) wet-bulb temperature along zone II

0.04 >0 for any TW4 > TW5 increases

0.4 >0 for TW4> TW5in some regions and forTW5> TW4in other regions

may increase or decrease




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4.0 >0 for any TW5 > TW4 decreases

Figure 2a:

Study of the variation of the function f(TW4, TW5) for a selected value of the parameter r= 0.04.

Figure 2b:





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Figure 2c:


ALGORITHM STRUCTURE

The algorithm developed in this work calculates the product flow rate and the exit airconditions for an adiabatic rotary dryer operating with granular, non-porous solids having

only unbound surface moisture.

The methodology assumes the knowledge of the following data:

dryer geometry:

- length (Z)

- internal shell diameter (D)

solids conditions:

- specific heat (CS)

- dry-basis moisture content at the inlet (Xi) and at the outlet (Xo)

- temperature at the inlet (TSi) and at the outlet (Tso )

drying air conditions:

- pressure (P)

- surrounding air temperature (T3) and relative humidity (WR3)

- temperature (T2) after the air heater

- mass flow rate (G)

stream arrangement: counter flow or parallel flow

Then the following assignments are done, based on the stream arrangement:




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variable counter flow parallel flow

SG +1 -1

TSi TS1 TS2

Xi X1 X2

TSo TS2 TS1

Xo X2 X1

The algorithm follows the steps:

1.Calculate:

- the air mass velocity GSby the expression G / (p D2/ 4).

- the volumetric heat transfer coefficient Uausing equation (10).

- the length of the transfer unit LTusing equation (9).

- the number of heat transfer unit NTusing equation (11).

2.With P, T3and WR3, calculate all psychrometric properties for the air at point 3 (ambientair) (psychrometric chart).

3.With P, T2and W2= W3, calculate all psychrometric properties for the air at point 2 (dryerinlet) (psychrometric chart).

4.Assume initial values of TW4and the solids mass flow rate L.

5.Calculate:

- W1using equation (1), the overall water balance.

- the solids enthalpy at point 2 HS2by the expression ( CS+ X2) TS2.

6.Assume TS4= Tw4(model hypothesis).

7.Calculate:

- HS4the solids enthalpy at point 4 by the expression ( CS+ X2) TS4.

- H4using equation (3), enthalpy balance between points 2 and 4.

8.Assume W4= W2(model hypothesis).

9.With P, H4 and W4, calculate all psychrometric properties for the air at point 4, and inparticular the recalculated value Tw4c(psychrometric chart).

10.Use the values of Tw4and Tw4cto calculate a new value of Tw4and return to step 6 untilconvergence is attained.

11.Assume the initial value of Tw5.

12.Assume TS5= TW5 (model hypothesis).

13.Calculate:




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- H5using equation (4) enthalpy balance between points 4 and 5.

14.Assume W5= W1(model hypothesis).

15.With P, H5 and W5, calculate all psychrometric properties for the air at point 5 and inparticular the recalculated value Tw5c(psychrometric chart).

16.Use the values of Tw5and Tw5cto calculate a new value of Tw5and return to step 12 untilconvergence is attained.

17.Calculate the Number of Heat Transfer Units for zones I, II and III.

- NTIIIusing equation (8).

- NTIIusing equation (7)

- NTIusing equation (12)

18.Assume the initial value of T1.

19.Calculate T1Cusing equation (6) in the form below:

20.Use the values of T1 and T1c to calculate a new value of T1and return to step 19 untilconvergence is attained.

21.With P, T1 and W1, calculate all psychrometric properties for the air at point 1, and inparticular H1(psychrometric chart).

22.Calculate:


- The reiterative value of the solids mass flow rate Lcusing equation (2) overall enthalpybalance.

23.Use the values of the solids mass flow rate L and Lc to calculate a new value of L andreturn to step 5 until convergence is attained.

SELECTION OF INITIAL VALUES FOR THE ITERATIVE LOOPS

The algorithm has four iterative loops (TW4, TW5, L and T1) and some precautions had to be

taken in the selection of the initial values in order to assure the convergence of the loops.The use of constant values would not assure convergence for any arbitrary set of operatingconditions. To overcome this difficulty, the initial values are calculated based on the analysisof physical processes occurring in the dryer.

In that sense, the first estimate for TW4 is made under the following assumptions (refer toFigure 3):

At any point in the dryer the temperature TSof the solids must be above the dew pointtemperature TDof the air in order to avoid condensation and in particular for point 4,(TS4> TD4).

1.

TS4= TW4as assumed by Tsao and Wheelock (1967).2.TD4= TD2since there is no change on moisture content between points 2 and 4 (the dew

point for the air inlet is represented by point 6 in Figure 3).

3.

Consequently, the air is cooled at constant moisture from point 2 to point 4, TW4< TW2.4.




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The first three conditions imply that TW4 > TD2, and condition 4 implies that TW4 < TW2.Therefore, the first estimate for TW4is taken as the average between TW2and TD2.

The first estimate for L is calculated from the global mass balance, choosing a value for W1,based on the following assumptions:

W1> W2because moisture is added to the air during the drying process.1.The value for TW4estimated as described above, determines the maximum outletmoisture attainable in an ideal drying process (shown in Figure 3as point 8). Then, itfollows that W1< W8.

2.

Therefore, W1 is taken as the average between W2 and W8, and the value of L is thencalculated.

Since TW5 is close to TW4, the initial value for TW5 is taken as equal to TW4 (as in fact thetemperatures would be the same if the change in the total enthalpy of the solids stream werenegligible in an adiabatic dryer).

The initial value of T1 is obtained from equation (6). But since it cannot be put in a formwhere T1 is isolated on one side, an iterative loop is used. Among the several forms ofrearranging equation (6) in order to have T

1on the right-hand side, we have chosen this

particular one with a unique root:

(17)

Figure 3:

First estimates for moist air wet-bulb temperature TW4and moist air absolute humidity W1.

Since T1must be greater than TS1, its initial value is taken slightly above TS1.

The use of these criteria on the selection of the initial values for the iterative loops assuresthat the algorithm works properly.

COMPARING THE RESULTS OF THE METHOD WITH PUBLISHED DATA

As we discussed above, this algorithm was developed by assuming an adiabatic dryer. In

reality, commercial rotary-dryers have heat losses to the environment, and therefore, theydo not present rigorous adiabatic behavior.

Page 10 of 15Brazilian Journal of Chemical Engineering - CALCULATING CAPACITY TREN...



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Nevertheless, a comparison between predictions from our algorithm and real world was madeto evaluate the difference between the ideal and real situations.

In order to perform such a comparison, a data set published by Perry (1984) (table 20-13,page 20-33) for seven rotary dryers operating in parallel flow was used. This table furnishes:

dryer length and diameterfor the solids:

- inlet and outlet temperatures

- inlet and outlet wet-basis moisture content

- outlet solids mass flow rate (wet-basis)

for humid air:

- inlet and outlet temperatures

evaporation rate

Table 1shows for the seven dryers the data needed to use the algorithm and the results

obtained.

The solids moisture content and its mass flow rate are presented in dry-basis as requested bythe algorithm.

The specific heat of the solids was not given, and we assumed: 1 kJ kg-1 oC-1, which is arepresentative value for several materials (Perry (1984) table 20-14).

The air mass flow rate was estimated using the evaporation rate and the followingapproximation:

(18)

where: E = evaporation rate in kg s-1

l = water latent heat of evaporation at (TS1+TS2)/2 in kJ kg-1

As can be seen from Table 1, the feed rates agree to within 10% and the air outlettemperatures agree to within 5%. The results obtained are lower consistently than the realones, but these do not affect the validity of an evaluation of trends on dryer operation. Inaddition, the agreement remains along a broad range of dryer sizes.

Table 1: Comparative results between published and calculated values for sevencommercial.

Rotary dryers

Experimental Data Used For Running the Program

Xi: 0.3333

Xo:: 0.005

CS: 1 kJ kg-1oC

TSi: 27oC

TSo: 65oC




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T2: 165oC

P: 101234 Pa

Dryer

#1 #2 #3 #4 #5 #6 #7

D m 1.219 1.372 1.524 1.839 2.134 2.438 3.048

Z m 7.62 7.621 9.144 10.698 12.192 13.716 16.767

G kg s-1 0.957 1.276 1.595 2.233 2.871 3.829 6.062

Feed Rate

Lexperim. kg s-1 0.113 0.144 0.176 0.263 0.351 0.451 0.715

Lcalculated kg s-1 0.108 0.136 0.174 0.244 0.316 0.416 0.650

Difference % 4.42 5.55 1.13 7.22 9.97 7.76 9.09

Air Outlet Temperature

T1 experim.o

C

71 71 71 71 71 71 71

T1 calculatedo

C

67.7 72.6 70.4 70.3 69.7 70.7 71.9

Difference % 4.67 -2.25 0.84 0.99 1.83 0.42 -1.27

THE EXECUTABLE PROGRAM

An executable program, which runs in a user-friendly environment, where all the operationscan be followed in a single screen page, is available upon request to the reader.

This program uses a previous algorithm developed by Pacheco (1995) which calculates themoist air properties and plays the role of the psychrometric chart. For the convergence of theiterative loops, the Wegstein method as exposed by Franks (1972) was used.

The executable program is composed of two files:

ROTDRYER.EXE - the executable fileDATA.DAT - a file containing the last set of data entered in the program

The program is started by typing ROTDRYER




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at the DOS prompt. The program screen, shown in Figure 4, is composed of threesets of lines:

13 lines which contain input data that can be changed by the user7 lines containing the results of the calculationsone line for modification of any of the inputs

The lines containing the results display the product flow rate, the exit air conditions, and the

accordance with the following guidelines given by practice:

0.1 D/Z 0.25

0.28 GS 13.9 kg s-1m-2

1.5 NT 2.0

To change one of the inputs, just type the number of the item and press . A new lineappears, showing the parameter to be changed.

Type the new value, and press . The program then calculates the new outputs.Repeat the process to modify other parameters.

1- pressure (Pa): 1012342- solid initial moisture content d.b. (%): 33.333- solid final moisture content d.b. (%): 0.504- solid initial temperature (oC): 27.05- solid final temperature (oC): 65.06- solid specific heat (kJ/kg/oC): 1.0007- ambient air temperature (oC): 27.08- ambient air relative humidity (%): 609- inlet air temperature (oC): 165.010- air flow rate (kg/s): 6.06211- length (m): 16.812- diameter (m): 3.0

13- flow (1= counter flow 2= parallel-flow ): 2Solid flow rate (kg/s): 0.650Outlet air temperature (oC): 71.9Outlet air relative humidity (%): 21.9Dew-point temperature at the outlet (oC): 39.9D/Z: 0.18 (0.1


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Among the several possibilities, here we will only discuss two solutions:

a) Increasing the air flow rate by 10% (from 6.062 to 6.668 kg s-1) and increasing the inlet

air tempertaure from 160 to 181 oC. The resulting product flow rate in this case would still be

0.65 kg s-1.

b) Increasing the air inlet temperature from 160 to 190 oC, keeping the air flow rate

constant. The resulting product flow rate is also 0.65 kg s-1.

Although both alternatives satisfy the condition of maintaining the production rate and theproduct specifications, the first one does the job with an NTof 1.41, while the second onedoes it with an NTof 1.53. This means that the second alternative uses energy in a moreefficient way, and therefore, should be chosen. In fact, the first option operates with an air

flow rate of 6.67 kg s-1 and gives an air exit temperature of 79.4 oC, while the second

operates with a lower air flow rate (6.06 kg s-1) and a lower air exit temperature of 78.4 oC.

This was an example of an application of the presented algorithm, but several others can bedevised.

CONCLUSIONS

This algorithm, in spite of being rather simple since it consists basically of a set of algebraicequations, permits a quick look at the performance of a rotary dryer.

However, several details had to be examined in order to assure that the algorithm wouldwork properly. The examination of these details actually uncovered rather interesting aspectsof the behavior of a rotary dryer.

The agreement between the algorithm and real data shows that our approach can help thepractical engineer to obtain a rapid diagnosis of the performance of a rotary dryer.

NOMENCLATURE

CGAir specific heat, kJ kg-1oC-1

CSSolids specific heat, kJ kg-1oC-1

CWWater specific heat, kJ kg-1oC-1

D Internal rotary dryer diameter, m

E Evaporation rate, kg s-1

G Air mass flow rate, dry-basis, kg s-1

GSAir mass velocity, dry-basis, kg s-1m-2

H Moist air enthalpy, kJ kg-1

HSSolids enthalpy, kJ kg-1

K Constant for the heat transfer coefficient

L Solids mass flow rate, dry-basis, kg s-1LTLength of a Transfer UnitNTNumber of Heat Transfer UnitsP System total pressure, PaPpWater vapor partial pressure, PaPss: Water vapor pressure at dry-bulb temperature, PaPsuWater vapor pressure at wet-bulb temperature, Par (G/L)/(CS+CWX5)SGCounter flow; parallel flow identification variable

T Moist air dry-bulb temperature, oC

TDDew point temperature,oC

TSSolids temperature,oC (Tsi- inlet; Tso- outlet)

TWMoist air wet-bulb temperature,oC




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UaVolumetric heat transfer coefficient, W m-3oC-1

W Moist air absolute humidity, kg/kgWRMoist air relative humidity, %/100X Solids moisture content, dry-basis, kg/kg (Xi- inlet; Xo- outlet)Z Dryer shell length, m

Greek letters

lWater latent heat of evaporation at (TS1+TS2)/2 (kJ kg-1)

REFERENCES

Baker,C.G.J., Cascading Rotary Dryers, in Advances in Drying, McGraw-Hill, Vol 2, ch 1(1983). [ Links]

Franks, R.G.E.,Modeling and Simulation in Chemical Engineering. Wiley-Interscience, (1972).[ Links]

Friedmann, S.J. and Marshall, W.R., Studies in Rotary Drying Part II - Heat and Mass

Transfer, Chem.Eng.Prog. Vol.45, pp.573-588 (1949). [ Links]

Pacheco, C.R.F., Carta Psicromtrica para Computadores Pessoais - PC for PC, RevistaBrasileira de Eng. Qumica, Vol.XV, No.2, pp.21-26 (Nov.,1995). [ Links]

Perry, R.H.; Green, D.W. and Maloney, J.O., Perry's Chemical Engineers'Handbook, McGraw-

Hill Book Co, 6thed., New York (1984). [ Links]

Tsao, G.T. and Wheelock, T.D., Drying Theory and Calculations. Chem.Eng. , pp. 201-214(June 19,1967). [ Links]

All the content of the journal, except where otherwise noted, is licensed under a Creative Commons License

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