MATHEMATICAL MODELS FOR THE EFFICIENCY OF FLOTATION PROCESS FOR

NORTH WAZIRISTAN COPPER

By

SARDAR ALI Ph.D. Scholar

UNIVERSITY OF EDUCATION LAHORE

PAKISTAN 2007

i

MATHEMATICAL MODELS FOR THE EFFICIENCY OF FLOTATION PROCESS FOR

NORTH WAZIRISTAN COPPER

By

SARDAR ALI University Registration No. 87-03

A thesis submitted to the University of Education Lahore (Pakistan) in partial fulfillment of the requirements for the award of degree of

Doctor of Philosophy in Mathematics

with specialization in Mathematical Statistics at the Division of Science and Technology, University of Education Lahore.

SUPERVISOR CO-SUPERVISOR

JUNE 2007

ii

In the Name of

Allah,

Most Merciful and Compassionate the

Most Gracious and Beneficent

Whose help and guidance I always solicit at

every step, at every moment.

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DEDICATED

To my Wife and Children

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(ACCEPTANCE BY THE THESIS OF EXAMINATION COMMITTEE)

Thesis entitled

MATHEMATICAL MODELS FOR THE EFFICIENCY OF FLOTATION PROCESS FOR NORTH WAZIRISTAN COPPER

Submitted by

MR. SARDAR ALI

Accepted by the Division of Science and Technology, University of Education, in partial fulfillment of the requirements for degree of Doctor of Philosophy in Mathematics with specialization in Flotation Process.

Thesis Examination Committee:

Director External Examiner Supervisor Member Member

Date: ___________

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ABSTRACT

The objectives of this research are to analyze empirically the effects of different

explanatory variables on recovery and grade of copper from ore found in North

Waziristan and to develop mathematical models for the enrichment of copper in

Pakistan.

This study is based on the primary data from flotation process experiment for

enrichment of copper. Seven variables were studied in experiments. The variable were

type and dosage of collector (X1g/ton) pH (X2), depressant sodium cyanide (X3 g/ton)

sulfidizer Na2S(X4g/ton), frother dosage (X5 g/ton), pulp density (X6 w/v) and

conditioning time (X7 minute) and consists of 31 observations. Flotation process

parameters were studied to concentrate the copper content of chalocopyrite the North

Waziristan copper ore. Mathematical models were developed using various model

selection procedures. Regression parameters were estimated by applying Ordinary

Least Squares (OLS) method for regression analysis and adopted general to simple

modeling procedure. In this study we found that the variables X1, X3, X4 and X6 of

equation (5.57) are statistically significant and concluded that an increase in these

variables there is increase in recovery of copper.

Maximum grade were obtained from equation (6.65) the combined significance

variable X1, X3, and X7.

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ACKNOWLEDGEMENTS

All praise and thanks for Almighty Allah, Who has given me power to

complete this report successfully.

I am extremely grateful to Dr. Ghulam M. Mustafa, Vice-Chancellor,

Education University Lahore for his expert guidance, incisive and scholarly advice

and very useful suggestion which were of great help in making this report.

I am also greatly thankful to my supervisor Dr. Mir Asad Ullah, COMSAT,

Abbottabad, for his constant help at each stage, with out which I probably would not

have been able to execute this project with such professional excellence.

Heartedly thanks are due to my Co-supervisor, Prof. Dr. Muhammad Mansoor

Khan, Dean N-W.F.P, University of Engineering & Technology, Peshawar.

My sincere thanks goes to Prof. Dr. Mian Izhar ul Haq, Director, Ph.D

Programme, Education University, Lahore, for his timely help, encouragement and

cooperation.

I am unable to find words for paying thanks to my wife and my children who

were so helpful and extending warm co-operation whenever called upon.

Last but not the least, I would like to thanks Mr. Syed Sajid, Alias (Doctor),

Supervisor, Words’ Masters, U.O.P, for compiling this stuff in such a short period of

time.

SARDAR ALI

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TABLE OF CONTENTS

Abstract v

Acknowledgements vi

List of Tables x

List of Figures xi

CHAPTER NO. 1: INTRODUCTION 1

1.1 Introduction 1

1.2 Why Mathematical Models are required in Flotation Process? 3

1.3 Benefits of the Present Research 4

1.4. Objectives of the Research 5

1.5 Scope of the Research 6

1.6 Sources of the data 7

1.7 Background of the Problem 8

1.8 Significance of the Research 8

1.9 Outline of the Study 9

CHAPTER NO.2: REVIEW OF LITERATURE 10

CHAPTER NO. 3: EXPERIMENTS 18

3.1 Previous Work 18

3.2 Geology of North Waziristan Copper 19

3.3 Location and Accessibility of North Waziristan Copper Ore 19

3.4 Uses of Copper 20

3.5 World Occurrences 21

3.6 World Mine Production and Reserves 22

3.7 More New Discoveries 25

3.7.1 Copper of Occurrences in Pakistan 25

3.7.2 Gilgit Agency (Northern Areas) 25

3.7.3 Punjab Province 25

3.7.4 Baluchistan Province 26

3.8 Basic Information About Chalcopyrite Mineral 26

3.9 Occurrences of North Waziristan Copper Ore 28

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3.9.1 Shinkai Area 28

3.9.2 Degan area 28

CHAPTER NO. 4: METHODOLOGY 29

4.1 The Principle of Least Squares 30

4.2 Estimation Techniques 32

4.3 Estimation Of Model Parameters 33

4.3.1 Least Squares Estimation of the Regression Coefficients 33

4.3.2 Properties of the Least-Squares Estimators 38

4.3.3 Estimation of 2 39

4.3.4 Test for Significance of Regression 40

4.3.5 Stepwise Regression Procedure 41

4.3.6 Studentized Residuals 41

4.3.7 Test Statistic for Skewness 42

4.3.8 Testing for Heteroscedasticity 42

4.3.9 The t-statistic - Normal Approximation 44

4.4 Collection of Copper Ore Samples and their Analysis for Pilot Scale Studies 45

4.5 Justification of the Explanatory Variables 46

4.5.1 Collector types & dosage 46

4.5.2 pH value 46

4.5.3 Depressant 47

4.5.4 Sulphidizer (Na2S) 47

4.5.5 Frothers Dosage 47

4.5.6 Frother 47

4.5.7 Effect of pulp density 48

4.5.8 Flotation time 48

CHAPTER NO. 5: MODEL BUILDING

5.1 General Model For Recovery: 51

5.2 General Description: 51

5.3 Mathematical Model For Optimum Recovery about the Data for recovery of copper 53

5.3.1 Effect of variation in collector dosage NaPX (X1). 53

5.3.2 Effect of variation in pH (X2) 55

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5.3.3 Effect of variation in depressant-NaCN (X3) 55

5.3.4 Effect of variation in Sulfidizer Na2S (X4) 55

5.3.5 Effect of Pine Oil (X5) 56

5.3.6 Effect of Pulp density (X6) 56

5.3.7 Effect of conditioning time (X7) 56

5.4 Modeling Effect Of Individual Variable For Recovery 59

5.5 Modeling Combined Effect Of Variables On Recovery 72

5.6 Forward selection procedure for model building or simple to general. 73

5.7 Backward elimination or general to simple procedure for model building 74

5.8 Best Subset For Recovery 75

5.9 Multiple Regression Model for Recovery 81

5.9.1 Jarque – Bera: A Combined Test: 85

5.9.2 Testing for Heteroscedasticity 86

5.10 Reduced model for recovery 87

5.10.1 Tests for basic assumptions: 89

5.10.2 Other tests for Normality 90

CHAPTER – 6: MATHEMATICAL MODEL FOR OPTIMUM GRADE 94

6.1 Mathematical model for optimum grade. 94

6.2 Modeling effect of individual Variable for grade 97

6.3 Modeling Combined Effect Of Variables On Grade 110

6.4 Forward Selection 110

6.5 Backward Elimination 112

6.6 Best Subset For Grade 112

6.7 Model development for grade: 120

6.8 General Remarks about the Model 123

6.8.1 Statistical Significance 123

6.8.2 Sample Size? 123

6.9 Specific Remarks about the Model 124

6.10 Discussion of size of coefficients and scientific judgment of coefficient 125

CHAPTER – 7: CONCLUSION 127

References 129 Appendix (1-7) 137

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LIST OF TABLES

Table No. Title Page

Table – 1: World Refined Copper Consumption 22

Table – 2: World copper mined production 23

Table – 3: Data for the recovery of Copper 54

Table – 4: Mathematical models involving one predictor variable for recovery of copper by flotation.

59

Table – 5: Coefficient Analysis and model fitness statistic 82

Table – 6: Analysis of Variance 82

Table – 7: Test for normality of residuals: 84

Table – 8: Skew ness E. Kurtosis Jarque-Bera 85

Table – 9: Coefficient Analysis and model fitness statistic for four variables

88

Table – 10: Analysis of Variance for four variables 89

Table – 11: Tests for skewness, kurtosis and Jarque bera for four variables 90

Table – 12: Bin Frequency 90

Table – 13: Mathematical models involving one predictor variable for grade of copper.

98

Table – 14: Primary data on grad of copper 111

Table – 15: Coefficient Analysis for grad and model fitness statistic for

Seven variables

119

Table – 16 OLS estimates121 for three significant variables 120

Table – 17 Analysis of Variance 121

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LIST OF FIGURES

Figure No. Title Page

Figure 1: Effect o1f collector (NaPX) on recovery of copper 57

Figure 2: Effect of pH on recovery of copper 57

Figure 3: Effect of depressant (NaCN) on recovery of copper 57

Figure 4: Effect of sulfidizer (Na2S) on recovery of copper 57

Figure 5: Effect of frother (pine oil) on recovery of copper 58

Figure 6: Effect of pulp density on recovery of copper 58

Figure 7: Effect of conditioning time on recovery of copper 58

Figure 8: (a). Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential and (f) two straight-line models fitted to the recovery of copper data from five levels of collector type and dosage in the flotation process.

63

Figure 9: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the recovery of copper data from five levels of pH of pulp in the flotation process.

65

Figure 10: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the recovery of copper data from five levels of depressant in the flotation process.

66

Figure 11: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the recovery of copper data from five levels of sulphidizer in the flotation process.

67

Figure 12: (a) Linear, (b) Logarithmic (c) power (d) and exponential models fitted to the recovery of copper data from three levels of frother dosage in the flotation process.

68

Figure 13: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the recovery of copper data from four levels of Pulp density in the flotation process.

69

Figure 14: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the recovery of copper data from four levels of conditioning time in the flotation process.

71

Figure 15: Effect of sodium cyanide (X3) on the recovery of copper. 77

Figure 16: Effect of sodium sulphide (X4) on the recovery of copper 77

Figure 17: Copper recovery (YR) response surface for sodium cyanide (X3) and sodium sulphide (X4).

78

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Figure 18: Copper recovery (YR) response surface for sodium sulphide (X4), and frother dosage (X5).

79

Figure 19: The figure (5.19) shows visual test for standard residuals of seven variables.

83

Figure 20: Histogram 84

Figure 21: Standard Residual Plot 86

Figure 22: Plot of residuals 89

Figure 23: Histogram 91

Figure 24: Standard Residual Plot 91

Figure 25: Effect of collector (NaPX) on grade of copper 96

Figure 26: Effect of pH on grade of copper 96

Figure 27: Effect of depressant (NaCN) on grade of copper 96

Figure 28: Effect of sulfidizer (Na2S) on grade of copper 96

Figure 29: Effect of frother (pine oil) on grade of copper 96

Figure 30: Effect of pulp density on grade of copper 96

Figure 31: Effect of conditioning time on grade of copper 97

Figure 32: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from five levels of collector use in the flotation process.

101

Figure 33: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from four levels of pH in the flotation process.

103

Figure 34: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from four levels of sulfidizer in the flotation process.

104

Figure 35: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from five levels of depressant in the flotation process.

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Figure 36: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from four levels of frother dosage in the flotation process.

106

Figure 37: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from four levels of pulp density in the flotation process.

107

Figure 38: (a) Linear, (b) Logarithmic (c) quadratic (d) power (e) exponential (f) and two straight-line models fitted to the grade of copper data from four levels of flotation time in the flotation process.

109

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Figure 39: Effect of sodium cyanide (X3) on the grade of copper. 114

Figure 40: Effect of sodium Sulphide (X4grams/ton) on the grade of copper.

114

Figure 41: Copper grade (YG) response surface for sodium cyanide (X3) conditioning time (X7).

116

Figure 42: Copper grade (YG) response surface for sodium Sulphid (X4) conditioning time (X7).

117

Figure 43 Histogram 121

Figure 44 Testing for heteroscedasticity 122

Figure 45 Residuals are normal. It qualifies the visual test of normality.

122

Figure 46 Conceptual general model for recovery and grad 126

CHAPTER – 1 INTRODUCTION

1.1 Introduction

Mineral processing is the art and science of processing ores to separate

valuable minerals from waste by physical means (crushing, grinding, screening,

gravity separation, magnetic separation and flotation) and of doing it for a profit. To

maximize profitability, accurate simulations of mineral processing unit operations

have been actively pursued as minerals processing technology has matured. The

modeling and simulation of mineral processing systems are inherently difficult

because they are multiphase (particles, fluids and air) and because the ore particles are

heterogeneous (size, shape, composition and texture). From the beginning, the rate of

development in mineral processing modeling was controlled by limitations in ones

understanding of the basic sub processes in each unit operation and in the

computational relationship required to solve model1 equations.

Mathematical statistics is an interdisciplinary subject aimed at developing

models and analytical methods for systems containing a substantial element of

random variation, often the motivation for the research is a practical problem

involving the development and analysis of a mathematical statistical model.

1 A model can simply be defined as a representation or a description of the physical phenomenon occurring in any activity.

2

A model explains the phenomenon either mechanistically (theoretically) or

statistically (empirically), which can be used for prediction of the phenomenon

(Pekkanen, 1998b). The development of a new process typically involves a lot of

experiments on several scales: Laboratory, bench scale and pilot stages.

This dissertation is mainly focusing on the state of mathematical modeling and

also to formulate mathematical models for grade and recovery of the North Waziristan

copper ore in Pakistan. The North-Waziristan copper ore is chalcopyrite. The ore is

of low grade within economic limit, therefore it must be upgraded before it can be

subjected to metallurgical treatment to obtain blister copper. The experimental work

was undertaken to upgrade the lean copper ore through flotation technique to make it

suitable for further metallurgical treatment to obtain blister copper. Extensive

flotation test work was carried out to investigate effects of various process variables

on recovery (YR) and grade (YG) of copper. Effects of collector dosage; pH,

Sulfidizer dosage; depressant levels, frother dosage, pulp density and conditioning

time were investigated in flotation tests. The results of the pilot scale studies showed

that the copper content in the ore can be upgraded from 0.9 % to 22-25 % in a staged

cleaning flotation with recoveries up to 80%. The grade can be further enhanced by

improving the machine efficiency and conducting more research on reagents.

The important information on some flotation results of copper were obtained

from the Department of Mining Engineering, N.W.F.P University of Engineering and

Technology, Peshawar, and used to develop mathematical models for grade and

recovery of copper.

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1.2 Why Mathematical Models are Required in Flotation Process?

Mathematical Models are required because;

More time and money consumes during systematic experimentation or by hit

and trial

Scientific way to improve the efficiency of the process is to develop a correct

mathematical model

It is the only course of action available for the improvement in the system

Once we successfully construct mathematical model for a process it can be

used in future for any alteration for improvement in the process

It will help to improve the process of extraction of copper from the copper ore.

The study of mathematical models, simulation and optimization are important

because of the following reasons.

Modeling reduce manufacturing costs

Reduce research and development expenditure and save time.

Increase efficiency.

Greater understanding of the problem

Decision support

Knowledge management

Ability to handle complex problems

Technology transfer

Improve the safety of the plants

4

Bring new products to market faster

Reduce waste in process development

Improve product quality

Reduce need for potentially hazardous experiments.

Gives precise and accurate results

Mathematical models have been used in mineral processing system design

optimization in control for more than 32 years. All major technological innovations

involve information technology and mathematical modeling, and apparently

computerized mathematical model play an increasingly decisive role within

engineering sciences, i.e. within industrial production, within planning and

economics, within mineral processing. Mathematical modeling activities are aimed at

methodologies enabling one to deal with today’s ever increasing quantities

information.

1.3 Benefits of the Present Research

Saving of foreign exchange

Meet the demand of indigenous industry

To utilize the 122 million ton of copper ore of North Waziristan area.

In order to upgrade the copper content by mathematical model to make it

suitable for metallurgical treatment

To improve the quality and quantity of copper ore in concentrate

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This model can be utilized specially in glass, ceramics industry, copper

concentrator, saindak (Baluchistan) process of bentonite clay, enrichement of

uranium, purification of soap, stone and fertilizer industries.

Optimization of flotation parameters.

1.4 Objectives of the Research

1. To develop mathematical models for the enrichment of copper in Pakistan.

2. To examine how to improve and increase the efficiency of the process by

scientific way.

3. To analyze empirically the effect of different explanatory variables2 on

recovery and grade of copper in the research area of Pakistan.

4. To give appropriate suggestions in the light of our findings.

The North Waziristan copper ore is chalcopyrite (CuFeS2). Chemistry of

chalcopyrite is such that it can be efficiently concentrated by the froth flotation from

associated gangue minerals. Flotation process parameters were studied using

chalcopyrite Copper ore of North Waziristan to obtain a copper concentrate suitable

for further metallurgical treatment. The important flotation variables examined were,

collector, depressant, pH, frothers, Sulphidizer (Na2S), pulp density and conditioning

time. By stage wise optimization of flotation variables, copper were upgraded from

0.9% to 10% and 20% in roughing stage and to as high as 22% in a cleaning stage

with recoveries up to 80 to 90% in experimental work done by the Department of

2 Seven important explanatory variables, e.g. type and dosage of collector (X1gms/ton) PH (X2), depressant sodium cyanide (X3gms/ton) sulfidizer Na2 S (X4gms/ton) frother dosage (X5gms/ton), pulp density (X6) and conditioning time (X7 minute).

6

Mining Engineering, NWFP University of Engineering & Technology Peshawar,

Pakistan.

To improve and increase the efficiency of the process by scientific way and to

develop a correct mathematical model, so that in future any alteration or change in the

process can be improved by utilizing the mathematical models. Now-a-days every

chemical, mechanical and electrical process is governed by mathematical or statistical

models. These 5.67 and 6.65 models are the first mathematical models, which have

been developed for the mineral industry in Pakistan and of course these will pave the

way to run our mineral based Industry. Using such mathematical models to improve

their products quantity and quality. These models can be utilized specially in glass,

ceramics industry copper concentrator Saindak (Baluchistan) process of bentonite

clay, enrichment of Uranium, purification of soap, stone and fertilizer industries.

1.5 Scope of the Research

Copper is one of the very essential minerals in modern industry. It is a good

conductor and is used in electrical networks, various equipments and weapons. The

United States, the world’s largest consumer (1999), uses between 2.5 and 3.0 million

tons of copper annually. Most wires and electrical equipment are made of pure copper

and considerable alloys of copper such as brass and bronze. The brasses are Cu – Zn

alloys (55%-99% Cu, 45%-1% Zn) and the bronzes are Cu – Sn – Zn (88% Cu, 10%

Sn and 2% Zn). There are also Ni, Al, and steel alloys of Cu; minor special alloys

utilize arsenic; beryllium, cadmium, chromium, cobalt, iron, lead, magnesium,

manganese and silicon. Copper sulphide deposits of North – Waziristan vary in grade

from 0.3% to as high as 1.0%. Due to its low grade it cannot be directly subjected to

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metallurgical treatment for producing blister copper. Pakistan is still meeting its

requirement through import from other countries. Successful development of

mathematical model will provide optimum parameters for the enrichment of copper in

the final product. In this way it will save cost for further experimentation and time to

achieve similar objectives.

1.6 Sources of the data

The study is based on primary data from flotation process experiments on

samples collected by the Department of Mining Engineering, NWFP University of

Engineering and Technology, Peshawar Pakistan with the assistance of the political

authorities of North Waziristan agency and Federally Administered Tribal Area

Development Corporation (FATA DC). Other relevant information about copper

deposits were also obtained from FATA DC. An inventory of the ore samples was

prepared and each sample was tagged with a number and weighted.

Both the chemical and mineralogical analysis of the samples were carried out

at Department of Mining Engineering Laboratories (MEL) and Mineral Testing

Laboratories, Sarhad Development Authorities (SDA), Peshawar. The mineralogical

investigations include X-Ray Diffraction, X-Ray Fluorescence and ore microscopy.

The chemical constituents were determined by classical and instrumental methods of

analyses. On site the samples were collected by blasting irregularly spaced holes

within the regularly spaced rows for minimum chances of errors. A total of 30 tons of

samples were collected, comprising of six sub samples weighing five tons each from

six different locations. The rows of holes drilled on each location were spaced at an

equal interval of 300 feet. The collected samples were transported to the NWFP,

University of Engineering and Technology Peshawar, through trucks.

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1.7 Background of the Problem

Extraction of copper from the available copper ore in Pakistan is important.

The metal has many uses and ranks fifth amongst the metals in tonnage consumed.

God has blessed Pakistan with abundant copper ore and its occurrences have been

reported throughout the country. However, the occurrences at Saindak and Ricodak in

Baluchistan and North Waziristan Agency in NWFP are of more importance. The

survey conducted by Federally administrated tribal areas (FATA) development

corporation has confirmed a minimum of 122 million tons of reserves of copper ore in

Boya-Datta Khel area (about 40km from Miran Shah), having copper contents better

than that found at Saindak at some places and in some layers.

Thus an extensive study of North Waziristan copper ore was carried out by the

Department of Mining Engineering, N.W.F.P, University of Engineering &

Technology, Peshawar, through a research proposal sponsored by Board of Advanced

Studies and Research (BOASAR). The laboratory evaluations of raw ore were made

in Phase-1 of the project and in order to confirm these evaluations, a study of flotation

process by a single stage pilot plant was carried out in Phase-II. These studies have

generated sufficient data for constructing mathematical models for the processes.

1.8 Significance of the Research

For obtaining optimum level of variables for efficient flotation process to

extract copper from raw ore, experimentation by systematic or by hit and trial

procedures takes a lot of time and costs enormous amount of money. The standard

scientific way to improve and increase the efficiency of the flotation process for

enrichment of copper ore is to develop a mathematical model for the process. It

9

should be remembered that in some cases mathematical modeling is the only course

of action available for the improvement in the system. Once we successfully construct

mathematical model for a process, it can be improved and used in future for any

alteration for improvement in the process. Thus mathematical models to be developed

will reduce the extent of further experimentation to achieve certain desired objectives.

These will help to improve the process of extraction of copper from the copper ore

and will save sizeable amount of money for the country.

1.9 Outline of the study

Outline or organization of the study is as follows. Chapter one deals with

introduction regarding mathematical modeling for the efficiency of mineral

processing of North Waziristan copper ore, benefits of the present study, main

objective, scope, background and significance of the study. Chapter two presents

review of literature. Chapter three explains experiments, previous work, geology of

North Waziristan copper ore, location and accessibility Waziristan copper ore, and

occurrences of North Waziristan copper ore. Chapter four deals with methodology,

justification of the explanatory variables and flotation process. Chapter five presents

building of mathematical models. Chapter six consist mathematical models for grade.

Final and the last chapter seven consist of summary, conclusion and

recommendations.

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CHAPTER – 2 REVIEW OF LITERATURE

In the literature, the construction of mathematical models for flotation process

in mineral processing has been approached in various ways depending on the

philosophy of the researcher as well as the expected usage of the model and the

allowable investment in personnel and time. The goal of using these models is to

undertake plant scale tests on batch laboratory evaluations. Flotation tests were

carried out on samples of a US porphyry ore (Pinto Valley, AZ) by Dowling et al.

(1985). The ore was tested using various collector and frother system to produce

different time recovery profiles these were used to calculate flotation rate and ultimate

recovery parameters for each model. The models were then evaluated statistically to

determine the over all fit of the calculated to the observed data and to test the range of

significance of the parameters in each model.

Each flotation model will have an associated error. This error will effect and

can be measured by both the fit to the observed data and the range of statistical

significance of each parameters. Two types of errors were found due to experiment

and due to model. The researcher found the model variance S2r and compared the

optimal model variance to the model variance from a given change in the parameter

being assessed and F-value calculated. To determine the optimal flotation model

parameters, a generalized parameters estimation computer program was used (Klimpl

1980”) by Dowling in his study. The criteria used for estimation of parameters value

11

is the minimization of the absolute sum of the square of deviation at a given time

between observed in calculated recovery.

Wills (1986) studied simple nodal sensitivity analysis in complex circuit

analysis were found by using matrices and statistical techniques. The researcher

worked to develop a best-fit material balance model. The method described makes use

of the minimum number of sampled streams and analysis of only one component,

such as metal assay or dilution ratio, on each stream one involved in a unit process.

He found that plant flow sheet was reduced to a series of nodes, where process either

join or separate. Simple nodes have either one input and two outputs (a separator ) or

two inputs and one output ( a junction ).

Munn (1998) investigated that metal recovery in mineral processing plants is

often linearly correlated with feed or concentrate grade, particularly in flotation. This

correlation can be used to analyze the data form plant trials in which two operating

conditions are being compared, such as different reagent regimes or circuit

configurations. The method involves the statistical comparison of the two linear

recovery –grade regression lines corresponding to the two operating conditions.

Although not as efficient as a formal experimental design, the method can be used

where such designs are impractical, or in the analysis of historical data.

Khan (1999) studied flotation process parameters to concentrate the copper

content of chalcopyrite, the North Waziristan copper ore, in pilot-scale to obtain a

copper concentrate suitable for further metallurgical treatment. The important

flotation parameters, e.g. type and dosage of collector, dosage of depressant, and

frother and conditioning time for collector were examined. During stepwise

optimization of flotation parameters, the copper content was upgraded from 0.9% to

12

20% in roughing stage, and to as high as 22% in a single-stage cleaning with

recoveries of over 83%. A flow sheet depicting different products of flotation, for an

industrial concentrator, has also been suggested.

Elzinga E.J. Van J.J.M and Swartjes. F.A. (1999) worked on General purpose

Freundlich isotherms for cadmium, copper and zinc in soils. They have tried to derive

generally applicable isotherm for cd, zn and zn using data from batch sorption

experiments on a wide range of soils and experimental conditions. They used a

linearized logarithmic transformation of the Freundlich sorption equation.

Freundlich derived equations for cd, zn and zn using multiple linear regression

on batch sorption data. The equations were based both total dissolved metal

concentrations and free metal activities in solution. He calculated free metal activities

from total metal concentration talking into account ionic activity. The logarithmic

transformation of the Freundlich constant for cadmium was regressed on the

logarithmic transformations of cation exchange capacity. He used Minitab for

statistically analysis.

They used step-wise forward regression by evaluating the t–ratios, stepwise

improvement of 2adjR could be attributed either to addition of an argument or to

reduction of data. A best model was selected based on 2adjR t-ratio and the number of

data points considered. The regression co-efficient of the best models were significant

at the P = 0.001 level. All sorption point were considered as independent

observations.

Sripriya et al. (2002) examined the kinetic model based on time recovery data,

which uses the extra dimension of rate and has been in vogue since time immemorial

13

for scaling up of laboratory data. The air flow number and the froth number were used

as a basis for scale up. The performance of the froth flotation circuit, an efficiency

parameter (co-efficient of separation c s) was used. The yield from the flotation circuit

improved, the froth ash reduced and the rejects ash went up. Various empirical and

kinetic models were evaluated.

Sripriya developed regression equations for predicting the combustible

recovery ash recovery and Ks for combustibles and ash. The effects of three most

important reagents for coal flotation namely sodium meta silicate, collector (kerosene)

and frother were studied using 23 full factorial design. The regression models were

developed using factorial experiment data to quantify the effect of sodium meta

silicate, collector and frother and to predict grade and recovery of combustible

material for different reagent conditions. The addition of sodium meta silicate

increased the recovery without affecting the grade significantly. The MIBC addition

reduce the surface tension at the liquid–vapor interface, which results in the

production of finer bubble size distribution and thus improves flotation rates and

recovery values. However, a finer bubble size is tribution also increases water

recovery, which results in a greater recovery of certain able ash bearing particles and

thus degradation of the product grade. The interaction between OH group of MIBC

and hydrated mineral matter improves floatability of high ash coal particles and

degrades the product grade further. The negative effect of kerosene and MIBC

interaction on both grade and recovery could be due to the recovery of high ash coal

particles in preference to low ash coal particles. The highest possible grade of product

is 94.19% combustibles with 25.33% recovery. A product with 91.11% combustibles

14

grade at 95.58% recovery was obtained at 0.1 g/kg sodium silicate, 0.4 g/kg collector

and 0.075 g/kg frother from the coal fines tested.

Ziyadanogullari (2003) worked on flotation of oxidized copper ore obtained

from Ergani Copper Mining Company in Turkey. The ore contained 2.03% copper,

0.15% cobalt and 3.73% sulfur. An effective processing method has not been found to

recover copper and cobalt from this ore, which has been stockpiled for 40-45 years in

a idled plant. It was established that recovery of copper and cobalt from this ore with

hydrometallurgical treatment is not economical, so using flotation to increase the

concentration of copper and cobalt was chosen. When flotation of the oxidized copper

ore was performed under standard operating conditions in the plant, good results were

not obtained. Because of this, the flotation of samples obtained from sulfurized

medium containing different ratios of H2S+ H2O gases was done under the same

conditions. Following flotation, it was seen that copper, cobalt and sulfur present in

the medium were concentrated. In this solution, concentration of copper and cobalt

were found five times higher than normal level.

Elemental sulfur produced by chloride leaching of sulfide ores or concentrates

contains selenium and tellurium usually too high to be used in various industrial or

agricultural uses. The sulfur in the leaching residue can be upgraded to 90% in grade

by froth flotation and the sulfur concentration can be followed by sulfur purification

and selenium and tellurium removal. The sulfur in the leaching is in a form of discrete

particles with a size range of 5 to 10 microns. The sulfur particles tend to agglomerate

in the pulp and hence mechanically entrap gangue minerals. With sodium silicate as

the dispersant as well as the depressant for siliceous material, a sulfur concentrate of

90% in grade and 90% in recovery can be obtained with a single-stage froth flotation.

15

The flotation reagent consumptions are minimum. The majority of chalcopyrite

remains in the sulfur flotation tailings and can be readily recovered by flotation with

different flotation reagents. When amyl xanthate is used, 85% of chalcopyrite can be

recovered with a copper grade of 14.5% in a single-stage froth flotation. The

chalcopyrite flotation concentrate can be sent back to chloride leaching circuits.

Cilek (2004) combined the classical first order kinetic model with a properly

built statistical model based on a factorial experimental design. In order to accurately

predict the rougher flotation efficiency for various flotation conditions, a three-level,

three factor experimental design was used to develop statistical model to predict each

of the kinetic model parameters as a function of the air flow rate, the feed grade and

the froth thickness. The statistical evaluation of the experimental results indicated that

the ultimate recovery, the rate constant and time correlation are not constant, but each

of these kinetic model parameters can be defined as a function of variables

considered. The rate of change in the kinetic parameters due to the feed grade

fluctuation and their effects on the metallurgical performance can accurately be

predicted by using the models developed. To reduce the detrimental feed grade

fluctuations on the metallurgical performance, the operating variables of the flotation

can be manipulated to obtain the desired concentrate grade. Cilek obtained the results

of the statistical evaluation; the rate data were used to build a statistical model

considering the variables.

Among all models the following models, which were built by using piece wise

(or breakpoint) linear regression method were selected.

Km = (0.4273-0.52 f+0.0051Qa+0.617Tf-0.05QaTf) , km2.24

Km= (3.565+0.38f-0.31Tf+0.104Qa+0.003QaTf), km<2.24,R2=0.9431........……….(1)

16

bm = (0.2012-0.09f+67.10-3Qa+0.074Tf+0.014 f Tf-0.011QaTf), b0.406

bm= (3.695-3.75f-0.08Qa+0.348Tf-0.694fTf+0.097QaTf),b<0.406, R2 = 0.8945.....…(2)

RIm= [27.973f-1.337Qa+0.89Tf(25.33-8.64f-Qa)-9.592Qa-0.549], RIm69.3

RIm= [75.284-4.811f+0.015Qa+0.654Tf(9.896+f-Qa)-0.115fQa],Im69.3,R2=0.9022..(3)

topt = [0.538+0.77f-0.112f Tf+0.02Qa(2.2+Tf-2.4f)] topt 2.32

topt=[4.097-0.867f+0.2fTf+0.051Qa(1.47-Tf+1.087f)],topt<2.32, R2=0.9087.........…..(4)

The high R2 values for all the responses reveal that the experimental data

provide evidence to indicate that the developed models satisfactorily predict the

Kinetic Parameters, where topt RI, k and b are the optimum flotation, time ultimate

recovery, the rate constant and the time correction factor also Tf, f, Qa denotes pulp

level, feed grade and factor respectively.

Barbaro and Piga (1998), adopted statistical approach to evaluate the Pb-Zn

selectivity of various organic collectors of the Mercaptobenzothiazole (MBT) and

aminothiophenol (ATP) types, in the flotation of lead and zinc minerals. Six

replicated tests were performed using each collector in order to obtain an estimate of a

statistical population characterized by an average and a variance. Comparison of these

statistical populations indicated the most selective collectors. The selectivity exhibited

by the collectors was then related to their molecular structure.

Horbstand and Potapov (2004), reported that mathematical simulation have

been used in mineral processing system design, optimization and control for more

than 30 years. Presently a new set of simulation tools based on the physics of the

underlying processes has been developed. Because these models provide accurate

micro scale simulations of equipment and process behavior, these high-fidelity

17

simulation (HFS) tools are deemed to constitute a radical innovation of great

importance to the mineral processing industry.

18

CHAPTER – 3

EXPERIMENTS

3.1 Previous Work

The commercial copper deposits occur in variable sizes. However, the ores

containing 0.3% and more copper are deemed feasible for exploitation on commercial

scale. With the state of the art mineral processing techniques the ores with lower

grades can be economically beneficiated as well. Undoubtedly large area of Federally

Administered Tribal Area (FATA) abounds in mineral resources. The survey

conducted by FATA development corporation has confirmed a minimum of 122

million tons of inferred reserves of copper ore varying in depth upto 30m in Boya-

Datta Khel area about 40 kms from Miran-Shah. The average content of this copper

ore is 0.3865%. The copper content increases with depth and at places it is 0.90%

which is better than that found at Saindak (Baluchistan). This low grade raw copper is

of little value unless it is enriched to a higher grade concentrate. The Department of

Mining Engineering through a research proposal (Beneficiation of North Waziristan

Copper Ore) sponsored by Board of Advance Studies and Research (BASAR) carried

out laboratory evaluation of raw ore. Based on the encouraging results of phase-I,

further work on the project was considered necessary. In the phase II of the project, it

was proposed to install a flexible single stage pilot plant for flotation process to study

the laboratory results at the pilot scale. The pilot plant, locally fabricated, has been

19

installed in the premises of NWFP University of Engineering and Technology,

Peshawar.

3.2 Geology of North Waziristan Copper Ore

North Waziristan area remained unexplored before seventies. It was in 1985,

that Federally Administered Tribal Area Development Corporation prepared a

geological base map of about 2350 sq. km on 1:50,000. Investigation resulted in

delineating certain prospective areas having copper mineralization. Detailed

topographical mapping on scales 1:10,000, 1:1000 and 1:500 were conducted in the

mineralized zones. Petrographic study of the representative rock samples was made

before starting pilot plant operations. Geochemical studies of grid samples had

already been carried out at laboratory scale. Probe core drilling in two copper

mineralized bodies has just been completed in collaboration with the technical

expertise of China. According to the estimates given by the Chinese geologist there

are 80 million tons of confirmed reserves of copper ore having an average copper

content of 0.8%. The grade increases with depth and at some places it is 1%,

sometimes approaching 2 to 5%.

3.3 Location and Accessibility of North Waziristan Copper Ore

The tribal belt lies at the Pak-Afghan Border. This belt is divided into seven

units (Agencies) namely Bajaur, Mohmand, Khyber, Aurakzai, Kurram, North

Waziristan and South Waziristan and four frontier regions attached to Peshawar,

Kohat, Bannu and Dera Ismail Khan.

Investigation in the southern region have revealed the presence of copper

mineralization at various places in North Waziristan. Boya, an important locality, is

20

located at longitude 69o 55/ 06// and latitude 32o 57/ 09// N on the right bank of Tochi

River. It lies at a distance of 19 km from Miranshah, the Agency Headquarters and the

local business center. Miranshah is fairly connected to the down districts. It is

accessible by about 270 km of Peshawar-Bannu-Miranshah mettalled road and also

through Peshawar-Tal-Miranshah road. Peshawar is connected by road, rail and air to

Islamabad, the distance being 167 km, similar communication links are available to

Bannu(District Headquarters), that falls at a distance of 61 km to the east of Miran

Shah. Bannu is also connected by about 141 km length of metalled road to D.I.Khan

(Divisional Headquarters) in the south. Tal and Bannu are connected by metalled

roads. The former is also connected to Kalabagh by a metalled road.

3.4 Uses of Copper

The tremendous growth in the use of copper is indicated by the fact that of the

total world production of copper during the last 100 year, about 80% was mined in the

last 25 years and more than one half of it in the last 12 years copper consumption by

major countries and regions is given in table-1. Annual world production ranges

around 20 million metric tons of metallic copper. In spite of the significant number of

closures in United States and Canada, Western world copper mine production rose

3.8% due to projects that came on stream in 1999. In Chile, the Pelambres project

came on stream, with its main impact to be felt during 2000. Similarly, Collahuasi

started up in late 1998 and reached full capacity in 1999, as did Andina expansion and

Escondida’s SX-EW operation. In Australia the Olympic dam expansion started up, as

did the Cuajone expansion in Peru, then Indonesia’s Batu Hijau mine began to

produce (Enrique, 2000). Copper ranks fifth among the metals in tonnage consumed.

It has a variety of uses and the important one is in electrical supply, use and

21

manufacturing industries due to its good conductivity that gives it an advantage over

most metals. It is extensively used in communications equipment including cables and

television transmitters and receivers. The second important use of copper is in

construction worm, particularly in plumbing and hardware and decorative purposes. It

is also a substance used in non-electrical industrial applications such as alloy with

nickel for tubing used in sea water desalination plants. It is used in heat exchangers,

pollution control, and liquid waste disposal. Automobile radiator cores are made of

copper; it is also used in air conditioners, heaters, gas and oil line, and bearings and

bushings.

Military uses of copper are fourth in rank, and the price usually go up during

periods of military spending. This is one of the incremental uses which can rapidly

increase consumption. Coinage, jewellery, chemicals, pigments, brass and bronze

wares and a multitude of minor uses also demand copper in variable amounts.

Copper is essential for plant growth, if copper content falls below 10ppm in

soils, good growth is not possible. On the other hand, if a large amount of copper is

present in the soil it is toxic to some plants.

3.5 World Occurrences

There are hundreds of copper minerals and dozens of settings for copper

deposits. By far the most important mineral is chalcopyrite and the large portion of

this mineral and of copper production comes from the porphyry deposits.

The term porphyry refers to a rock which has an intergrowth of distinctly large

and small crystals. Porphyries are considered to have intruded as molten rock or

magma from depth of ten to hundreds of kilometers. To form the texture of porphyry,

22

it should have approached to within about 3km of the surface before crystallizing as

rock. The texture of mixed coarse and fine crystals is brought to indicate fairly rapid

cooling. Copper deposits the world over can be classified according to the nature of

the deposits.

3.6 World Mine Production And Reserves

Reserves and reserve based estimates for Australia, Chile, China and Poland

have been revised upward based on new information from official country sources.

Revisions to other countries were based on updated tabulations of resources reported

by companies or individual proprietors. Table 2 shows the world mine production and

reserves of Copper.

Table 1: World Refined Copper Consumption (in 1000 tons of Cu)

Area 1994 1995 1996 1997 1998 1999 (e)

Western Europe 3341 3388 3345 3536 3751 3710

Africa 123 117 115 118 110 115

Japan 1375 1415 1480 1441 1255 1260

Other Asia 1833 1955 2126 2240 2148 2420

Canada 199 190 218 225 245 270

United States 2560 2534 2621 2790 2905 2935

Latin America 503 511 619 734 828 820

Oceania 148 174 170 166 161 160

Total 10082 10283 10691 11250 11403 11690

Annual Growth (%) 8 2 4 5 1 3

Source: (Enrique, 2000), (e) Estimate

23

Other Asia includes China, Taiwan, India South Korea, Thailand, Malaysia,

and Philippines.

North America

There are some greatest concentrations of copper in Arizona and Cordilleran

parts of the United States, Canada and Mexico and includes all the well known North

American “porphyry coppers” and a host of other famous districts. All the ores are

associated with felsic types of intrusions. There is another very productive copper

province in Montana at Butte. Other areas where copper can be found include the

Appalachian, the fruitful take Superior district, and Cascadian – Coast Range belt

extending from Yukon Territories through northern British Columbia to the state of

Washington.

Canada

Copper deposits extend from Manitoba to New Brunswick includes the

Hudson Bay, Sodbury, Noranda, heath Steele, Kidd Creek, and other deposits.

Table 2: World Copper Mine Production (in 1000 tons of Cu) (Enrique, 2000)

Area 1994 1995 1996 1997 1998 1999 (e)

Western Europe 304 323 290 317 305 260

Africa 647 618 585 577 563 491

Asia 674 806 798 810 1054 1034

Australasia 614 598 660 625 730 871

Latin America 2853 3198 3884 4280 4658 5316

North America 2399 2534 2537 2544 2495 2211

Total 7491 8077 8754 9153 9805 10183

(e) Estimate

24

South America

Andean copper belt is the most renown in this region. It extends from Chile to

Panama and includes large deposits like Chuquickamata, Braden, Potrerillos, El

Salvador, Cerro Colorado, Rio Blanca, Toquepala, Cerro de Pasco deposits, and many

more. The copper deposits found in these areas are normally associated with

monzonitic intrusives.

Central Africa

The central Africa province constitutes the most concentrated copper belt in

the world and includes the most productive mines of Zambia and adjacent Zaire.

These are strata bound deposits where the metals precipitated from sea with the

sediments.

Several new copper porphyries have been discovered in New Zealand, Fiji,

New Hebrides, Buganville, British Solomon Island protectorate, the territory of

Papua, New Guinea, and West Irian. Some of the copper producing deposits is

Penguna, Ok Tedi, Frieda, and the high grade deposits of Carstenz.

Other copper belts include Uralian province of Russia, the outer Japanese

Island arc, Spain – Portugal (Rio Tinto), Bor in Yugoslavia, Mansfeld in Germany,

Outokumpu in Finland, and Boliden in Sweden.

In Australia there are various copper centers such as Mount Lyell, Mount

Morgan, Mount Isa, Cobar, Tennant Creek, and Mount Oxide.

25

3.7 More New discoveries

Few more new discoveries have been made in Australia, Chile, Peru, British

Columbia, Panama, New Guinea, Fiji, New Idria, Brazil, Puerto Rico, New

Brunswick, Philippines, Solomon Islands, North western Brazil (John, 1984).

3.7.1 Copper of Occurrences in Pakistan

Several copper occurrences have been reported in Pakistan. They are available

in numerous geological settings and contain a variety of copper minerals. However,

the occurrences at Saindak (Baluchistan) and North Waziristan (FATA) are of much

importance. Minor occurrences have been reported from various other places of the

country.

Investment oriented study on Minerals and Mineral based Industries, Expert

advisory cell, Ministry of Industries & Production, Govt. of Pakistan. April, 2004

3.7.2 Gilgit Agency (Northern Areas)

Copper minerals have been located in quartz veins in the northeastern regions

of the area. Similarly chalcopyrite has been reported in alluvial sands in Indus, Gilgit,

Nagar and Hunza rivers.

3.7.3 Punjab Province

Small occurrences of copper have also been reported in Northern Punjab at

Kattha, Mussa Khel and Nilawahan Gorge in salt range. In these areas oxide copper

minerals are found in sandstone beds with malachite and cuprite as the major copper

minerals. Up till now these findings has no economic value. No detail exploration

work has been carried out to access the potential of these deposits.

26

3.7.4 Baluchistan Province

Copper mineralization has been reported in Chagai (Saindak), Loralai and

Zhob districts. Both copper sulphide and oxide minerals as well as native copper have

been reported in Chaghi district. Copper minerals in these areas occur mainly as

sulphides, oxides, silicates and carbonates. The minerals are found in disseminated

form in association with quartz, galena, hematite, siderite and monazite (Zaki, 1969).

In chaghi the amount of potential deposits of copper is reported as 729mt (0.64% Cu

and 0.39g/t Au (Malik,2003). Saindak copper deposits comprise of copper carbonate,

chalcopyrite, chalcocite and malachite minerals. These copper minerals are in

disseminate form and present in association with hydrothermally altered shales,

volcanic tuffs and shales, and limestone of Eocene age.

3.8 Basic information about Chalcopyrite Mineral

Chalcopyrite looks like, and is easily confused with pyrite and is also one of

the minerals referred to as “Fool’s Gold” because of its bright golden color, but it is

brittle, dissolves in acid and has a dark green streak. It is distinguished from pyrite by

ease of scractching, and by copper tests. The color is slightly more yellow than that of

pyrite or is often tarnished in brilliant iridescent hues, which is also called “peacock

copper ore”. Pyrite will frequently show striated cubes or pyritohedra, whereas

chalcopyrite, if not massive, has the characteristic sphenoidal or disphenoid crystals.

Chalcopyrite is the primary minerals, which by alteration and successive

enrichment with copper produces the series starting with chalcopyrite and going

through bornite (Cu5SFeS4), covellite (CuS), chalcocite (Cu2S), and ending rarely as

native copper (Cu). Its structure is so closely related to that sphalerite that it forms

intergrowths with mineral, and isolated free-growing crystals perched on crystals of

27

sphalerite are all parallel. The same face on all the chalcopyrite gives simultaneous

reflections. (It Sparkles) from the Greek words chalkos, “copper” and pyrites, “strike

fire”

Following are the various physical properties of chalcopyrite ore

Composition: CuFeS2 (34.5% Cu, 30.5% Fe, 35% S)

Class: Sulfides

Group: Chalcopyrite

Crystal system: Tetragonal

Fracture: Conchoidal and brittle

Hardness: 3.4-4

Specific gravity: 4.2

Luster: Metallic

Streak: Dark green

Cleavage: Poor in one direction

Color: Brassy yellow, greens, yellows and purples.

Transparency: Opaque

Associated Minerals: Barite, calcite, fluorite, galena, pyrite, pyrrhotite, quartz

and Siderite. Sphalerite and tetrahedrite are a few of the

most Common.

Chalcopyrite is usually massive, but crystals are also common. Often they are

large and the faces usually are somewhat uneven or may have striations on most

crystal faces. Chalcopyrite is often tarnished in brilliant iridescent hues. Spheroidal

28

crystals are common. Also common are disphenoid crystals, which are like two

opposing wedges that resemble a tetrahedron. Crystals are sometimes twinned and can

also be botryoidal.

On charcoal, chalcopyrite fuses to magnetic black globule, touched with HCl tints

flame with blue flash. Solution with strong nitric acid is green; ammonia precipitates

red iron hydroxide and leaves a blue solution.

3.9 Occurrences of North Waziristan Copper Ore

3.9.1 Shinkai Area

The host rocks in this area are normally breccia composed of fragments of

lava flow, ultra basic and intermediate igneous rocks. At some places breccia occurs

with altered andesitic, rhyodacitic, granodioritic, ultrabasic, doleritic and jesperitic

rock fragments. Still at some other places it is a simple breccia. The associated rocks

commonly found are ultrabsic, dolerite, volcanic, andesite, and lava flow. Jerositic is

the gossan type found in the area. The associated minerals are malachite, azurite,

pyrite, chalcopyrite and chalcocite. The extension of mineralized bodies varies from

960-17400 square meters.

3.9.2 Degan area

The host rocks in this area are normally composed of breccia. The associated

rocks commonly found are lava flow, andesite, interbedded limestone, shale, diorite,

and dolerite. Jerositic is the gossan type found in the area, the same as found in the

shinkai area. The associated minerals are malachite, azurite, pyrite, chalcopyrite and

manganese. The extension of mineralized bodies varies from 2604-108800 square

meters.

29

CHAPTER – 4 METHODOLOGY

Data from the following seven variables of flotation process were used to

develop mathematical models for recovery and enrichment of copper from North

Waziristan copper ore.

S. No. Name of Variable Level used

1. Propylxanthate 50, 100, 150, 200, 250

2. pH 10, 10.3, 11, 11.58, 12

3. Sodium Cyanide 10, 15, 20 ,25, 30

4. Sodium Sulphide 10, 30, 40, 50, 60

5. Frother (Pineoil) 25, 46, 70

6. Pulp Density 15, 25, 30, 35

7. Conditioning time 10, 13, 16, 18

Experiments conducted by the Department of Mining Engineering NWFP

University of Engineering and Technology Peshawar. The least square fitting

procedure is used for data analysis as purely descriptive technique. Computer

algorithms Minitab statistical software and Microsoft Excel were used for developing

best mathematical models for the efficiency of seven process variables.

30

Mathematical models were developed to observe the effects of process

variable on recovery and grade of copper. Simple and multiples regression were used

for developing the models.

The most general type of linear mathematical models can be described with

variables X1, X2, -----, Xp. in the form as follows where stands for variation caused

by other than X1, X2, -------.

Y= βo + β1 X1 + β2 X2 + ………. βp Xp + €

4.1 The Principle of Least Squares.

The principle of least squares (LS) consists of determining the values for the

unknown parameters that will minimize the sum of squares of errors (or residuals)

where errors are defined as the difference between observed values and the

corresponding values predicted or estimated by the fitted model equation.

The parameters values thus determined, will give the least sum of the squares

of errors and are known as least squares estimates. The method of least squares that

gets its name from the minimization of a sum of squared deviations, is attributed to

Gauss (1777-1855) some believed that the method was discovered at the same time by

Legendre (1952-1833).

Laplace (1749-1827) and other mar Kov’s name is also mentioned in

connection with its further development this method is used as one of the important

methods of estimating the population parameters.

The best regression line is the one, which minimizes the sum of the squares of

the vertical deviations between the observed values yi and the corresponding values yi

(hat) predicted by the regression model iioi exy 1ˆˆ –––––––––––– (4.1)

31

The set of observations (xi, yi), i = 1,2,...n, where yi are the values of y

randomly drawn from a population and xi and fixed values. Then the observed yi may

be expressed in a linear form of the population parameters as

iii xy

or in terms of sample data iioi exy 1ˆˆˆ …… (4.2)

Where 0 and 1 are the least-squares estimates of and , commonly called

residual is the deviation of the observed yi from its estimate. Provided by

iiii xy ….. (4.3)

In general the response y may be related to k regressor or Predictor variable.

The model y = 0 + 1x1 + 2x2,+ 3x3+,……….+ pxp … (4.4)

is called a multiple linear regression model with P regressors. A regression model

that involves more than one regressor variables. The parameters j, j = 0,1,2,----,p are

called the regression coefficients.

This model describes a hyper plane in the k – dimensional space of the

regressor variables xj. The parameters j represents the expected change in the

response y per unit change in xj when all of the remaining regressor variables xi

(i j) are held constant.

For this reason the parameters j, j = 1, 2, …..p, are often called partial

regression coefficients. Multiple linear regression models are often used as empirical

models or approximating functions. That is the true functional relationship between y

and x1, x2, …., xp known, but over certain ranges of the regressor variables the linear

regression model is an adequate approximate to true unknown function.

32

Models that are more complex in structure than equation (4.4) may often still

be analyzed by multiple linear regression techniques.

4.2 Estimation Techniques

The following techniques and test statistics were use in this study.

1. Ordinary Least Square Method (OLS) for parameters estimation

2. Coefficient of Determination (R-squared)

3. Adjusted R-squared

4. Standard Error Test

5. F-statistics

6. Stepwise regression procedure

7. Correlation Matrix

8. Visual normal test for standard residuals

9. Histograms

10. Test for Jarquebera

11. Test Statistic for Skewness

12. Test Statistic for Kurtosis

13. Testing for Heteroscedasticity

14. The Gold feld-Quandt Statistic (GQ-Test)

15. The t-statistic - Normal Approximation

33

4.3 Estimation of Model Parameters

4.3.1 Least Squares Estimation of the Regression Coefficients:

The method of least squares can be used to estimate the regression coefficients

in eq. (4.2) suppose that n>k observations are available, and let yi denote the i-th

observed response and xij denote the i-th observation or level of regressor xj. The data

is given in table 4.1. Assuming that the error term in the model has E() = 0, Var

() = 2 and that the errors are uncorrelated.

Data for multiple linear regression

Observation

I

Response

Y

Regression

x1, x2, …… xp

1 Y1 x11 x12 x1p

2 Y2 x21 x22 x2p

. . . . .

. . . . .

. . . . .

. . . . .

N YN xn1 xn2 xnp

Assume that the regressor variables x1, x2, ….xp, are fixed. (i.e., mathematical

or nonrandom) variables, measured without error. All the simple linear regression

models of our results are valid for the case where the regressors are random variables.

This is certainly important, because when regression data arises from an observational

study, some or most of the regressors will be random variables. When the data result

from a designed experiment.

34

It is more likely that the x’s will be fixed variables. When the x’s are random

variables it is only necessary that the observations on each regressor be independent

and that the distribution not depend on the regression coefficients (the ’s) or on 2.

When the testing hypotheses or constructing confidence intervals, Assume that the

conditional distribution of y given x1, x2, ….. xp be normal with mean

yi = 0 + 1x1 + 2x2,+ 3x3,-----------+ pxp and variance 2.

The sample regression model corresponding to equation (4.2) as

yi = 0 + 1xi1 + 2xi2,+ 3xi3 +,----+ pxip + i = 0 +

p

j 1

jxij, + i ––– (4.5)

i = 1,2,….n

The least-square function is

S (0, 1, ……,p) =

n

i

p

jijji

n

ii xy

1

2

10

1

2 )( –––––––––––––––––– (4.6)

The function S must be minimized with respect to 0, 1, …..,p. The least –

squares estimations of 0, 1, …..,p must satisfy.

0ˆˆ21 1

0ˆ,.......ˆ,ˆ0

10

n

iij

P

jji xy

S

p

–––––––––––––––––––––– (4.7)

and

0ˆˆ21 1

0

ˆ,.......ˆ,ˆ10

ij

n

iij

P

jji

j

xxYS

p

for p = 1,2,......p –––––– (4.8)

Simplifying equation (4.8) obtaining the least square normal equations.

35

n

i

n

iiipp

n

ii

n

ii yxxxn

1 1122

1110

ˆ..........ˆˆˆ

n

i

n

iiiipip

n

iii

n

ii

n

ii yxxxxxxx

1 111

1212

1

211

110

ˆ..........ˆˆˆ

– – –

– – –

n

i

n

iiipipp

n

iiip

n

iiip

n

iip yxxxxxxx

1 1

2

122

111

10

ˆ..........ˆˆˆ ––––––––– (4.9)

These k = (p +1) equations are called the normal equations, one for each of

the unknown regression coefficients. The solution to the normal equations will be the

least-square estimators ( p ˆ,......,ˆ,ˆ10 ). It is more convenient to deal with multiple

regression models, if they are expressed in matrix notation. This allows a very

compact display of the model, data, and results. In matrix notation, the model given

by Equation (4.5) is

Y = X +

where

ppn

nPn

P

P

y

y

y

Y

xx

xx

xx

X

.

.

.,

.

.

.,

.

.

.,

.........1

:::

.........1

..........1 2

1

1

0

2

1

1

221

111

36

In general Y is an n x 1 vector of the observations, X is an n x K matrix of the

levels of the regressor variables. is a k x 1 vector of the regression co-efficient and

is an (n x 1) vector of random errors.

To find the vector of least-square estimators. , that minimizes

S() =

n

ii XyXy

1

2 )()(

S() may be expressed as

S() = XXXyyXyy = XXyXyy 2

Since yX is 1 x 1 matrix and its transpose ( yX )/ = Xy is the same

scalar. The least square estimator must satisfy.

0ˆ22ˆ

XXyXS

which become

yXXX –––––––––––––––––––––––––––––––––––––––––––––– (4.10)

Equation (4.10) are the least-squares normal equations. To solve the normal

equations, multiply both sides of (4.10) by the inverse of X/X. Thus the least-square

estimator of is;

yXXX 1)( ––––––––––––––––––––––––––––––––––––––––––– (4.11)

provided that the inverse matrix (X/X)-1 exists. The (X/X)-1 matrix will always exist if

the regressors are linearly dependent, that is, if no column of the X matrix is a linear

combination of the other columns.

37

The matrix form of the normal equation (4.10) is identical to the scalar form

(4.9).

The normal equation can be written as

n

i

n

iIP

n

i

n

iiPiPiiPiiP

n

i

n

iIPiPiii

n

ii

n

ii

n

iIP

n

ii

n

ii

xxxxxxx

xxxxxxx

xxxn

1 1 1 121

1 1121

11

11

112

11

........

....

....

....

....

..........

P

ˆ

.

.

ˆ

ˆ

1

0

=

n

iiiP

n

iii

n

ii

yx

yx

y

1

11

1

.

.

If the indicated matrix multiplication is performed the scalar form of the

normal equation (4.9) is obtained. In this display we see that X/X symmetric matrix

and X/y is a k x *1 column vector. The special structure of the X/X matrix. The

diagonal elements of X/X are the sums of squares of the elements in the columns of X,

and the off-diagonal elements are the sums of cross products of the elements in the

columns of X. The elements of X/y are the sums of cross products of the columns of X

and the observations yi.

The fitted regression model corresponding to the levels of the regressors

variables x/ = [1, x1, x2, …, xp] is

p

jjj xxy

10

ˆˆˆˆ

The vector of fitted value iy corresponding to the observed values yi is

38

HyyXXXXXy 1)(ˆˆ –––––––––––––––––––––––––––––––– (4.12)

The n x n matrix H = X (X/X)-1X/ is usually called the hat matrix. It maps the

vector of observed values into a vector of fitted values. The hat matrix and its

properties play a central role in regression analysis.

The difference between the observed value yi and the corresponding fitted

value iy is the residual iii yye ˆ . The n residuals may be conveniently written in

matrix notation as:

e = y – ŷ ––––––––––––––––––––––––––––––––––––––––––– (4.13a)

There are several other ways to express the vector of residuals e that will

prove useful, including

yHIHyyXye )(ˆ –––––––––––––––––––––––––––– (4.13b)

4.3.2 Properties of the Least-Squares Estimators

The statistical properties of the least-squares estimators may be easily

demonstrated. Consider first bias:

E( ) = E[(X/X)-1X/y] = E [(X/X)-1X/(X + )]

= E[(X/X)-1X/X +(X/X)-1X/]=

since E() = 0 and (X/X)-1 X/X = I. Thus, is an unbiased estimator of .

The variance property of is expressed by the covariance matrix.

Cov ( ) = E{[ -E( )][ - E ( )]/}

39

Which is a k x k symmetric matrix whose j-th diagonal element is the variance

of j and whose (ij)th off-diagonal element is the covariance between i and j.

the covariance matrix of is

Cov ( ) = 2(X/X)-1

Therefore, if we let C = (X/X)-1, the variance of j is 2Cjj and the covariance

between i and j is 2Cjj. The least-square estimator is the best linear unbiased

estimator of (the Gauss-Markov theorem).

4.3.3 Estimation of 2

As in simple linear regression, we may develop an estimator of 2 from the

residual sum of squares

eeeyySSn

ii

n

iiis

1

2

1

2Re )ˆ(

substituting e = y - X , we have

SSRes = (y - X )/(y-X )

= y/y – /X/y – y/X + /X/X

= y/y –2 /X/y + /X/X

since X/X = X/y, this last equation becomes

SSRes = y/y – /X/y ––––––––––––––––––––––––––––––––––––– (4.14)

40

The residual sum of squares has n – k degrees of freedom associated with it

since p parameters are estimated in the regression model. The residual mean squares

is

MSRes = kn

SS s

Re –––––––––––––––––––––––––––––––––––––––––– (4.15)

The expected value of MSRes is 2, so an unbiased estimator of 2 is given by

sMSRe2ˆ –––––––––––––––––––––––––––––––––––––––––––– (4.16)

In the simple linear regression case, this estimator of 2 is model dependent.

4.3.4 Test for Significance of Regression:

The test for significance of regression is a test to determine if there is a linear

relationship between the response y and any of the regressor variables

x1, x2, ……, xk. This procedure is often thought of as an overall or global test of model

adequacy. The appropriate hypotheses are:

H0: 1 = 1 = ….. = p = 0

H1: j 0 for at least one j

Rejection of this null hypothesis implies that at least one of the regressor

x1, x2, …..xp contributes significantly to the model.

The test procedure is a generalization of the analysis of variance used in

simple linear regression. The total sum of squares SST is partitioned into a sum of

squares due to regression, SSR, and a residual sum of squares, SSRes. Thus,

SST = SSR + SSRes

41

If the null hypothesis is true, then SSR/2 follows a 2i distribution, which has

the same number of degrees of freedom as number of regressor variables in the

model. Also SSRes/2 ~ 21knX and that SSRes and SSR are independent. By the

definition of an F statistic.

s

R

s

R

MS

MS

pnSS

pSSF

ReRe0 )1/(

/

follows the Fk, n – p – 1 distribution

also E(MSRes) = 2

E(MSR) = 2 + 2

/ **

p

XX cc

4.3.5 Stepwise Regression Procedure

Stepwise regression procedure is one of the most popular algorithms of

Efroymson (1960). Stepwise regression is a modification of forward selection in

which at each step all regressors entered into the models are tested.

A regressor added at an earlier step may now be redundant because of the

relation ship between it and regressor now in the equation. If the partial F-statistic for

a variable is less than Fout that variable is dropped from the model. Stepwise

regression requires two cut off values, FIN and Fout. Some analysists prefer to choose

FIN = Fout, although this is not necessary. Frequently choosing FIN > Fout, making it

relatively more difficult to add a regressor than to delete one.

4.3.6 Studentized Residuals

Studentized residuals are helpful in identify outliers which do not appear to be

consistent with the rest of other data the hat matrix is used to identify “high leverage”

42

points which are outliers among the independent variables, the two concepts are

related.

In the case of studentized residuals, large deviations from the regression line

are identified since the residuals from a regression will generally not be independently

distributed (even if the disturbances in the regression model are), it is advisable to

weight the residuals by their standard deviations.

4.3.7 Test Statistic for Skewness

Let r*=(r(1),…,r(T)) be the vector of OLS residuals. Since the mean of the

OLS residuals is zero, the test statistic for skewness can be written as:

SK(r*) =

T

t SER

Tr

T 1

3)(1

We want to test if this is close enough to 0. To do this, we need to know the

usual range of variation of SK(r*) under the null hypothesis that the regression errors

are normal. If the observed statistic falls within the usual range of variation, we will

accept the null hypothesis of normal errors. If it falls outside the usual range then we

will reject the null hypothesis.

4.3.8 Testing for Heteroscedasticity

The assumption that the errors all have the same distribution (identical

distributions) also needs to be tested. The basic lesson is this: we must make sure that

our assumptions about the error term are valid. One assumption we have already

discussed earlier is that of homoscadasticity. We now study violations of this

assumption in greater detail.

43

Whenever one of our assumptions fails in a regression model, we say that we have

a misspecified model. There are generally three goals in misspecification analysis:

1. What happens if a misspecification occurs and is ignored?

2. How do we detect when this misspecification occurs?

3. How do we fix problems created by this misspecification?

In the standard regression model, we assume that the error terms are i.i.d. with

common distribution N(0,2). The assumption that all errors have the same variance is

called homoscadasticity. What happens when this assumption is violated? It is

possible that the t-th error (t) has variance 2 (t). When the errors have different

variances, we say that the errors are heteroscadastic. In this situation, the OLS

estimates continue to be unbiased. They are also consistent - this means that as the

sample size increases to infinity, the OLS estimates will converge to the true

parameters. However, the SER for the regression, and the SE’s for the parameters

(and therefore the t-statistics) are incorrectly computed and hence misleading.

We have discussed how OLS analysis is damaged by the presence of

heteroscedasticity. Next we consider the issue of how we can detect if

heteroscedasticity is present. In the type of case under discussion, where

heteroscedasticity increases with X(t), it is relatively easy to detect. One simple test is

the Goldfeld-Quandt test. This consists of splitting the sample into two halves, and

estimating the regression separately on both halves. Let SER(1) and SER(2) be the

Standard Error of Regression for the first half and the second half of the data set

respectively. If the ratio SER(1)/SER(2) is close to 1 then the SE’s on both halves of

the data set are similar. If the ratio is far from 1 than the two SE’s are different (which

44

is what we expect in the case of heteroscedasticity). Next, the issue is: how do we find

the critical values? That is, at what point can we say that the ratio is too far from 1 for

the null hypothesis of equal variances in both halves to be valid?

The Goldfeld-Quandt statistic is based on the ratio of variances (not SE’s):

GQ = [SER(2)/SER(1)]2

4.3.9 The t-statistic - Normal Approximation

A very important issue in the regression model is to find out whether a

regressor has any effect on the dependent variable or not. Consider the regression

model

Y(t) = 1 + 2 X2 + 3 X3 + … + k Xk + (t)………….. (4.17)

The null hypothesis that 3=0 says that X3 does not belong in the regression

equation. Equivalently, it says that X3 has no effect on the dependent variable Y(t).

This situation arises when we do not know which variables have an effect of Y

and which do not. We often have a list of variables all of which are potential

candidates for explanatory variables for Y. In this case, we are genuinely interested in

the null hypothesis, and wish to find out whether or not X3 affects Y. The t-test

provides a way of doing this. In such situations, it is often the case that if we find out

that X3 is not significant, we take it out of the regression. We might then try some

other variable. There are a number of ways of including and excluding regressors on

the basis of t tests to try to arrive at a particular set of best explanatory variables. Such

procedures are called “stepwise regression” procedures.

45

4.4 Collection of Copper Ore Samples and their Analysis for Pilot

Scale Studies

This research work is based on primary data. The representative samples were

collected by the Department of Minerals Engineering, NWFP University of

Engineering and Technology, Peshawar, with the assistance of the political authorities

of North Wazirsitan agency and Federally Administrated Tribal Area Development

Corporation (FATA DC). Other relevant information about copper deposits was

obtained from FATA DC. The survey conducted by FATA DC his conformed a

minimum of 122 million tons of inferred reserves of copper ore Boya-Datta Khel area

about 40 kms from Miran Shah. The average content of this copper ore is 0.3865%

varying from 0 to 100 feet. The Copper content increases with depth and at places it is

.90% which is better than that found at Saindak (Baluchistan) (Badshah 1983, 1985,

1996). An inventory of the ore samples was prepared. Each sample was tagged with a

number and weighed.

Both the chemical and mineralogical analyses of the sample were carried out

at Mining Engineering Laboratories (MEL) and Mineral Testing Laboratories (SDA),

Peshawar. The mineralogical investigation include X-Rays Diffraction, X-Rays

Fluorescence and ore microscopy. These chemical constituents were determined by

classical and instrumental methods of analyses. On site the samples were collected by

blasting the irregularly spaced holes within the regularly spaced rows for minimum

chances of errors. A total of 30 tons of sample was collected, comprising of six sub

sample weighing five tons each from six different locations. The rows of holes drilled

on each location were spaced at an equal interval of 300 feet. The collected samples

46

were transported to the NWFP, University of Engineering and Technology, through a

truck.

4.5 Justification of the Explanatory Variables

4.5.1 Collector dosage (X1)

Collectors, some time called promoters, are organic substance. Collectors are

the organic chemicals, which are able to selectively adsorb onto the mineral surface,

and render the mineral surface hydrophobic. Commercial collectors should ideally

possess the following character ristics:

1. They can be easily produced from broadly available materials.

2. They are cheap and convenient for users to handle them

3. They are well soluble, less toxic, and chemically stable

4. They have strong collecting capability

5. They provide higher selectivity, being expected to adsorb only one specific

minerals.

4.5.2 pH value (X2)

Control of solution pH is one of the most widely used methods, for regulating

complex separations in flotation. The depressant action of alkalis results from an

increase in the rate of dissolution or oxidation of the mineral surface. Pulp pH value

plays a significant role in flotation through its influence both on mineral flotability

and reagent function.

47

4.5.3 Depressant (X3)

Chemical used to modify the surface of gangue to prevent it from

hydrophobacity.

4.5.4 Sulphidizer (Na2S) (X4):

Sodium sulphide is a major modifier used for the activation of oxide minerals.

It is a salt produced from reaction between strong alkali and weak acid.

Na2S + H2O 2Na+ + H2S + OH-

As a result of hydrolysis, hydroxide ions and hydrogen sulphide appear in

solution. The latter is disassociated with the formation of hydrosulphide ion.

4.5.5 Frothers Dosage (X5):

Chemical used to modify the surface of copper to make it suitable for

hydrophobicity.

4.5.6 Frothers:

The function of frother is to disperse air into fine bubbles and to form a stable

froth. Frothing action is thus due to the ability of the frother to adsorb on the air water

interface because of its surface activity and to reduce the surface tension. Thus

stabilizing the air bubble interface. Bubbles undue merge or breakage is harmful to

flotation through destroying bubble-particle attachment and dropping the collected

valuables back to pulp before the froth carrying them is removed.

Bubbles strength i.e. their stability is required which can be realized by

increasing aeration and frother proper frother. Frother acts entirely in liquid phase and

does not influences, the state of the mineral surface.

48

1. Prevent coalescence of separate air bubbles

2. Decrease the rate at which air bubbles rise in the flotation machine to the

surface of the pulp.

3. Affect the action of the collectors

4.5.7 Effect of pulp density (X6):

An increase in pulp density the recovery and grade curves show an upward

trend due to hindered setting conditions upto 30%.

4.5.8 Flotation time (X7):

Increasing flotation time increases recovery but at the cost of decrease in

concentrate grade. Flotation time is dependent on mineral floatability, grinding

fineness, reagent scheme and other conditions.

In flotation process the raw ore is ground with water, the thick pulp 30% water

is prepared by adding various reagents having specific purposes, the copper particles

are floated and/recovered materials is called % recovery and % copper content in it is

called % grade.

Mathematically if C is the concentrate, c is the metal weight in the

concentrate, if f is the average feed and F is the feed weight then % recovery =

(Cc/Ff) x 100

Flotation is one of the most important mineral concentration techniques. It is

known that the appearance of the froth in the flotation cells tells much about the state

of the flotation process. A machine vision measurement device was used to compute

dynamical, morphological and colour variables of the froth on the top of a flotation

49

cell during a set of experiments. In order to examine the dependencies between the

state of the process and the separated image variables, a set of experiments was

carried out. The behaviour of the froth state variables imply that these image variables

are useful in the control and monitoring of the complex process. In process industry

one of the most frequently use method for separation of valuable substances from the

waste is flotation. Especially in the mining industry flotation is widely used. Flotation

means the use of air bubbles to concentrate small mineral grains from the ore

suspension. Relatively heavy mineral grains attach themselves to the air bubbles due

to surface chemical phenomena and are transported to the froth. Concentrated froth is

collected for further treatments as it flows over the shoulder into the gutter.

Information of the state of a flotation process can be seen from the appearance of the

froth layer on the top of the flotation cell. Operators at the flotation plant shave

applied this information in manual control of the flotation process for ages. They use

the colour, speed and shape information of the froth layer. Development of image

processing methods has made it possible to acquire real-time numerical data of the

froth for control purposes. The possibility of utilizing image information in mineral

flotation has aroused a lot of interest in the mineral engineering community. Up to

now, however, the research has been mainly concentrating on image analysis

problems, i.e. how to extract a certain image feature from the froth images. To really

investigate whether the image data can be utilized in the monitoring and control of

flotation process or not, a set of experiments was designed, carried out and analyzed.

As a result information would be obtained about the appearance of the froth and the

behaviour of a flotation process in different control circumstances. Experiments were

carried out in the zinc flotation circuit of the flotation plant at Pyhäsalmi, Finland, in

October 1998.

50

In flotation the efficiency of mineral enrichment is determined by properties of

the minerals; these properties can be modified by the use of suitable chemical

treatment. Chemicals that are used in the flotation process can be roughly divided into

three different categories: collectors, frothing agents and regulators. The task of

collector chemicals is to make valuable minerals hydrophobic. The frothing agent is

used for lowering the surface tension of water; this makes froth, which forms on the

top of the flotation cell, viscous and stable enough. The regulator chemicals control

the selectivity of the flotation process. These chemicals are divided into two

subgroups depending on whether they ease flotation of certain minerals, in which case

they are called activators, or make more difficult the flotation of unwanted minerals,

in which case they are called depressants.

51

CHAPTER – 5 MODELS BUILDING

5.1 General Model For Recovery:

General to simple strategy was used to construct mathematical models to

maximize the efficiency of flotation process for the recovery (YR) and grade (YG) of

copper ore. The response variables YR and YG were regressed on seven variables X1,

X2, …. X7 as shown in the following model:

So that Y = 0 + 1X1 + 2X2 +……….+7X7 +

(Where Y = YR and Y = YG)

The above is a multiple linear regression model because more than one

regressor is involved when Xi are called the independent variable or response

variables. The adjective linear is employed to indicat that the model is linear in the

parameters 0, 1, …… 7 not because YR and YG is a linear function of the Xi’s. An

important objective of regression analysis is to estimate the unknown parameters in

the regression model. This process is also called fitting the model to the data.

5.2 General Description:

Brief Discussion About The Model

Models are often used to decide issues in situations marked by uncertainty.

However, statistical inferences about data depend upon assumptions about the process

52

which generated the data. If the assumptions do not hold, the inferences may not be

reliable. The limitation is often ignored by the applied workers who fail to identify

crucial assumptions or subject them to any kind of empirical testing. In such

circumstances, using statistical procedures may compound the uncertainty. Therefore

developing models by checking some of the assumptions.

To fit a model, to a set of data, one or both of the following methods are employed.

1) Start with the general model for YR (the dependent variable) that contains all

available independent variables, then simplify the model by eliminating the

independent variables that do not contribute significantly to the variability in

the dependent variable;

2) Start with a simple model and elaborate on it by adding additional independent

variables. The variable highly correlated with dependant variable is used to

develop simple one variable model, the partial correlations of the remaining

independent variables with dependent variable are calculated and the variable

with highest partial correlation is then included in the simple model and so on.

Here we have employed both the methods in our study.

53

Mathematical Model For Optimum Recovery

5.3 About the data for recovery of copper

The data given in Table – 3 were recorded in a series of seven experiments

with a total of 31 different treatments. The experiments were carried out in the

Department of Mining Engineering, NWFP University of Engineering and

Technology, Peshawar.

The data consist of values for grade and recovery as affected by the different

values of seven flotation process variables; collector NaPX (X1), PH(X2), depressant-

NaCN (X3), sulphadizer Na2S (X4), frother pine oil (X5), pulp density (X6), and

conditioning time (X7). The whole data are made up of seven sub groups. In each sub

group only one of the process variables was varied and others were kept constant.

5.3.1 Effect of variation in collector dosage, NaPX (X1).

The first experiment was conducted to investigate the effect of five levels of

collector, sodium propylxanthate, on the recovery of copper, while keeping all the

other six variables constant. Effect of collector dosage on recovery is given in Figure

1. The trend of recovery presented in Figure 1 shows that with an increase in level of

sodium propylxanthate up to 200 g/ton of feed, there was a corresponding increase in

recovery of copper; with further increase in the level of collector there was a slight

decrease in recovery. This decrease might be due to the nonspecific absorption of

collector by the gangue particles. Therefore, 200g of Prophylxanthate per ton feed is

the optimum level for recovery of copper.

54

Table – 3: Primary Data On Recovery (YR) Of Copper As Affected By Seven Flotation Process Variables (X1 to X7)

X1 X2 X3 X4 X5 X6 X7 YR

50 11 0 0 75 30 10 32

100 11 0 0 75 30 10 32.7

150 11 0 0 75 30 10 38

200 11 0 0 75 30 10 41.5

250 11 0 0 75 30 10 41

200 10 0 0 75 30 10 35

200 10.3 0 0 75 30 10 36

200 11 0 0 75 30 10 42

200 11.58 0 0 75 30 10 45.2

200 12 0 0 75 30 10 40

200 11.58 10 0 75 30 10 49

200 11.58 15 0 75 30 10 50

200 11.58 20 0 75 30 10 55

200 11.58 25 0 75 30 10 63

200 11.58 30 0 75 30 10 60

200 11.58 25 10 75 30 10 60

200 11.58 25 30 75 30 10 63

200 11.58 25 40 75 30 10 67

200 11.58 25 50 75 30 10 73

200 11.58 25 60 75 30 10 59.4

200 11.58 25 50 25 30 10 70

200 11.58 25 50 46 30 10 74

200 11.58 25 50 70 30 10 71.56

200 11.58 25 50 46 15 10 56

200 11.58 25 50 46 25 10 64

200 11.58 25 50 46 30 10 75

200 11.58 25 50 46 35 10 68

200 11.58 25 50 46 30 10 69.77

200 11.58 25 50 46 30 13 73.3

200 11.58 25 50 46 30 16 68

200 11.58 25 50 46 30 18 64

55

5.3.2 Effect of variation in pH (X2)

The next experiment was conducted to investigate the effect of variation in pH

of pulp (X2) on the recovery of copper, while keeping collector level at the optimum

found in experiment one, and all the other five variables constant. Five levels of pH

were used in the experiment. It is evident from Figure 2 that increase in pH from 10 to

11.58 increased recovery, the fourth level of pH studied gave the highest recovery and

thus pH of 11.58 is optimum for better recovery because pH beyond 11.58, decreased

recovery. This decrease is due to the deactivation of NaOH on the copper minerals.

5.3.3 Effect of variation in depressant, NaCN (X3)

The third experiments was conducted to investigate the effect of variation in

depressant NaCN (X3) on the recovery of copper, while keeping all other variables

constant. The effect of depressant on recovery of copper ore is shown in Figure 3. The

figure clearly shows that recovery increased with increase in depressant dosage upto

25g/ton. Levels of depressant higher than 25 g/ton decreased recovery of copper. This

decrease might be due to the deactivation of copper particles in the pulp by sodium

cyanide as complex cyanides. From the figure 3 it is clear that the optimum dosage of

depressant is 25g/ton.

5.3.4 Effect of variation in sulfidizer, Na2S (X4)

The fourth experiment was conducted to investigate the effect of variation in

sulfadizer (Na2S) on the recovery of copper, while keeping the first three variables at

the optimum levels found in the previous experiments and the other three variables

constant. The effect of sulfidizer on recovery of copper ore is shown in Figure 4. The

curve in Figure 4 shows that recovery was maximum at 50g sulfidizer per ton of feed;

56

with further increase in the level of suplfidizer, there was a drastic decrease in

recovery of copper. This decrease may be attributed to depressive action of sodium

sulphide.

5.3.5 Effect of Variation in Frother Pine Oil (X5)

The next experiment was conducted to investigate the effect of variation in

pine oil (X5) on the recovery of copper, while keeping the first four variables at the

optimum levels found in first four experiments, and variables six and seven constant.

The effect of frother dosage investigated in the fourth experiment is shown in Figure

5. Frother imparts stability to the mineral froth and helps in achieving maximum

recovery.

The optimum frother dosage was found to be 46g of pine oil per ton of feed.

5.3.6 Effect of Pulp density (X6)

Four levels of pulp density were studied in the sixth experiment to investigate

the effect of variation in pulp density (X6) on the recovery of copper. The first five

variables were kept at the levels giving highest recoveries in the previous experiments

while variable seven, conditioning time was kept at 10 minute. Figure 6 shows that

with an increase in pulp density the recovery showed an upward trend due to hindered

setting condition up to 30% Pulp density. Recovery decreased with further increase in

pulp density beyond 30%. The recovery showed marked decrease with the highest

pulp density of 35% due to the entrapped fine slime particles.

5.3.7 Effect of conditioning time (X7)

The next experiment was conducted to investigate the effect of variation in

conditioning time (X7) on the recovery of copper, while keeping all other variables at

57

the optimum levels which were found in the previous experiments. The conditioning

time was varied between 10 to 18 minutes. Effect of conditioning time on recovery of

copper is presented in Figure 7. It is evident from the Figure 7 that 13 minutes

conditioning time was optimum, since beyond this, recovery markedly decreasing due

to dissolution of copper xanthate ions in the equilibrium system.

(a)

10

15

20

25

30

35

40

45

50 100 150 200 250 300

PropylXanthate (g/ton)

% R

eco

very

(b)

20

25

30

35

40

45

50

9 10 11 12 13

pH%

Rec

ove

ry

Figure1: EFFECT OF COLLECTOR

(NaPX) ON RECOVERY OF COPPER Figure-2: EFFECT OF pH ON RECOVERY

OF COPPER

(c)

30

40

50

60

70

0 5 10 15 20 25 30 35

Sodium Cyanide(g/ton)

% R

eco

very

(d)

20

40

60

80

0 20 40 60 80

Sodium Sulphide (g/ton)

% R

eco

very

Figure 3: EFFECT OF DEPRESSANT (NaCN) ON RECOVERY OF COPPER

Figure-4: EFFECT OF SULFIDIZER (Na2S) ON RECOVERY OF COPPER

58

(e)

10

30

50

70

90

0 20 40 60 80

Pineoil (g/ton)

% R

eco

very

(f)

20

30

40

50

60

70

80

10 20 30 40

Pulp Density (%wt/vol)

% R

eco

very

Figure-5: EFFECT OF FROTHER (PINE OIL) ON RECOVERY OF COPPER

Figure-6: EFFECT OF PULP DENSITY ON RECOVERY OF COPPER

(g)

20

30

40

50

60

70

80

0 5 10 15 20

Conditioning time (minutes)

% R

eco

very

Figure-7: EFFECT OF CONDITIONING TIME ON RECOVERY OF COPPER

59

5.4. Modeling effect of individual variable for recovery

To develop mathematical models for recovery of copper involving one

variable, four to six models were fitted for each of the seven variables. The models

fitted were linear, logarithmic, quadratic, power, exponential and two straight-line for

each of the seven independent variables; quadratic model was not used for variable

five because it will lead to over fitting and will pass through all the three points.

Minitab statistical analysis was used to fit the forty mathematical models in

single predictor variable. The fitted equations and co-efficient of determination are

given in Table 4 below.

Table – 4: Mathematical Models Involving One Predictor Variable For Recovery Of Copper By Flotation.

YR = 0.0536X1 + 29, R2 = 0.8897 (5.1)

YR = 6.5631Ln(X1) + 5.0807, R2 = 0.8619 (5.2)

YR = -0.0001X21 + 0.0896X1 + 26.9, R2 = 0.9053 (5.3)

YR = 15.289X10.1805, R2 = 0.8678 (5.4)

YR = 29.541e0.0015X1, R2 = 0.8884 (5.5)

YR = 27.6 + 0.06 X – 3.5X’ R2 = 0.9503 (5.6)

Graphical representation of above six equations are given in figure-8

YR= 3.815X2 –2.237, R2= .5757 (5.7)

YR = 42.465Ln(X2) - 61.994, R2 = 0.5940 (5.8)

60

YR = -5.1611X22 + 117.21X2 - 622.2, R2 = 0.8568 (5.9)

YR = 2.9069X21.0898, R2 = 0.6161 (5.10)

YR = 13.468e0.0979X2, R2 = 0.5975 (5.11)

YR = -33.52 + 6.81 X – 8.27 X’ R2 = 0.9868 (5.12)

Graphical representation of above six equations are given in figure-9

YR = 0.7X3 + 41.4 R2 = 0.8210 (5.13)

YR = 12.69Ln(X3) + 18.277 R2 = 0.8123 (5.14)

YR = -0.0143X23 + 1.2714X3 + 36.4 R2 = 0.8330 (5.15)

YR = 28.043X30.2311 R2 = 0.8302 (5.16)

YR = 42.746e0.0127X3 R2 = 0.8360 (5.17)

YR = 37.8 + 0.94X – 6X’ R2 = 0.9176 (5.18)

Graphical representation of above six equations are given in figure-10

YR = 0.0897X4 + 61.07 R2 = 0.0938 (5.19)

YR = 3.2386Ln(X4) + 53.2 R2 = 0.1652 (5.20)

YR = -0.0107X24 + 0.8295X4 + 51.563 R2 = 0.4242 (5.21)

YR = 54.228X40.048 R2 = 0.1634 (5.22)

61

YR = 61.129e0.0013X4 R2 = 0.0889 (5.23)

YR = 55.62 + 0.31 X – 14.9 X’ R2 = 0.9222 (5.24)

Graphical representation of above six equations are given in figure-11

YR = 0.0314X5 + 70.376 R2 = 0.1233 (5.25)

YR = 1.8784Ln(X5) + 64.781 R2 = 0.2327 (5.26)

YR = 65.03X50.0264 R2 = 0.2391 (5.27)

YR = 70.347e0.0004X5 R2 = 0.1283 (5.28)

Graphical representation of above four equations are given in figure-12

YR = 0.76X6 + 45.8 R2 = 0.6694 (5.29)

YR = 18.224Ln(X6) + 7.0532 R2 = 0.7168 (5.30)

YR = -0.0509X26 + 3.2691X6 + 17.8 R2 = 0.7773 (5.31)

YR = 25.854X60.2881 R2 = 0.7542 (5.32)

YR = 47.704e0.012X6 R2 = 0.7040 (5.33)

YR = 37.0 + 1.20 X – 11.0 X’ R2 = 0.9258 (5.34)

Graphical representation of above six equations are given in figure-13

YR = -0.7931 X7 + 80.07 R2 = 0.5153 (5.35)

62

YR = -9.8924Ln(X7) + 94.811 R2 = 0.4352 (5.36)

YR = -0.3245X27 + 8.274X7 + 19.746 R2 = 0.9533 (5.37)

YR = 100.95X7-0.1463 R2 = 0.4463 (5.38)

YR = 81.159e-0.0117X7 R2 = 0.5269 (5.39)

YR = 88.29 – 1.85 X + 9.16X’ R2 = 0.9988 (5.40)

Graphical representation of above six equations are given in figure-14

Information presented in Figure 8, reveals that all the models gave good fit to

the data. Two straight line model had the highest R2 of 0.9503, followed by quadratic

model, the simple linear regression model also gave good fit as it had the third highest

R2. Two straight line model is best because the X-maximum calculated from the

quadratic model, 448 gram per ton, is much out of the range used in the study. The

recovery of copper increased at the rate of 0.0536 per gram increase in sodium

propylxanthate considering linear model. The equation for two straight lines show that

recovery increased at the rate of 0.0676% per one gram increase in collector up to

200g/ton, there after the recovery remained the same upto 250 g/ton of collector.

63

(a)

y = 0.0536x + 29

R2 = 0.889720

30

40

50

40 90 140 190 240 290

PropylXanthate (g/ton)

% R

ecov

ery

(b)

y = 6.5631Ln(x) + 5.0807R2 = 0.8619

0

10

20

30

40

50

0 100 200 300

PropylXanthate

% R

ecov

ery

(c)

y = -0.0001x2 + 0.0896x + 26.9R2 = 0.9053

0

10

20

30

40

50

0 100 200 300

ProphylXanthate

% R

eco

very

(d)

y = 15.289x0.1805

R2 = 0.8678

0

10

20

30

40

50

0 50 100 150 200 250 300

ProphylXanthate

% R

eco

very

(e)

y = 29.541e0.0015x

R2 = 0.8884

20

30

40

50

40 90 140 190 240 290

ProphylXanthate (g/ton)

% R

eco

very

Figure-8: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL AND (f) TWO STRAIGHT-LINE MODELS FITTED TO THE

RECOVERY OF COPPER DATA FROM FIVE LEVELS OF COLLECTOR TYPE AND DOSAGE IN THE FLOTATION PROCESS.

20

30

40

50

40 90 140 190 240Sodium propylxanthate (g/ton)

% R

eco

very

(f)

Y = 27.6 + 0.0676X - 3.5X'R2 = 0.9503

64

Two straight lines gave the best fit followed by quadratic model gave best fit

followed by power model, and the other three models also gave good fit to the data

for copper recovery as affected by pH of pulp in flotation process (Fig. 9) the X-max

calculated from the quadratic equation show that 11.3 pH of the pulp will result in

maximum recovery, however, the two straight lines model show that pH of 11.6 is the

joining point with the highest recovery.

(a)

y = 3.8154x - 2.2376

R2 = 0.5757

20

30

40

50

9.8 10.3 10.8 11.3 11.8

pH

% R

eco

very

(b)

y = 42.465Ln(x) - 61.994

R2 = 0.594

20

30

40

50

9.8 10.3 10.8 11.3 11.8

pH

% R

eco

very

(c)

y = -5.1611x2 + 117.21x - 622.2

R2 = 0.8568

20

30

40

50

9.8 10.3 10.8 11.3 11.8

pH

% R

eco

very

(d)

y = 2.9069x1.0898

R2 = 0.6161

20

30

40

50

9.8 10.3 10.8 11.3 11.8

pH

% R

eco

very

65

(e)

y = 13.468e0.0979x

R2 = 0.5975

20

25

30

35

40

45

50

9.8 10.3 10.8 11.3 11.8

pH

% R

eco

very

Figure-9: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE RECOVERY OF COPPER DATA FROM FIVE LEVELS OF PH OF PULP IN

THE FLOTATION PROCESS.

The two straight lines model gave best fit. All the other five models gave good

fit to the data for copper recovery as affected by depressant (Fig. 10) the differences

in the R2’s of the models are very small. Though the original data points show

maximum recovery at 25, the quadratic X-max is beyond the range used, the

maximum recovery may be obtained around the last 2 data points i.e. 25 and 30 g/ton

of depressant. The two straight lines model gave in figure 10 show a joining point at

25 g/ton of depressant with decrease on both sides thus 25g depressant per ton is

optimum for recovery of copper.

20

30

40

50

9.8 10.3 10.8 11.3 11.8pH

% R

eco

very

Y = -33.52 + 6.8165X - 8.2751X'R2 = 0.9868

(f)

66

(a)

y = 0.7x + 41.4

R2 = 0.821

30

40

50

60

70

8 12 16 20 24 28 32

Sodium Cyanide(g/ton)

% R

eco

very

(b)

y = 12.69Ln(x) + 18.277

R2 = 0.8123

30

40

50

60

70

8 12 16 20 24 28 32

Sodium Cyanide(g/ton)

% R

eco

very

(c)

y = -0.0143x2 + 1.2714x + 36.4R2 = 0.8330

30

40

50

60

70

8 12 16 20 24 28 32

Sodium Cyanide(g/ton)

% R

eco

very

(d)

y = 28.043x0.2311

R2 = 0.8302

30

40

50

60

70

8 12 16 20 24 28 32

Sodium Cyanide(g/ton)

% R

eco

very

(e)

y = 42.746e0.0127x

R2 = 0.836

30

40

50

60

70

8 12 16 20 24 28 32

Sodium Cyanide(g/ton)

% R

eco

very

Figure-10: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

RECOVERY OF COPPER DATA FROM FIVE LEVELS OF DEPRESSANT IN THE FLOTATION PROCESS.

Equations and coefficient of determinations given in Figure 11 show that two

straight lines model which gave best fit followed by quadratic model which gave good

fit to the copper recovery data as affected by sulphidizer in the flotation process X-

30

40

50

60

70

8 12 16 20 24 28 32Sodium Cyanide (g/ton)

% R

eco

vry

(f)

Y = 37.8 + 0.94X - 6X'R2 = 0.9176

67

max calculated from the quadratic equation is 38 g/ton. However, the two straight

lines model show that the joint point at 50 g/ton of sulphidizer will give maximum

recovery of copper.

(a)

y = 0.0897x + 61.07

R2 = 0.0938

20304050607080

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

(b)

y = 3.2386Ln(x) + 53.21

R2 = 0.1652

20304050607080

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

8

(c)

y = -0.0107x2 + 0.8295x + 51.563

R2 = 0.4242

20

30

40

50

60

70

80

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

(d)

y = 54.228x0.0489

R2 = 0.1634

20

30

40

50

60

70

80

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

(e)

y = 61.129e0.0013x

R2 = 0.0889

20

30

40

50

60

70

80

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

Figure-11: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE RECOVERY OF COPPER DATA FROM FIVE LEVELS OF SULPHIDIZER

IN THE FLOTATION PROCESS.

20

30

40

50

60

70

80

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium Sulphide (g/ton)

% R

eco

very

Y = 55.62 + 0.3114X - 14.914X'R2 = 0.9222

(f)

68

Power and logarithmic models gave fair fit to the copper recovery data on

frother dosage from flotation process. The other two models gave poor fit which is

given in (Figure-12). It seems that maximum recovery will be obtained when the

dosage of frother (pine oil) is around 50 (vol/wt). Further data is needed in this case,

as R2 is low.

(b)

y = 1.8784Ln(x) + 64.781

R2 = 0.232769

70

71

72

73

74

75

0 20 40 60 80Frother (g/ton)

% R

eco

very

(c)

y = 65.03x0.0264

R2 = 0.239169

70

71

72

73

74

75

0 20 40 60 80Frother (g/ton)

% R

eco

very

(d)

y = 70.347e0.0004x

R2 = 0.128369707172737475

0 20 40 60 80

Frother (g/ton)

% R

eco

very

Figure-12: (a) LINEAR (b) LOGARITHMIC (c) POWER (d) AND EXPONENTIAL MODELS FITTED TO THE RECOVERY OF COPPER

DATA FROM THREE LEVELS OF FROTHER DOSAGE IN THE FLOTATION PROCESS.

Figure 13 show that two straight lines, quadratic and power function gave fit

than the other models in case of copper recovery data as affected by the pulp density

in the flotation process for enrichment of copper ore. Two straight lines give the next

(a)

y = 0.0314x + 70.376R2 = 0.1233

69

70

71

72

73

74

75

0 20 40 60 80Frother (g/ton)

% R

eco

very

69

best fit quadratic equation also gave good fit and X-max was 32 showing that pulp

density of 32 will gave maximum recovery, though the trend of other functions show

that recovery increased with increase in pulp density.

(a)

y = 0.76x + 45.8

R2 = 0.6694

20304050607080

10 20 30 40

Pulp density (%wt/vol)

% R

eco

very

(b)

y = 18.224Ln(x) + 7.0532

R2 = 0.7168

20

30

40

50

60

70

80

10 15 20 25 30 35 40

Pulp density(%wt/vol)

% R

eco

very

(c)

y = -0.0509x2 + 3.2691x + 17.8

R2 = 0.777320304050607080

10 20 30 40

Pulp density(%wt/vol)

% R

eco

very

(d)

y = 25.854x0.2881

R2 = 0.7542

10

20

30

40

50

60

70

80

10 15 20 25 30 35 40

Pulp density(%wt/vol)

% R

eco

very

(e)

y = 47.704e0.012x

R2 = 0.70420

30

40

50

60

70

80

10 20 30 40Pulp density(%wt/vol)

% R

eco

very

Figure-13: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE RECOVERY OF COPPER DATA FROM FOUR LEVELS OF PULP DENSITY

IN THE FLOTATION PROCESS.

20304050

607080

10 20 30 40Pulp Density (%wt/vol)

% R

ec

ov

ery

Y = 37.0 + 1.20X - 11.0X'R2 = 0.9258

(f)

70

The coefficient of determination for the six models given in Figure 14 show

that all the models gave good fit to the data on recovery of copper as affected by

conditioning time. The recovery decreased with increase in the flotation time, beyond

13 minutes. Quadratic model gave better fit the X-max from quadratic equation is

about 11 minutes. However the two straight line model gave best fit with R2 = 0.9988;

there was not much effect of conditioning time in the range of 10-12 minutes beyond

13 minutes the recovery decrease.

(a)

y = -0.7931x + 80.07

R2 = 0.515320

30

40

50

60

70

80

8 12 16 20

Conditioning time (minute)

% R

eco

very

(b)

y = -9.8924Ln(x) + 94.811

R2 = 0.435220

30

40

50

60

70

80

8 12 16 20

Conditioning time (minute)

% R

eco

very

(c)

y = -0.3245x2 + 8.274x + 19.746

R2 = 0.953320

30

40

50

60

70

80

8 12 16 20

Conditioning time (minute)

% R

eco

very

(d)

y = 100.95x-0.1463

R2 = 0.4463

20

30

40

50

60

70

80

8 12 16 20

Conditioning time (minute)

% R

eco

very

71

(e)

y = 81.159e-0.0117x

R2 = 0.5269

20

30

40

50

60

70

80

8 12 16 20

Conditioning time (minute)

% R

eco

very

Figure –14: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

RECOVERY OF COPPER DATA FROM FOUR LEVELS OF CONDITIONING TIME IN THE FLOTATION PROCESS.

Summing up the results of modeling individual variable effects, the graphs

reveals that the predicted values are with in the range when not extrapolated. The

most suitable models for the effect of the seven individual variables on recovery for

enrichment of copper are models 5.6,5.12,5.18,5.24,5.27,5.34 and 5.40 for X1, X2, ----

-, X7 respectively based on R2. In other words, two straight line equation give best fit

for X1,X2, X3, X4 and X7 while power model give good fit for X5, and exponential

give best fit for variable X6.

20.0

30.0

40.0

50.0

60.0

70.0

80.0

8 12 16 20

Conditioning Time

Gra

de

of

co

pp

er

(%)

Y = 88.29 - 1.8526X + 9.16X'R2 = 0.9988

20

30

4050

60

70

80

8 12 16 20Conditioning Time (minutes)

% R

eco

very

Y = 88.29 - 1.8526X + 9.16X'R2 = 0.9988

(f)

72

5.5. Modeling combined effect of variables on recovery

The data from all experiments were used to construct models for recovery of

copper ore. The following four strategies were followed for model selection.

1. Forward selection procedure or simple to general model building strategy.

2. Back ward elimination procedure or general to simple model building strategy.

3. Stepwise selection procedure

4. Best subset procedure

5. Modeling effect of pairs of variables with interaction from best subset

procedure.

6. Multiple regression model with testing Apt mess of model and checking the

assumptions.

73

5.6 Forward selection or simple to general procedure for model

building

The forward stepwise selection procedure was used to select variables for

modeling recovery of copper using the seven variables. The equations selected at each

step are given below. The details of each step are given in appendix-1

YR =38.254+1.101 X3 (5.41)

YR=38.493+0.837 X3+0.168 X4 (5.42)

YR =15.65 +0.835 X3+0.185 X4+0.76 X6 (5.43)

YR=6.132 +0.054 X1+0.779X3 +0.188X4+0.76 X6 (5.44)

YR =-35.026+0.053X1+3.7X2+0.688 X3+0.191 X4+0.76X6 (5.45)

In the forward selection procedure or simple to general model building

strategy, the model involving X3 was the first model fitted. The program then selected

X4 and the next model involved X3 and X4. The process was continued till no more

variable met the criteria of entering in the model. The final model had intercept and

five variables, X1, X2, X3, X4, and X6. In five variables model intercept and X2, are

not important so we drop intercept and X2. Thus model involving X1, X3, X4, and X6

without intercept is the best model.The improvement in R2 of model five over model

four is very meager to 0.

When stepwise procedure was used for model selection, it gave the same

result as forward selection.

74

However, when no intercept option was used the stepwise first included X2

then X3, X4, X6 and X1 but finally removed X2 as its probability was greater than

alpha to remove and thus the final model from stepwise procedure was the same as

forward selection with out intercept option.

5.7 Backward elimination or general to simple procedure for

model building

The backward elimination procedure or general to simple model building

strategy was used to select best model for recovery of copper ore.

The following equations, were fitted at each step. The details of each step are

given in appendix – 2.

YR=-25.20+0.053X1+3.7X2+0.69X3+0.158X4-0.098X5+0.83X6-0.45X7 (5.46)

YR=-29.93+0.053X1+3.7X2+0.69X3+0.155X4-0.080X5+0.79X6 (5.47)

YR=-35.03+0.053X1+3.7X2+0.69X3+0.191X4+0.76X6 (5.48)

YR = 6.132+0.053X1+0.779X3+0.188X4+0.762X6 (5.49)

By adopting backward elimination procedure or general to simple procedure

the full model with seven variables was fitted first Variable X7 had not much

contribution so it was eliminated first, X5 was eliminated next. In model including five

variables intercept, X2 were not statistically significant so we drop intercept and X2

from this model and we obtained the best fit model as given in the forward selection

procedure.

75

5.8 Best subset for recovery

The total all possible regression models involving seven variable are 128, it is

very difficult to check all these models so the best subset procedure was used to select

two best models involving one, two, three, four, five and six, variables. Using best

subset method (Minitab), we got thirteen models, two in each subset and the full

model for recovery of copper. The summary of best subset models are given below.

Models with one variable are:

M1: YR = 0 + 1X3

M2: YR = 0+ 1X4

Models with two variable are:

M3: YR = 0+ 1X3 + 2 X4

M4: YR = 0 + 1X3 + 2 X5

Models with three variable are:

M5: YR = 0 + 1X1 + 2 X3 + 3 X4

M6: YR = 0 + 1X3 + 2 X4 + 3 X6

Models with four variable are:

M7: YR = 0 + 1X1 + 2 X3 + 3 X4 + 4 X6

M8: YR = 0 + 1X2 + 2 X3 + 3 X4 + 4 X6

M9: YR =1X2 + 2 X3 + 3 X4 + 4 X6

Models with five variables are:

76

M10: YR = 0 + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X6

M11: YR = 0 + 1X1 + 2 X3 + 3 X4 + 4 X6 + 5 X7

Models with six variables are:

M12: YR = 0 + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X6 + 6 X7

M13: YR = 0 + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X5 +6 X6

Models with seven variables are:

M14: YR = + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X5 +6 X6 + 7 X7

Among the subset with single predictors.

The regression equations with single predictors for recovery of copper

obtained from least square analysis are as follows:

YR = 41.4 + 0.7 X3 ………………..…(5.50)

YR = 61.07 + 0.089 X4 ………………(5.51)

The R2 show that the first equation explained 85.2% of the variation and the

second equation explained 67.3% variation in the recovery of copper using flotation

process. The coefficients of equation (5.50) are different from coefficients of equation

(5.13), though both have X3 as independent variable. The differences in coefficients of

the two equations for X3, are due to the fact that equation (5.13) is based on the data

from one experiment and equation (5.50) is based on combined data from seven

experiments. Similarly, the differences in coefficient of equation (5.19) and (5.51) are

due to the same reason as above; equation (5.19) is based on data from one

experiment and equation (5.51) is based on data from seven experiments.

77

010203040506070

0 10 20 30 40Sodium Cyanide (g/ton)

% R

eco

very

58596061626364

0 20 40 60 80Sodium Sulphide (g/ton)

% R

eco

very

Figure-15: EFFECT OF SODIUM CYANIDE (X3)ON THE RECOVERY OF

COPPER.

Figure-16: EFFECT OF SODIUM SULPHIDE (X4)ON THE RECOVERY OF

COPPER.

In a single predictor variable models involving X3 and X4 were found as better

model than other five models. Based on R2, F-value, t-statistics and P-value (as shown

in the output, given in appendix – 3) model having X3 is better than model having X4.

The first model show that thirty g/ton of depressant gave maximum recovery 71.3%

and Recovery increased at the rate 1.10% per one-gram increase in depressant. The

second model show that recovery increased at the rate of 0.465% per one gram

increase in sulfidizer.

Among the twenty-one models in the subset with two predictor variables, the

two best regression equation involving two predictor variables are:

YR = 38.5 + 0.837 X3 + 0.168 X4……………………………………(5.52)

YR = 50.7 + 0.991 X3 - 0.164 X5…………………………………….(5.53)

The equation (5.52) involving X3 and X4 explained 89.2% and the equation

(5.53) involving X3 and X5 explained 87.5% of the variation in the recovery of

copper.

78

Response surfaces were developed for the variables involved in the above two

equations.

Figure -17: COPPER RECOVERY (YR) RESPONSE SURFACE FOR SODIUM

CYANIDE (X3) AND SODIUM SULPHIDE (X4).

The combine response surface for sodium cyanide (X3 g/ton) and depressant

sodium sulphide (X4 g/ton) on the recovery of copper reveals that the maximum peak

of surface shows the estimated maximum recovery of 73.69% with 30 gram per ton of

sodium cyanide and 60 gram per ton of sodium sulphide.

X4

X3

YR

79

Figure-18: COPPER RECOVERY (YR) RESPONSE SURFACE FOR SODIUM

SUPHIDE (X4), AND FROTHER DOSAGE (X5).

The combine response surface for sodium sulphide (X4 g/ton) and frother

dosage (X5 g/ton) on the estimated recovery of copper is given in the Figure 18. The

maximum peak of surface show the maximum recovery of 76.33% with 30 gram per

ton of sodium sulphide and 68-70 grams per ton of frother.

The best subset program picked the following two best regression equations

involving three predictor variables among the fifty-five, 3-variable models in the

subset with 3-predictors:

YR = 29.0 + 0.053 X1 + 0.782 X3 + 0.171 X4 ………………………………….(5.54)

YR = 15.6 + 0.835 X3 + 0.185 X4 + 0.762 X6 ………………………………….(5.55)

The two equations (5.54) and (5.55) explained 91.7% and 90.7% of the total

variation in recovery of copper.

X3

X5

YR

80

Among the next subset with 4 predictors, the following two best regression

equations involving four predictor variables were selected by the program.

YR = 6.13 + 0.0539 X1 + 0.779 X3 + 0.188 X4 + 0.762 X6 ……………………..(5.56)

YR = 0.0615 X1 + 0.777 X3 + 0.191 X4 + 0.918 X6 ……………………………..(5.57)

YR = - 26.6 + 3.83 X2 + 0.741 X3 + 0.189 X4 + 0.762 X6 …………………………………..(5.58)

Equation (5.56) explained 93.2% of the total variation in recovery; This model

all variables are collectively important except intercept. Equation (5.58) explained

92.4% of the variation in the data for recovery of copper, but the intercept and X2 are

not statistically significant so we drop this model. As intercept in equation (5.56) was

not significant a model with no intercept (model equation 5.57) was fitted to the data

which gave very good fit.

The following two best regression equations involving five predictor variables

were selected by the program:

YR = - 35.0 + 0.0534 X1 + 3.74 X2 + 0.688 X3 + 0.191 X4 + 0.762 X6………(5.59)

YR = 8.56 + 0.054 X1 + 0.778 X3 + 0.196 X4 + 0.782 X6 - 0.305 X7 ………..(5.60)

In both equations (5.59) and (5.60), the inclusion X2 and X7 did not improve

the fit significantly.

The improvement in R2 from equations with five predictor variables

(equations 5.59 and 5.60) over equations with four predictors (equations

5.56,5.57,5.58) are very small and not significant, so the models with four predictors

sufficiently explained the variation in copper recovery.

81

The two best regression equations involving six predictor variables are given

below:

YR = - 32.6 + 0.053 X1 + 3.75 X2 + 0.687 X3 + 0.200 X4 + 0.783 X6 -0.306 X7 ….. (5.61)

YR = -29.9 + 0.053X1 + 3.73X2 + 0.694X3 + 0.155X4 - 0.079X5 + 0.795X6 ……(5.62)

There improvements with equations (5.61) and (5.62) as compared to

equations (5.59) and (5.60) are not statistically significant. The full model is given

below, the improvement in R2 is very small over model (5.61) and (5.62).

YR = - 25.2 + 0.053 X1 + 3.74 X2 + 0.693 X3 + 0.158 X4 -0.0982 X5 + 0.833 X6 -

0.450 X7………………………………………………………………………….(5.63)

From the information provided by the best subset procedure, it is concluded

that equations with four predictor variables explains sufficient variation in the

recovery of copper ore data from all the experiments. The best equation involves

variables sodium propylxanthate, sodium sulphide, sodium cyanide and pulp density.

The second best equation involves pH, sodium sulphide, sodium cyanide and pulp

density. The equation without intercept involving the variables sodium

propylxanthate, sodium sulphide, sodium cyanide and pulp density explains almost

99% of the variation in recovery of copper.

5.9 Multiple Regression Model for Recovery

An other approached followed was that the full model was fitted and based on

significance of the parameters reduced model was fitted excluding the variables with

probability greater than 0.05 and the detail study was made about assumption of the

full and reduced model.

82

The recovery data was regressed on the seven flotation process independent

variables. The Excel out put is given Tables 5 and 6 are given below.

Regression output of copper recovery on seven independent variables

Table 5: Coefficient Analysis And Model Fitness Statistic For Seven Variables

Predictor Coef SE Coef T P Standard Error 3.752,

Constant -25.20 25.81 -0.98 0.339 R-Square 94.6%,

X1 0.053 0.021 2.53 0.019 (Adjusted) R-Square 92.9%

X2 3.735 2.162 1.73 0.097 Press 680.517,

X3 0.693 0.104 6.67 0.000 Observation 31%

X4 0.158 0.051 3.05 0.006

X5 -0.098 0.069 -1.42 0.168

X6 0.833 0.237 3.51 0.002

X7 -0.450 0.412 -1.09 0.287

Table 6: Analysis Of Variance

Source DF SS MS F P

Regression 7 5629.65 804.24 57.14 0.000

Residual Error 23 323.71 14.07

Total 30 5953.35

The residuals are independent identically normally distributed (i.i.d), so here

we use t-statistic. The t-statistic, and its probability show that intercept, X2, X5, and

83

X7 are not statistically significant, therefore, we dropped these insignificant variables,

and fitted a reduced model check the assumptions of the full model tests to nosuality

of residuals give some description.

Visual normal test for standard residuals for seven process parameters

0

0.2

0.4

0.6

0.8

1

1.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Number of Observation

Re

sid

ua

ls

Figure-19: VISUAL NORMAL TEST FOR STANDARD RESIDUALS FOR SEVEN PROCESS PARAMETERS MODEL.

The Figure (19) shows visual test for standard residuals of seven variables and

it has little deviation from 45-degree line yet it does not give vital evidence against

the normality.

84

Table – 7: Test for normality of residuals:

Bin Frequency Cumulative %

-2 1 3.33%

-1 2 10.00%

0 14 56.67%

1 7 80.00%

2 6 100.00%

3 0 100.00%

More 0 100.00%

Histogram

0

2

4

6

8

10

12

14

16

-2 -1 0 1 2 3 More

Bin

Fre

qu

ency

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Co

mm

ula

tive

%Frequency Cumulative %

Figure-20: HISTOGRAM OF SEVEN VARIABLES

It is obvious from the histogram that the distribution of the error terms is

symmetric but not normal. In this study the co-efficient of skewness for standard

residual is –0.47, which is inside the 96% confidence interval. Thus the data is note

skewed and therefore satisfies one of the normality conditions. Also E.Kurtosis is

1.107, which is inside the 96% confidence interval and hence satisfies normality

condition.

85

5.9.1 Jarque – Bera: A Combined Test:

Instead of using the two tests separately, one can use a linear combination of

the two. The Jarque – Bera test was devised as an optimal test against a certain class

of alternative to the null distribution. The test statistic is:

JB = T{EK2/24 + (SK)^2/6}

In this study the value of Jarque – Bera (JB) is 2.747, in the table given below

here are calculated values of different tests and also there critical values calculated by

simulation.

EK=kurtosis of any distribution –3= k-3

Kurtosis is measure of heaviness of tail K for normal distribution.

Skew ness (SK) a non-symmetric distribution is known as skewed distribution.

Table – 8: Skew ness E. Kurtosis Jarque-Bera for seven variables

Test Calculated Lower critical

value

Upper critical

value

Results

Skew ness -0.47 -.7 .7 Pass

E.Kurtosis 1.1 -.99 1.55 Pass

Jarque-Bera 2.7 N.A 462 Pass

86

Standard Residual Plot

-4

-3

-2

-1

0

1

2

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Figure-21: STANDARD RESIDUAL PLOT FOR SEVEN VARIABLES

From the standard residual plot it is clear that approximately 68% data is

inside the interval [-1.1] and approximately 95% data is in the interval [-2, 2], which

is a sign of normality. If we look it more deeply we can see that the variation in the

first half is different than the second half, which is indicating heteroscedasticity

problem.

5.9.2 Testing for Heteroscedasticity

Let SER(1) and SET(2) be the Standard Error of Regression for the first half

and the second half of the data set respectively. If the ratio SER(1)/SER/2) is close to

1 then the two SE’s are different. The Goldfled-Quandt statistic is based on the ratio

of variances (not Se’s)

2[ (2) / (1)]GQ SER SER

Now in study Var1 (Variance of first half)=0.654, Var2 (Variance of second

half) =1.298, GQ test=0.254, p-value=0.995. The value of GQ test is 0.28, which is

87

very much different from 1. This can also be seen from the p-value of GQ test.

Standard residuals are normal but are not identically distributed, so it fails to be i.i.d

random variables. Thus R2 is meaningless.

Table-5 show’s that R2 of the full model is high and have a low Standard

Error. It means that 94% variation in YR can be explained by the regressors with ± 7

units with 95% confidence if the residuals are identity independent Normal. But

residuals are not identity independent Normal so these statistics are meaningless.

Regression coefficients, their standard errors, t-values and probability are

given in Table-5.

From Table-5 we have write the following equation

YR = -25.2049 + 0.053 X1 +3.73 X2 + 0.69 X3 + 0.15 X4 -0.09 X5 + 0.833 X6 –0.45X7

…………………………………………………………………………………. (5.64)

T-statistics and probability given in the Table-4 shows that intercept, X2, X5

and X7 are not significant at 5% level so a reduced model excluding these was tried.

5.10 Reduced model for recovery

We discard intercept, X2, X5 and X7, which are not significant therefore our

reduced model is

YR = δ1X1+ + δ3X3+ δ4X4+ δ6X6+ R’ ………………………………………… (5.65)

Here are true parameters that we want to conjecture and residuals is identity

independent Normal with mean 0 and variance σ2.

Note that the “stepwise regression” procedures is a little bit risky procedure

there is a chance of loosing some valuable information but over all performance can

88

be tested by different ways. First we check the basic assumptions about the model and

then will compare both the models, for example By F-test etc. Also before discarding

variables we must consult the theory. Note that when we say that we are discarding

one agent it does not mean that that agent is not involved in the process. It simply

means that it is kept constant, usually on its critical value because its variation is not

effecting the dependent variable and its effect is very slight. Using the reduced model

given above, the recovery was regressed on X1, X2 X3 X4 and X6 with no intercept.

The Excel out put of the fitted model is given in the table 6 to 8. The information in

table 9, show that R2 of (5.65) is slightly less and standard error is slightly more then

model (5.65). It is important to examine the optness of the model (5.65).

Table – 9: Coefficient Analysis And Model Fitness Statistic For Four Variables

Predictor Coef SE Coef T P Standard Error 3.9

Constant 0.0 0.0 0.0 0.0 R-Square 99.60%

X1 0.061 0.019 3.16 0.004 (Adjusted) R-Square 99.55%

X3 0.776 0.093 8.34 0.000 Press 526.202,

X4 0.191 0.043 4.44 0.000 Observation 31

X6 0.917 0.118 7.78 0.000

89

Table -10: Analysis of Variance for four variables

Source DF SS MS F P

Regression 4 102807 25702 1688.93 0.000

Residual Errors 27 411 15

Total 31 103218

5.10.1 Tests for basic assumptions:

(i) F-test of the P- value suggests that all variables are collectively important. As

compared to the full model the F-value is more showing better fit.

The regression equation is

YR = 0.061 X1 + 0.777 X3 + 0.191 X4 + 0.918 X6

(ii) The t-statistics

The P-value of t-statistic of each variable shows that all variables are

individually important.

iii. Test for normality of residuals

0

0.2

0.4

0.6

0.8

1

1 4 7 10 13 16 19 22 25 28 31

No. of observations

resi

du

als

Figure 22: PLOT OF RESIDUALS OF REDUCED MODEL

90

Figure 22 shows normal test for standard residuals of seven variables. Though

it has little deviation from 45 degree line yet it does not give vital evidence against the

normality.

5.10.2 Other tests for normality:

i) Tests for skewness, kurtosis and Jarque bera for four variables

Table 11:

Test Calculated

value

Lower critical

value

Upper

critical value

Result

Skewness -0.6 -.7 .7 Pass

Kurtosis E-.89 -.99 1.55 Pass

Jarque bera -1.8 N.A 4.62 Pass

ii) Histogram

Table-12:

Bin Frequency

-3 0

-2 1

-1 4

0 9

1 11

2 6

3 0

More 0

91

Histogram

01

4

9

11

6

0 00

2

4

6

8

10

12

-3 -2 -1 0 1 2 3 More

Bin

Fre

qu

ency

Figure 23: HISTOGRAM FOR FOUR SIGNIFICANT VARIABLES

Frequency table and histogram shows that data is normal.

Another way of checking the normality is as follows:

iii) Standard Residual Plot

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1 3 5 7 9

11 13 15 17 19 21 23 25 27 29 31

Figure 24: STANDARD RESIDUAL PLOT

From the standard residual plot it is clear that approximately 68% data is in

side [-1, 1] and 95% data is in the interval [-2, 2] which is the evidence that data is

normal.

92

iv) For identical distribution we again use the GQ test

Var-1 Var-2 GQ-Test P-value

0.8 1.21 0.43 0.94

It qualifies the GQ test. So residuals are identically distributed

v. Structural stability:

Chow Test 2.015

P. significance 0.048

So Model is structurally stable.

Next natural question is that can we really drop few variables and whether our

new model is better than before.

We can perform F-test to see if the removal of the three theoretically least

important regressors X2, X5, and X7 has made any significant difference.

The later of F-statistic can be computed as follows:

F = [(CRSS-URSS)/2]/[URSS/(T-K)]

= [(382.5-311.05)/2]/[311.05/(31-4] = 3.1004,

where CRSS=Constrained residuals sum of square.

and URSS=Unconstrained residuals sum of square, T=Number of observations=31

K = Number of regressors = 4

This has degrees of freedom 2 and 27

93

Using FDIST (3.1,2,27) we get the p-value 0.0641. Which is not significant.

This means that collectively these three regresors X2, X5, X7 have no collective

importance. It has a simpler theoretical structure. By using this model one can

estimate YR within +3.704 with 68% probability and U+ 7.4 with 9.8% probability

result, drop these variables one by one t-statistics suggest that they are not important.

So our final model is

YR = 0.061*X1 + 0.776*X3 + 0.191*X4 + 0.917*X6 …………………………..(5.67)

It is clear from the model that curve passes through the origin it is obvious

from this model that if we increase one unit of X1, YR will increase 0.061 unit keeping

all other variables constant. We can define the other entire coefficient in the same

fashion. These coefficients (slopes) give partial value.

94

CHAPTER – 6 MODEL BUILDING FOR GRADE

6.1 MATHEMATICAL MODEL FOR OPTIMUM GRADE.

PREVIOUS STUDIES

First experiment was carried out to investigate the effect of variation in

collector propylxanthate (X1) on the grade of copper, while keeping all the other

regressors constant. The response of propylxanthate to grade of the chalcopyrite ore is

shown graphically in Figure 25 with increase in dosage of propylxanthate there was a

corresponding increase in grade of copper. Beyond the dosage of 200g/ton of

propylxanthate there was no increase rather decrease in grade occurred as shown in

Figure 25. Next experiment was conducted to investigate the effect of variation in

collector pH (X2) on the grade of copper, while keeping all other variables constant

Figure 26, shows that pH (X2) at 11.58 gives the maximum grade. But pH greater then

11.58 decreased the grade. The third experiment was carried to investigate the effect

of variation in depressant sodium cyanide (X3), on the grade of copper, while keeping

all other variables constant. The results are given in Figure 27. Grade increase within

increase in sodium cyanide up to 25/gon, however, further increase in depressant

caused decrease in grade. Grade was maximum at 50g/ton dosage of sulfidizer (X4)

and beyond that dosage there was a slight decrease in grade of copper as shown in

Figure 28. The next experiment was conducted to investigate the effect of variation in

pine oil (X5) on the grade of copper, while keeping all other regressors constant. The

95

maximum grade of copper was noted when frother, the pine oil, was used at the rate

of 46 g/ton. Beyond 46g/ton of pine oil the grade was reduced as shown in Figure 29

graphically. The optimum pulp density (X6) is of great importance, as in general the

more dilute the pulp, the cleaner the separation. The effect of pulp density on the

grade of chalcopyrite ore has been shown in Figure 30. The curve shows that

maximum values of grade was obtained at 30% solids by weight. However, beyond

that level of pulp density, the grade markedly decreased due to the entrapped fine slim

particles. Next test was conducted to find out the effect of conditioning time of

collectors ranging from 10 to 18 minutes on grade. The graph in Figure 31 indicates

that 13 minutes conditioning time was optimum for obtaining better grade of copper.

Conditioning time greater than 13 minutes reduces the grade due to dissolution of

copper xanthate ions in the equilibrium system.

96

(a)

5

7

9

11

13

15

17

0 100 200 300

PropylXanthate (g/ton)

% G

rad

e

(b)

12.513

13.514

14.515

15.516

9.5 10 10.5 11 11.5 12 12.5

pH

% G

rad

e

Figure-25: EFFECT OF COLLECTOR ROPYLXANTHATE ON GRADE OF

COPPER

Figure 26: EFFECT OF PH ON GRADE OF COPPER

(c)

13.5

14

14.5

15

15.5

16

16.5

17

0 10 20 30 40

Sodium Cyanide (g/ton)

% G

rad

e

(d)

15

15.5

16

16.5

17

17.5

18

0 20 40 60 80

Sodium Sulphide (g/ton)

% G

rad

e

Figure-27: EFFECT OF DEPRESSANT ON GRADE OF COPPER

Figure-28: EFFECT OF SULFIDIZER ON GRADE OF COPPER

(e)

16

16.5

17

17.5

18

18.5

0 20 40 60 80

Pineoil (g/ton)

% G

rad

e

(f)

579

1113151719

0 10 20 30 40

Pulp density (%wt/vol)

% G

rad

e

Figure-29: EFFECT OF FROTHER (PINE OIL) ON GRADE OF COPPER

Figure-30: EFFECT OF PULP DENSITY ON GRADE OF COPPER

97

(g)

5

7

9

11

13

15

17

19

5 10 15 20

Conditioning time(minutes)

% G

rad

e

Figure-31: EFFECT OF CONDITIONINGTIME ON GRADE OF COPPER

6.2 Modeling effect of individual variable for grade

For developing single variable mathematical models for the grade of copper,

forty one models were fitted to select suitable models based on F-test, and R2 of the

model and t-test of the model parameters.

The models and their R2 are given in Table 13 and graphically presented in

Figures (1 to 7). Mathematical models 6.6, 6.13, 6.18, 6.24, 6.29, 6.35 and 6.41 were

best for X1, X2, X3, X4, X5, X6 and X7 respectively among the forty one single predictor

models.

98

Table-13: Mathematical models involving one predictor variable for

grade of copper.

YG = 0.0182X1 + 10.67 R2 = 0.65 (6.1)

YG = 2.4687Ln(X1) + 1.3788 R2 = 0.7728 (6.2)

YG = -0.0002X21 + 0.0756X1 + 7.32 R2 = 0.9017 (6.3)

YG = 5.1995X10.1929 R2 = 0.7965 (6.4)

YG = 10.748e0.0014X1 R2 = 0.6687 (6.5)

YG = 9.45 + 0.03X – 3.05X’ R2 = 0.9421 (6.6)

Graphical representation of above Six equations are given in figure-32

YG = -0.1495X2 + 16.04 R2 = 0.0169 (6.7)

YG = -1.4068Ln(X2) + 17.767 R2 = 0.0124 (6.8)

YG = -1.7788X22 + 38.933X2 - 197.63 R2 = 0.6541 (6.9)

YG = 16.534e-0.0128X2 R2 = 0.0248 (6.10)

YG = 19.32X2-0.1236 R2 = 0.0194 (6.11)

YG = 5.014 + 0.90X – 2.90 X’ R2 = 0.9868 (6.12)

Graphical representation of above six equations are given in figure-33

YG = -0.044X3 + 16.38 R2 = 0.1066 (6.13)

YG = -0.5273Ln(X3) + 17.042 R2 = 0.0461 (6.14)

99

YG = -0.0177X23 + 0.6646X3 + 10.18 R2 = 0.7114 (6.15)

YG = 17.339X3-0.039 R2 = 0.0574 (6.16)

YG = 16.471e-0.0031X3 R2 = 0.1233 (6.17)

YG = 14.49 + 0.082 X – 3.15 X’ R2 = 0.9808 (6.18)

Graphical representation of above six equations are given in figure-34

YG = 0.0158X4 + 15.516 R2 = 0.1047 (6.19)

YG = 0.5068Ln(X4) + 14.352 R2 = 0.1459 (6.20)

YG = -0.0013X24 + 0.1032X4 + 14.393 R2 = 0.2709 (6.21)

YG = 14.469X40.0306 R2 = 0.1460 (6.22)

YG = 15.529e0.0009X4 R2 = 0.1025 (6.23)

YG = 14.696 + 0.049 X – 2.248 X’ R2 = 0.7835 (6.24)

Graphical representation of above six equations are given in figure-35

YG = -0.0154X5 + 18.011 R2 = 0.201 (6.25)

YG = -0.4706Ln(X5) + 19.058 R2 = 0.0992 (6.26)

YG = 19.207X5-0.0282 R2 = 0.1067 (6.27)

YG = 18.03e-0.0009X5 R2 = 0.2111 (6.28)

YG = 17.94 – 0.02 X + 1.40 X’ R2 = 0.3567 (6.29)

100

Graphical representation of above five equations are given in figure-36

YG = -0.1829X6 + 21.6 R2 = 0.4213 (6.30)

YG = -3.6902Ln(X6) + 28.686 R2 = 0.3196 (6.31)

YG = -0.0329X26 + 1.4391X6 + 3.5 R2 = 0.9115 (6.32)

YG = 36.157X6-0.2406 R2 = 0.326 (6.33)

YG = 22.761e-0.0119X6 R2 = 0.4276 (6.34)

YG = 17.6 + 0.0171 X – 5X’ R2 = 0.9973 (6.35)

Graphical representation of above six equations are given in figure-37

YG = -0.9143X7 + 28.229 R2 = 0.8491 (6.36)

YG = 11.961Ln(X7) + 46.689 R2 = 0.7890 (6.37)

YG = -0.1557X27

+ 3.436 X7 –0.7142 R2 = 0.9741 (6.38)

YG = 129.66X7-0.8222 R2 = 0.7805 (6.39)

YG = 36.521e-0.063X7 R2 = 0.8436 (6.40)

YG = 32.3473 – 1.444X – 5 X’ R2 = 0.9994 (6.41)

Graphical representation of above five equations are given in figure-38

The various models with R2 for collector are presented in Figure 32. Though

quadratic model had better fit, all other models also gave good fit based on R2. The X-

max calculated from the quadratic equation was 189 gram per ton which shows that

collector level of about 190 gram per ton will gave the highest grade of copper. The

101

two straight line model give best fit, showing that the joining point at 200 g/ton of

collection will give the highest grade.

(a)

y = 0.0182x + 10.67

R2 = 0.65

10

11

12

13

14

15

16

40 90 140 190 240

PropylXanthate (g/ton)

% G

rad

e

(b)

y = 2.4687Ln(x) + 1.3788

R2 = 0.7728

10

11

12

13

14

15

16

40 90 140 190 240 290ProphylXanthate (g/ton)

% G

rad

e

(c)

y = -0.0002x2 + 0.0756x + 7.32

R2 = 0.9017

10

11

12

13

14

15

16

40 90 140 190 240 290

PropylXanthate (g/ton)

% G

rad

e

(d)

y = 5.1995x0.1929

R2 = 0.7965

10

11

12

13

14

15

16

40 90 140 190 240 290PropylXanthate (g/ton)

% G

rad

e

(e)

y = 10.748e0.0014x

R2 = 0.6687

10

11

12

13

14

15

16

40 90 140 190 240 290ProphylXanthate (g/ton)

% G

rad

e

Figure-32: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

GRADE OF COPPER DATA FROM FIVE LEVELS OF COLLECTOR USE IN THE FLOTATION PROCESS.

10111213

141516

40 90 140 190 240Propylxanthate (g/ton)

% G

rad

e

(f)

Y = 9.45 + 0.0304X - 3.05X'R2 = 0.9421

102

The R2’s of the different models given in Figure 33 show that only quadratic

model gave good fit. The X-max calculated from the function is 10.9 showing that pH

of 10.9 that will give the highest grade of copper. The two straight lines gave much

better fit with the highest R2, it show that pH of 11.5 will give the highest grade of

copper.

(a)

y = -0.1495x + 16.04

R2 = 0.01691011121314151617181920

9.8 10.8 11.8 12.8

pH

% G

rad

e

(b)

y = -1.4068Ln(x) + 17.767

R2 = 0.01241011121314151617181920

9.8 10.8 11.8 12.8

pH

% G

rad

e

(c)

y = -1.7788x2 + 38.933x - 197.63

R2 = 0.65411011121314151617181920

9.8 10.8 11.8 12.8

pH

% G

rad

e

(d)

y = 16.534e-0.0128x

R2 = 0.024812.5

13

13.5

14

14.5

15

15.5

16

9.5 10 10.5 11 11.5 12 12.5

pH

% G

rad

e

103

(e)

y = 19.32x-0.1236

R2 = 0.01941011121314151617181920

9.8 10.8 11.8 12.8

pH

% G

rad

e

Figure-33: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

GRADE OF COPPER DATA FROM FOUR LEVELS OF PH IN THE FLOTATION PROCESS.

Coefficient of determinations given in Figure 34 show that quadratic equation

give good fit while the other models gave poor fit. The calculated X-max show that 18

gram per ton of sulfidizer will give the highest grade of copper. The two straight lines

model gave excellent fit with an R2 of 0.9808 indicating that it explained 98% of the

variation in data for grade of copper.

(a)

y = -0.044x + 16.38

R2 = 0.10661011121314151617

8 12 16 20 24 28 32

Sodium Cyanide (g/ton)

% G

rad

e

(b)

y = -0.5273Ln(x) + 17.042R2 = 0.0461

1011121314151617

0 10 20 30 40Sodium Cyanide (g/ton)

% G

rad

e

10

11

12

13

14

15

16

9.8 10.3 10.8 11.3 11.8pH

% G

rad

e

(f)

Y = 5.0419 + 0.9056X - 2.9091X'R2 = 0.9868

104

(c)

y = -0.0177x2 + 0.6646x + 10.18

R2 = 0.7114

1011121314151617

8 12 16 20 24 28 32

Sodium Cyanide (g/ton)

% G

rad

e

(d)

y = 17.339x-0.039

R2 = 0.0574

1011121314151617

8 12 16 20 24 28 32Sodium Cyanide (g/ton)

% G

rad

e

(e)

y = 16.471e-0.0031x

R2 = 0.1233

1011121314151617

8 12 16 20 24 28 32

Sodium Cyanide (g/ton)

% G

rad

e

Figure-34: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

GRADE OF COPPER DATA FROM FOUR LEVELS OF SULFIDIZER IN THE FLOTATION PROCESS.

Except the two straight line model, none of the models presented in Figure 35

gave good fit to the observed data for grade as affected by levels of depressant. The

original data points show that highest grade of copper was obtained when 50 g per ton

of depressant was used though X-max from quadratic function was about 40 gram per

ton of depressant. The two straight line model also how that X-max is 50 g/ton.

1011121314151617

8 12 16 20 24 28 32Sodium Cyanide (g/ton)

% G

rad

e

(f)

Y = 14.49 + 0.082X - 3.15X'R2 = 0.9808

105

(a)

y = 0.0158x + 15.516

R2 = 0.1047

10111213141516171819

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium sulphide (g/ton)

% G

rad

e

(b)

y = 0.5068Ln(x) + 14.352

R2 = 0.1459

10111213141516171819

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium sulphide (g/ton)

% G

rad

e

(c)

y = -0.0013x2 + 0.1032x + 14.393

R2 = 0.2709

10111213141516171819

5 10 15 20 25 30 35 40 45 50 55 60 65

Sodium sulphide (g/ton)

% G

rad

e

(d)

y = 14.469x0.0306

R2 = 0.146

10111213141516171819

5 10 15 20 25 30 35 40 45 50 55 60 65Sodium sulphide (g/ton)

% G

rad

e

Figure-35: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (c) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE

GRADE OF COPPER DATA FROM FIVE LEVELS OF DEPRESSANT IN THE FLOTATION PROCESS.

None of the models given in figure 36 gave good fit to the observed data on

grade of copper as affected by frother dosage. The original data points show that 46

gram per ton of frother gave maximum grade of copper, the two straight lines who

show the same result regarding X-max. The two straight line gave much better fit than

the other models.

(a)

y = -0.0154x + 18.011

R2 = 0.201

10111213141516171819

20 30 40 50 60 70 80Frother (g/ton)

% G

rad

e

(b)

y = -0.4706Ln(x) + 19.058

R2 = 0.0992

10111213141516171819

20 30 40 50 60 70 80

Frother (g/ton)

% G

rad

e

106

(c)

y = 19.207x-0.0282

R2 = 0.1067

10

11

12

13

14

15

16

17

18

19

20 30 40 50 60 70 80

Frother (g/ton)

% G

rad

e

(d)

y = 18.03e-0.0009x

R2 = 0.2111

10111213141516171819

20 30 40 50 60 70 80Frother (g/ton)

% G

rad

e

Figure-36: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE GRADE OF COPPER DATA FROM FOUR LEVELS OF FROTHER DOSAGE

IN THE FLOTATION PROCESS.

The model for grade of copper as effected by pulp density are given Figure 37.

Quadratic regression gave best fit followed by linear regression to the data on grade of

copper as affected by pulp density. However two straight lines give excellent fit with

fit with an R2 of 0.9993. X-max from quadratic equations was 21.8 g per ton X-max

from two straight lines is 30 g per ton.

10111213141516171819

20 30 40 50 60 70 80Frother (g/ton)

% G

rad

e

(e)

Y = 17.943 - 0.0297X + 1.4046X'R2 = 0.3567

107

(a)

y = -0.1829x + 21.6

R2 = 0.4213

10111213141516171819

10 20 30 40Pulp density (%wt/vol)

% G

rad

e

(b)

y = -3.6902Ln(x) + 28.686

R2 = 0.3196111213141516171819

10 20 30 40Pulp density (%wt/vol)

% G

rad

e

(c)

y = -0.0329x2 + 1.4391x + 3.5

R2 = 0.9115

1011121314151617181920

10 20 30 40

Pulp density (%wt/vol)

% G

rad

e

(d)

y = 36.157x-0.2406

R2 = 0.32610111213141516171819

10 20 30 40

Pulp density(%wt/vol)

% G

rad

e

(e)

y = 22.761e-0.0119x

R2 = 0.4276

10111213141516171819

10 20 30 40

Pulp density (%wt/vol)

% G

rad

e

Figure-37: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE GRADE OF COPPER DATA FROM FOUR LEVELS OF PULP DENSITY IN

THE FLOTATION PROCESS.

The models for grade of copper as affected by conditioning time are present in

Figure 38. Coefficients of determination of models presented in Figure 38 show that

10111213141516171819

10 20 30 40Pulp Density

% G

rad

e

(f)

Y = 17.6 + 0.01714X - 5X'R2 = 0.9973

108

all the models give good fit to data of copper grade as affected by flotation time.

Grade of copper decreased with increase in time of flotation. Two straight line model

gave excellent fit with an R2 of 0.9994 showing that it explained almost all the

variation in data for grade. The graph for two straight line show that there was no

significant difference in grade when conditioning time was 10 to 13 minutes but

conditioning time greater that 13 minute reduce grade of copper.

(a)

y = -0.9143x + 28.229

R2 = 0.8491

1011121314151617181920

8 12 16 20Conditioning time (minute)

% G

rad

e

(b)

y = -11.961Ln(x) + 46.689

R2 = 0.7891011121314151617181920

8 12 16 20Conditioning time (minute)

% G

rad

e

(c)

y = -0.1557x2 + 3.436x - 0.7142

R2 = 0.97411011121314151617181920

8 12 16 20

Conditioning time (minute)

% G

rad

e

(d)

y = 129.66x-0.8222

R2 = 0.780510

11

12

13

14

15

16

17

18

19

20

8 12 16 20Conditioning time (minute)

% G

rad

e

109

(e)

y = 36.521e-0.063x

R2 = 0.84361011121314151617181920

8 12 16 20Conditioning time (minute)

% G

rad

e

Figure-38: (a) LINEAR (b) LOGARITHMIC (c) QUADRATIC (d) POWER (e) EXPONENTIAL (f) AND TWO STRAIGHT-LINE MODELS FITTED TO THE GRADE OF COPPER DATA FROM FOUR LEVELS OF FLOTATION TIME

IN THE FLOTATION PROCESS.

10111213141516171819

8 12 16 20Conditioning Time (minutes)

% G

rad

e

(f)

Y = 32.3473 - 1.4447X - 5X'R2 = 0.9994

110

6.3 Modeling Combined Effect Of Variables On Grade

The data from all experiments were used to construct models for grade of

copper ore. The following four strategies were followed for models selection.

1. Forward selection procedure or simple to general strategy

2. Back ward elimination procedure or general to simple strategy

3. Best subset procedure

4. Modeling effect of pairs of variables with interaction from best subset

procedure

6.4 Forward Selection

The forward stepwise selection procedure was used to select variables for

modeling grade of copper using seven independent variables. The details of steps are

given in appendix – 4. The equations selected at each step are given below.

YG = 14.01 + 0.085 X3 (6.42)

YG = 19.26 + 0.104 X3 –0.53 X7 (6.43)

YG = 15.88 + 0.019 X1 + 0.086 X3 - 0.52 X7 (6.44)

YG = 20.16+ 0.019 X1+0.06X3 -0.044 X5 - 0.63 X7 (6.45)

Equation (6.42) in one predictor variable, equation (6.43) in two-predictor

variable, equation (6.44) in three-predictor variable, equation (6.45) in four-predictor

variables were selected for the grade of copper by forward selected procedure.

111

Table–14: Primary Data On Grade (YG) Of Copper As Affected By Seven Flotation Process Variables (X1 To X7).

X1 X2 X3 X4 X5 X6 X7 YG

50 11 0 0 75 30 10 11

100 11 0 0 75 30 10 12.1

150 11 0 0 75 30 10 14.7

200 11 0 0 75 30 10 15.2

250 11 0 0 75 30 10 14

200 10 0 0 75 30 10 14.2

200 10.3 0 0 75 30 10 14.2

200 11 0 0 75 30 10 15.1

200 11.58 0 0 75 30 10 15.5

200 12 0 0 75 30 10 13

200 11.58 10 0 75 30 10 15.4

200 11.58 15 0 75 30 10 15.5

200 11.58 20 0 75 30 10 16.3

200 11.58 25 0 75 30 10 16.5

200 11.58 30 0 75 30 10 13.8

200 11.58 25 10 75 30 10 15.51

200 11.58 25 30 75 30 10 15.78

200 11.58 25 40 75 30 10 16.2

200 11.58 25 50 75 30 10 17.7

200 11.58 25 60 75 30 10 15.4

200 11.58 25 50 25 30 10 17.2

200 11.58 25 50 46 30 10 18.1

200 11.58 25 50 70 30 10 16.56

200 11.58 25 50 46 15 10 17.8

200 11.58 25 50 46 25 10 18.2

200 11.58 25 50 46 30 10 18

200 11.58 25 50 46 35 10 13.2

200 11.58 25 50 46 30 10 17.9

200 11.58 25 50 46 30 13 18.2

200 11.58 25 50 46 30 16 13.7

200 11.58 25 50 46 30 18 11

112

6.5 Backward elimination

Backward elimination procedure or general to simple model building strategy

was used to select model for grade of copper using data combined over all

experiments. The following models were fitted at each step. The details are given in

appendix – 5.

YG = 21.58 + 0.0194X1 + 0.07X2 + 0.038X3 + 0.018X4 -0.024X5 - 0.130X6 - 0.60X7….. (6.46)

YG = 22.30 + 0.019X1 + 0.04X3 + 0.018X4-0.024X5-0.130X6 - 0.60X7……………….. (6.47)

YG = 20.55 + 0.019X1 + 0.039X3 + 0.028X4 - 0.142X6 -0.57X7 ……………………… (6.48)

YG = 20.43 + 0.0216X1 + 0.041X4 - 0.141X6 - 0.57X7………………………………. (6.49)

YG = 16.48 + 0.021X1 + 0.045X4 - 0.60X7…………………………………………… . (6.50)

Equation (6.46), (6.47), (6.48) and (6.49), are not statistically significant these

models are not good fit. While equation (6.50), is a best fit model.

The forward selection and backward elimination selected at different

equations. So we will look at the best subset procedure and the model from that

procedure which correspond to any of the model from forward or backward will be

the approximate model-1.

6.6 Best subset for grade

It difficult to fit and test all the possible regression models involving seven

variables therefore the best subset procedure was used to select models involving one,

two, three, four, five and six variables. The best subset procedure (Minitab), produced

the following thirteen models two in each subset and the full model for grade of

copper.

113

Models with one variable are:

M1: YG = + 1X3

M2: YG = + 1X4

Models with two variable are:

M3: YG = + 1X4 + 2 X7

M4: YG = + 1X3 + 2 X7

Models with three variable are:

M5: YG = + 1X1 + 2 X4 + 3 X7

M6: YG = + 1X1 + 2 X3 + 3 X7

Models with four variable are:

M7: YG = + 1X1 + 2 X3 + 3 X5 + 4 X7

M8: YG = + 1X1 + 2 X4 + 3 X6 + 4 X7

Models with five variable are:

M9: YG = + 1X1 + 2 X3 + 3 X4 + 4 X6 + 5 X7

M10: YG = + 1X1 + 2 X3 + 3 X5 + 4 X6 + 5 X7

Models with six variable are:

M11: YG = + 1X1 + 2 X3 + 3 X4 + 4 X5 + 5 X6+6 X7

M12: YG = + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X6+6 X7

114

Models with seven variable are:

M13: YG = + 1X1 + 2 X2 + 3 X3 + 4 X4 + 5 X5+ 6 X6+7X7

The regression equations for single predictors for recovery of copper obtained

form least square analysis are as follows:

YG = 14.0 + 0.0854 X3………………………………………………………. (6.51)

YG = 14.5 + 0.0374 X4………………………………………………………. (6.52)

The R2 show that the first equation explained 24.3% of the variation and the

second equation explained 20.7% variation in the grade of copper using flotation

equation process. The coefficient of equation (6.51) are different then coefficient of

(6.13), because the equation (6.13) is based on data from on experiment and equation

(6.51) is based on data from all experiments. Similarly the coefficients of equations

(6.19) and (6.52) are different because one is based on data from a single experiment

having five treatments and the other is based on combined data from seven

experiment having thirty-one treatment.

15

15.5

16

16.5

17

17.5

18

0 10 20 30 40

Sodium Cyanide (g/ton)

% G

rad

e

0

5

10

15

20

0 20 40 60 80

Sodium Sulphide (g/ton)

%Grade

Figure-39: EFFECT OF SODIUM CYANIDE (X3) ON THE GRADE OF

COPPER.

Figure-40: EFFECT OF SODIUM SULPHIDE (X4) ON THE GRADE OF

COPPER.

115

In the subset with single predictor variable, models involving X3 and X4 were

found as better model than others for grade of copper. Based on R2, F-value, t-

statistics and P-value (as shown in the output), model having X3 is better than model

having X4.

Grade increased at the rate 0.0854% per one gram increase in depressant.

Sixty gram per ton of sulfidizer gave maximum grade 19.12%. Grade increased at the

rate of 0.0854% per one gram increase in sulphidizer.

Among the twenty-one models in the subset with two predictor variables, the

two best regression equations involving two predictor variables are:

YG = 20.5 + 0.0518 x4 - 0.599 X7 (6.53)

YG = 19.3 + 0.104 X3 - 0.527 X7 (6.54)

The equation involving X4 and X7 explained 46.1% and the equation involving

X3 and X7 explained 45.1% of the variation in the grade of copper.

Response surfaces were developed for the variables involved in the above two

equations.

116

Figure 41: COPPER GRADE (YG) RESPONSE SURFACE FOR SODIUM CYANIDE (X3) AND CONDITIONING TIME (X7).

The combine response surface for sodium cyanide and conditioning time on

the grade of copper is shown in Figure 41. The maximum peak of surface shows the

estimated maximum grade of 16.42% with 28 gram per ton of sodium sulphide and 11

minutes conditioning time.

X3

X7

YG

117

Figure 42: COPPER GRADE (YG) RESPONSE SURFACE FOR SODIUM SULPHID (X4) AND CONDITIONING TIME (X7).

The combine response surface for sodium sulphide and conditioning time on

grade of copper is shown in Figure 42. The maximum peak of surface shows the

estimated maximum grade of 17.36% with 55 gram per ton of sodium sulphide and 10

minutes conditioning time.

The best subset program picked the following two best regression equations

involving three predictor variables among the 55, 3-variable models.

YG = 16.5 + 0.0216 X1 + 0.0448 X4 - 0.598 X7 …………………………... (6.55)

YG = 15.9 + 0.0191 X1 + 0.0859 X3 - 0.525 X7 …………………………… (6.56)

The two equation explain 58.6% and 54.3% of total variation in grade of

copper.

X4

X7

YG

118

Among the next subset with four predictors, the follow two best regression

equations involving four predictor variables were selected by the program:

YG = 20.4 + 0.0216 X1 + 0.0411 X4 - 0.141 X6 - 0.570 X7 (6.57)

YG = 20.2 + 0.0193 X1 + 0.0601 X3 - 0.0437 X5 - 0.626 X7 (6.58)

Equation (6.57) explained 62.7% but in this model the contribution X6 is not

important. Equation (6.58) explained 61.1% of the variation in the data, all variables

in this model are collectively important so this is a good fit model for grade of copper.

The following two best regression equation involving five predictor variables

were selected by the program:

YG = 20.5 + 0.0194 X1 + 0.0386 X3 + 0.0281 X4 - 0.142 X6 - 0.567 X7 (6.59)

YG = 23.3 + 0.0193 X1 + 0.0591 X3 - 0.0371 X5 - 0.133 X6 - 0.593 X7 (6.60)

The improvement in R2 from equations with five predictor variables over

equations with four predictor variables is small so the models (6.57) and (6.58) are

better for data on grade of copper.

Both the above models are not significant different then model (6.57) and

(6.58).

The two best regression equations involving six predictor variables are given

below:

YG = 22.3 + 0.019 X1 + 0.039 X3 + 0.018 X4 - 0.023 X5 - 0.130 X6 - 0.602 X7…(6.61)

YG= 19.8+ 0.019 X1 +0.068 X2 +0.036 X3 +0.028 X4 -0.142 X6-0.567 X7…….. (6.62)

Both the above models are not statistically significant.

119

The full model involving seven predictor variables is:

YG=21.6+0.0194X1+0.066X2+0.0383X3+0.0182X4-0.0236X5-0.130X6-0.602X7 (6.63)

Full model was fitted for the grade of copper the excel out put is given below:

Table:– 15: Coefficient Analysis for Grad And Model Fitness Statistic For Seven

Variables

Table – 16: Analysis of Variance

Source DF SS MS F P

Regression 7 82.926 11.847 6.39 0.000

Residual Error 23 42.637 1.854

Total 30 125.563

From Statistical Analysis from Table (15), the regressors X2, X3, X4, X5, and X6 are

insignificant but collectively we cannot exclude these all regressors.

Predictor Coef SE Coef T P Standard

Error

1.362

Constant 21.577 9.366 2.30 0.031 R-Square 66.0%

X1 0.019390 0.007662 2.53 0.019 R-Square

(Adjusted)

55.7%

X2 0.0658 0.7846 0.08 0.934 Press 129.905

X3 0.03834 0.03773 1.02 0.320

X4 0.01824 0.01885 0.97 0.343

X5 -0.02360 0.02505 -0.94 0.356

X6 -0.13009 0.08607 -1.51 0.144

X7 -0.6017 0.1499 -4.02 0.001

120

Subset F (4,23) = 1.998 [0.1296] so we can exclude X2, X4, X5, and X6 on statistical

basis. So our new significant models is YG=15.9+0.019X1+0.0859X3-0.525X7.

Subset F (3,27) = 10.68 [0.0000] all regressors are collectively important.

6.7 Model development for grade:

To find out the best fit model for grade, a full model was filled first that

contain all available candidates as predictor, then the model was simplify by

discarding the variables that did not contribute to explaining the variability in the

dependent variable.

After working with different full and reduced models the following final

model was selected. The model gave good fit as judged by the adjusted R2 and it has

the required criteria– independence, normality etc of residual to some extent:

YG = 15.9 + 0.0191 X1 + 0.0859 X3 - 0.525 X7 ………………………………… (6.64)

The excel output for the model is given in table 15 and 16.

Table 16: OLS estimates for three significant variables

Predictor Coef SECoef T P S.E 1.458

Constant 15.876 2.135 7.44 0.000 R.Sq 54.3%

X1 0.019072 0.008204 2.32 0.028 R-Sq(adj) 49.2%

X3 0.08593 0.02450 3.51 0.002 Press

X7 -0.5249 0.1502 -3.49 0.002

121

Table – 17: Analysis of Variance

Source DF SS MS F P

Regression 3 68.147 22.716 10.68 0.000

Residual Error 27 57.416 2.127

Total 30 125.563

T-state in column 4 of table 15 for each coefficient suggests that each variable

is individually significant.

F statistics given below tells that collectively, all regressors are important

Significance F = 0.0043

Graphical Analysis:

Histogram of Residuals

Figure 43

The histogram looks normal

122

Testing for heteroscedasticity using squares

Figure 44

Residuals are heteroscedastic

Residuals are normal. It qualifies the visual test of normality.

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure-45: RESIDUALS ARE NORMAL

123

6.8 General remarks about the model

For each regressor, a regression produces

A point estimate

A standard error

The point estimate helps us determine economic significance.

6.8.1 Statistical significance

A particular result is explained by chance if a coin is flipped five times in a

row and comes up heads each time, should I suspect that the coin is not a fair one with

equal chance of heads and tails?

In engineering data, we usually don’t have the luxury of adding more

observations to help us decide whether a result is due to chance. So we perform a

hypothesis test.

First, statistical significance. The difference is statistically significant, by

definition, if the 95% confidence interval does not overlap zero, or if the p value for

the effect is less than 0.05. Values of 95% or 0.05 are also equivalent to a type one

error rate of 5%: in other words, the rate of false alarms in the absence of any

population effect will be 5%. It has to be 5%, or less preferably, but like most

researchers I opted for 5%.

6.8.2 Sample Size?

The traditional approach for estimation of sample size is based on statistical

significance of outcome measure. We have to specify the smallest effect we want to

124

detect, the Type I and type II error rates, and design of the study. But unfortunately

we have limited observation.

Few agents who are not included are important for conducting experiment so

their constant values are present that are clear from the intercept.

Usually we need to exercise judgment to decide whether an estimated effect is

economically significant but in our case we can produce high quality copper grade by

just looking the model.

Usually if an estimated effect is economically and statically significant, we

need to weigh our results against those of other researchers but again it is our

misfortune that we don’t have any mathematical model to compare the results.

In analysis few times we face a problem that data are unusual. Because you

did things in a slightly different way than did others? If so, is our method one that is

knowledge and time? But in our case data is from control experiment so it can’t be

unusual.

6.9 Specific Remarks about the Model

We have the model

YG = 16.41 + 0.019 X1 + 0.048 X4 – 0.58 X7 ………………………………..… (6.65)

Here we have a linear model with the constraints of upper limits, which are

obvious from the data. We are choosing variable for inclusion solely on the basis of

statistical significance because we are not removing any regressor entirely. All or

their few combinations are necessary for experiments.

125

Here we have a statistically significant intercept in the model and it is

justifiable. If X1, X4, X7 all are collectively zero even though we can get copper grade

because other four regressors (here working as a constant) are presents.

In the model we have a negative coefficient of regressor X7 conditioning time

theory suggests the maximum values of grade 18.2 g/ton at 13 minutes can be

obtained. However, beyond that, the grade has markedly decreased due to the

entrapped of fine slim particles. That if we increase the optimum pulp density we will

get poor copper grade.

6.10 Discussion of size of coefficients and scientific judgment of

coefficient

If we generate the data from the model then this generated data will be

meaningless, as it can’t be compared with any other set of data because no such data

is available.

Though we have tested a lot of assumptions about the validity of the model

still we are uncertain about its results.

To overcome this problem one can randomly removed few observations from

the given set of data and again built model over same assumptions and then predict

the values that he had removed.

By simulation we checked that the coefficients are reasonable.

126

Figure 46: CONCEPTUAL GENERAL MODEL FOR RECOVERY / GRADE

If out of the seven inputs only one is varied then its variation will effect the

recovery/Grade YR/YG. Above figure demonstrates a conceptual model of this

phenomena only when input X is varied.

X

Recovery / Grade

PROCESS

127

CHAPTER-7

CONCLUSION

I understand that in Pakistan this is the first ever attempt to develop

Mathematical models both for recovery (YR) and grade (YG) of copper from the

copper ore.

Mathematical model for recovery

YR = 0.061X1+0.776X3 +0.191X4+0.917X6

Indicate that out of seven variables X1,X2, --, X7 only four of them Propylxanthate

(X1), Sodium Cyanide (X 3), Sodium Sulphide (X4) and pulp density (X6) are significant.

This model not only gives overall picture of the variables but also shows that X6 and

X3 play dominant role.

For Grade (YG) the mathematical model

YG = 0.019X1+0.0858X3-0.525 X7+15.9

Indicate that out of seven variables only three variables Propylxanthate (X1),

Sodium Cyanide (X3) and conditioning time (X7) are significant. Conditioning time

(X7) is the most dominant variables.

For Grade (YG) the variable X7 gives optimum results at 12 minute. For this fixed

value of X7, YG is further modified and YG = 0.019X1+0.0858X3+9.6

These mathematical models for recovery and grade, are strictly based on the

data provided by the Department of Mining Engineering N.W.F.P University of

Engineering and technology Peshawar. These models may not be valid for another

data if that do not conform to over data.

128

It is suggested that more experimental data may be generated against which

the above mathematical models and conclusions drawn from the models may be

verified.

A proposal amounting to Rs. 200 million was prepared by Prof. Dr.

Muhammad Mansoor Khan, Dean, Faculty of Engineering and was submitted for

approval to the Pakistan Science Foundation for enrichment and production of 99.9%

pure copper of North Waziristan Copper Ore.

129

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137

APPENDIX-1

General model for Recovery and Grade of copper using E-view

software:

YR = -25.2 + 0.05334*X1 + 3.735*X2 + 0.6932*X3 + 0.1583*X4

(SE) (25.8) (0.0211) (2.16) (0.104) (0.0519)

- 0.09816*X5 + 0.8331*X6 - 0.4502*X7 ---------- (a)

(0.069) (0.237) (0.413)

Since the t-ratio of following regressors are insignificant so we now try to observe

whether we can collectively drop all these variables or not.

Test for excluding:

[0] = Constant

[1] = X2

[2] = X5

[3] = X7

Subset F (4,23) = 1.5485 [0.2216]

So we can collectively drop them

Now our new model will be

YR = + 0.06148*X1 + 0.7765*X3 + 0.1911*X4 + 0.9176*X6 ---------- (b)

(SE) (0.0194) (0.0932) (0.0431) (0.118)

T-ratio shows that all above regressors are individually important.

Subset F (4,27) = 1688.9 [0.0000]**

So collectively they are important.

R2 = 99.60

138

R2 is very high but we know that t-ratio, F-stat and R2 etc give meaningful values if

residuals are identically independent normal. So we check that whether these

variables are identically independent normal or not.

Graphical Analysis For recovery of Copper:

Figure (A)

It looks almost normal.

Normality test for Residuals

Observations 31

Mean 0.043489

Std. Devn. 3.6404

Skewness -0.78933

Excess Kurtosis 1.1999

Minimum -11.304

Maximum 6.2073

Asymptotic test: Chi^2(2) = 5.0787 [0.0789]

Normality test: Chi^2(2) = 4.8027 [0.0906]

So residuals qualify normality test.

139

Actual and fitted values

Figure (B)

Standard Residual graph

Figure (C)

Residuals look independent.

140

Autocorrelation of residuals

Figure (D)

No significant autocorrelation present

Therefore residuals are identically independent normal.

General Model for Grad of Copper

YG = 21.58 + 0.01939*X1 + 0.06566*X2 + 0.03842*X3 + 0.01821*X4

(SE) (9.37) (0.00766) (0.785) (0.0377) (0.0188)

- 0.02359*X5 - 0.1301*X6 - 0.6017*X7 ------------- (c)

(0.025) (0.0861) (0.15)

Test for excluding:

[0] = X2

[1] = X3

[2] = X4

[3] = X5

[4] = X6

Subset F (5,23) = 4.4172 [0.0058]**

141

We can’t collectively exclude the all above regressors

Test for excluding:

[0] = X2

[1] = X4

[2] = X5

[3] = X6

Subset F (4,23) = 1.9908 [0.1296]

So we can exclude these regressors on totally statistical basis

YG = + 15.88 + 0.01907*X1 + 0.08596*X3 - 0.525*X7 ----------- (d)

(SE) (2.13) (0.0082) (0.0245) (0.15)

R2 = 0.542911

Subset F (3,27) = 10.68 [0.0000]**

All variables are collectively important.

Figure (E)

142

It looks normal

Normality test for Residuals

Observations 31

Mean 0.00000

Std.Devn. 1.3607

Skewness -0.41587

Excess Kurtosis 0.62105

Minimum -3.3897

Maximum 3.1853

Asymptotic test: Chi^2(2) =1.3918 [0.4986]

Normality test: Chi^2(2) = 3.1105 [0.2111]

Residuals qualify normality test

Figure (F)

143

Testing for heteroscedasticity using squares

Chi^2(8) = 29.292 [0.0003]** and F-form F (8,18) = 38.580 [0.0000]**

Residuals are heteroscedastic.

Now we take other possible model

Test for excluding:

[0] = X2

[1] = X3

[2] = X5

[3] = X6

Subset F (4,23) =1.2639 [0.3127]

So we can exclude these variables

YG = + 16.48 + 0.02159*X1 + 0.04478*X4 - 0.598*X7 -------- (e)

(SE) (2.06) (0.00758) (0.0111) (0.147)

T-ratio shows that all variables are individually important.

F (3,27) = 12.73 [0.000]**

F stat shows that all variables are collectively important

R2 = 0.585837

T-ratio, F-stat and R2 etc give meaningful values if residuals are identically

independent normal. So we check that whether these variables are identical

independent normal or not.

144

Graphical analysis of residuals

Testing for heteroscedasticity using squares

Chi^2(6) = 5.8926 [0.4353] and F-form F (6,20) = 0.78232 [0.5937]

So residuals are homoscedastics

Normality test for Residuals

Normality test for Residuals

Observations 31

Mean 0.00000

Std.Devn 1.2952

Skewness -0.65212

Excess Kurtosis 1.1558

Minimum -3.8576

Maximum 2.9363

Asymptotic test: Chi^2(2) = 3.9228 [0.1407]

Normality test: Chi^2(2) = 4.8522 [0.0884]

Residuals histogram

Figure (G)

145

Actual fitted values.

Figure (H)

Standard residuals for above model

Figure (I)

Residual looks independent

146

APPENDIX – 2

FORWARD SELECTION

Output from forward model selection procedure used for data on copper

recovery by the flotation process with seven variables (Alpha-to-enter: 0.25, N = 31)

Table – A

Step 1 2 3 4 5 6

Constant 38.254 38.493 15.650 6.132 -35.026 -29.933

X3 1.101 0.837 0.835 0.779 0.688 0.694

T-Value 12.94 7.52 8.41 8.29 6.55 6.65

P-Value 0.000 0.000 0.000 0.000 0.000 0.000

X4 0.168 0.185 0.188 0.191 0.155

T-Value 3.18 3.90 4.29 4.53 2.97

P-Value 0.004 0.001 0.000 0.000 0.007

X6 0.76 0.76 0.76 0.79

T-Value 2.87 3.12 3.23 3.38

P-Value 0.008 0.004 0.003 0.003

X1 0.054 0.053 053

T-Value 2.44 2.50 2.52

P-Value 0.022 0.019 0.019

X2 3.7 3.7

T-Value 1.71 1.72

P-Value 0.099 0.098

X5 -0.080

T-Value -1.19

P-Value 0.247

S 5.50 4.80 4.28 3.94 3.80 3.77

R-Sq 85.25 89.17 91.69 93.24 93.95 94.28

R-Sq(adj) 84.74 88.39 90.77 92.20 92.74 92.85

C-p 35.4 20.8 12.1 7.6 6.6 7.2

Press 1000.40 787.710 672.060 565.64 537.56 632.9

R-Sq(Pred) 83.20 86.77 88.71 90.50 90.97 89.37

147

APPENDIX – 3

BACKWARD ELIMINATION

Minitab out from backward elimination model selection procedure used for the

combined data for copper recovery in the flotation process with seven variables

(Alpha-to-Remove: 0.1, N = 31)

Table – B

Step 1 2 3

Constant -25.20 -29.93 -35.03

X1 0.053 0.053 0.053

T-Value 2.53 2.52 2.50

P-Value 0.019 0.019 0.019

X2 3.7 3.7 3.7

T-Value 1.73 1.72 1.71

P-Value 0.097 0.098 0.099

X3 0.69 0.69 0.69

T-Value 6.67 6.65 6.55

P-Value 0.000 0.000 0.000

X4 0.158 0.155 0.191

T-Value 3.05 2.97 4.53

P-Value 0.006 0.007 0.000

X5 -0.098 -0.080

T-Value -1.42 -1.19

P-Value 0.168 0.247

X6 0.83 0.79 0.76

T-Value 3.51 3.38 3.23

P-Value 0.002 0.003 0.003

X7 -0.45

T-Value -1.09

P-Value 0.287

148

S 3.75 3.77 3.80

R-Sq 94.56 94.28 93.95

R-Sq(adj) 92.91 92.85 92.74

C-p 8.0 7.2 6.6

Press 680.517 632.917 537.569

R-Sq(pred) 88.57 89.37 90.97

149

APPENDIX – 4

Minitab out from best subset procedure used for the combined data for copper

recovery by the flotation process with seven variables.

i. One variable best model YR = 38.254 + 1.101X3

Table – A: S = 5.504, R-Sq = 85.2%, R-Sq(adj) = 84.7%, Press = 1000.40

Predictor Coef SECoef T P

Constant 38.254 1.691 22.62 0.000

X3 1.101 0.085 12.94 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 1 5075.0 5075.0 167.55 0.000

Residual Errors 29 878.4 30.3

Total 30 5953.4

ii. One variable second best model YR = 44.9 + 0.465X4

Table – A: S = 8.19, R-Sq = 67.3%, R-Sq(adj) = 66.1%, Press = 2236.53

Redictor Coef SE Coef T P

Constant 44.919 2.058 21.83 0.000

X4 0.464 0.060 7.72 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 1 4004.3 4004.3 59.58 0.000

Residual Errors 29 1949.0 67.2

Total 30 5953.4

150

iii. Two variables best model YR = 38.5 + 0.837 X3 + 0.168X4

Table – A: S = 4.800, R-Sq = 89.2%, R-Sq(adj) = 88.4%, Press = 787.710

Predictor Coef SE Coef T P

Constant 38.493 1.477 26.07 0.000

X3 0.837 0.111 7.52 0.000

X4 0.168 0.052 3.18 0.004

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 2 5308.4 2654.2 115.22 0.000

Residual Errors 28 645.0 23.0

Total 30 5953.4

iv. Two variables second best model YR = 50.7+ 0.991 X3 –0.164 X5

Table – A: S = 5.153, R-Sq = 87.5%, R-Sq(adj) = 86.6%, Press = 910.345

Predictor Coef SE Coef T P

Constant 50.675 5.736 8.84 0.000

X3 0.991 0.093 10.61 0.000

X5 -0.164 0.072 -2.25 0.032

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 2 5209.8 2604.9 98.09 0.000

Residual Errors 28 743.6 26.6

Total 30 5953.4

151

v. Three variables best model YR = 29.0+ 0.053 X1 + 0.782 X3 + 0.171 X4

Table – A: S = 4.526, R-Sq = 90.7%, R-Sq(adj) = 89.7%, Press = 681.242

Predictor Coef SE Coef T P

Constant 28.983 4.702 6.16 0.000

X1 0.053 0.025 2.12 0.044

X3 0.781 0.108 7.23 0.000

X4 0.170 0.049 3.42 0.002

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 3 5400.2 1800.1 87.87 0.000

Residual 27 553.1 20.5

Total 30 5953.4

vi. Three variables second best model YR=15.6+ 0.835X3 + 0.185X4 + 0.762 X6

Table – A: S = 4.280, R-Sq = 91.7%, R-Sq(adj) = 90.8%, Press = 672.060

Predictor Coef SE Coef T P

Constant 15.650 8.078 1.94 0.063

X3 0.834 0.099 8.41 0.000

X4 0.185 0.047 3.90 0.001

X6 0.761 0.265 2.87 0.008

152

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 3 5458.8 1819.6 99.34 0.000

Residual Errors 27 494.5 18.3

Total 30 5953.4

vii. Four variables best model YR=6.13+0.0539X1+0.779X3 +0.188X4+0.762 X6

Table – A: S = 3.935, R-Sq = 93.2%, R-Sq(adj) = 92.2%, Press = 565.649

Predictor Coef SE Coef T P

Constant 6.132 8.392 0.73 0.472

X1 0.053 0.022 2.44 0.022

X3 0.779 0.094 8.29 0.000

X4 0.187 0.043 4.29 0.000

X6 0.761 0.244 3.12 0.004

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 4 5550.7 1387.7 89.61 0.000

Residual Errors 26 402.6 15.5

Total 30 5953.4

153

viii. Four variables second best model with out intercept YR = 0.0615X1 +

0.777X3 +0.191X4 + 0.918X6

Table – A: S = 3.9, R-Sq = 99.60%, R-Sq(adj) = 92.74%, Press = 526.202

Predictor Coef SE Coef T P

Constant 0 0 0 0

X1 0.061 0.019 3.16 0.004

X3 0.776 0.093 8.34 0.000

X4 0.191 0.043 4.44 0.000

X6 0.917 0.118 7.78 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 4 102807 25702 1688.93 0.000

Residual Errors 27 411 15

Total 31 103218

ix. Four variables third best model YR=-26.6+3.83X2+0.741X3+0.189X4+0.782X6

Table – A: S = 4.16, R-Sq = 92.4%, R-Sq(adj) = 91.3%, Press = 63.69

Predictor Coef SE Coef T P

Constant -26.57 27.57 -0.96 0.344

X2 3.831 2.398 1.60 0.122

X3 0.741 0.112 6.57 0.000

X4 0.189 0.046 4.09 0.000

X6 0.761 0.258 2.95 0.007

154

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 4 5503.0 1375.8 79.43 0.000

Residual Errors 26 450.3 17.3

Total 30 5953.4

x. Five variables best model YR = -35.0+ 0.0534X1 + 3.74X2 +0.688X3 +

0.191X4 + 0.762X6

Table – A: S = 3.79, R-Sq = 93.9%, R-Sq(adj) = 92.7%, Press = 537.569

Predictor Coef SE Coef T P

Constant -35.03 25.38 -1.38 0.180

X1 0.053 0.021 2.50 0.019

X2 3.744 2.188 1.71 0.099

X3 0.688 0.105 6.55 0.000

X4 0.191 0.042 4.53 0.000

X6 0.761 0.235 3.23 0.003

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 5 5592.9 1118.6 77.59 0.000

Residual 25 360.4 14.4

Total 30 5953.4

155

xi. Five variables second best model YR = 8.56+ 0.054X1 + 0.778X3 +0.196X4

+ 0.826X6-0.305X7

Table – A: S = 3.972, R-Sq = 93.4%, R-Sq(adj) = 92.0%, Press = 617.656

Predictor Coef SE Coef T P

Constant 8.563 9.118 0.94 0.357

X1 0.053 0.022 2.41 0.023

X3 0.778 0.094 8.20 0.000

X4 0.195 0.045 4.30 0.000

X6 0.782 0.248 3.15 0.004

X7 -0.305 0.423 -0.72 0.478

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 5 5558.9 1111.8 70.47 0.000

Residual Error 25 394.4 15.8

Total 30 5953.4

xii. Six variables second best model YR = -32.6+ 0.053X1 + 3.75X2 +0.687X3 +

0.200X4 + 0.783X6 –0.306X7

Table – A: S = 3.831, R-Sq = 94.1%, R-Sq(adj) = 92.6%, Press = 589.660

Predictor Coef SE Coef T P

Constant -32.61 25.81 -1.26 0.219

X1 0.053 0.021 2.48 0.021

X2 3.746 2.207 1.70 0.103

X3 0.687 0.106 6.48 0.000

X4 0.199 0.043 4.54 0.000

X6 0.782 0.239 3.27 0.003

X7 -0.306 0.408 -0.75 0.461

156

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 6 5601.18 933.53 63.62 0.000

Residual Error 24 352.17 14.67

Total 30 5953.35

xiii. Six variables best model YR = -29.9+ 0.053X1 + 3.73X2 +0.694X3 + 0.155X4

+ 0.079X5 –0.795X6

Table – A: S = 3.766, R-Sq = 94.3%, R-Sq(adj) = 92.9%, Press = 632.917

Predictor Coef SE Coef T P

Constant -29.93 25.54 -1.17 0.253

X1 0.053 0.021 2.52 0.019

X2 3.735 2.170 1.72 0.098

X3 0.693 0.104 6.65 0.000

X4 0.154 0.052 2.97 0.007

X5 -0.079 0.067 -1.19 0.247

X6 0.794 0.235 3.38 0.003

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 6 5612.92 935.49 65.95 0.000

Residual Error 24 340.43 14.18

Total 30 5953.35

157

xiv. Seven variables i.e full model YR = -25.2+ 0.053X1 + 3.74X2 +0.693X3 +

0.158X4 - 0.0982X5 +0.833X6-0.450X7

Table – A: S = 3.752, R-Sq = 94.6%, R-Sq(adj) = 92.9%, Press = 680.517

Predictor Coef SE Coef T P

Constant -25.20 25.81 -0.98 0.339

X1 0.053 0.021 2.53 0.019

X2 3.735 2.162 1.73 0.097

X3 0.693 0.104 6.67 0.000

X4 0.158 0.051 3.05 0.006

X5 -0.098 0.069 -1.42 0.168

X6 0.833 0.237 3.51 0.002

X7 -0.450 0.412 -1.09 0.287

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 7 5629.65 804.24 57.14 0.000

Residual Error 23 323.71 14.07

Total 30 5953.35

158

APPENDIX – 5

FORWARD SELECTION FOR GRADE

Minitab output from forward model selection procedure used for data on grade

by the flotation process with seven variables. (Alpha – to – enter 0.1, N=31)

Table - A

Step 1 2 3 4 5

Constant 14.01 19.26 15.88 20.16 23.33

X3 0.085 0.104 0.086 0.060 0.059

T-Value 3.05 4.19 3.51 2.31 2.34

P-Value 0.005 0.000 0.002 0.029 0.028

X7 -0.53 -0.52 -0.63 -0.59

T-Value -3.26 -3.49 -4.21 -4.05

P-Value 0.003 0.002 0.000 0.000

X1 0.0191 0.0193 0.0193

T-Value 2.32 2.50 2.57

P-Value 0.028 0.019 0.017

X5 -0.044 -0.037

T-Value -2.14 -1.83

P-Value 0.042 0.080

X6 -0.133

T-Value -1.58

P-Value 0.126

S 1.81 1.57 1.46 1.37 1.33

R-Sq 24.31 45.12 54.27 61.11 64.66

R-Sq(adj) 21.70 41.20 49.19 55.13 57.59

C-p 24.3 12.2 8.0 5.3

159

APPENDIX – 6

BACKWARD ELIMINATION. ALPHA-TO-REMOVE: 0.1

Minitab output from backward elimination model selection procedure used for

the combined data for copper on grade in the flotation process with seven variables.

(Alpha–to–remove 0.1,N = 31)

Table - A

Step 1 2 3 4 5 Constant 21.58 22.30 20.55 20.43 16.48 X1 0.0194 0.0194 0.0194 0.0216 0.0216 T-Value 2.53 2.59 2.59 2.95 2.85 P-Value 0.019 0.016 0.016 0.007 0.008 X2 0.07 T-Value 0.08 P-Value 0.934 X3 0.038 0.040 0.039

T-Value 1.02 1.25 1.21

P-Value 0.320 0.223 0.237

X4 0.018 0.018 0.028 0.041 0.045 T-Value 0.97 0.99 1.84 3.76 4.05 P-Value 0.343 0.334 0.078 0.001 0.000 X5 -0.024 -0.024 T-Value -0.94 -0.96 P-Value 0.356 0.345 X6 -0.130 -0.130 -0.142 -0.141

T-Value -1.51 -1.54 -1.71 -1.68 P-Value 0.144 0.136 0.100 0.105 X7 -0.60 -0.60 -0.57 -0.57 -0.60 T-Value -4.02 -4.10 -3.99 -3.98 -4.07 P-Value 0.001 0.000 0.001 0.000 0.000 S 1.36 1.33 1.33 1.34 1.39 R-Sq 66.04 66.03 64.72 62.65 58.59 R-Sq(adj) 55.71 57.54 57.67 56.90 53.99 C-p 8.0 6.0 4.9 4.3 5.1 Press 129.905 124.851 118.041 116.114 73.6739

160

APPENDIX – 7

Minitab out from best subset procedure used for the combined data copper

grade by the flotation process with seven variables.

I. One variable best model YG = 14.0 + 0.0854 X3

Table – A: S = 1.810, R-Sq = 24.3%, R-Sq(adj) = 21.7%, Press = 108.113

Predictor Coef SE Coef T P

Constant 14.0078 0.5563 25.18 0.000

X3 0.08539 0.02798 3.05 0.005

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 1 30.522 30.522 9.31 0.005

Residual Error 29 95.041 3.277

Total 30 125.563

ii. One variable second best model YG = 14.5 + 0.0374 X4

Table – A: S = 1.853, R-Sq = 20.7%, R-Sq(adj) = 17.9%, Press = 114.609

Predictor Coef SE Coef T P

Constant 14.4922 0.4652 31.15 0.000

X4 0.03741 0.01361 2.75 0.010

161

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 1 25.938 25.938 7.55 0.010

Residual Error 29 99.625 3.435

Total 30 125.563

iii. Two variables best model YG = 20.5 + 0.0518 X4 - 0.599 X7

Table – A: S = 1.554, R-Sq = 46.1%, R-Sq(adj) = 42.3%, Press = 87.837

Predictor Coef SE Coef T P

Constant 20.472 1.689 12.12 0.000

X4 0.05177 0.01208 4.29 0.000

X7 -0.5994 0.1647 -3.64 0.001

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 2 57.924 28.962 11.99 0.000

Residual Error 28 67.640 2.416

Total 30 125.563

iv. Two variables second best model YG = 19.3 + 0.104 X3 - 0.527 X7

Table – A: S = 1.569, R-Sq = 45.1%, R-Sq(adj) = 41.2%, Press = 90.8320

Predictor Coef SE Coef T P

Constant 19.256 1.681 11.45 0.000

X3 0.10437 0.02494 4.19 0.000

X7 -0.5266 0.1616 -3.26 0.003

162

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 2 56.653 28.327 11.51 0.000

Residual Error 28 68.910 2.461

Total 30 125.563

v. Three variables best model YG = 16.5 + 0.0216 X1 + 0.0448 X4 - 0.598 X7

Table – A: S = 1.388, R-Sq = 58.6%, R-Sq(adj) = 54.0%, Press = 73.673

Predictor Coef SE Coef T P

Constant 16.480 2.058 8.01 0.000

X1 0.021587 0.007575 2.85 0.008

X4 0.04479 0.01106 4.05 0.000

X7 -0.5979 0.1471 -4.07 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 3 73.563 24.521 12.73 0.000

Residual Error 27 52.000 1.926

Total 30 125.563

vi. Three variables second best model YG = 15.9 + 0.0191 X1 + 0.0859 X3 - 0.525 X7

Table – A: S = 1.458, R-Sq = 54.3%, R-Sq(adj) = 49.2%, Press = 80.260

Predictor Coef SECoef T P

Constant 15.876 2.135 7.44 0.000

X1 0.019072 0.008204 2.32 0.028

X3 0.08593 0.02450 3.51 0.002

X7 -0.5249 0.1502 -3.49 0.002

163

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 3 68.147 22.716 10.68 0.000

Residual Error 27 57.416 2.127

Total 30 125.563

vii. Four variables best model YG = 20.4 + 0.0216 X1 + 0.0411 X4 - 0.141 X6 -

0.570 X7

Table – A: S = 1.343, R-Sq = 62.7%, R-Sq(adj) = 56.9%, Press = 116.114

Predictor Coef SE Coef T P

Constant 20.435 3.082 6.63 0.000

X1 0.021614 0.007331 2.95 0.007

X4 0.04112 0.01092 3.76 0.001

X6 -0.14119 0.08395 -1.68 0.105

X7 -0.5700 0.1433 -3.98 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 4 78.666 19.666 10.90 0.000

Residual Error 26 46.898 1.804

Total 30 125.563

164

viii. Four variables second best model YG = 20.2 + 0.0193 X1 + 0.0601 X3

- 0.0437 X5 - 0.626 X7

Table – A: S = 1.370, R-Sq = 61.1%, R-Sq(adj) = 55.1%, Press = 73.70

Predictor Coef SE Coef T P

Constant 20.163 2.836 7.11 0.000

X1 0.019256 0.007710 2.50 0.019

X3 0.06011 0.02600 2.31 0.029

X5 -0.04375 0.02046 -2.14 0.042

X7 -0.6264 0.1489 -4.21 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 4 76.735 19.184 10.22 0.000

Residual Error 26 48.828 1.878

Total 30 125.563

ix. Five variables best model YG = 20.5 + 0.0194 X1 + 0.0386 X3 + 0.0281 X4 -

0.142 X6 - 0.567 X7

Table – A: S = 1.331, R-Sq = 64.7%, R-Sq(adj) = 57.7%, Press = 118.04

Predictor Coef SE Coef T P

Constant 20.547 3.056 6.72 0.000

X1 0.019410 0.007490 2.59 0.016

X3 0.03855 0.03182 1.21 0.237

X4 0.02809 0.01526 1.84 0.078

X6 -0.14219 0.08320 -1.71 0.100

X7 -0.5671 0.1420 -3.99 0.001

165

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 5 81.267 16.253 9.17 0.000

Residual Error 25 44.296 1.772

Total 30 125.563

x. Five variables best model YG = 23.3 + 0.0193 X1 + 0.0591 X3 - 0.0371 X5 -

0.133 X6 - 0.593 X7

Table – A: S = 1.332, R-Sq = 64.7%, R-Sq(adj) = 57.6%, Press = 120.67

Predictor Coef SE Coef T P

Constant 23.325 3.405 6.85 0.000

X1 0.019264 0.007496 2.57 0.017

X3 0.05909 0.02528 2.34 0.028

X5 -0.03712 0.02032 -1.83 0.080

X6 -0.13327 0.08416 -1.58 0.126

X7 -0.5926 0.1464 -4.05 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 5 81.187 16.237 9.15 0.000

Residual Error 25 44.377 1.775

Total 30 125.563

166

xi. Six variables best model YG = 22.3 + 0.019 X1 + 0.039 X3 + 0.018 X4 - 0.023

X5 - 0.130 X6 - 0.602 X7

Table – A: S = 1.333, R-Sq = 66.0%, R-Sq(adj) = 57.5%, Press = 124.58

Predictor Coef SE Coef T P

Constant 22.300 3.562 6.26 0.000

X1 0.0194 0.007 2.59 0.016

X3 0.039 0.031 1.25 0.223

X4 0.018 0.018 0.99 0.334

X5 -0.02360 0.02453 -0.96 0.345

X6 -0.13009 0.08427 -1.54 0.136

X7 -0.6017 0.1467 -4.10 0.000

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 6 82.913 13.819 7.78 0.000

Residual Error 24 42.650 1.777

Total 30 125.563

xii. Six variables second best model YG= 19.8+ 0.019 X1 +0.068 X2 +0.036 X3

+0.028 X4 -0.142 X6-0.567 X7

Table – A: S = 1.358, R-Sq = 64.7%, R-Sq(adj) = 55.9%, Press = 123.102

Predictor Coef SE Coef T P

Constant 19.796 9.152 2.16 0.041

X1 0.019400 0.007644 2.54 0.018

X2 0.0682 0.7828 0.09 0.931

X3 0.03690 0.03761 0.98 0.336

X4 0.02816 0.01559 1.81 0.084

X6 -0.14218 0.08491 -1.67 0.107

X7 -0.5671 0.1450 -3.91 0.001

167

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 6 81.281 13.547 7.34 0.000

Residual Error 24 44.282 1.845

Total 30 125.563

xiii. Seven variables best model YG = 21.6 + 0.0194 X1 + 0.066X2+0.0383

X3+0.0182 X4-0.0236X5 -0.130 X6 - 0.602 X7

Table – A: S = 1.362, R-Sq = 66.0%, R-Sq(adj) = 55.7%, Press = 129.905

Predictor Coef SE Coef T P

Constant 21.577 9.366 2.30 0.031

X1 0.019390 0.007662 2.53 0.019

X2 0.0658 0.7846 0.08 0.934

X3 0.03834 0.03773 1.02 0.320

X4 0.01824 0.01885 0.97 0.343

X5 -0.02360 0.02505 -0.94 0.356

X6 -0.13009 0.08607 -1.51 0.144

X7 -0.6017 0.1499 -4.02 0.001

Table – B: Analysis of Variance

Source DF SS MS F P

Regression 7 82.926 11.847 6.39 0.000

Residual Error 23 42.637 1.854

Total 30 125.563

PAPER PUBLISHED

2

* Sardar Ali, M. Izhar ul Haq

** M. Mansoor Khan

***Mir Asad ullah

MATHEMATICAL MODELS FOR THE EFFICIENCY OF FLOTATION PROCESS FOR THE RECOVERY OF NORTH

WAZIRISTAN COPPER.

Abstract

Mathematical models were developed to give an insight to see the effect of process

variables propylxanthat (X1 g/ton), pH (X2), sodium sulphide (X3,g/ton) and sodium

cyanide (X4g/ton) on the recovery (YR) of copper.

The optimum recovery (YR) 62.95%, at X4=60g/ton were obtained (6,7)

Introduction

The simulation of mineral processing system design optimization of flotation

parameter and control is used for the least 30 years. (1-5, 9-11, 14, 20-23). Federally

Administrated Tribal Area Development Corporation carried out exploration work

and confirmed 122 million tons of estimated reserves.

The North – Waziristan ore is a sulphide ore body, which contains chalcopyrite as the

ore mineral. The ore is of low grade within economic limit (6,8,17) therefore it must

be upgraded before it can be subjected to metallurgical treatment to obtain blister

copper. Extensive floatation test work were carried out to investigate effects of

various process parameters on recovery of copper. Effects of collector type and

dosage; pH, sulfidizer dosage; depressant were investigated during flotation test. The

results of the pilot scale studies showed that the copper content in the ore was

upgraded from 0.9% to 22-25% in a staged cleaning flotation with recoveries up to

80%. The recovery can be further enhanced by improving the machine efficiency and

conducting more research on reagents. (7,19,24).

______________________ * University of Education Lahore ** Department of Mining Engineering , N.W.F.P, U.E.T. Peshawar. *** Department of Mathematics, COMSAT,Abbottabad.

3

EXPERIMENTAL WORK

Twenty tests were carried out to evaluate the flotation response, using different

dosages and type of collector. Five tests were conducted out each to investigate the

effect of individual parameter such as the collector type dosage NapX, pH, depressant

and sulphidizer on the grade and recovery of final concentrate copper.

Methodology

Applying Regression Analysis for enrichment of copper ore experiments were

conducted to study the effect of the collector type dosage, depressant, sulphidizer and

frother dosage on recovery of North Waziristan copper ore.

The most general type of linear mathematical model can be described with variables

Z1,Z2, ….., Zp in the form as follows where € stands for variations caused by other

than Z1, Z2 …..,

Y = βoZo + β1 Z1 + β2 Z2 + ……….. βp Zp + € …….. (1*)

Zo = 1 and stands for effects of the regression model

However, it is some times convenient to have a Zo in the model.

The following four mathematical models were used to estimate recovery of copper ore

in the final product based on first order, second order, logarithmic and exponential.

1. First Order Mathematical Model

1. If p = 1 and Z1 = X in eq. (1*)

we get the simple first-order mathematical model with one predictor variable.

Y = βo + β1 X + € …………………. (2*)

2. Second-order Mathematical Model:

We obtain a second order-variable mathematical model with one predictor

variable:

Y = βo + β1 X + β11 X2 +€……………….. (3*)

4

3. The Logarithmic Transformation:

By taking P = 2, Z1 = In X1, Z2 = In X2 eq. (1*) becomes

Y = βo + β1 In X1 + β2 In X2 ………….. (4*)

4. The Exponential Mathematical Models:

Y = eβo + β1

X1

+ β2

X2 + € ……………taking natural logarithms of both sides we get

In Y = βo + β1X1 + β2 X2 + In € …………….(5*)

Results and discussions

Fifteen mathematical models were developed by using statistical techniques for

recovery of North Waziristan copper.

The following first order and second order mathematical model were derived with one

predictor variable i.e. Linear, logarithmic, polynomial and exponential.

YR = 0.0536X1 + 29, R2 = 0.8897 Eq. (1)

YR = 6.5631Ln(X1) + 5.0807, R2 = 0.8619 Eq. (2)

YR = -0.0001X21 + 0.0896X1 + 26.9, R2 = 0.9053 Eq. (3)

YR = 29.541e0.0015X1, R

2 = 0.8884 Eq. (4)

Graphical representation of above five equations are given in figure-1

YR= 4.0591X2 –5.1084, R2= 0.5837 Eq.(5)

YR = 45.116Ln(X2) - 68.534, R2 = 0.6005 Eq. (6)

YR = -5.0413X22 + 114.83X2 - 610.69, R2 = 0.8239 Eq. (7)

Graphical representation of above five equations are given in figure-2

YR = 0.7X3 + 41.4, R2 = 0.821 Eq. (8)

YR = 12.69Ln(X3) + 18.277, R2 = 0.8123 Eq. (9)

YR = -0.0143X23 + 1.2714X3 + 36.4, R2 = 0.833

Eq(10)

YR = 42.746e0.0127X3, R

2 = 0.836 Eq(11)

5

Graphical representation of above five equations are given in figure-3

YR = 0.0720X4 + 58.4, R2 = 0.9348 Eq(12)

YR = 1.8153Ln(X4) + 54.903, R2 = 0.7821 Eq(13)

YR = 0.0013X24 + 0.0212X4 + 59.544, R2 = 0.9938 Eq(14)

YR = 59.467e0.0012X4, R

2 = 0.9367 Eq(15)

Graphical representation of above five equations are given in figure-4

Using X1, in equations 1,2,3 and 4 we obtained equation 3 is the appropriate model is

equation 10 which is quadratic.

Using X2 the models 5,6 and 7 we get the suitable fit model 7 which is again quadratic

using X3 in equations 8,9,10 and 11 the comparatively better quadratic model in using

X4 in equations 12,13,14 and 15we obtained the best fit model 14.

CONCLUSION

Suitable models for the effect of individual variable X1,X2,X3, and X4 on the recovery

YR, for the enrichment of copper are the equations 3,7,10 and 14 with high value of

R2. These all equations are quadratic one predictor variable. It was concluded with

high degree of confidence that the effect of processes parameters can be predicated by

these equation within the given rang. Maximum copper recovery were obtained when

the value of X1 is 200mg/ton, X2 is 11.58, X3 is 30gm/ton and the value of X4 is 60

gm/ton. Comparing the results of recovery of all above four parameters, the best

model is which gives the maximum recovery among all the parameters with high

value of R2 and is significant at the level of probability. However it will be more

appropriate if further models may be derived to have combined effect of these

parameters on the recovery of copper concentrate in the treatment of copper ore by

flotation process. Optimum copper recovery were obtained when X1 = 200g/ton, X2 =

11.58, X3 = 30 g/ton and X4 = 60 g/ton.

X4 gives the maximum recovery 62.95%. More models will be derived to have

combine effect of these parameters.

Figure-1: Showing the effect of NaPX on recovery.

y = 0.0536x1 + 29R2 = 0.8897

01020304050

0 100 200 300

%R

Dosage gm/ton

figure a figure by = 6.5631Ln(x1) + 5.0807

R2 = 0.8619

01020

304050

0 50 100 150 200 250 300

Dosage gm/ton

%R

figure cy = -0.0001x2

1 + 0.0896x1 + 26.9

R2 = 0.9053

0

10

20

30

40

50

0 50 100 150 200 250 300

Dosage gm/ton

R%

figure d y = 29.541e0.0015x

R2 = 0.8884

0

10

20

30

40

50

0 100 200 300

Dosage gm/ton

R%

Figure-2: Showing effect of pH on recovery

figure a y = 4.0591x2 - 5.1084R2 = 0.5837

0

10

20

30

40

50

9.5 10 10.5 11 11.5 12 12.5

pH

R%

figure b y = 45.116Lnx2 - 68.534

R2 = 0.6005

0

10

20

30

40

50

9.5 10 10.5 11 11.5 12 12.5

pH

R%

figure c y = -5.0413x22 + 114.83x2 - 610.69

R2 = 0.8239

0

10

20

30

40

50

9.5 10 10.5 11 11.5 12 12.5

pH

R%

7

Figure-3: Showing the effect Na2S on recovery

figure a y = 0.7x3 + 41.4

R2 = 0.821

0

20

40

60

80

0 10 20 30 40

Na2S gm/ton

R%

figure b y = 12.69Lnx3 + 18.277

R2 = 0.8123

010203040506070

0 10 20 30 40

Na2S gm/ton

R%

figure cy = -0.0143x3 + 1.2714x3 + 36.4

R2 = 0.833

010203040506070

0 10 20 30 40

Na2S gm/ton

R%

figure d y = 42.746e0.0127x

R2 = 0.836

010203040506070

0 10 20 30 40

Na2S gm/ton

R%

Figure-4: Showing the effect NaCN on recovery

figure a y = 0.072x4 + 58.424

R2 = 0.9348

58596061626364

0 20 40 60 80

NaCN gm/ton

R%

figure b y = 1.8153Lnx4 + 54.903

R2 = 0.7831

58596061626364

0 20 40 60 80

NaCN gm/ton

R%

figure c y = 0.0013x42 - 0.0212x4 + 59.544

R2 = 0.9938

59

60

61

62

63

64

0 20 40 60 80

NaCN gm/ton

R%

figure d y = 58.467e0.0012x

R2 = 0.9367

58596061626364

0 20 40 60 80

NaCN gm/ton

R%

8

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