1

ACKNOWLEDGEMENTS

The University of Zambia (UNZA), Institute of Distance Education (IDE) wishes to thank DR. LORDWELL K. WITIKA for writing this module , MET 5139

SELECTED TOPICS IN MINERAL PROCESSING MODULE.

2

INTRODUCTION.

Welcome to Module on Special Topics in Mineral Processing (MET 5139). It is very

interesting to study how Ores are processed in minerals of interest and subsequently

further made into useful metallic products by value addition for our daily use.

Have you ever experienced how it look like, that the ordinary sand we step on every

single day may be transferred in something you may not associate with processing of

materials around us? Silicon Valley, has used it’s potential to make it the most

industrious part of the USA which uses sand to make electronic chips. Human race

depend on All mineral resources have potential to make our life ease. The folk, knife

and all the other things we use not only in the kitchen are made from natural resources

around us. A car, airplane, submarines, computers, etc are all made from natural

mineral resources after being processed properly.

Such questions provide the gist of what Processing and beneficiation is all about, this

is a science that seeks to explain the phenomena which is involved in the enhancing

the understanding of the use of natural minerals resources.

In this module we will begin by consolidate the fundamental understanding of the

selected processing techniques and will go on to define what is meant by value

addition to mineral processing products. Examples will be cited for those areas of

direct relevance to this subject.

3

Aim

The aim of this module is to introduce you to the origin, development, nature of

Mineral Processing, value addition and subsequently environmental awareness and to

understandinghowwecanmakeournaturalresourcesusefulwithoutimpairing

theenvironment.Becauseof theirphysical significance they findapplication in

many areas of the process industries. In this course, the students will be

provided with the necessary skills to tackle problems involving Mineral

Processingtechniquesingeneralandvalueadditionasanultimategoal.

Objectives At the end of the course, the student is expected:

(a) To gain an appreciation of the Mineral processing advanced principles and downstream processing techniques with emphasis on the similarities and differences methods used in various processing techniques.

(b) To discuss some applications of the some practical principles within

the process, metallurgical and chemical, petrochemical industries, and pharmaceutical and other related industries.

Assessment Your work in this module will be assessed in the following three ways:

• Two Assignments each 10%

• Two tests each worth 15% each

• A written examination set by the University of Zambia at the end of the

module 60%.

Continuous Assessment: 40% 2 Assignments – 10% each

2 tests – 15% each

Final Examination: 60%

4

RECOMMENDED READINGS.

1. .C. Chatfield.” Statistics for Technology : A Course in applied Statistics.” Second Edition, J. W. Arrowsmisth Ltd, Bristol, 1979.

2. B. A. Wills, "Mineral Processing Technology". Pergamon 4th Edition, 1988

3. T. J. Napier-Munn, S. Morrell, R. D. Morrison and T. Kojovic, " Mineral Comminution Circuits: Their Operation and Optimisation" .Series: JKMRC monograph series in mining and mineral processing, No. 2., 1999.

4. Flotation Science and Engineering, Editor K. A. Matis, Marcel Dekker, Inc, 1995.

5. Innovations in Flotation Technology, Editors P. Movros and K. A. Matis, NATO Series: Applied Sciences Vol. 208, 1992, Kluwer Academic Publishers.

6. J. A. Finch and G. S. Dobby, "Column flotation". Pergamon Press, 1990. 7. Flotation of sulphide minerals, Development in Mineral Processing, Editor

K. S. E. Forssberg, Elserview, 1985. 8. Gupta A. and Yan. D. S, “Mineral Processing Design and Operation”

(January 2006), Perth, Australia.

SUPPLEMENTARY TEXTBOOKS.

1. D. J. Shaw, " Introduction to Colloid and Surface chemistry". 3rd Edition, spottiswoode, Ballantyne Ltd, 1980.

2. L. Leja, “ Surface Chemistry of Froth Flotation”. Plenun Press, New York, 1982.

3. R. D. Crozier, “ Floation: Theory, Reagents and Ore Testing”, 1st Edition, Pergamon Press Plc, Headington Hill Hall, Oxford, England, 1992.

4. S. K. Jain,” Mineral Processing”. 2nd Edition CSB Publishers and Distributors, New Delhi, India, 2001

You may find the recommended and supplementary readings provided at the beginning of the module useful, but you could also explore other sources of information, particularly the Internet which has a lot of websites with invaluable information.

Time frame

You are expected to spend at least 60 hours of study time on this module. In addition, there shall be arranged contacts with lecturers from the University from time to time

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during the course. You are requested to spend your time judiciously so that you reap maximum benefit from the course. Study Skills You may not have studied by distance education before. Here are some simple tips for you to follow which will help you do better in your learning and keep you focused-

1. Set goals such as: I will succeed in this course. At the beginning of the module, break the lessons into manageable chunks. You might not have time to do a full lesson in one night, so plan how much you can do, then stick to it until you are done.

2. Establish a regular study/learning schedule 3. Determine what time is best for you to study 4. Have a dedicated study place with all the supplies you might need 5. Tell people what you are doing because only then are you more likely to stick

to a course. 6. Ask someone to proofread your work before you submit it. 7. Reward yourself with whatever work for you, along the way. 8. If you do not understand something ask your local learning centre or your

tutor, who will be able to help you. 9. Search for the meaning of principles and concepts instead of just memorizing

Need help? In case you have difficulties during the duration of the course, please get in touch with the Director, Institute of Distance Education, or the resident lecturer in your province. All enquiries in connection with the payment of fees should be directed to the Director, Institute of Distance Education:

The Director, Institute of Distance Education, University of Zambia, P. O. Box 32379, 10101 Lusaka Coordinator, Learner Support Services (Land Cell): +260 978772248 Senior Administrative Officer (Programme Development & Production) +260 977639993 IDE Land Line: +260 211 290719 IDE Fax: +260 211 290719 IDE E-mail: director-ide@unza.zm http://www.unza.zm

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UNIT 1

1.0 SAMPLING OF BULK MATERIALS. 1.1 Introduction. In this Unit we will do a revision of statistical fundamentals: mean, variance, and mode, standard deviation, discrete and continuous distributions. Confidence limits/intervals, significant testing, tests on sample means, comparing two sample means using student's t test, chi-square and F tests. The use of the Gy’s formula will be introduced for sampling purposes and subsequently various sampling techniques will be presented. Relevance to mineral and chemical processing systems will enable you to appreciate the subject from a Metallurgical point of view as you generally go through this unit.

1.2 Objectives

By the end of this unit you should be able to;

1. Define the term discrete and continuous distributions as applied mineral processing techniques.

2. Describe the meaning of the terms used in chemical thermodynamics. 3. Define the use of the Gy’s formula for sampling purposes and how

subsequently various sampling techniques are applicable in mineral processing

1.3 Reflection

Have you ever thought about the use of statistics as used in mineral processing? Well, whether you have ever considered it or not, think about how you can use this tool in this course. Did you think `mineral processing is just about the processing of a ore to make or obtain the valuable metal for sale. Statistics help in developing new process techniques or routes which are more viable and economical.

1.4 Sampling. Sampling methods depends on what is required to be looked into and gives an outlook of properties of the bulk material. Hence the main purpose of sampling is to draw a portion of a larger entity from which measurements can be taken in order to infer the characteristics of the large entity called a population which is simply the aggregate from which a sample is taken and it contains all the possible outcomes. It is important to distinguish between target and sampled population. Target population will describe the available material in say a concentrator, whilst sampled population is the portion drawn from the target population. In sampling, any portion taken for examination is referred to as a sample. Sample in general may be of any of the following type:

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a) Representative Sample - Sample taken in such a manner those measurements taken on it can be assumed to produce some answers as would be obtained if measurements were made on the aggregate.

b) Systematic samples -Samples taken according to a fixed cyclic procedure in time and space.

c) Simple random sample - samples taken in such way as to give all possible values of properties to have an equal chance of acceptance.

d) Stratified samples - samples taken in aggregates which are sub divided in smaller portions called strata which are then sampled independently but making sure that the strata do not overlap but homogeneous.

In order to draw a sample a sampling unit is required which is simply a small portion of aggregate assembled to make a sample using a flame which is a instrument for collecting sample.

1.5 PRESENTATION OF DATA

The two main methods presenting data are:

a) Pictorial method - This may take a form of a histogram, curve etc.

i) Bar Chart Plot of frequency distribution versus property under measurement discrete data only.

Figure 1.1 : Shows a version of bar / Histogram.

ii) Histogram- Similar to a bar chart but the areas of the rectangles are proportional to frequency of groups – this can be used for continuous data.

iii) Cumulative Frequency - Useful for finding out if a particular value

is less than or (greater than) a given value where observation are arranged in ascending order of magnitude.

iv) Pie Chart- A circular representation data adding up to 100%

0

2

4

6

Category1Category2Category3Category4

Series1

Series2

Series3

8

Figure 1.2 : shows a pie chart. b) Arithmetic method A statistic in any quantity calculated from data. There are two types of statistics:

i) that deal with measure of central tendency i) Mean for a population

∑=

=++++

=n

i

in

nx

nxxxx

1

321 .....µ

For grouped data

∑=

=+++

++++=

n

i i

ii

n

nn

fxf

fffxfxfxfxf

121

332211

.........

µ

Where the variant xi occurs in frequency fi

ii) Median For a set of observations arranged in order of magnitude. The middle value or arithmetic mean of mid values ie the median 1, 3, 5, 7, 9, 13, 15 median = 7 1, 3, 5, 7, 9, 13 median = 5 + 7/2 = 6 This can also be defined as the value of variant which divides the area of the histogram into two equal parts.

For grouped data

Median = L1 + ( ) CfN⎥⎦

⎤⎢⎣

⎡ − ∑ 12

L1 = Lower Class boundary of median class N = Number of items in data (total frequency) ( )1f∑ = Sum of frequencies of all classes lower than median class. C = Size of the median class.

iii) Mode

Sales1stQtr

2ndQtr

3rdQtr

4thQtr

9

Value of variant, which occurs most, common or with greatest frequency.

For group data

Mode = L1 + C⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ+Δ

Δ

21

1

L1 = Lower class boundary of model class 1Δ = Excess of modal frequency over frequency of next lower class ( )lm ff −

2Δ = Excess of modal frequency over frequency of next higher class ( )mu ff − C = Size of modal class Most important is the mean. For a symmetric distribution, mean, medium and mode coincide. For skewed distributions they are all different.

ii) Measures of Spread Need to know variability of data.

a) Range - Difference between largest and smallest values - only useful for samples of equal size. 1.6 Variance and standard deviation.

1.6.1 Population Variance ( ) ( ) ( )n

xxx n1.... 22

22

12 µµµσ −++−+−=

= 2

1⎟⎠

⎞⎜⎝

⎛ −∑= n

xin

i

µ

( )

∑−

=n

i

nnx 22

2 µσ

σ is the STANDARD DEVIATION σ is preferred to σ 2 since if you use σ 2 the units are also square. It is hence appropriate to use square roof σ so that measure of spread and mean ( )µ are of the same units.

c) Coefficient of Variation

= σµ

advantage is that it has no units

1.6.2. DISCRETE PROBABILITY DISTRIBUTION.

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If a random variable X can assume a set of values nXXXX ,..., 32,1 with

probability nPPPP ,....,, 321 Where Pi =∑ 1, then we say that a discrete probability distribution has been defined. e.g. Tossing a pair of fair dice, X denotes the sum obtained.

X 2 3 4 5 6 7 8 9 10 11 12 P 1 2 3 4 5 6 5 4 3 2 1 In 900 trials we expect to get sum = 5 in 100 cases. 1.6.3 BINOMIAL DISTRIBUTION. If p is probability that an even occurs (Prob. of success) and q = 1 – p is probability that it will not occur (prob. of failure), then probability of r successes out of n tries is

( )( )

( )

( )

P C P P nr n r

P q

r n

rn

rn r r n rr

= − =−

=

− −1

0 1 2

!! !

.

, , ,......,µ

σ

= nP mean

= npq variance2

1.6.4. POISSON DISTRIBUTION

( )P rer

rr

= =

=

=

−λ

µ λ

σ λ

λ

!, , ,....0 1 2

1.6.5. CONTINOUS PROBABILITY DISTRIBUTIONS. A continuous random variable has a probability of zero of assuming any of these values. The probability distribution is defined over a continuous sample space. Hence graph of f(x) will be continuous.

Probability (a < x < b) = ( )f x dxa

b

∫

Where f(x) ≥ 0 for all values of x

Furthermore ( )f x dx =∞

∞

∫ 1

Total area under the curve has to be unity.

11

1.6.6 NORMAL DISTRIBUTION Mathematical equation has two important parameters µ σand For the probability density function

( )

∑−

⎟⎠

⎞⎜⎝

⎛=

⎥⎦

⎤⎢⎣

⎡ −−

∑=

=

−

n

inxi

x

i

nx

where

eaN

1

2

21

2

2

21,,

µσµ

πσσµ σ

µ

x may be expressed in terms of standard normal deviation, Z

Z = σµ−x

( )[ ]

N Z, eZ

µ σσ π

, =−1

2

12

2

Tables of normal distribution are presented in standardised form with µ σ= =0 1and Example Diameter of grinding balls in a batch is known to be normally distribution with µ σ= =50 3mm and mm what is the probability of finding a ball > 60 mm.

Z x=

−=

−= =

µ

σ

60 503

103

333.

From the tables of encountering a ball less than 60 mm is 0.9996. Hence probability of finding one > 60 mm is 0.0004 = 0.04%. 1.7. POPULATION AND SAMPLE DATA. Statistical inference is concerned with getting information from a sample of data about population from which a sample is drawn. However it is important to distinguish between a sample and population characteristics. For example if a sample of n observations x1,x2,x3, …..xn is drawn from a

population (normal distribution) of mean µ and standard deviation σ then −

x is an intuitive estimate of µ and sample standard deviation

( )∑− −

−==

n

i

i

nxx

SSD1

2

1 For grouped data

12

( )11

2

1

22

−

−=

−

−== ∑∑

= nxxf

nxnx

SSD iin

i

i

SD is an estimate of σ Note that if n is used instead of (n – 1), a biased estimate of σ 2 will be obtained. Denominator is referred to as the number of degrees of freedom. As µ is not known a linear constraint on values of ( )x x− is that can only be independently compared

with (n –1) observations hence (n – 1) degrees of freedom. For samples

( )

( )

( )

E X

E S

E S

µ

σ

σ

=

=

=2 2

1.8 SAMPLING DISTRIBUTIONS Given a set of data x x x x xn1 2 3, , ,.... , and s2 can be computed. If we take another sample of size n from the same population, slightly different value of x and s2 will result. For repeated samples from the same population, statistic of interest can be regarded as a random variable. Its distribution is called the Sampling Distribution. If the population has mean µ and standard deviationσ , distribution of means will have

same mean µ but a much smaller standard deviation given by σ

n. The standard

deviation is the standard error of Mean (SEM)

( )

( )

E

En

SEM sn

x

x

µ µ

σσ

=

=

=

Example - The copper content of flotation concentrate is measured by a series of samples. Measurements are free from systematic errors and have a standard deviation of 2%. How many measurements must be taken so that Cu content is < 0.6%. Assume n measurements.

13

SEM

n nn or samples are required

= = <

>

σ 2 06%

111 12

.

.

1.8.1 CENTRAL LIMIT THEOREM. If random samples of size n are taken from any distribution the sampling distribution

of x will be approximately normal with mean µ and standard deviation of σn

.

Approximation improve with n. Example: The following recoveries from flotation are obtained,

76.48, 78.12, 78.05, 74.26, 75.99, 74.36, 74.34, 77.12

What is SEM in estimating mean assay.

x S

In absence of SEM Sn

= =

=

= =

76 343 1464

14648

0518%

. , .

,

. .

σ

1.8.1 CONFIDENCE LIMITS AND INTERVALS. It is usually desirable to have an interval estimate so that we have a certain confidence that unknown parameter is contained in the stated interval

Zx

n

=−µσ

Prob

%95,

nsxZ

knownnotisifn

ZXn

ZX

µ

σ

σµ

σ

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+<<−

95% of the standard normal distribution lies between + 1.96 Therefore

14

Prob − <

−<+

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=196 196 095. . .xsn

µ

This implies

nsxand

nsx 96.196.1 +<<− µµ

Hence Prob (x - 1.96 95.0)96.1 =+<<nsx

ns

µ

The interval x + 1.96 ns is called the 95% confidence interval for µ (ie 95%

confident that lies in the interval). The end points are the confidence limits.

Generally degree of confidence is given by χ + z ns with confidence level equal

to value of Z from tables. The limit can be single or double sided. Convection formula

%2

100100 ⎥⎦

⎤⎢⎣

⎡ −−=

αβ

Calculate the two sided 95% confidence limit for 50 Zn assays given χ = 12.50% S = 1.00% n = 50 If using single sided tables

%5.97295100100 =⎥⎦

⎤⎢⎣

⎡ −−=β

From the tables Z = 1.96

Confidence limit µ = ±

= ± = ±

Z snx196 10050

0 28%. .

.

We are 95% confident that Zn estimate lies between 12.22% and 12.78%. 1.8.2 SIGNIFICANT TESTS. Statistical procedure for comparing quantities in such a way to judge significance of the observed differences. Based on formulating an hypothesis whose validity is tested. NULL hypothesis (Ho); there is no significant difference between quantities:

15

Ho ; µ µ

ο=

µ o is some standard value. This is usually assumed until proved wrong. Any other hypothesis is alternative hypothesis µ µ µ µ µ µ≠ > <=o o o, , 1.8.2.1 LEVEL OF SIGNIFICANCE. This is the probability of getting a result which is more extreme than one obtained. A result which is unlikely to occur if Ho is true is called a SIGNIFICANT RESULT. 1.8.2.2 Student’s t - DISTRIBUTION.

Is the appropriate alternative to normal distribution for small samples (n < 30) or where σ has to be estimated from the data.

ns

t µχ −=

The confidence limits are nts

±χ

If we are interested in values of χ significantly > µ o (or less) a one tailed test is used (departures in one direction). For departures in two directions a two tailed test is used. 1.8.2.3 Student t-Test on sample mean Example 1 - According to Mineralogical calculations the Zn content of an ore is 12.1%. 9 samples are analysed to check the calculations; with the following results. 11.7, 12.2, 10.7, 11.4, 11.3 12.0, 11.1, 10.7, 11.6 is the sample mean significantly lower that 12.1% χ χ µ= < =1143 0 49. , .o S degree of freedom = 8.

1.449.0

1.1243.11−=

−=

−=

nns

t µχ

From table t 0.95,8 = 1.86 I t I > t0.95,8

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Probability of observing more extreme value is 0.05 at least. Result is significantly different at 95% confidence. a) Single tailed test appropriate b) Negative sign significant c) 95% (sometimes 99%)

Confidence limit normally assumed for rejection of Ho. Example 2 A DMS cyclone has a guaranteed Ep value of <0.05. 5 measurements are done to check the performance with results. 0.048, 0.062, 0.055, 0.054, 0.06. is guaranteed satisfied? χ = 0.0558, S = 0.0055 n = 5 degree of freedom = 4 We are interested in finding out whether χ µ> o , so this is a single tailed test.

36.2

50055.0

05.00558.0=

−=

−=

ns

t oµχ

From tables t0.95,4 = 2.13. Hence the difference is significant with 95% confidence. 1.8.2.4 Student t – Test for two sample means. Tests the significance of the difference between χ 1 and χ 2 for sample sizes n1 and n2 with standard deviations S1 and S2 respectively. The null hypothesis assumed

21

21

21

21

2,11210

1111

;,;

nn

S

nn

St

HH

PP

+

−=

+

−=

≠=

χχχχ

χχχχ

Sp - pooled variance

( ) ( )( ) ( )

Spn S n S

n n=

− + −

− + −

1 12

2 22

1 2

1 1

1 1

Example Two assay procedures for Cu concentrates are given OLD NEW χ 25.169 25.310 S 0.316 0.280

17

n 10 10 Do the two assay procedures give same results? 2 tailed test is used degree of freedom = (n1 – 1) + (n2 – 1)

(10 – 1) + (10 – 1) = 18.

Since n = n1 = n2

SS S

S

t

=+

=+

= =

=−

= −

+

12

22 2 2

20 316 0 28

2

0 089128 0 299

25169 253100 299

105

110

110

. .

. .

. ..

.

From the table t0.80, 18 = 0.862 from single tailed tables. Double sided confidence limit is α= 60%

( )( ) ( )

β α

α

= − −

= − − = − −

100 100

80 100 100 100 100 602

2

/ %

Example 2 - Significant with 60% confidence. Hence insignificant two impellers give the following measurements. Impeller 1 Impeller 2 χ 3.6 2.13 S 0.62 0.44 n 4 7 Does impeller 1 give a significantly higher line velocity? n1 = 4 n2 = 7 degree of freedom = (4 – 1) + (7 – 1) = 9

18

( ) ( )

( ) ( )

71

4111

56.3507.013.226.3

507.09

44.0662.03

211

21

21

22

21

222

211

++

=−

=−

=

=+

=

−+

−+−=

nn

St

S

nnSnSnS

χχ

t0.995, 9 = 3.25 from single tailed tables. This shows that impeller 1 give a significantly higher line velocity. 1.8.2.5. Student t - Test for paired comparison. Null hypothesis is that each pair of results is equal Ho; X1,i = X2,i each difference di = 0 For n pairs

( )∑= −

−=

=

n

i

d

nddisdwhere

nSdt

1

2

1

Example: Two identical flotation circuits are fed with usual collector A and few collector B. Experiments are run for 2, 6 day weeks and recoveries are monitored daily. Does the new collector give a significantly improved recovery? OLD NEW Day A B di di 1 84.2 86.2 1.6 2.0 2 71.9 75.6 3.1 3.7 3 77.8 76.6 -1.2 -1.2 4 80.1 82.6 2.4 2.5 5 75.6 77.8 2.1 2.2 6 70.8 70.0 -0.8 -0.8 7 81.3 85.2 3.7 3.9 8 82.6 80.6 -2.0 -2.0 9 78.5 77.1 -1.4 -1.4

19

10 79.2 81.7 2.3 2.5 11 80.6 82.4 1.6 1.8 12 75.2 75.4 -0.8 0.2 d = 117. The test is single sided since we are interested in improving the recovery.

( )

54.1

1298.188.0

98.11

11deg1298.188.0

1

2

==

=−

−=

=

==

==

∑=

t

nddsd

nSdt

freedomofreesnsdd

n

i

i

d

From the tables t0.90, 11 = 1.363 Hence t is significant with only 90% confidence in such marginal cases if A is unattractive (ie inefficient, costlier etc) Decision would be to continue testing and see whether significance would improve. 1.9 THE CHI-SQUARED TEST (χ2). Data can often be grouped in K(pure numbers) Mutually exclusive classed. χ2 - test determines whether observed frequencies in each class are significantly different from the expected if some hypothesis were true.

( )∑=

−=

k

n i

ii

EEO

1

2

2χ

with k - 1 degree of freedom Oi - observed frequency Ei - Expected frequency Example - 3 makes of pumps are installed on 3 identical hydrocyclones circuits and their frequencies of break downs monitored to determine any differences in reliability. After 3 months the results are as follows: Pump 1 2 breakdowns Pump 2 6 breakdowns Pump 3 7 breakdowns

20

Total 15 breakdowns Assuming the reliability of pumps is identical. The expected frequency of breakdowns are 15 = 5 breakdown per pump 3

( ) ( ) ( )χ2

2 2 22 55

6 55

7 55

28

=−

+−

+−

= .

For 2 degree of freedom the tables gives

0 75 2

22 77

. ,.χ =

Hence the observed difference are only 75% significant (insignificant). However, over a longer period the frequencies will increase with the same degree of freedom and allow for a better decision. Hence by continuing testing for 3 more months. If relative rate of breakdowns is constant χ2 will double. χ2 = 5.6 for two degrees of freedom is significant with nearly 95% confidence.

10. F – TEST. This is used widely to compare different estimates of variance S1

2 and S22. Ideally

S12/S2

2 should be close to unit. When S12 > S2

2 The distribution

F = SS

is the F distribution12

22 −

As before if S1

2/S22 > F V V0 05 1 2. , , the observed difference are significant (at 5%

confidence limits Where 2

11 deg SforfreedomofreeV =

11

deg

22

11

222

−=

−=

=

nVnV

SforfreedomofreeV

Example: Two alternative crusher feeder arrangements are tested are tested to see which one gives more consistent feed rate. Tonnes treated in several hour shift are Feeder 1 126 138 101 112 146 129 118 122 128 Feeder 2 136 129 120 138 131 126 131 Which feeder is more consistent?

21

Χ

Χ

1 12

2 22

1 2

0 05 8 6

124 44 179 53

13014 36 48

179 5336 48

4 92

8 6

415

= =

= =

= =

= =

=

. .

. .

..

.

.. , ,

S

S

F

V V

F

1.11 Sampling: Gy's Theory. The formula given in this paper expresses the variance to the expected in the assay value when a given weight of material is taken as a sample from bulk quantity. Assumptions

1) Normal distribution It is assumed that if a large number of similar samples were taken from the bulk, then the frequency distribution of the assays would follow a normal curve.

2) Equal chance of acceptance It is assumed that every separate particle and type of component has an equal chance of being selected.

3) Freedom from bias It is assumed that the sampling techniques and measurements are free from bias or systematic errors Notice that in the case of gold, diamond and some sedimentary deposits, the assay frequency distribution is likely to depart from normal and use should be made of the central limit theorem to take samples in such a way that the grouped mean grade will follow a normal distribution. Basic formula

3'

2 11 fgmldppa ⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠

⎞⎜⎝

⎛σ

fglmwhereCCdppa

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠

⎞⎜⎝

⎛ 3'

2 11σ

where

22

a = mean metal content of the lot (%) 2σ = variance of the grade distribution

p' = weight of the sample (kg) p = weight of the lot to be tested (kg) f = shape factor g = size factor m = mineralogical index (kg/m3 )

d = size of the largest piece in the lot. That is the sieve aperture which retains between 5 to 10% of the original lot.(m)

l = liberation index Determination of C = fglm

(a) f shape factor let d1 = the length of the side of the square mesh through which the particle will always pass ρ = particle density

then ρ13diclemassofpartf =

f varies from 0 to 1. For plate like minerals, f = 0.1 (mica, scheelite), but in most cases, f lies between 0.3 and 0.7. The error is generally small if 0.5 is taken. (b) g the size factor.

Dj = a size parameter of a size range Wj = the fraction of the total in size range j ( wt) D = maximum particle size

Then ∑=

n

jj

j WDD

1

The following values can be used in practice Unclassified material g is between 0.1 and 0.5 Scalped feed(Oversize removed) g is between 0.15 and 0.5 Deslimed material g is between 0.2 and 0.5 Deslimed and classified g is between 0.3 and 0.8 Most ordinary cases are covered by taking g = 0.25 © l the liberation index let a = the approximate grade of the lot being sampled (% metal) a' = the assay of the richest piece in the largest size fraction (% metal) α = the metal assay of the pure mineral being sought (% metal)

23

then aaal

−

−=α' =

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−

α

ααa

aa

1

'

the average value of l is as follows Type of material Completely homogeneous 0 since a' = a Completely heterogeneous 1 since a' = α Very heterogeneous 0.8 Mixed particles 0.2 Moderately disseminated 0.1 Finely disseminated 0.05 Run of mine 0.1 – 0.4 Reduced to liberation mesh 0.8 a’ ≈ a Concentrates 0.2 high grade 0.4 low grade Tailings 0.1 0.05 Finely disseminated ---------------------- m the mineralogical index let ρv = density of valuable component ρg = density of gangue

αa = Ratio of average grade to assay of pure mineral

then m = gva

a

a

ρα

ρ

α

α ⎟⎠

⎞⎜⎝

⎛ −+

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −1

12

When The values of m vary from 30 for a 90% concentrate to about 107 for a 0.1% tailing. Thus m is very much more important than f, g or l and usually, approximations in these latter quantities, do not affect the result as do the values of m. Approximation for a large lot. as P becomes larger !

! → o

Thus for a large lot

24

1.11.1 Successive Sampling for assay.

The value of 2

⎟⎠

⎞⎜⎝

⎛aσ is somewhat more complex when successive

Sampling is required. It is seldom possible to do assay tests on the whole of a primary sample and thus it is necessary to reduce the amount. Rearranging Gy’s equation we have

p cd a' =3 2

2σ

P1 sample weight is influenced by the value of d3 – size of the largest particle in the population. Thus the size or amount of sample is very dependent on the size of the largest pieces. If the primary sample is crushed, even without varying the value of C significantly, the size requirement can be very much reduced. Before the amount of reduction can be satisfactorily determined we must know how to combine the variances of successive sampling procedures. If there are a number of distributions with variance σ j

2

Then σ σ σ σ2 222

32overall etcI= + +

Apart from special considerations then it is usually preferable to choose each sampled portion to give equal variance. 1.11.2 PRACTICAL SAMPLING SYSTEM

The material to be tested may be in one of several different forms and the methods of taking the samples will be adapted to the characteristics of the form of the material and to the purpose for which the test is undertaken.

The following classifications are relevant:

1. Pure liquids 2. Suspensions and slurries 3. Wet Solids 4. Dry solids Each of these forms may be flowing or static. 1.11.2.1 Flowing Materials 1.11.2.1.2 Pure Liquids

In general it may be assumed that the composition of a pure liquid (including solutions and mixtures are miscible liquids, but not mixtures of immiscible liquids), remains constant over the cross section of the flowing form. Thus it is reasonably justifiable to take as a sample a fraction of the cross section. This can be achieved by a take off pipe in the flow as shown in the sketch. This must be so arranged that it does

25

not cause stagnant areas to occur in the vicinity of the take off as this might tend to mask changes in composition with time. The amount of sample taken, i.e. the sampling fraction, is often quite large (1/5 – 1/20) and the bulk is reduced by taking a second cut and returning the reject to the main flow. This in general gives a faster response and probably a better distribution.

Notice that this is a volume split and if the density varies the weight taken will not be constant in equal increments of time. This is true generally and most samples are taken as increments of constant (or nearly constant) volume not as increments of constant weight. 1.11.2.1.3 Slurries, Wet and Dry Solids In these cases we cannot assume that the distribution of the variable we are examining (assay, size distribution density etc) remains uniform across a cross section. It is very unlikely that some segregation will not take place and thus the whole of the cross section must contribute to a sample. Assuming that variations in the direction of flow are random and not cyclic, the most reliable method of cutting samples from a flowing stream is to move an aperture of fixed area across the material at a constant rate. The constant rate ensures that each element of the stream of a given thickness is collected in equal increments of time and thus represents equal increments of volume. Thus, on a volume basis, the principle of equi probability is followed. That is, each volume element of the material has an equal chance of being collected. Again, in order to avoid large variations in the sample it is often advisable to take a fairly large primary sample and reduce its value by subsequent sampling when the bulk will be moving more slowly and large apertures can be used as before. There are a number of different types of machines for taking samples automatically and they have the advantage of being consistent.

1.12 SAMPLERS.

1.12.1 Linear Samplers. The aperture of the cutter is adjusted to be such that the largest members of the feed can easily pass the jaws without piling up. The rate of traverse is then adjusted to give the right amount of sample. However, it does seem that larger particles could be rejected more often than smaller ones when the aperture is relatively narrow and this could give rise to sampling errors. Cutters are available, for most types of material wet or dry, from the various manufacturers and the literature should be studies before a particular method is chosen.

26

1.12.2 Rotary Samplers.

1.12.2.1 Vezin Type. This sampler is used mainly for slurries and wet solids. In this case the width of the sample cutter is about the same width as the feed pipe and thus as it rotates the cutter takes effectively a complete slice from the stream. The frequency of cut should be chosen to give the required sampling rate.

1.12.2.2 Synder Type.

This method uses a horizontally driven cone in the sides of which are suitable apertures. It is not very suitable for materials with wide size distributions or very variable density.

1.12.2.3 Conveyor Sampling.

It is usually most convenient to sample conveyor borne material at the point of input to or output from the conveyor, but where this is not possible a complete section may be pushed off the belt from time to time by a moving cross blade. This is not very suitable for troughed conveyors and has limited application in the mineral industry.

27

1.12.2.4.1 The Jones Riffle. This sampler can be used for fixed dumps by manual feeding or for flowing materials . The stream is divided into two parts by interleaved sections. It is widely used but care is given cut and thus the principle of equi probability is not completely satisfied. Any lateral stratification of the stream will introduce bias into the sampling system.

1.13 General Comments on Flowing Materials.

When systematic sampling systems are used it is most important to test the stream for cyclic variations which have the same frequency as the sampler. For example, if the rate of revolution of a Vezin sampler is a simple sub multiple of the frequency of a Vibratory feeder, it is quite possible that only belt inequality puts the load to one side of a conveyor in a cyclic runner, the rate of a linear sampler must be chosen so that the two frequencies do not coincide.

1.13.1 Size of Sample in Flowing Materials

When the assay or test procedure requires a static sample, the rate of sampling is governed by the test sample and the total amount of sample as predicted by the appropriate formula. The variability of the material must therefore be known in order to avoid taking more sample than necessary. In continuous measurements however, an effective volume must be determined for the test machine. In the case of an X-ray analyser for instance the effective sample volume is the amount that passes through the cell while the machine is taking one reading. As this volume may be much greater than the cell volume, an extra error is introduced due to the variation of the stream value while the reading is being taken.

1.13.2 Frequency of Cut. It may in some cases, be possible to take samples too frequently when a fixed volume is the limiting factor. Thus although a large number of samples are taken their distribution will tend to be that of the parent distribution and not that of the means. The standard distribution, and therefore the sampling error, will be larger than that of a sample taken over a longer period.

1.13.3 Fixed or Static Material.

Especially where ore bodies or large dumps are concerned Mahomet must go to the mountain. A great deal of work has been done on taking samples from mineral deposits of various kind s and the literature should be consulted when particular examples are needed. In most cases a grid pattern of drilling is employed but particular attention should be paid to the characteristics of the drill and to the type of ground being worked. Where the bore is flushed with water, segregation can easily occur

28

and if the ground is soft or crumbly or wet, material from one level can contaminate lower sections. Dumps present special problems because, due to the nature of the method of formation, segregation will almost certainly have taken place. Large pieces will roll down to the base and fines may be washed down or away. Chemical action takes place in dumps of certain types and this leads to variation in composition throughout the mass. These considerations suggest that all parts of the dump should be tested; although this is sometimes expensive it is, in general, desirable for accurate results. When the size of samples has been found by, for example, using Gy’s method, it is advisable to assemble the total quantity by collecting a number of samples from all parts of the bulk being tested. In order to ensure that each particle in a sample has an equal chance of being selected, it is necessary to mix the material very thoroughly and to take cuts in such a manner that the random principle is preserved. Rolling thoroughly mixes the particles but cannot prevent segregation. The material is therefore formed into a cone which has circular symmetry and selection taken from opposite quarters or alternatively, smaller amounts may be removed from the periphery at equal angles. This is supplemented by a sample from the centre.

1.14 DESIGNING EXPERIMENTS. In the laboratory, pilot-plant or industry, engineers and researchers seek relationship between inputs and outputs, which lead to help to run the process better and in more efficient way. Whilst in the design section, engineer’s analyses performances, cost and safety data to help them make decisions about new plant and operations, in the administration offices, they are looking at trends in supply and demand prices and cost of running the business Designing industrial experiments require a good understanding of environmental constraints use is made of the factorial design approach to determine the number of minimum trials to be undertaken for a given set of parameters. An industrial environment offers a considerable variation in many conditions of industrial experiments: raw materials varying, operators varying from shift to shift, undesirable peaks of energy inputs, machinery performance variations throughout the work shift etc.

29

UNIT 2. 2. PARTICLE SIZE REDUCTION (COMMINUTION) 2.1 Introduction. In this Unit we will review the theories of particle breakage according to Von Rittinger’s, Kick’s and Bond’s theories. Mechanics of particle breakage, microprocesses of comminution. Comminution efficiency. Breakage function and breakage as a rate process. Introduction to mathematical modelling of comminution process. We will touch on the latest developments in comminution machines and processes, autogenous and semi-autogenous grinding mills, impactors, vibrating mills, jet mills, thermal breakage and use of microwave technology in grinding. 2.2 Objectives

By the end of this unit you should be able to;

1. Define the terms used in particle breakage and how this relates to various theories of breakage as applied in mineral processing techniques.

2. Introduce breakage functions and mathematical modelling of comminution

2.3 Reflection

Have you ever thought about the importance of size reduction in mineral processing? Well, whether you have ever considered it or not, the is what is very cardinal in liberation of the minerals of interest from the gangue material. Did you think `mineral processing is just about the processing of a ore to make or obtain the valuable metal for sale. This forms the core to developing new process techniques or routes which are more viable and economical.

2.4 Mineral Comminution. Distribution of sizes in breakage products, particle –size distribution functions. Purpose of particle size reduction.

1. To produce material of a size which is:

a) convenient for transportation and storage b) suitable for use with no further treatment

2. Liberation of mineral values 3. Exposure of values to chemical attack 4. Production of a given surface area

30

The mechanism which cause most rock breakage are those of nature phenomena such as action of water through weathering and wind but in most cases these are so slow that to speed these processes, intensive energy application may be required. Because large energy is required to achieve substantial breakage, the efficiency of comminution is important because its cost will be one of the factors determining whether low grade mineralisation constitute an orebody. 2.2 VARIOUS TERMINOLOGY USED IN PARTICLE SIZE REDUCTION Reduction Ratio = Size before Comminution = R Size after Comminution This expression is meaningless in practice as a wide range of sizes and shapes. More useful to specify the top size. In practice we define a more precise reduction ratio as Limiting Reduction Ratio R2 = Sieve Size Passing 100% feed before Comminution Sieve Size Passing 100% of Product In actual sense we use 80% passing ratio, R80 because it’s more practical than 100% passing.

R80 = Sieve Size Pas g of feedSieve Size Pas g of oduct

sinsin Pr

80%80%

D80 for both for both the particle size of the feed and product can be determined from plots of size analysis. Mean Reduction Ratio: Rm = Average Particle of feed Average Particle of Product This is the best expression for Reduction ratio but the average particle size is difficult to measure. The specific surfaces of feed and product can be used to measure the size

RS

SSurface area of feed unit VolumeSurface area of oduct unit Volumes

v f

v p

= =/

Pr /

The surface areas can be obtained by permeability and absorption methods. This is unsuitable for crushing products. Apparent Reduction Ration,

RUpper size it for receiving feed GapeEffective setting of the disch e opening setA =

lim ( )arg ( )

This is commonly used for crushing equipment and specific maximum possible size reduction in a pass. Thus one need to know what is meant by reduction ratio. In describing broken material the 80% passing ratio is used commonly. For crushing equipment RA is commonly used.

31

2.3 PROPERTIES OF MATERIALS Solids are broken by changing their shapes beyond certain limit. By testing specimen to known forces, resistance to fracture can be measured and the results when plotted on stress – strain diagram forms the basis for classification material. 2.4.0 CLASSIFICATION OF MATERIALS HARDNESS - The material has high stress limit. Hardness is measured in several ways: a) Mohr’s scale - resistance to scratchability b) Vicker’s scale - resistance to penetration of a given object of known dimensions BRITTLENESS Ease of breakage by impact DUCTILITY - Ability of a material to flow under continous stress e.g. metallic Bonds are weak TOUGHNESS- Ability of material to resist crack propagation due to stress Relieving capacity by increasing the radius of curvature of Crack tip. This relieves stress concentration. FRAIBILITY - Ease with which the material may be broken by abrasion. 2.5.0 BEHAVIOUR OF ROCK UNDER STRESS This can be investigated by compression tests The rock behaves differently from metals except at very small sizes when limit of grindability is reached, the rock behaves as a brittle material Both ε σand failure are much higher for rentals. Energy absorbed and stored in unit volume is area under the curve.

Stained Energy = σ εxForce x deformationArea x length

=

= Work Volume

2.6.0 NATURE (MECHANISM OF BREAKING For a particle to fracture, we have to apply a stress higher than the fracture strength. The way by which a particle breaks depends on: a) Nature of particle b) The manner in which a particle is subjected to the applied force (rate of

application) c) Type of force (Shear, compression etc). 2.6.1 ABRASION - Insufficient force to significantly break the particle as a whole. Localised stressing results in a distribution of fine particles.

32

2.6.2 CLEAVAGE - Occurs under conditions of slow compression (“Real Crushing”). Note that compression in itself does not fracture the rock. Fracture is a result of local tensile forces which result

2.6.3 SHATTER (IMPACT) – Occurs when the energy applied is far in excess of

that required for fracture. A large number of particles usually result with a wide size distribution diagrammatically.

Usually all three are present Roll Crushers - mainly compression Hammer mills - Impact Jaw and Gyratory Crushers - more compression than impact Cone Crushers - More impact compression Grinding Mills - Impact and abrasion Feed rate (Choke fed) More compression than abrasion 2.7.0 MECHANICS OF FACTURE The resulting product of fracture is a distribution of the particle size. However, each particle break as a result of stresses applied to it alone. The theory of fracture can be treated in terms of single particle fracture. GRIFFITHS (1920) who investigated the mechanics of fracture of brittle particles looked at the effects of cracks on strength of particles. Weakness in the particles are due to minute discontinuities (Griffiths Cracks); Stresses concentration at cracks, in particular at crack tips.

σT Tensile stress perpendicular to the major axis in maternal remote from crack−

L - Crack length r - radius of curvature of crack tip

It showed that at crack tip, ( )21

2 rLtoincreasedis TT σσσ = . The crack propagate if

( )21

2 rL

Tσ is greater than molecular bond strength or crack propagate if strain energy

is sufficient for creation of new surfaces. At the point of Crack growth

( )σσ

εTErL

E N m=⎛

⎝⎜

⎞

⎠⎟ =

212

2

Π

E = Young’s Modulus r = Surface energy/unit Area This is the Griffiths Criterion. The most important idea from Griffith’s work is that cracks multiply an applied stress to overcome local molecular cohesion. Although little work is required to bring about actual fracture energy must be supplied to propagate the cracker.

33

2.8.0 ENERGY CONSUMED IN COMMINUTION The supplied energy is absorbed partly by the material (ie surface energy) partly by the equipment (drive) and also surroundings (noise). Qualification of these energies is difficult. Several attempts have been made to determine minimum energy requirements for estimation of comminution efficiency. In most cases commercial machines have efficiencies of < 10%, sometimes even <1%. Hence comminution is a highly energy intensive process. Energetically, the rate of application of the force is also important. For instance, a small force maintained overlong period (slow compression) is more efficient than impact breaking. 2.8.1.0 ENERGY SIZE REDUCTION RELATIONSHIPS These relationships are useful in Equipment design. They can be summarised by the so-called comminution “Laws” – These are mostly empirical. 2.8.1.1 Von Rittinger’s Hypothesis This hypothesis is based on the surface theory and it states that the Energy required for the size reduction is directly proportional to the new surfaces created.

E SdR vp

α α1

Assuming we have a spherical particle with diameter dp Specific Surface = Area Unit volume of mass

= Π

Πp

p

dd

2

343 8

= Sr = 6d p

This shows that the energy required to achieve same reduction for a given quantity of

solids varies inversely proportional to the particle size. Sdvp

α1⎛

⎝⎜⎜

⎞

⎠⎟⎟ . Small particles

require more energy for the reduction mathematically this can be represented as

( )12 VVR SSKE −= or ⎥⎦

⎤⎢⎣

⎡−=

12

11XX

CE R which translates in

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

1

1XRCE R

34

E - energy input/unit volume or weight for the reduction in size from X X1 2→ X1, X2 = Initial and final particle size respectively Sv1, Sv2 = Initial and final specific surfaces of the particles R = Reduction Ratio KR, CR = are constants 2.8.1.2 KICK’S HYPOTHESIS This hypothesis is based on the volume theory. Kicks proposed that the equivalent geometrical change in size of particles require equal energy ie Energy is proportional to the volume of the particles.

E = Kk log XX1

2

= Kk log R

For the same reduction, E is a constant comparison of Rittinger’s and Kick’s hypothesis can be shown by way of an example. Consider stage size reduction 1000 100; 100 10; 10 1 mm (1) (2) (3) According to Rittinger’s hypothesis - Stage (3) requires 100 times the energy as is required for stage (1). But Kick’s maintained that the energies for all the 3 stages are the same: Rittinger's

⎥⎦

⎤⎢⎣

⎡−=

21

11XX

KE R

Stage 1 for 1000 100 ⎥⎦

⎤⎢⎣

⎡ −=1001

10001

1 RKE 9 x 10-3 KR

Stage 3 for 10 1 ⎥⎦

⎤⎢⎣

⎡ −=11

101

2 RKE 9 x 10-1 KR

100100.9100.9

3

1

1

2 ==−

−

xx

EE

Hence E2 = 100 E1 which contradicts kick’s hypothesis. 2.8.1.3 BOND’S HYPOTHESIS Using this hypothesis Bond expanded Griffith’s model and suggested that the total work (includes previous work) done on the particle must be considered. Absorption of applied stress and production of new surfaces can be volume dependent. Hence comminution is concerned with both volume and surfaces.

35

The overall formulation was that the total useful work input which has been applied of broken material is inversely proportional to the square root of the product particle. Mathematically this can be expressed as

pKE B

B =

EB = WB = Total Work required to reduce a particle from infinite size to produce size P. Bond’s theory is completely empirical. WALKER - Proposed that all these 3 theories can be derived from one general equation of the kinetic form

dE CdXxn= −

Where E the energy per unit weight of material X - particle size n - order of process and is a positive number If n = 2

sRittingerEXX

CXdxCdE R

E X

XR '11

0 122

2

1

==⎥⎦

⎤⎢⎣

⎡−=−=∫ ∫

for n = 1

sKickRKXXCE

XXCdE KKK

E X

X

'loglnln2

1

0 1

22

1

====−=∫ ∫

if n = 1.5

⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−==−=∫ ∫

12120 23

111122

1XX

KXX

CEX

dxCdE BBB

E X

X

if X 0

sBondX

KE BB '1

2

=

HUKKI (1961) plotted all the 3 equations on a composite curve. From the diagram it is apparent that Rittinger’s equation holds for fine grinding Kick’s relation is valid for course crushing (> 10 cm). For intermediate range (100 mm – 10 cm) which is normal comminution range, Bond’s equation can be used as an approximate. The integrated form of walker’s equation is

⎥⎦

⎤⎢⎣

⎡−

−=

−− 11

12

111 nn XXn

CE

36

Holmes, Charles and others have used a size distribution in energy – size reduction equation. The general form of equation

( ) ndKE −= *

d* = size modules in Gaudin – Schulmann equation K - Constant

37

UNIT 3 3 PARTICLE SIZE ANALYSIS 3.1 INTRODUCTION. In this Unit we are going to learn about particle size, shape and distribution, particle-size distribution functions. Interaction of solid particles and fluids in gravitational and centrifugal fields. Methods of sub-sieve size analysis, sedimentation methods, elutriation methods, cyclosizer, coulter counter, pore-size and volume determination, permeametry, laser light scattering method. On-stream particle-size analysis 3.2 Objectives

By the end of this unit you should be able to; 1. Define the terms used in particle size characterisation for better

liberation 2. Define various equivalent diameters 3. Various methods used in sub sieve size analysis

3.3 Reflection

Have you ever thought about the importance of particle size analysis in mineral processing and pharmaceutical applications? Well, the particle size is what determines the degree of liberation after comminution. This is particularly important for reactions involving surface phenomena and more especially in briquetting pills or capsules in the pharmaceutical industries

3.4 Particle size analysis The evaluation of particle characteristics is a critical component of mineral processing as it is used to estimate efficiencies of comminution equipment or processes of separation. More important, however, is the fact that characterisation of particles allows for effective control of unit processes. In size reduction for example, the aim is to liberate the mineral values from the gangue. However, over or under liberation results in problems of separation so that it is important to achieve the correct degree of particle size reduction. In reality, however, particle size is not easy to measure because typical products of comminution have a wide range of characteristics and consideration has to be given to

a) the size of individual particles b) the average size of all the particles c) the shape of the particles d) the size distribution of particulate systems e) the density and surface characteristics of the particles.

These and other aspects of particle size analysis are considered in the following notes.

38

3.5 TERMINOLOGIES. Particle: Discrete portion of matter, small in relation to surroundings. The

absolute size is not important nor is the physical state i.e. not necessarily solid.

Particulate Material: This consists of a number of particles which retain their individuality i.e. they do not adhere together. Powder: Consists of discrete particles of material with a maximum size of about

100µm. Particulate System: Consists of a particulate material dispersed in a continuous

phase of another material. Sizing: Refers to the division of a particulate system into groups of nearly the

same size. Classification: Operation in which differences in density and size of particles are

exploited to obtain grading by settling through a fluid. The fluid is normally water or air.

Sedimentation: The process of settling of particles in fluid by (usually) gravity. Elutriation: Washing away of finer particles from a particulate system by rising

currents of fluid. 3.4 PARTICLE SIZE 3.4.1 Equivalent Diameters

Only regular geometric shapes can have their size conveniently described because it is difficult to define any linear size without reference to the particle shape. As the shape becomes more complex the problems of defining the particle size increase e.g. Sphere: only 1 dimension will uniquely define size For a cylinder: 2 dimensions will uniquely define size For a cuboid: 3 dimensions will uniquely define size Broken material, on the other hand have irregular shapes and their sizes are therefore difficult to quantify. It is convenient to express the size of such particles in terms of the diameter of a sphere which is equivalent to the particle with regard to some stated property. The advantage is that the size of the particle can be stated by a single number.

a) Projected area diameter (da): Diameter of the sphere having the same projected area as the particle when viewed in a direction perpendicular to the plane of greatest stability.

39

21

4⎟⎟⎠

⎞⎜⎜⎝

⎛=

πp

a

Ad , where Ap is the projected area of the particle

b) Volume diameter (dv): Diameter of a sphere of the same volume as the particle.

( )31

31

66 ⎟⎠

⎞⎜⎝

⎛==π

πVVdv

where V is the volume of the particle

c) Surface diameter (ds): Diameter of a sphere having the same surface as the

particle.

( )21

21

⎟⎠

⎞⎜⎝

⎛==π

πSSds

where S is the surface area of the particle

d) Free falling diameter (df): Diameter of a sphere having the same density and free-fall velocity as the particle in the same fluid medium.

e) Stokes’ diameter (dst): The free-falling diameter in the laminar flow regime

dst = ( )g

V

ls

t

ρρη

−

18

where Vt is the terminal velocity of a particle falling in fluid with a viscocity of η. ρs and ρl

are the densities of the solid and the liquid and g is the gravitational pull.

f) Sieve diameter (dA): Diameter corresponding to a square aperture of side A through

which the particle just passes. It can be seen that various methods measure different diameters. It is therefore vital

whenever mentioning particle size to define the nominal diameter used. Microscopic measurements give da. Sedimentation and elutriation methods measure df or dst. 3.5 Presentation of Sizing Data

3.5.1 Tabular Form The most precise method of data presentation is the tabular form. Results can be tabulated in the form in which they are obtained e.g. screening results as a weight frequency distribution. Screen Size, um Weight % + 212 0.4 -212 + 150 1.0 -150 + 106 6.8 -106 + 75 11.6 -75 80.2 100

40

The frequency figures can be converted to a cumulative distribution, i.e. Particle size, um % by wt. > particle % by wt < particle Size size 212 0.4 99.6 150 1.4 98.6 106 8.2 91.8 75 19.8 80.2 3.5.2 Graphical Presentation These methods have several advantages over tables.

a) Give a quick visual picture b) Several size distribution data can be given in one graph. This facilitates comparisons c) Yield numerical values describing distribution e.g. median, mode etc d) Enable the fitting and determination of an equation describing the distribution and

hence the parameters of the distribution. Conventionally the particle size is plotted as the abscissa and the measured quantity as the ordinate. Histograms: This is the simplest method of the graphical techniques. Fig. 1 shows a histogram of weight % retained against screen size dA. Histograms give a useful visual picture. Size Frequency Curves: Provided the size intervals are small the histograms can be presented as a continuous curve drawn through the mid points of the ranges. This is shown in Fig. 2. Area under the curve approximates the total weight of the particles. The mode and the distribution of size (i.e. dispersion, skewedness) are readily distinguished in the continuously plotted area. Cumulative Plots: These are useful for determining the amount of material greater or smaller than a specified size. The median and also the % between any two sizes are easily read off. The difference between the two curves gives the size frequency plot. The main disadvantage, however, is that the sigmoidal shape (see Fig 3) is difficult to express mathematically. Furthermore, the ends of the vertical scale have insufficient spread. It is sometimes useful to plot size frequency and cumulative curves on linear scales to have an intuitive picture of the spread of the material over the entire size range. This, however, compresses the size scale at the fine end. Hence a logarithmic size scale is more often used to spread the data along the abscissa. 3.5.3 Particle Size Distribution Functions The above methods of presenting particle size distribution data are normally adequate for quantitative description of sizing data. However, for analytical reasons it is desirable that the size distribution data conform to some mathematical distribution law to enable, for example, extrapolation to a different size range.

41

Several attempts have been made to arrive at such mathematical functions. Although it would appear that some of them can be mathematically derived, they are all based on empirical relationships which over the years have been modified to improve their description of comminution products. The equations are al of the general form

⎟⎠

⎞⎜⎝

⎛=*ddW f nf

where Wf = cumulative fraction or % finer than d d* = a reference size. d* is referred to as the size modulus. The equation also include an exponent to indicate the dispersion of the data and this is appropriately called the distribution modulus. Additional parameters have been suggested to account for skewness of the size distribution but this introduces additional complexity which is hardly justifiable. The most commonly used functions in mineral processing are the Gaudin –Schuhmann and Rosin-Rammler functions. 3.5.3.1 Gaudin-Schuhmann (G.S.) function. This was introduced by Professor Gaudin (1926) and modified by Professor Schuhmann (1940). The function is defined by

mf

m

f kdW

kdW ⎟

⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛=100

,100

k is the size modulus and defines the maximum size for which the function is valid. For d = k, wf = 100% m indicates the dispersion in the range in which the function is valid. For a particular sample, m and k will be constants. Taking logs of the G.S. equation Log (wf/100) = m log d – m log k If the function therefore fits the data a straight line will result if log Wf is plotted against log d. This is useful because it allows for extrapolation of the line to cover particle size ranges outside those used to establish the graph (e.g. into the sub-sieve range) – However, actual data often deviates from a straight line particularly at the fine particle size range as shown in Fig. 4 and at very coarse sizes. Nevertheless the distribution is widely used in mineral processing because of its simplicity. The weight frequency curve can easily be obtained as the derivative of the G.S. function, but only for values of m > 1, for m < 1, d 0 as dWf/dp increases. 3.5.3.2 Rosin Rammler (R-R) Function It was introduced by Rosin in 1933 and can be defined in terms of cumulative weight % retained (or finer), Wr

42

n

dd

roew⎟⎟⎠

⎞⎜⎜⎝

⎛−

=100 do = absolute size constant or characteristic particle size. n = dispersion coefficient or distribution modulus Wr = 100 - Wf If d = do Wr = 100/e = 36.8% Hence do is the size at which 36.8% of the particles is retained. Taking logs twice,

In n

r

n

r

dd

Wnor

ddW

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟

⎠

⎞⎜⎝

⎛

00

1001100

1n (100/Wr) = n logd - nlog do Hence if the data fits the R-R equation a plot of log 1n (100) Wr against logd will give a straight line of slope n. If n is small the particles are spread over a wide size range but as all the particles approach the same size, n tends to infinity. 3.5.3.2 Comparison of the G-S and R-R functions

Both the G-S and R-R functions are only suitable for cumulative weight data. The R-R function, however, expands both ends of the vertical scale unlike the G-S function. However, its widespread use appears to be limited by the scarcity of suitable graph paper. In practice it is not easy to state with certainty which function suits the data best. At fine particle sizes the G-S and R-R functions became similar. From the G-S equation

−−−⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛−⎟⎠

⎞⎜⎝

⎛−=

⎟⎠

⎞⎜⎝

⎛ −=

⎟⎠

⎞⎜⎝

⎛−=⎟⎠

⎞⎜⎝

⎛=

mmm

mr

mr

mf

kd

kd

kd

kdWnHence

kdWor

kdW

3

31

2

21

1ln100

1

1100

,100

Neglecting the higher order terms of the expansion will give an equation similar to the R-R equation. In practice it may also be advantageous to use d80 rather than d100 in place of k in the G-S function. This is because of the deviation of the equation at coarse sizes.

43

UNIT 4 4 DETERMINATION OF PARTICLE CHARACTERISTICS. 4.1 Introduction In this unit some of the methods used to measure particle characteristics are reviewed. The particle characteristics considered are particle size and surface area. Particle size distribution functions are introduced to enable the extrapolation to sub sieve size to characterise the size distribution of particle where it is not possible measure the size using mathematical modelling. 4.2 Objectives

By the end of this unit you should be able to; 1 Define the terms used in particle size characterisation for better liberation 2 Define various methods used in sub sieve size analysis

4.3 Reflection

Have you ever thought about the importance of particle size characteristics and the shapes analysis in mineral processing and pharmaceutical applications? Well, the particle characteristics and sizes are important in such areas as making pellets, tablets in pharmacy and packing things like metal final products as in the case of bars, sheets or granules and in many forms. These properties are also very important in such as classification and in the monitoring various useful properties in quality maintenance and control.

4.4 MEASUREMENT OF SIZE DISTRIBUTION 4.4.1 Methods depending on geometrical similarity

(i) Sieving (dA)

This is the simplest and most widely used sizing method. Woven wire sieves of square apertures are normally used down to about 45 um. Punched plates are used where larger apertures, of a few um upwards, are required whereas for smaller apertures where small tolerances (e.g. + 2 um) are required electroformed (micromesh) sieves are utilised. However, the latter are expensive and fragile and are not normally used for routine laboratory work.

Standard Meshes A Wide variety of meshes is currently in use. The main ones are:- U.S. sieve series American Tyler series British Standard series German Standard and French series. The International Standards Organisation has recommended and international standard series corresponding to the U.S. series. This series is based on 1mm sieve

44

with successive apertures following a (2)1/4 progression. Apertures greater than 1mm are expressed in mm, those smaller than this are in um. Wet Sieving: A common problem in sieving is the adherence of very fine particles to larger particles or to each other by electrostatic attraction or surface tension. These fine particles are more quickly eliminated by wet sieving. Wet sieving is also used for pigments and usually helps to disperse aggregates. “Endpoint” of a sieving test: Main aim is to ensure that a “near mesh” particle which can only pass through if favourably presented is given sufficient time to pass through. This is normally defined by the condition at which the rate of passage of the particles is reduced to a specific weight. One way of ensuring this is by shaking for five minutes and noting the amount passing in subsequent two minutes. If this amount > 0.2% of sample weight the five minutes state is repeated followed by another retest of two minutes. Weight of sample: Sieves must not be overloaded. Usually up to 100g is sufficient (for 8 inch screens) if more than 50% of sample is retained.

ii) Microscopic Methods (da) This is the most direct method of measuring particle size and has the advantage that qualitative information about shape can also be obtained. The main method used is usually the direct measurement technique in which the sample is directly examined under the microscope. A slide is usually prepared with the particles uniformly and randomly distributed in their stable orientation and are viewed in transmitted light. Because it is impractical to measure the size of every particle on the slide representative areas are chosen in a predetermined fashion. The actual number of particles need to be measured. With a typical size distribution of ground material more than 600 particles are usually measured. A conventional optical microscope with a standard graticule is normally adequate. The method is not suitable for particles < 0.8 um, and should preferably be restricted to particles less than 75 um. A big problem of microscopic particle size measurement is deciding what diameter to measure. Several diameters can be defined.

a) Martin’s diameter: M is the length of the line which bisects the image of the particle.

b) Feret’s diameter: F is the distance between two tangents on opposite sizes of the particle, parallel to some fixed direction.

c) Projected Area diameter: da is generally found to be the most satisfactory although originally M or F was measured. Hence da is the most often measured using manual or semi-automatic techniques. Measurement can easily be achieved by, for example, counting squares in a calibrated graticule.

45

In the semi-automatic methods settings of the microscope are made usually by the operator and the instrument records the results. Notable is the Zeiss-Ender microscope. They have the advantage that the operator can decide on the treatment of aggregates and distinguish foreign bodies e.g. human hair. The automatic methods on the other hand can count many more particles. They are usually based on a moving spot of light ( or slit), recording the uninterrupted light falling onto a photocell. Polished mounted and Thin sections are not usually used to determine particle size although they are frequently used to determine grain size and liberation sizes. The information obtained however has to be stereologically transformed in order to relate the apparent sizes measured to true sizes.

4.4.2 Methods Based on Hydrodynamic Similarity The methods are based on the settling rate of particles in a fluid. It is a characteristic of the methods therefore that they measure dst. In stoke’s equation, which is written as

( ) ( ) 2121

tan,,,18

2

thkVkd

systemgivenafortsconsaregandp

gdV

st

ls

ls

==∴

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

ρρη

ηρρ

Where h is the distance (m or cm) fallen in time t (s). The assumptions made in the derivation of Stoke’s law are:

a) laminar flow i.e. Re = 2.0<ηVd

b) the particle is smooth and spherical c) the particle is not influenced by both inter-particle or wall affects d) the terminal settling velocity has been reached.

Condition (a) is easy to fulfil because the critical dst diameter (i.e. largest diameter to which Stoke’s law would apply) can be calculated. Condition (c) is normally satisfied if the diameter of the vessel is 100d, and if the particle concentration is < 2% by volume. The general method consists of measuring the terminal settling velocity of the particle and calculating dst from the above relation. Particles must be properly dispersed. For particles > dst critical, sieving is normally used. It is possible, however, to use hydrodynamic methods by employing a more viscous fluid or using non-laminar flow equations. These, however, are both difficult alternatives. In practice hydrodynamic methods fall into two categories: Sedimentation: fluid is relatively motionless and particles fall thorough it by gravity or centrifugal forces.

46

Elutriation: grading of particles effected by upward moving current of fluid.

i) Sedimentation Methods These methods are non-fractionating except for decantation technique. They can be subdivided into two groups:

a) Cumulative Methods in which the total mass fraction sedimented from a definite height is measured as a function of time.

b) Incremental Methods in which the concentration of solids at a given level is periodically determined. Cumulative Methods The most popular of the methods is the sedimentation balance method. This consists of a pan suspended near the base of a sedimentation chamber with the pan connected to some device for recording mass gain. A disadvantage of the method is that it is prone to errors arising from the immersion of the pan in the suspension which varies in concentration as sedimentation proceeds. Sedimentation columns are another method. These are much simpler and are glass columns designed so that the sediment can be drawn off at the bottom at required time intervals. Although they are not susceptible to disturbance during sample withdrawal, errors may arise from the taper at the bottom of the tube (e.g. hang-ups). In the interpretation of the data, it is necessary to realise that the total weight of the particles which settle during time t includes those particles not large enough to fall the full distance h, but which started part of the way down. If dt is the particle size of those particles whose settling rate is just sufficient to fall through h in time t, the basic equation relating to cumulative methods is:

dtdptPW −=

Where W = fractional weight of particles of size greater than dt deposited in time t. P = fractional weight deposited in time t. In principle (p) and (t) are measured and (w) calculated from the equation. The cumulative % oversize can also be obtained graphically from a plot of p against t. Graphical approach: Avoids problems associated with integration of the basic cumulative equation. Let x be the size of particles which could have theoretically fallen the whole distance h in time t. At time t, all particles > x have settled but at this time particles < x are

47

settling at the rate dtdp . In time t the fractional amount of particles < x that settled is

given by dtdpt

∴ Total weight of particles settled during t is

dtdptWP +=

A sedimentation curve of P against t is plotted as shown in figure 12. For a specific fall time tx, through h, we can find the corresponding theoretical particle size from the relation

d x k htstx

, = ⎛⎝⎜ ⎞

⎠⎟12

In order to obtain Wx, the fraction of size > dst,x a tangent is drawn to the sedimentation curve at tx and extrapolated to the ordinate. Wx is given by the intercept on the ordinate as shown in the figure. The proportion of oversize particles (> x) in the original sample is given by: W/Pmax Where Pmax is the settled weight if the smallest particle in the suspension has been given sufficient time to settle. The cumulative oversize (weight %) is given by: Y = W/Pmax x 100 Pmax is approximately the original sample weight. The cumulative weight % undersize is given by (100-V) Thus a conventional cumulative weight % finer (or greater) that particle size can be plotted. Furthermore, the weight frequency curve can be obtained from the two cumulative curves. Repeated Decantation is another cumulative method which is widely used because it does not require special equipment other than a set of beakers. The sediment in beaker A is freed of particles > dst by decanting into a fresh beaker. The re dispersed suspension is allowed to settle again prior to further decantation. The procedure is repeated several times with different settling times producing different size fractions. The method is however often used to prepare close size fractions (e.g. for chemical analysis) than as an analytical method. For cumulative methods, the concentration of solids required is lower than for incremental methods with balance techniques being the most attractive.

Incremental Methods The principle is that at any given depth in the sedimentation vessel the maximum possible particle size instantaneously present at that depth can easily be calculated from a knowledge of the fall height and the time of fall, using stoke’s equation. If the

48

suspension concentration is therefore measured at that depth and after different time intervals the different mass fractions collected will be devoid of particle sizes greater than the calculated dst values. Hence a plot of C/Co against dst is effectively a cumulative undersize mass fraction. C is the suspension concentration at time t and Co is the initial particle concentration. The different incremental methods available differ only in the way the concentration is measured. One of the most widely used is the Andreasen pipette shown below:

49

This is a fixed depth pipette in which a known volume of suspension is withdrawn at known times after stirring has ceased. The sample is evaporated to dryness and the solids weighed.

Calculation of results

The results obtained are directly cumulative weight % undersize which are plotted as such.

The basic calculation is:

pww

x VV

xnn

s f

= 100

Where pn = cumulative weight % of particles smaller than each selected dst value Wn = weight of solids in sample (g) Ws = weight of test sample in suspension (g) V = volume of sedimentation vessel to the mark (ml) Vf = volume of pipette (ml) In the Andreasen pipette h changes with sample withdrawal. As settling is still taking place during sample withdrawal (should take 20s) the height is usually taken as the mean of the height before and after sample withdrawal. Calculation of time of settling: Stokes law is used to calculate the initial time from a knowledge of the limiting value of dst (e.g. 53 um if from sieving test). A 2: 1 time progression is then normally used after this. The advantages of the Andreasen pipette are that it is simple and of low cost. However, the main disadvantages, sample withdrawal can disturb the suspension. Some of the sample is also left behind in the pipette stem for the next sample. With care, however, reproducible results can be obtained. An alternate method is the Measurement of density of the suspension. Some of the sample is also left behind in the pipette stem for the next sample. With care, however, reproducible results can be obtained.

Disadvantages

a) Sample withdrawn can disturb the suspension b) Sample left behind add to the next sample drawn

An alternate method is the measurement of density of the suspension. Two alternatives are:

a) measurement at known levels using hydrometers b) measurement of density gradient using “divers”.

50

Particles settle onto hydrometers and it is also difficult to know exactly where the density is being measured. Divers give more accurate results especially if they can be electronically sensed. Those are small glass vessels calibrated or adjusted to a known density (see fig. 15).

Centrifugal Sedimentation With gravitational forces only particles greater than about 5 um can be analysed. For particles less than 5 um, prolonged settling times occur due to convection currents, Brownian motion and particle surface charge. Centrifugation extends the size range down to 0.02 um. However, because the centrifugal force varies radially, the analysis of particle size and behaviour is far more difficult than the above methods.

Elutriation Methods The apparatus consists of a series of cylindrical vessels having conical bottoms and arranged in order of increasing diameter. The same flow Q of fluid passes through each column. Stoke’s law is normally assumed to apply. The sizes separated d1, d2, d3 etc are therefore calculated from

( ) 21

Vkdst = and V = Rate of flow Q (cm3/s) Cross sectional area of column (cm2) The main advantages of elutriation are

i) Fractions of particles are produced ii) Apparatus is simple iii) Fluid can be liquid or gas

The disadvantages are:

i) To achieve laminar flow, tall columns are required and sharp bends must be avoided. ii) Fluid flow across the tube is parabolic because of wall drag. In reality particles are

normally dragged into the fastest flow region so that they tend to be cut by the maximum fluid velocity. The effect is also less significant with larger elutriation.

iii) Process is lengthy and it is not easy to maintain a constant flow over the necessary period of elutriation.

51

The Cyclosizer The elutriation columns are replaced by specially designed hydrocyclones which are mounted “upside-down” and a closed box is fitted to each apex nozzle. Not all the solids which would normally leave via the apex do so. They are continuously carried back into the conical body. This repeated sorting produces a sharp separation as well as affording additional opportunity for the small particles to leave through the vortex finder. The cyclosizer shown below ,comprises a set of five, 3” diameter hydrocyclones in series with decreasing diameter feed inlets and vortex finders, giving an increasing centrifugal force.

The sample is therefore divided into five fractions. The cyclosizer can be used for particles in the range 8-50mm and has the advantage that a run takes only 10-30 minutes.

5.1.3. Streaming Methods These measures particle size by measurement of individual particles in a flowing stream of fluid passing a suitable detector. It is essential that the stream concentration be so low that only one particle is detected at a time. One of the more successful applications is the coulter counter shown below.

52

The device has particles suspended in an electrolyte, and their presence produces a change in resistance as they pass between two electrodes. This signal is amplified and counted. By setting suitable detection limits, only particles greater than a certain size can be detected. Further, by successive adjustment of the limits, a cumulative size distribution can be built-up. Since the signal depends on the particle volume, this method measures dv.

5.1.4 On-Stream Particle size analysis On-stream analysers have considerable advantages in mineral processing, e.g. in the control of grinding circuits.

53

The size distribution of the product from a grinding operation often plots as an approximately straight line (e.g. on log-log paper). When the product from a mill changes, it will often do so in such a way that a family of parallel straight lines is produced. Hence the deviation can be defined by determining only one point on the curve. This obviously simplifies the problem. Sampling, however, presents problems in on-stream analysis because an analyser usually requires a uniform rate a flow much lower than that in the process stream which may vary in flow rate, solids concentration, etc. Several analysers have been designed. In one of the more common ones, the sample is pumped into a rectangular section tube which is bent through one turn of a helix. Centrifugal forces cause the particles, initially uniformly dispersed, to form a concentration gradient across the tube. The concentration profile is measured using a transmission beta radiation gauge. Calibration of the gauge enables the difference in signals to be interpreted in terms of e.g. weight % greater than a particular size.

54

UNIT 6 6.0 Measurement of surface area. 6.1 Introduction. In this Unit, the importance of surface area and specific surface are introduced to appreciate their importance in understanding chemical surface reactions. 6.2 Objectives

By the end of this unit you should be able to;

1. Define the various terms used in particle surface characterisation in terms of common shapes.

2. Define various equivalent surface characterisation methods which include gas and liquid mono layer and multi layer adsorption.

3. Introduce the use of BET method to determine the specific surface of powders for purpose of determining the adsorption characteristics.

6.3 Reflection

Have you ever thought about the importance of the absorption of various chemicals or regents on other things we use in many applications? For instance, the adsorption of various detergents in washing and cleaning of domestic utensils is important to have an idea the quantities used for planning purposes. In extractive metallurgical application such as leaching and solvent extraction, distillation and many industrial applications, surfaces play a pivotal role determining the rate of reactions taking place on the surfaces more especially when adsorption is the critical step. This is also very important in the synthesis of everyday commonly widely used chemicals.

The methods available include:

(a) Calculation from the size distribution and a knowledge of shape factor (b) Permeability methods using a gas (c) Monomolecular layer adsorption

Gas permeability methods The specific surface is determined by measurement of the resistance to flow of fluid ( usually gas) through a settled bed of particles. For laminar flow, the pressure drop, P in the fluid due to frictional loose across a bed of particles can be expressed by the Kozey-Carman equation:

3

2

εη LSVK

P Vo=Δ

K= proportionality constant dependent on the shape of particles η = viscosity of fluid Vo = superficial velocity of the fluid ( velocity over the open area without particles)

55

SV = specific surface ( surface area per unit volume) of the bed of particles) L= depth of bed of particles ε = porosity ( void fraction ) of the bed of particles For most powder the value of K is usually taken as 5. Vo of fluid is obtained from the flow rate:

where Q = volumetric flow rate through the bed per unit time

A = cross sectional area of the bed

The method is really only suitable for comparative work as a result depend on the degree of packing achieved. It is applicable to powder in the range 2 to 50 m with a voidage of 0.4 to 0.7 6.4 Adsorption methods 6.4.1 Adsorption from liquids If a powder is shaken with a solution containing a solute which is readily abstracted on the surface, an equilibrium will be established between the adsorbed solute and that remaining in solution. As long as the adsorbed mono-molecular layer does not have further affinity for more solute from solution, the maximum amount of solute abstracted can be assumed to result from complete coverage of the surface by monolayer of solute. If the value of surface occupied by one molecule of the solute is known, the surface area can be computed from the total amount of solute adsorbed. The method is simple and can be used for comparative work. It is not as accurate as gas adsorption, however, because

(i) The area covered by each molecule is not known with certainty let alone the configuration of the adsorbed solute molecules

(ii) Large molecules are usually used. Area does not therefore include small cracks. (iii) There is always competition between solute and solvent with the solvent being co-

adsorbed in amounts dependent on e.g the polarity of the surface. 6.4.2 Adsorption of Gases The methods consists of measuring the quantity of gas physically (reversibly ) adsorbed onto the solid surface. As the quantity adsorbed increases with decrease of temperature. The measurements are done at low temperatures ( e.g in liquid N2 at 77o K). The result are made at constant temperature and plotted as the volume adsorbed against pressure at which adsorption takes place. These plots are called adsorption isotherms.

AQVo =

56

There are many shapes of isotherms. Two common ones are:

(i) Monolayer or Langmuirian adsorption. This is obtained when adsorption is restricted to a monolayer. Few solids ( e.g chalcoal) exhibit this type of isotherm although it is common with adsorption at liquid-gas and liquid-liquid interfaces. Langmuir developed an equation of this isotherm

aPaPV

V m

+=1

V = volume of gas adsorbed at pressure P Vm = monolayer capacity A = an affinity constant If P tends to infinity

mm V

aPaPV

V =→

Hence Vm is the limiting amount adsorbed. Rearranging gives

mm VP

aVVP

+=1

A plot of VP against P will result in a straight line from which Vm can be obtained as

reciprocal of the slope.

(ii) Multi-layer adsorption isotherm The isotherm is sigmoidal and is common with adsorption at the solid-gas interface ( e.g N2 onto most solids) At low pressure mono-layers are formed with multi-layers forming at higher P values Brunauer, Emnet and Teller derived an equation for the multiplayer type of isotherm. This is referred to as the BET equation i.e

( ) ( )( )o

o

m

PPcPP

cPVV

11 −+−=

where Po = the vapour pressure of the liquefied gas c = an affinity constant

A straight line is obtained when ( )PPVP

o − is plotted against

oPP with

cVm

1 as the

intercept, from which Vm can be calculated.

57

( )( ) ( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−+

−=⎥

⎦

⎤⎢⎣

⎡−+

PPVP

cVc

cVPPVcPV

PPc

ommo

m

o

11,11

Determination of surface area from Vm The surface area is calculated from monolayer capacity from the equation

WMVN

SV

mW

σ=

Sw = specific surface area (m2/g) N = Avogadro's number = 6.023x 1023 σ = Area occupied by one adsorbate molecule = 16.2x 10-20 m2 for N2 Mv = Volume of 1 mole of N2 molecule W = Weight of sample (g) Experimental Methods The most common method is the volumetric method.

The sample and apparatus are initially degassed at high temperature and low pressure to remove surface impurities. The volume of “dead space”, which is constant for a given apparatus, is determined using helium which is not absorbed. Measurements

are then made over the range of interest (usually 7.01.0 −=oPP ), by admitting a

known volume of gas in the adsorbent atmosphere at a known pressure. The BET equation is then used to obtain Vm. Equipment (“single point” apparatus) is now commonly available giving values to within a few percent of those obtained by the full BET method. These have greatly reduced the time for each determination.

58

6.5 Choice of Method of Measurement There are many factors which influence the choice of method. The main one may be summarised as the:- (a) size range of the material concerned (b) purpose of analysis e.g. whether routine product quality control or research © frequency with which analyses are required (d) cost of measurement which in general is regulated by the economy of scale of overall operations. The key word is careful interpretation of the data. This requires that the method of measurement should be selected so as to yield the relevant data for specific situations. This is especially so with measurements of particle size due to the many different nominal diameters that can be measured. There are also a number of techniques that can be automated with the aim of providing a large number of analyses free from operator bias. Although the former is achieved, it should be remembered that the operator experience may be critical (e.g. for removing foreign objects). The cost of automation may not also justify the scale of operations. Finally an essential prerequisite for meaningful results is sampling and sample preparation in the case of fine particles agglomeration due to inter-particle forces often takes place leading to erroneous results (and conclusions) e.g. if surface area is measured. The correct sampling procedure is therefore a critical component of the determination of particle characteristics.

59

UNIT 7.

7.0 CHARGED INTERFACES 7.1 Introduction. In this Unit we are going to learn about various ways surface charge is generated on particles when insert or placed in a fluid. The interaction of solid particles and fluids in various flotation and other concentration processes will be discussed and various phenomena introduced. Methods of surface charge determination will be introduced how they can be exploited in surface chemistry. The definitions of electrophoresis, electro-osmosis streaming and sedimentation potentials will be given and their applicability outlined. 7.2 Objectives By the end of this unit you should be able to;

1 Define the terms used in surface chemistry such as those responsible for surface charge generation.

2 Define various electrokinetic phenomena, such as electrophoresis, electro-osmosis, streaming and sedimentation potentials.

3 Know how to determine the surface charge widely used in surface chemistry of mineral particles.

4 Relate surface charge to what is known as Zeta potentials and iso electric points and points of Zero charge

7.3 Reflection

Have you ever thought about the importance of surface charge in most processes in everyday life activities? Imagine how a cloth sticks to your own body through electrostatic attraction. What makes a collector or in general, most reagents, such paints, dyes, greases, lotions and many cosmetic products stick to various substances? Most of these phenomena, occurs due to surface charge generated on them when subjected to various different environments. Well, the surface particle charge is in most cases responsible for adsorption mechanisms of various reagents through chemisorptions or columbic or electrostatic attraction. This is particularly important for reactions involving surface phenomena in many mineral and chemical processing processes.

7.4 The Electric Double Layer Almost all known substances acquire some surface electric charge when brought into contact with an aqueous medium, resulting from possibly charging mechanisms such as ionisation, preferential ion adsorption and ion lattice dissolution. This surface charge influences the distribution of nearby ions in the polar medium. Ions of opposite charge (counter-ions) are attracted towards the surface and ions of like charge (co-ions) are repelled away from the surface. This, together with mixing tendency of thermal motion, leads to the formation of an electric double layer made up of the charged surface and a neutralising excess of counter-ions over co-ions distributed in a diffuse manner in the polar medium. The theory of the electric double

60

layer deals with this distribution of ions and hence, with the magnitude of the electric potentials which occur in the locality of the charged surface. 7.4.1 Origin of the Charge at Surfaces 7.4.2 Ionisation Proteins acquire their charge mainly through the ionisation of carboxyl and amino groups to give COO- and NH +

3 ions. The ionisation of these groups, and so the net molecular charge, depends strongly on the pH of the solution. At low pH a protein molecule will be positively charged and at high pH it will be negatively charged. The pH at which the net charge (and electrophoretic mobility) is zero is called the iso-electric point. 7.4.3 Preferential Ion Adsorption A net surface charge can be acquired by the unequal adsorption of oppositely charged ions. This may involve positive or negative charged ions. Surfaces in contact with aqueous media are more often negatively charged than positively charged. This is a consequence of the face that cations are usually more hydrated than anions and so have the greater tendency to reside in the bulk aqueous medium, whereas the smaller, less hydrated and more polarising anions have the greater tendency to be specifically adsorbed. 7.4.4 Preferential Ion Dissolution Ionic substances can acquire a surface charge by virtue of unequal dissolution of the oppositely charged ions of which they are composed. Silver iodide particles in aqueous suspension are in equilibrium with a saturated solution of which the solubility product aAg+a1

- is about 10-16 at room temperature. With excess I- ions, the silver iodide particles are negatively charged; and with sufficient excess Ag+ ions, they are positively charged. The zero point of charge is not at pAg 8 but is displaced to pAg 5.5. (pH 10.5), because the smaller and more mobile Ag+ions are held less strongly than the I-ions in the silver iodide crystal lattice. The silver and iodide ions are referred to as potential-determining ions, since their concentrations determine the electric potential at the particle surface. Silver iodide sols have been used extensively for testing electric double layer and colloid stability theories. In a similar way, hydrogen and hydroxyl ions are potential-determining for metal oxide and hydroxide solutions. 7.5 Adsorption and Orientation of Dipoles Adsorption of dipolar molecules will not contribute to a net surface charge, but the presence of a layer of orientated dipolar molecules at the surface may make a significant contribution to the nature of the electric double layer.

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7.5.1 The Diffuse Double Layer The electric double layer can be regarded generally as consisting of two regions: an inner region which may include adsorbed ions, and a diffuse region in which ions are distributed according to the influence of electrical forces and random thermal motion. The diffuse part of the double layer will be considered first. Quantitative treatment of the electric double layer presents an extremely difficult and in some respects unsolved problem. The simplest quantitative treatment of the diffuse part of the double layer is that due to Gouy (1910) and Chapman (1913), which is based on the following model: (1) The surface is assumed to be flat, of infinite extent and uniformly charged (2) The ions in the diffuse part of the double layer are assumed to be point charges

distributed according to the Boltzmann distribution. (3) The solvent is assumed in influence the double layer only through its dielectric

constant, which is assumed to have the same value throughout the diffuse part. (4) A single symmetrical electrolyte of charge number z will be assumed. This

assumption facilitates the derivation while losing little owing to the relative unimportance of co-ion charge number.

7.6 ELECTROKINETIC PHENOMENA Electrokinetic is the general description applied to four phenomena which arise when attempts are made to shear off the mobile part of the electric double layer from a charged surface.

62

If an electric field is applied tangentially along a charged surface, a force is exerted on both parts of the electric double layer. The charged surface (plus attached material) tends to move in the appropriate direction, while the ions in the mobile part of the double layer show a net migration in the opposite direction, carrying solvent along with them, thus causing its flow. Conversely, an electric field is created if the charged surface and the diffuse part of the double layer are made to move relative to each other. Let the electric potential be oψ at the flat surface and ψ at a distance x from the surface in the electrolyte solution. Taking the surface to be positively charged and applying the Boltzmann distribution,

⎥⎦

⎤⎢⎣

⎡−=+ T

zenn o κψexp and ⎥⎦

⎤⎢⎣

⎡=− T

zenn o κψexp where +n and −n are the respective

number of positive and negative ions per volume at points where the potential is ψ(i.e where the electric potential energy is zeψ , respectively) and on is the corresponding bulk concentration of each ionic species. The net volume charge density ρ at points where the potential is ψ is therefore, given by

( )−+ −= nnzeρ , ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡−

Tze

Tzezeno κ

ψκψ expexp

= Tzezeno κψsinh2−

ρ is related to ψ by poisson's equation, which for flat double layer takes the form

ερψ

−=2

2

dxd

where ε∗ is the permittivity Combination of equation 1 and 2 gives

Tzezen

dxd o

κψ

εψ sinh22

2

= The solution of this expression with the boundary conditions

(Ψ=Ψo when x = 0; and Ψ = 0, dΨ/dx = 0 when x = ∞ ) can be written in the form

[ ][ ]⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

−+=

xx

zeT

κγκγκ

ψexp1exp1ln2 when simply this becomes

∗ The permittivity of a material is the constant in rationalised expression221

4 rqqF

πε=

where F is the force between charges q1 and q2 separated by a distance r.

63

[ ]xo κψψ −= exp which shows that at low potentials, the potential decreases exponentially with distance from the charged surface. The potential Ψo can be related to the charge density at the surface by equation the surface charge with the net space charge in the diffuse part of the double layer

( ) ∫∞

−=0

. dxei o ρσ and applying the Poisson-Boltzmann distribution. The result

expression is ( )T

zeTn ooo κ

ψεκσ

2sinh8 2

1 which at low potentials reduce to

oo εκψσ =

The surface potential Ψo, therefore depends on both the surface chargeσ o and (through κ) on the ionic composition of the medium. If the double layer is compressed (i.e κ increased), then either chargeσo must increase, or Ψo must decrease or both. The four electrokinetic phenomena are as follows: (1) Electrophoresis - the movement of a charged surface plus attached material

(i.e. dissolved or suspended material) relative to stationary liquid by an applied electric field.

(2) Electro-osmosis - the movement of liquid relative to a stationary charged surface (e.g. a capillary or porous plug) by an applied electric field (i.e. the compliment of electrophoresis). The pressure necessary to counterbalance Electro-osmotic flow is termed the Electro-osmotic pressure.

(3) Streaming potential – the electric field which is created when liquid is made to flow along a stationary charged surface (i.e. the opposite of electro-osmosis).

(4) Sedimentation potential – the electric field which is created when charged particles move relative to stationary liquid (i.e. the opposite of electrophoresis).

Electrophoresis has the greatest practical applicability of these electrokinetic phenomena and has been studies extensively in its various forms, whereas electro-osmosis and streaming potential have been studied to a moderate extent and sedimentation potential rarely owing to experimental difficulties. 7.7 Electrophoresis A number of techniques have been developed for studying the migration of colloidal material in an electric field. 7.7.1 Particle (microscope) electrophoresis If the material under investigation is in the form of a reasonably stable suspension or emulsion containing microscopically visible particles or droplets, then electrophoretic behaviour can be observed and measured directly. Information relevant to soluble

64

material can also be obtained in this way if the substance is adsorbed on to the surface of a carrier, such as oil droplets or silica particles. The electrophoresis cell usually consists of a horizontal glass tube, of either rectangular or circular cross-section, with an electrode at each end and sometimes with inlet and outlet taps for cleaning and filling. Platinum black electrodes are adequate for salt concentrations below about 0.001 mol dm-3 to 0.01 mol dm-3, other wise appropriate reversible electrodes, such as Cu CuSO4 or Ag AgCl, must be used to avoid gas evolution. Electrophoretic measurements by the microscope method are complicated by the simultaneous occurrence of electro-osmosis. The internal glass surfaces of the cell are usually charged, which causes an electro-osmotic flow of liquid near to the tube walls together with (since the cell is closed) a compensating return flow of liquid with maximum velocity at the centre of the tube. This results in a parabolic distribution of liquid speeds with depth, and the true electrophoretic velocity is only observed at locations in the tube where the electro-osmotic flow and return flow of the liquid cancel. For a cylindrical cell the ‘stationary level’ is located at 0.146 of the internal diameter from the cell wall. For a flat cell the ‘stationary levels’ are located at fractions of about 0.2 and 0.8 of the total depth ratio. If the particle and cell surfaces have the same zeta potential, the velocity of particles at the centre of the cell is twice their true electrophoretic velocity in a cylindrical cell and 1.5 times their true electrophoretic velocity in a flat cell.

Cylindrical cells are easier to construct and thermostat than flat cells and dark-field illumination can be obtained by the ultramicroscopic method of illuminating the sample perpendicular to the direction of observation. The volume of dispersion required is usually less for cylindrical cells than for flat cells and owing to the relatively small cross-section, it is more often possible to use platinum black rather than reversible electrodes with cylindrical cells. However, unless the capillary wall is extremely thin, an optical correction must be made with cylindrical cells to allow for the focusing action of the tube, and optical distortion may prevent measurements from being made at the far stationary level. Cylindrical cells are unsatisfactory if any

65

sedimentation takes place during the measurement; if a rectangular cell is adapted for horizontal viewing, sedimenting particles remain in focus and do not deviate from the stationary levels. The electrophoretic velocity is found by timing individual particles over a fixed distance (c. 100µm) on a calibrated eyepiece scale. The field strength is adjusted to give timings of c. 10s – faster times introduce timing errors and slower times increase the unavoidable error due to Brownian motion. Timings are made at both stationary levels. By alternating the direction of the current, errors due to drift (caused by leakage, convection or electrode polarisation) can largely be eliminated. The electrophoretic velocity is usually calculated from the average of the reciprocals of about 20 timings. The potential gradient E at the point of observation is usually calculated from the current I, the cross-sectional area of the channel A and the separately determined conductivity of the dispersion ko – i.e. E=I/koA. Particle electrophoresis studies have proved to be useful in the investigation of model systems (e.g. silver halide sols and polystyrene latex dispersions) and practical situations (e.g. clay suspensions, water purification and detergency) where colloid stability is involved. In estimating the double-layer repulsive forces between particles, it is usually assumed that ψ d is the operative potential and that (calculated from electrophoretic mobilities) are identical. Particle electrophoresis is also a useful technique for characterising the surfaces of organisms such as bacteria, viruses and blood cells. The nature of the surface charge can be investigated by studying the dependence of electrophoretic mobility on factors such as pH, ionic strength, addition of specifically adsorbed polyvanlent counter-ions, addition of surface-active agents and treatment with specific chemical reagents, particularly enzymes.

66

UNIT 8

8.0 FLOTATION KINETICS.

8.1. Introduction In this Unit we review the elementary aspects of flotation kinetics, such as hydrodynamics of bubbles and particles interaction, theory of collision efficiency and many other related processes, mechanism of particle capture, induction time, residence time distribution, influence of particle and bubble size upon flotation rate among other related phenomena.. Flotation will be treated as a rate reaction like in chemical thermodynamics. 8.2 Objectives By the end of this unit you should be able to;

1 Define the terms used in reaction kinetics in flotation 2 Define various probabilities encountered in the process of flotation. 3 Relate effluence of various parameters in the flotation kinetics using a

modified Hallimond tube. Refinery.

8.3 Reflection

Have you ever realised the importance of kinetics in most of important processes in mineral processing and other metallurgical processes such as leaching, solvent extraction, metal purification processes such smelting and refining which can either be pyrometallurgically or electrometallurgically carried out? Well, this area enable us to understand the various mechanisms which take place to separate various mineral/metal values from the impurities to produce marketable products. By determining the speed or rate at which various minerals are separated, you are able to concentrate valuable minerals/metals in different fraction for different uses.

Flotation is a macroprocess composed of a number of microprocesses taking place simultaneously and successively in time and space. In order to model this process, there is need to make very empirical simplification to describe the process more clearly. The kinetic study of the flotation process include the determination of all the factors that influence the rate of the concentration process. Various ways may be use to express the this rate, but for most practical purposes, more especially when dealing with mineral ores, it is generally measured as absolute recovery in a given time. This kinetic process is normally studied use a Hallimond tube shown below at laboratory scale.

67

Since most of the recovery of minerals of interest occur in cells of some kind, it is important to understand the various physico-chemical processes, which take place during this process. The following takes place in a cell

Ø Homogenous suspension of the pulp Ø Introduction of air as small-dispersed bubbles Ø Turbulence agitation of the pulp to enhance bubble-particle interaction Ø Stable froth formation and transfer mineral values in the lauders

The theory of froth flotation is a complex and is still not completely understood and the process can only be applied to relative fine particles.

The attachment of solid particles to air bubbles is the most important stage of flotation. As early as 1932, Frumkin (1933) paid particular attention to the kinetics of thinning of the intervening film between an air bubble and a mineral surface and suggested this film thinning as means of interpreting the mechanism of froth flotation.

For the successful bubble-particle attachment, three elementary steps are important:

i. Thinning of an intervening liquid film to a thickness hcr ii. Rapture of an intervening liquid film and formation of a three-phase

contact (TPC) and iii. Expansion of TPC line from the critical radius to form a stable wetting

perimeter.

The attachment of a particle to a gas bubble in a flotation system is usually described as a result of a series of probabilities described below.

i. Probability of bubble- particle collision in the pulp (Pc)

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ii. Probability of bubble-particle adhesion (Pa) iii. Probability of non-detachment between particle and bubble (Pd)

Then the overall probability of a successful flotation can be expressed as a product of the above probabilities as below.

Pf = Pc.Pa.Pd

The empirical simplification used to model a flotation process will take into account the steps mentioned above and the simplest form of kinetic equation describing flotation is the evaluation of mineral particle number per unit time in a given volume of apparatus.

𝐝𝐧𝒅𝒕

= -z.np.nb.Pc.Pa.Pst.Ptpc

Where Pc, Pa, Pst, and Ptpc are probabilities of bubble-particle collision, bubble-particle attachment, stable froth and three-phase contact respectively. The np and nb denote the number of particles and bubbles respectively.

The configuration of the cell such as height/surface ratio, impeller diameter/cell length ratio etc will play a very important role in determining the nature of the flotation process. It is well established that the transfer probability of a mineral particle from the pulp to the froth is a product of the various probabilities, which may not easily be quantified for proper use to control an industrial flotation process. In it’s simplest form, the flotation process maybe be treated as a chemical reaction, but the collision between gas molecules and mineral particles should lead to a successful reaction. The approach used by many scientists is to relate the mineral quantity transferred to the froth as a direct function of its concentration when most parameters are maintained constant ( type of mineral, pulp density, pH reagent addition, aeration,, etc):

- 𝒅𝑪𝒅𝒕

= k(C)n k is the rate constant and C is the mineral concentration. The exponent is considered as the order of a chemical reaction and its value may take any value (0,1,2.). In two most widely cases, n, take the values of 1 and 2 for first and second order rate reaction. When n = 2

69

𝒅𝑹𝒅𝒕

= k(R*-R)2 Where R is the recovery at time t and R* is the ultimate recovery for an infinite time. Integrating with R = 0 when t = 0, it becomes: R = (R*)2kt/1 + R*kt or t/R = 1/(R*)2k + t/R* When drawing t/R in terms of t, the slope permit the calculation of R* and the ordinate, k. However, the most widely used expression is the first order equation, which take the form: − 𝒅𝑪𝒅𝒏

= R = kC where C is the mineral concentration in the sink, giving the final equation as: C = Coe-kt The following general steps should be recognized in order for flotation to occur: 8.2 Probability of adhesion of collector specific mineral particles This is usually determined by the conditioning time, which is related to the residence time of the particles in the pulp before flotation. It is implicitly agreed that collector adhesion is very rapid, but it is established from actual plant results that flotation results in the first in the first cells of a bank can be erratic due to conditioning going on for a certain time. Probability of collision between the mineral particles and bubbles and this collision resulting is a stable aggregate, which can be transferred into the stable froth layer This depends on complex physico-chemical and hydrodynamic phenomena that may be difficult to isolate. Not all particles, which are involved in a collision, which lead to an adhesion to form a stable aggregate. For particle moving in a projection path Rb, they can only be involved in a collision if they are within a streaming tube of limiting collision radius Rc. 8.3 Probability of the hold-up in the aggregation This will be determined by the impact processes, which will lead to deformation and either repulsion on attachment and the sliding action. The hold-up will depend critically on the duration of contact (Collision time, or sliding time) in relation to the film drainage time until it’s rupture. Therefore, the time required for attachment, must be greater than, or equal to the induction time. This is principle consist of both the adhesion probability by collision and the adhesion probability by sliding.

70

8.3 Application of flotation kinetics Many scientist and Engineers are using flotation kinetics to study in depth and with more details the influence of many parameters on this process. The most widely studied are the influence of particle size of the minerals, air flow rate, bubble size and reagent addition although of late the configuration of the cell is also attracting a lot of interest. 8.3.1 Influence of particle size. It is widely accepted that the maximum flotation rate is obtained in the intermediate size range, and that the rate decreases on either side of this range. However, it is difficult to establish whether the flotation rate constant k varies directly with particle size or with its logarithm on the fine side of the maximum. 8.3.2 Influence of bubble size. Many flotation scientist believe that, the smaller the bubble size, the more they are distributed in the bulk, bigger bubble lead to quicker adhesion to mineral particle hence increasing the kinetics. 8.3.3 Influence of reagent concentration. It is generally desired to use starvation concentrations of different regents especially where the cost is prohibitive as is the case in sulphide flotation. 8.3.4 Influence of pulp pH. Most minerals have their flotation kinetic very much dependent of the pulp pH. It generally recognised that Oxidic minerals faster reasonably fast at lower pH, whilst the case in different for sulphide minerals who require slightly higher pH 8.3.5 Influence of cell geometry and agitation mechanism. It is generally assumed that the better the agitation and cylindrical vessels or cells give higher flotation rates.

71

UNIT 9 9.0 Mineral processing flowsheet development and process design. 9.1 Introduction. In this Unit, the procedure for the establishment of physical and chemical properties of ores and mineral products, selection, applicability and limitations of processes for the preparation and separation of minerals which forms an integral part of flowsheet development and process design will be introduced. 9.2 Objectives By the end of this unit you should be able to;

1 Define the various steps to be taken for flowsheet development 2 Define various terminologies used to design a process. 3 Relate effluence of various parameters in the mineral process design.

9.3 Reflection

Have you ever thought about the importance of discovering new mineral deposits which may require specific treatment routes from the classical ones? The use of various treatment stages can be used for newly identified mineral deposits or raw material requiring specific upgrading treatment. By understanding the mineralogical, chemical and physical surface properties, various techniques can be used to upgrade various minerals.

In order for one to come with a process route for the treatment of any unknown ore, various steps are important to carry out such as:

• Identify/characterize the unknown ore (mineralogical and chemical composition)

• Come up with the better and more economic way of treating it • Process Flow Diagram (sheet) development • Design a process treatment route for the ore

A typical process design will be defined in accordance with a simplified as shown in figure 1 below;

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Figure 1 : A general typical flowsheet design.

Depending of the finding of the about, ores can be treated differently. The most common ores and minerals occur as oxides or sulphides in general. However other classes of minerals, may occur as Silicates, Carbonated or mixed forms. Depending on the minerals of interest, the components may be treated as waste or gangue. The treatment of common sulphide minerals , oxides mineral and silicates are well documented in the earlier courses and have been well documented elsewhere. The use of the JKSimMet software as a tool for analysis and simulation of mineral processing plant data has also been used to carry out modelling of process plants.

A process development must be simple in technology and for intended benefits. It’s technical efficiency should achieve meaningful results which are near to the expected results leading to viable and optimised process parameters

At laboratory stage, the modelling and simulation will determine the processing methods to be used at Pilot plant scale which can be scaled up to industrial level

The process streams can include the various general stages:

Ø Comminution and screening followed by chemical, mineralogical and petrographic analyses

Ø Classification Ø Gravity/heavy media separation Ø Magnetic separation Ø Flotation and dewatering

DE SIGN O B J E CT IV E -TH E N E E D

DA TA CO L L E CT IO N

GEN E RA T IO N O F PO SSIB L E D E SIGN SO LUT IO N S

SE L E CT IO N &E V A LUA T IO N(O PT IMISA T IO N )

F IN A L D E SIGN

COMMER CIAL OPPOR TUNITY

PossibleProcesses,equipmentperformance,

Testingsetc

DesignerwillprefertriedandtestedMethods.

Designernarrowsdowntothe"best"designfor

thepurpose.

73

Ø Granulation Ø Leaching and Solvent Extraction Ø Refining Ø Product dispatch.

Available Gravity separation methods can be used to separate the heavy minerals and the rare earth minerals from for instance apatite and dolomitic minerals taking advantage of differences in their specific densities and gravties. Magnetic separation utilises the differences in magnetic properties such as the magnetite and other rare earth minerals form non-magnetic minerals.

The surface chemistry properties of various minerals can be manipulated and changed by the presence of different ionic species to create an enabling environment to effect separation of minerals. A typical example of a flowsheet for the treatment an ore containing silicate minerals is shown below in figure 2;:

74

Flowsheet design of a Mineral processing plant

The ore to be treated contains Mica, Feldspar, Quartz and Fe Oxides iep’s of minerals are pH 1.5, 2.8, 2.8. 6.8 respectively.

H2S04

Amine

Frother

Conc1

Frother

H2S04 Ca(OH)2

PetroleumSulphonate Na2(Si0)4

Frother OleicAid

Conc2

H2S04 Conc2

PetroleumSulphonate

Frother

Conc3 Quartz,Feldspar

Tails

(a) (b)

ORE

C

Float,pH2.5

C

Float,pH3-4

C

Float,pH2.5

C

FloatpH8-9

C

75

Flowsheet design of a Mineral processing plant of a ore to be treated containing silicate minerals: Mica, Quartz, Feldspar and Iron minerals whose iep’s are at pH ’s 1.5, 2.8, 2.8. 6.8 respectively. This may be applied to any ore known such as oxide and sulphide minerals and various known mixtures of minerals. Descriptions. First you have to have a sketch of the iep in pH Vs zeta potential diagrams, outline the roles of the various reagents from your knowledge of how they interact with minerals in question. Then explain the flotation mechanisms taking place. This flowsheet involves the treatment of an ore containing Mica, Feldspar, Quartz and Iron Oxides. Process (a) Stage 1. The reagents in the first stage are: H2SO4 for pH adjustment to the acidic atmosphere pH= 2.5 Amine is a cationic collector for flotation Frother for froth stability, froth coalescence and surface tension reduction. Process description. At pH 2.5 only Mica will be positively charged and hence the cationic Amine will adsorb on the it’s surface though electrostatic attraction and hence make mineral floatable. Hence Con 1 will be predominately Mica. Stage 2. The reagents in the second stage are: H2SO4 for pH adjustment to the acidic atmosphere pH= 3-4 Petroleum Sulphonate is an anionic collector for flotation Frother for froth stability, froth coalescence and surface tension reduction. Process description. At the pH range 3-4, both Feldspar and Quartz will be negatively charged and whilst the Iron minerals will be positively charged and hence the presence of petroleum sulphonate will act on the surface of Iron minerals to make them hydrophobic and hence will form Con 2 Stage 3. The reagents in the third stage are: H2SO4 for pH adjustment to the acidic atmosphere pH= 2.5 Petroleum Sulphonate is an anionic collector for flotation Frother for froth stability, froth coalescence and surface tension reduction.

76

Process description. At the pH 2.5, both Feldspar and Quartz will be positively charged and hence the presence of petroleum sulphonate will act on the surface of both minerals to make them hydrophobic and hence will form Con 3 Process (b) If you replace stage 2 as shown in process (b) The reagents in the second stage are: Ca(OH)2 for pH adjustment to the alkaline atmosphere pH= 8-9 Oleic Acid is a collector collector for flotation Frother for froth stability, froth coalescence and surface tension reduction. Na(SiO)4 is a depressant for silicate minerals. Process description. In the pH range 8-9, the iron minerals negatively charge, however the presence of Na(SiO)4 is a depressant for silicate minerals, Mica and Feldspar, which are silicates will be depressed. In the presence of Oleic Acid will react with iron minerals form Ferric Oleate which is known to be hydrophobic and hence will form Con 2 of iron minerals as was the case with Process (a)