The determination of floatability distribution from laboratory batch cell tests

Minerals Engineering 83 (2015) 1–12

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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng

The determination of floatability distribution from laboratory batch cell tests R.L. Pascual b, W.J. Whiten a,⇑ a b

4 Magnet Cl, Riverhills 4074, Australia C/o University of Queensland, SMI, JKMRC, 40 Isles Road, Indooroopilly 4068, Australia

a r t i c l e

i n f o

Article history: Received 27 April 2015 Revised 22 July 2015 Accepted 10 August 2015

Keywords: Froth flotation Flotation rate distribution Batch flotation

a b s t r a c t The determination of the floatability distribution for froth flotation from laboratory batch cell tests is ill conditioned. Tikhonov regularisation with accuracy estimates, such as repeat tests for the recovery data, provides a way to produce solutions. Two approaches are examined: a continuous approximation of the distribution using Laguerre polynomials, and using finely spaced discrete components. It is shown that for data generated with a known floatability distribution, the distribution can be recovered. For experimental data from the literature both methods give similar floatability distributions. The accuracy of the calculated floatability distributions is estimated. Repeat data is important in determining the regularisation parameter and thus insuring the data is not over fitted or over smoothed. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Froth flotation is a widely used method of separation in the mineral processing industry. It relies on surface tension properties of small ore particles to selectively attach them to bubbles, and is the preferred method of mineral recovery for many of important minerals. Billions of tons of ore are processed by flotation annually (King, 2001 pp. 289, Lynch et al., 2010 p. 177). Many of our metals and mineral raw materials would be prohibitively expensive without the development of this process (Fuerstenau, 1999). The simulation and optimisation of industrial flotation processes depends on knowledge of the rate at which the particles of interest transfer from the cell to the concentrate, which is quantified as the flotation rate or floatability distribution. The particles in the cell vary in a continuous manner with size, composition, shape, and other variables, so it is reasonable to expect a distribution of flotation rates. Realistic models of industrial flotation cells need this distribution, which is often obtained from batch cell data. In spite of the importance of the froth flotation process in industry, mechanistic models of flotation have not been adequately developed. It is not due to lack of research or effort, but is due to inherent complexity of the flotation process. Although individual basic physical and chemical laws are known, the complexity of all the micro-processes involved in the flotation processes make ⇑ Corresponding author. E-mail addresses: [email protected] (R.L. Pascual), billwhiten@tpg. com.au (W.J. Whiten). http://dx.doi.org/10.1016/j.mineng.2015.08.007 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.

the development of quantitative predictive models extremely difficult. Schulze (1983) in his treatment of flotation from elementary physical and chemical processes identified fifty variables that affect flotation ranging from particle size to double layer potential. This is indeed a staggering amount of physical and chemical properties to construct a model from. A considerable number of these quantities cannot even be measured in the laboratory, let alone in the normal industrial setting. Mechanistic models for the flotation rate (Chander and Polat, 1994; King, 2001; Yoon et al., 2012; Heiskanen, 2013) make many assumptions to simplify the combined effect of the various physico-chemical processes involved. Bloom and Heindel (2003) and Fichera and Chudacek (1992) discuss the difficulties of determining the rate constants for the various sub-processes. An alternative way of modelling flotation is to use an empirical model where the different effects are combined into parameters that can be obtained from easily measured data such as a standard batch test. This approach constructs a model by an analogy with some well-known and understood physical processes. A number of adjustable parameters are incorporated into the model and fitted to make the model prediction as close as possible to some set of experimental data. This kind of model depends very heavily on accurate data, with known error properties, and statistical methods expressed as a computer algorithm. The most successful of these models to date are those based on the first order rate equations similar to those describing the rate of chemical reactions (Lynch et al., 1981; Jameson et al., 1977; Woodburn, 1970). The first order rate equation can be stated as the rate at which the

2

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

floatable particles decrease in the batch flotation cell (that is arrive at the concentrate) is proportional to the amount of floatable particles in the cell. The main experimental data used in the determination of the flotation rate distribution in the empirical approach is the batch cell test (Bazin et al., 1995; Wills, 1992). It is well known that conventional batch flotation tests have difficulty in giving reproducible results. Firstly the feed must be prepared in a reproducible manner (see Lotter et al., 2014 and references there in), and then the test conditions must be carefully reproduced. In particular the operator must insure the area and depth of the froth removed is reproduced (Roberts et al., 1982) as well as reagent additions, air addition and the amount of agitation. To remedy this problem various researchers have modified the standard batch cell test to reduce the variability in the measured recovery data. Roberts and co-workers (Roberts et al., 1982) modified standard Denver flotation cell geometry by means of a perspex insert. They concluded that new cell design reduced variability due to operator-dependent froth removal techniques. Luttrell and Yoon (1983) equipped a commercial laboratory flotation machine (Denver Model D-12) with automated features for ease of operation and improved recovery data reproducibility. The group at J. Roy Gordon Research Laboratory has implemented several improvements over the previous work on the conventional batch flotation test to increase the reliability of this procedure (Cherevaty and Agar, 2004). The least accurate of measurement in their chalcopyrite data has a relative standard deviation of 2.3%. This shows that a careful and automated measurement should be able to reduce the error in the measured recovery values to allow for predictive use of the batch flotation results. The determination of the flotation rate distribution from the mass fraction remaining in the cell as a function of time, measured in a batch cell can be treated as an inverse Laplace transform which belong to a general class of equations called the Fredholm equation of the first kind. This problem is an ill-posed mathematical problem. Ill-posed does not mean that a meaningful approximate solution cannot be computed, but it means that ‘‘elementary” standard methods such as the least squares method cannot be used in a straight forward manner to get the solution. In order to make calculations more manageable in industrial flotation modelling a discrete flotation rate distribution is very often assumed to be a sum of two or three Dirac delta functions leading to a sum of negative exponentials giving the mass fraction remaining in the cell (Kelsall, 1961; Imaizumi and Inoue, 1963; Lynch et al., 1981; Runge et al., 1998, 2001; Harris et al., 2002; Greet et al., 2013; Jameson and Payne, 2013; Nogueira et al., 2013; Ofori et al., 2013). The Dirac delta functions form, although simple, has several difficulties. The particles are given flotation rates of only two or three values with nothing in between them, that is, it is a discontinuous function. This is counterintuitive as most variables affecting flotation rate are continuous such as size, liberation, and surfactant cover. When the particles are divided into size fractions, or even size and liberation fractions, there is still a range of particle properties within these fractions. In most cases the determination of both the amplitude and the flotation rate for discrete distributions are calculated using a nonlinear least squares method. It is known for the fitting of a sum of negative exponentials, that there are many sets of parameters that will fit the experimental data and the parameters calculated can be sensitive to the initial estimate (Lanczos, 1959 pp 272–280). Others simplify the problem by assuming that the flotation rate distribution is described by some continuous analytic functions whose Laplace transforms are also analytic (Woodburn and Loveday, 1965; Loveday, 1966; Harris and Chakravarti, 1970; Kapur and Mehrotra, 1974). For example Harris and Chakravati used the gamma and bimodal gamma function as the assumed

floatability distribution. They concluded that a Gamma function distribution of rates leads to a simple recovery-time relationship and its two parameters are readily obtained by a simple graphical procedure using the log reciprocal vs. log grid. A rectangular distribution has also been used (Mattsson et al., 2013). Cherevaty and Agar (2004) noted that the fit using a rectangular distribution deviated slightly from the data. The shortcoming of these methods is a predefined form for the distribution is difficult to justify on experimental or theoretical grounds. Similar to the discrete rate case, nonlinear least squares can be used to evaluate the parameters. Once the assumed distribution is given more than two or three parameters, it is likely that the calculated parameters are not well defined by the available data. Kapur and Mehrotra, 1974 started from the flotation rate distribution being the inverse of a Laplace transform and used numerical integration and constrained least squares minimisation to calculate the cumulative rate distribution. Mehrotra and Kapur, 1974 showed how their technique can be used to investigate how the flotation rate changes with aeration, particle size and pulp density. In this paper we show how the floatability distribution from measured laboratory batch cell test result can be estimated using Tikhonov regularisation (Tikhonov and Arsenin, 1977; Arsenin and Tikhonov, 1995). Two methods of describing the flotation rate distribution are examined: a sum of weighted Laguerre functions (Abramowitz and Stegun, 1965); and closely spaced Delta functions to approximate a continuous distribution. In the second case non negative least squares (Lawson and Hanson, 1995) is used together with the regularisation. Repeated test data is an important part of the estimation of the flotation rate distribution. As noted by Napier-Munn, 2013; Lotter et al., 2014, and others, without repeat information it is impossible to tell if the difference between tests is real or just due to random variation. The repeat data provides estimates of the test repeatability, but due to the number of tests required for a large increase in accuracy by averaging, repeat data does not replace the need for accurate test work. 2. Batch flotation equations In this section it is shown how the simple batch flotation rate equation extends to a distribution of rates, both as a continuous function and as large number of discrete rates. In both cases the fraction remaining in the cell is expressed as a sum of known functions multiplied by unknown coefficients. The difference between these expressions and the available data give a linear regression problem which is solved by appending additional terms to the regression that ensure the coefficients are constrained to reasonable values. The magnitude of these extra terms, given by the parameter L, determines the amount of smoothing applied to the distribution of the flotation rates. In the case of the discrete rates an additional constraint that the coefficients are positive is applied. For the simple case of one component with a single flotation rate k the fraction cell content remaining at time t is:

CðtÞ ¼ expðktÞ In practice the particles in a flotation cell vary in size, shape and composition, and thus it is unlikely that a single flotation rate is sufficient to accurately describe batch cell behaviour. A more realistic alternative is to use multiple discrete rates or a distribution function for the rates. Thus the determination of the flotation rate distribution F(k) from batch data relies on inversion of the formula relating the initial flotation rate distribution F(k) to the cell content C(t) as a function of time:

Z

1

CðtÞ ¼

FðkÞ expðktÞdk 0

ð1Þ

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

FðkÞ ¼

3

1 X ai f i ðkÞ 0

where fi(k) are known functions and ai are determined from the experimental cell content values. For the choice of n discrete rates ki plus a non floating fraction (a0 with k0 = 0):

FðkÞ ¼

n X ai dðk  ki Þ 0

Substituting this into (1) gives:

CðtÞ ¼

n X

ai expðki tÞ

ð2Þ

0

Fig. 1. Distribution of tail (one minus cumulative distribution) for test of fit to amount remaining in cell.

Sufficiently closely spaced discrete rates give a good approximation to a continuous distribution of rates. For a continuous distribution ranging from zero to infinity, a natural choice (Davies and Martin, 1979) is Laguerre polynomials (Arfken, 1970; Gautschi, 1990; Nevai, 1979) defined by:

L0 ðxÞ ¼ 1; Thus C(t) is the Laplace transform of F(k) (Harris and Chakravarti, 1970). Given the measured values C(tj) for some values of tj it is desired to determine F(k). This is an inverse problem that is known to be ill conditioned (Davies and Martin, 1979; Kirsch, 1996). To determine F(k), it can be expressed as a series:

L1 ðxÞ ¼ 1  x

Li ðxÞ ¼ fð2i  1  xÞLi1 ðxÞ  ði  1ÞLi2 ðxÞg=i and these provide the orthogonal functions over the range zero to infinity:

Li ðkÞ expðk=2Þ

Fig. 2. Fit of Laguerre series to data generated using the rectangular distribution 0–2.

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R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

Table 1 Laguerre fit to a rectangular distribution. L

Error

Rel. error

Probability

Area

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0010 0.0010 0.0010 0.0016 0.0047 0.0143

0.82 0.82 0.88 1.35 3.96 12.13

1.00 1.00 0.95 0.01 0.00 0.00

1.00 1.00 1.00 1.00 1.00 0.98

Table 3 Fraction remaining in batch cell for chalcopyrite and pentlandite each with three repeats. Data from Cherevaty and Agar (2004).

RMS of data standard deviations 0.0012.

Table 2 Discrete rate fit to a rectangular distribution 0–2.

Time

Chalcopyrite

0.25 0.50 1.00 1.75 3.00 5.00 8.00 12.00

0.853 0.700 0.508 0.353 0.227 0.133 0.101 0.081

Pentlandite

0.848 0.693 0.500 0.359 0.214 0.127 0.095 0.077

0.846 0.691 0.497 0.356 0.218 0.126 0.092 0.076

0.889 0.745 0.551 0.386 0.251 0.143 0.101 0.081

0.886 0.737 0.540 0.386 0.244 0.138 0.098 0.076

Li ðkÞ expðkÞ

L

Error

Rel. error

Probability

Area

were found to be more satisfactory. For this expression:

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0011 0.0011 0.0012 0.0013 0.0015 0.0026

0.95 0.96 0.99 1.10 1.25 2.20

0.69 0.65 0.48 0.19 0.04 0.00

1.00 1.00 1.00 1.00 1.00 1.00

Z

1



Li ðkÞ expðkÞdk ¼

0

0

when i > 0

Putting:

FðkÞ ¼ a0 dðkÞ þ

RMS of data standard deviations 0.0012.

1 when i ¼ 0

n X ai Li1 ðkÞ expðkÞ i¼1

But these functions were found to give too much emphasis to large flotation rate values which are poorly defined in typical flotation data. The functions:

into Eq. (1) gives:

CðtÞ ¼ a0 þ

Z n X ai i¼1

1

Li1 ðkÞ expðkð1 þ tÞÞdk

0

Fig. 3. Fit of discrete rates to data generated using the rectangular distribution 0–2.

0.884 0.740 0.542 0.393 0.243 0.135 0.094 0.074

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

5

the cell is halfway between the initial and final value, multiplied by 1.5 has been found suitable. For both the discrete case and continuous case, the amount remaining in the cell, C(t) is expressed as a sum unknown coefficients times known functions ci(t):

CðtÞ ¼

n X

ai ci ðtÞ

0

where the ci(t) functions come from Eq. (2) for the discrete rates and from Eq. (3) for the Laguerre functions. The difference between the measured values C*(tj) and C(tj) is:

C  ðtj Þ 

n X ai ci ðt j Þ 0

Fig. 4. Cumulative distribution for test of equality of standard deviations.

Evaluating the integrals gives:

CðtÞ ¼ a0 þ

n X ai t i1 =ð1 þ tÞi

ð3Þ

i¼1

As the time can be measured in arbitrary units, a scale factor is needed to bring the function values into correspondence with the data times. The value of the time when the amount remaining in

and the fitting of flotation rates now requires minimising the sum of squares of these differences with respect to the linear coefficients ai which is a linear least squares problem. Unfortunately this fitting of coefficients is ill conditioned (see Lanczos, 1959; Arsenin and Tikhonov, 1995; Hansen, 1998) in that small changes to C*(tj) can result in large changes to ai. Further the number of data points (tj) is typically smaller than the number n of coefficients ai. A regularisation, done by appending the extra terms L ai to the above error terms before minimising the sum of squares, is necessary to obtain sensible results. The generalised singular value decomposition (GSVD: Van Loan, 1976, 1985; Paige, 1986; Hansen, 1992, 1998; Golub and Van Loan, 1996) provides an efficient method on solving for ai with multiple values of L. For the

Fig. 5. Laguerre series fit to chalcopyrite data from Cherevaty and Agar (2004).

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Table 4 Fit to the chalcopyrite data using Laguerre functions.

Table 5 Fit to the chalcopyrite data using discrete rates.

L

Error

Rel. error

Probability

Area

L

Error

Rel. error

Probability

Area

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0038 0.0042 0.0049 0.0058 0.0075 0.0214

0.84 0.93 1.08 1.28 1.66 4.73

1.00 0.75 0.23 0.03 0.00 0.00

0.99 1.00 1.00 0.98 0.96 0.91

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0045 0.0045 0.0045 0.0046 0.0050 0.0065

0.99 0.99 1.00 1.02 1.11 1.44

0.51 0.51 0.49 0.40 0.18 0.01

1.00 1.00 1.00 1.00 1.01 1.00

RMS of data standard deviations 0.00452.

RMS of data standard deviations 0.00452.

discrete rates case the additional requirement that ai all be positive, provides an additional degree of regularisation and insures a positive rate distribution is obtained. The algorithm NNLS from Lawson and Hanson (1995) provides the method of constraining least squares fits to positive coefficients.

3. Fit criteria Having introduced the regularisation parameter L, it is necessary to determine its value. Several methods of determining the appropriate value of L are available. Repeated data allows statistical criteria, that compare the error in the fit to the error in the data, to be developed for the determination of L. The distribution of the ratio of the errors, assuming the data are a correct fit, can be

generated by simulation, thus producing a test statistic for the fit to the data. As L is increased the sum of squared errors increases, and the integral (or equivalently sum) of the rate distribution, which should be one for normalised data, decreases. When these values start to change indicates an appropriate value of L and thus the extent of regularisation. The L curve, a plot of log L versus log of the residual, is often used to determine when the residual starts to change (Hansen, 1992, 1998). As the cases considered in this paper have repeats, statistical criteria can be developed. The data for the cases considered, consist of eight values of fraction remaining at different times with these being repeated three times. Each of the eight fraction remaining values is based around some unknown value, but the standard deviation, which measures the error of the data around the mean, can be calculated for each of eight values. Assuming the eight

Fig. 6. Discrete rates fit to chalcopyrite data from Cherevaty and Agar (2004).

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

values come from a similar distribution an overall the standard deviation can be calculated for comparison with the errors in predicting the data. So a comparison is of the root mean squares (rms) of the errors in predicting the data, with the root mean square of the eight standard deviations. Clearly the larger this value the worse the fit to the data. A Monte Carlo simulation can generate the distribution of this statistic under the assumption of a correct fit to the data. An eight by three array of random Gaussian samples with mean zero (so that the correct predicted value of zero is known) is generated, then the root mean square (rms) of the 24 values is divided by the root mean square of the eight standard deviations of the rows i.e.:

rmsðr i;j ; i ¼ 1 : 8; j ¼ 1 : 3Þ rmsðstdðr i;j ; j ¼ 1 : 3Þ; i ¼ 1 : 8Þ where r is an eight by three array of Gaussian sample values with mean zero and unit standard deviation. This is repeated 10,000 times to generate the distribution of the statistic as shown in Fig. 1. The same formula with the error values replacing r, is compared with this distribution. A conservative probability target of between 0.2 and 0.3 (20–30% of the calculated distribution greater than value from the errors in calculated fit) should avoid over fitting and still give a sufficiently accurate fit. Unlike the other criteria above this gives a value to aim for, rather than looking for the start of a change in the criteria. As the above statistic is the ratio of the prediction errors to the standard deviation of the data, its value is easily understood.

7

4. Fit to rectangular rate distribution As a demonstration of the calculation of a rate distribution, simulated data are generated from a known rate distribution and known errors. It is found both the continuous distribution and the discrete rates give a fit to the data with the expected accuracy, and a moderate reproduction of the rate distribution. The probability statistic works well for determining the regularisation value L. The data are generated from a rectangular distribution, from zero to two, of rates using

ð1  expð2tÞÞ=ð2tÞ for t = 0, 0.5, 1, 2, 4, 8, 16, 32 with a random Gaussian samples with standard deviation 0.001, i.e. N(0,0.001) added. The results of fitting rate distributions using the first 60 Laguerre expressions for different values of L are shown in Fig. 2 and Table 1. In Fig. 2 the Laguerre fit proportions are calculated at intervals of 0.1 and normalised to correspond with the discrete rates fit also done using intervals of 0.1. The probability values indicate the small values of L over fit the data. The area under the distribution curve starts at one and only starts to reduce for largest value of L. The initial over fitting makes it difficult to determine L using the error values. The probability indicates a value of L between 0.01 and 0.03. The graphs show a moderate agreement with the original rectangle with L between 0.01 and 0.03. Table 2 and Fig. 3 show the result of fitting discrete rates at intervals of 0.1. In this case the probability indicates only slight

Fig. 7. Laguerre series fit to pentlandite data from Cherevaty and Agar (2004).

8

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

Fig. 8. Discrete rates fit to pentlandite data from Cherevaty and Agar (2004).

over fitting and the area stays at 1.00. The graphs initially favour three discrete rates and approximate the rectangular distribution with L about 0.03. Both the Laguerre and discrete rate fits smooth the upper edge of the rate distribution as might be expected given the known ill conditioning of fitting decaying exponentials. The probability statistic is easier to use than the error values or the area. The Laguerre fit is not limited to positive rate values (although negative values could have been set to zero) and allows more flexibility to over fit, while the discrete rates corresponding to a positive sum of decaying exponentials is more constrained to a realistic rate distribution and fraction remaining curve.

5. Experimental data To demonstrate the rate distribution calculation on actual data, data from Cherevaty and Agar, 2004 are used. These data comes from a batch cell developed to give repeatable results, and they have provided data with three repeats so that estimates of the accuracy can be made. Table 3 gives values from Cherevaty and Agar converted to fraction remaining in cell for chalcopyrite (CuFeS2) and pentlandite (NiFeS2). To use the distribution given in Fig. 1 to test fits to these data, it is necessary to confirm these data are compatible with the assumption of equally sized errors for each of the timed samples. Given the symbol sd represents the vector of standard deviations of the data rows (each sample time) the statistic calculated

is rms(sd-rms(sd))/rms(sd) (rms is root mean square) which gives a measure of how much the row standard deviations vary from the overall standard deviation. Fig. 4 plots the cumulative distribution of 10,000 samples of this statistic and also shows the values for the two data sets. Both data sets are low on this distribution showing there is no evidence of variation in the standard deviations with time. Thus the distribution in Fig. 1 which assumes the distribution of the errors is constant with time, can be used to compare fits to this experimental data. If the standard deviations are not constant it may be possible to find a trend and thus estimate the individual standard deviations. Given an assumed trend the above statistic can be extended to include this case. If the standard deviations at the different times are assumed independent the fit statistic (by extending Section 3) with only three repeats for each value has a long tail and it is not easy to use to determine L.

6. Chalcopyrite data This section demonstrates the determination of the rate distribution for the chalcopyrite data from Cherevaty and Agar (2004). After an adjustment to the test start time an excellent fit to the data is obtained with the rate distribution indicating two spread peaks and a small amount of non floating component. Similar to Cherevaty and Agar (2004) a time offset was introduced to analysis their chalcopyrite data, due to the need to allow for the startup behaviour of batch flotation. To determine an

R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12 Table 6 Fit to the pentlandite data using Laguerre functions. L

Error

Rel. error

Probability

Area

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0034 0.0041 0.0046 0.0053 0.0069 0.0210

0.84 0.99 1.12 1.30 1.70 5.13

1.00 0.49 0.15 0.02 0.00 0.00

0.99 1.00 1.00 0.99 0.97 0.93

RMS of data standard deviations 0.00409.

Table 7 Fit to the pentlandite data using discrete rates. L

Error

Rel. error

Probability

Area

1.0e03 3.0e03 1.0e02 3.0e02 1.0e01 3.0e01

0.0043 0.0043 0.0043 0.0044 0.0049 0.0064

1.04 1.04 1.05 1.08 1.19 1.56

0.31 0.31 0.29 0.23 0.08 0.00

1.00 1.00 1.00 1.00 1.00 1.00

9

subtracting 0.08 reduced the sum to one, and so the adjusted times were used for analysis. A non floating component and the first sixty Laguerre functions (Li(x) exp(x)) were fitted to the data with the result shown in Fig. 5, a summary of the results is in Table 4. Fig. 6 shows the fit on discrete rates at intervals of 0.1 min1 and Table 4 shows the result of fit for different values of L and Table 5 summarises these results. Both the Laguerre rates and the discrete rates indicate nonfloating, fast and slow floating components. The Laguerre fits show evidence of over fitting initially. This is possible as the standard deviation is calculated using two degrees of freedom, which allows the root mean squares of the fitted errors calculated without a degrees of freedom correction, to become smaller than the standard deviations of the data. The discrete rates for low regularisation show two discrete rates and a non floating component. As the amount of regularisation increases the peaks become spread to become similar to the Laguerre function case at the probability measure of about 0.2.

RMS of data standard deviations 0.00409.

7. Pentlandite data

effective start time the zero time fractional content was not fitted. Unlike the simulated data it was found, using the times given, the sum of the mass of the flotation components was greater than one. Subtracting an amount from the given times reduces this sum, and

The pentlandite data shows similar behaviour to the chalcopyrite. A slightly higher time offset of 0.11 was found by adjusting the initial time to give the integral of the calculated rates as one. This higher value may be related to the chalcopyrite having a higher flotation rate and suppressing initial pentlandite flotation.

Fig. 9. Fit using discrete rates to individual flotation tests for chalcopyrite and pentlandite, data from Cherevaty and Agar (2004).

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R.L. Pascual, W.J. Whiten / Minerals Engineering 83 (2015) 1–12

The repeatability of the fits can be investigated by adding some random variation to the data. Gaussian distributed samples distributed with the standard deviation calculated from the data (chalcopyrite 0.0045 and pentlandite 0.0041) were added to the fraction remaining data, and the rate distributions recalculated 100 times. This procedure adds the expected error to the data which also contains errors, thus the resulting variation is about 1.4 times what might be in the calculated values. The middle graphs in Fig. 10 show the original fit and the repeat values. The perturbed repeat values all show the same type of fit with a non floating component and two spread peaks. The final two graphs in Fig. 10 shows the one standard deviation size of the variation in the fitted rate distribution.

Fig. 7 shows the flotation rate distribution for the Laguerre function fit and Table 6 summarises the effect of difference L values. Table 7 and Fig. 8 show the results fitting a discrete rate distribution. Again two peaks and a non-floating discrete component can be seen. Behaviour is very similar to that of the chalcopyrite. 8. Repeatability of fits It is not sufficient to calculate the flotation rate distribution without also determining an accuracy of the distribution. The rate distributions calculated for each of the three repeats are found to be similar to that calculated for the combined distribution calculated in the previous sections. To further investigate the accuracy of the fitted rate distribution the data are perturbed to determine the variation in the rate distribution. Both the chalcopyrite and pentlandite data have three repeats. Having determined an L value of about 0.06 as giving a reasonable fit to the data, the three tests can be fitted individually. Fig. 9 shows theses fits using discrete rates. It can be seen the form of the calculated rate distribution is similar in each case. The fit to the fraction remaining curves is close, and it is difficult to see the differences. The top graphs in Fig. 10 compares the errors in the fit to the standard deviations calculated from the data. The probabilities for the fits shown in Fig. 10 are calculated as 0.28 and 0.15. The errors seen in these graphs can be compared with the standard deviations used to perturb the data as described in the next paragraph.

9. Conclusions Determining the rate distribution from batch test data is difficult, due to the way different rate distributions can give virtually the same fraction remaining curve. It is easy to over fit the data, letting the randomness in the fraction remaining effect the calculated rate distribution. To get good results it is necessary to have both accurate data to reduce the variation in the calculated rate distribution, and repeated data to provide an estimate of the data accuracy. Even with these there usually are possible alternative interpretations of the data. Batch flotation data are often fitted with two or three discrete rates, but it is difficult to believe that particles of different sizes,

Chalcopyrite L=0.06 P=0.30

Pentlandite L=0.06 P=0.16 0.01

0.005

0.005

0

0

-0.005

-0.005

Error

0.01

-0.01

-0.01 0

5

10

0

5

Time Chalcopyrite L=0.06 Rpts 100 Distributuion

0.2

Pentlandite L=0.06 Rpts 100 0.2

Fitted Bootstrap

0.15

Fitted Bootstrap

0.15

0.1

0.1

0.05

0.05

0

0 0

2

4

6

0

2

Distribution one sd error

Rate Chalcopyrite L=0.06 Rpts 100

0.01

0.01

0.005

0.005

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6

Pentlandite L=0.06 Rpts 100

0.015

0

4

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10

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6

0

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Fig. 10. Top row: errors in discrete rate fit to data from Cherevaty and Agar (2004), error bars are plus and minus one standard deviation from the experimental data for each time interval. Middle row: fit distribution using discrete rates and 100 bootstrap repeats. Bottom row: one standard deviation values for bootstrap repeats.

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shapes, and composition such as composite particles, are all floating at the same rate. It is believed that a distribution of the rates gives more reliable information for understanding the nature of the ore and for making predictions of plant performance. Even when the ore is divided into size and composition fractions, the particles within a fraction are far from identical and a spread of flotation rates can be expected. Fitting with a continuous function such as the Laguerre function used in this paper, or a dense distribution of discrete rates can provide similar distributions of flotation rates given an appropriate amount of regularisation. It is felt the discrete rates are more flexible, avoid negative rate values, are less prone to over fitting, and the exponential decay of the rate components is more compatible with the observed behaviour. The extra regularisation condition of positive coefficients provides more stability to the discrete rates result. Further the discrete rates can be directly applied to continuous flotation simulations, while it is normally necessary to convert a continuous rate distribution to discrete rates for plant simulations. Provided there are repeat values the distribution for the errors in fitting can be calculated. This is very useful in determining the amount of regularisation required. However too few repeats will cause a long tail on the distribution of the errors, making it more difficult to determine a level for selection of the regularisation parameter L. The use of the linear model where the content of the cell is a sum of components that behave independently according to first order kinetics, can be questioned. However a more complex model, for instance including bubble overloading, introduces more parameters and increases the difficulty of calculating the rate distribution. An open question is how can more information about the rate distribution can be obtained. Does refloating of the concentrate give useful information, or does data from a pilot plant. What is clear is accurate data are required, and repeat data are needed to determine the accuracy of the data to avoid over fitting. An important question, that is also open, is how accurate are predictions made from the rate distributions. This usually depends not only on the accuracy of the fit to the data, but also assumptions made about the range of application of the model which assumes variables as reagent additions, bubble concentration, and unit scale are constant. Author contributions Ricardo Pascual examined the literature and developed the Laguerre techniques (Pascual, 2010). Bill Whiten extended the work to include the fitting of discrete rates. Acknowledgements The authors gratefully acknowledge the quality data published by Cherevaty and Agar (2004). The AMIRA P9 project at the Julius Kruttschnitt Mineral Research Centre supported the earlier part of this work. Extensions to the work were undertaken following the retirement of Bill Whiten. Suggestions to improve the text from the referees are acknowledged. References Abramowitz, M., Stegun, L.A., 1965. Handbook of Mathematical Functions, Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (reprinted 1972 by Dover Publication, New York). Arfken, G., 1970. Mathematical Methods for Physicists. Academic, New York, pp. 414–416.

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