First-order flotation kinetics models and methods for estimation of the true distribution of flotation rate constants

Int. J. Miner. Process. 58 Ž2000. 145–166 www.elsevier.nlrlocaterijminpro

First-order flotation kinetics models and methods for estimation of the true distribution of flotation rate constants M. Polat a

a,1

, S. Chander

b,)

Mining Engineering Department, Dokuz Eylul ¨ UniÕersity, BornoÕa, 35100 Izmir, Turkey PennsylÕania State UniÕersity, 115 Hosler Building, UniÕersity Park, PA 16802, USA

b

Received 2 July 1999; received in revised form 13 September 1999; accepted 30 September 1999

Abstract To improve their versatility, many first-order flotation kinetics models with distributions of flotation rate constants were redefined so that they could all be represented by the same set of three model parameters. As a result, the width of the distribution become independent of its mean, and parameters of the model and the curve fitting errors, became virtually the same, independent of the chosen distribution function. For the modified three-parameter models, the curve fitting errors were much smaller and their robustness, measured by PRESS residuals, was much better when compared to the corresponding two-parameter models. Three different methods were compared to perform flotation kinetics analysis and estimate model parameters. In Method I, recovery vs. time data were used to obtain model parameters. No significant insight into the distribution of rate constants could be obtained because the distributions were presupposed. In Method II, the froth products were fractionated into several size fractions and the data for each fraction were fitted to a model. This task was easy to perform and the method could describe the flotation kinetics reasonably well. In method III, flotation products were fractionated into many size-specific gravity fractions. The procedure involved a large amount of time and effort and it generated relatively large errors. Based on the analysis presented in this article, it was concluded the smallest errors were obtained with Method II. The overall distribution of flotation rate constants could be obtained from a weighted average of the distributions of individual size fractions. The distributions so obtained were demonstrated to be less sensitive to the choice of the model used to represent the kinetics of individual size fractions, and therefore can be assumed to

) 1

Corresponding author. Fax: q1-814-865-32-48; E-mail: [email protected] Fax: q1-232-373-82-89; E-mail: [email protected].

0301-7516r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 Ž 9 9 . 0 0 0 6 9 - 1

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be ‘‘true’’ representation of the flotation rate distribution. q 2000 Elsevier Science B.V. All rights reserved. Keywords: first-order flotation kinetics models; methods for estimation; flotation rate constants

1. Introduction Micro- and macro-scale models have often been used to describe the kinetics of flotation. In the micro-scale models, the sub-processes of the flotation system are identified and used to determine the cause and effect relationships between the system variables. However, identification of all sub-process and the corresponding cause and effect relationships is extremely difficult because of the interactions between physical and chemical parameters. The reader is referred to articles by Mika and Fuerstenau Ž1969., Schulze Ž1977., and Chander and Polat Ž1995., for a discussion of this approach. In macro-scale modeling, the overall response of the flotation system is related to various operating parameters through a set of mathematical equations. The macro-scale models can be divided into two categories, namely, empirical and phenomenological ŽWoodburn, 1970; Lynch et al., 1981; Herbst and Basur, 1984.. In empirical models, the amount of material floated is related to input and output variables through suitable mathematical equations. Statistical methods are used to relate dependent and independent variables and to estimate the curve fitting parameters. The parameters obtained by such an analysis usually do not have any physical significance. On the other hand, phenomenological models provide correlations between cause and effect relationships through the use of equations that are related to the physics of the process. Phenomenological models can be further classified as kinetic, probabilistic and population balance types. Probabilistic models are based on the relative occurrence of various sub-processes such as collision, adhesion and detachment and can serve as a bridge to integrate micro and macro-scale models. Kinetic models that invoke the chemical reactor analogy and consider flotation as a reaction between bubbles and particles have received most attention in the literature. These models can be readily adopted for development of control strategies in industrial applications. Some such models were modified as a part of this investigation in order to increase their versatility. The general rate equation for flotation may be written as: dCp Ž t . dt

s yk Ž t . Cpm Ž t . Cbn Ž t .

Ž 1.

where Cp Ž t . and Cb Ž t . are the concentrations of the particles and bubbles at time t, respectively. The exponents, m and n are the respective orders for particles and bubbles, and k Ž t . is a pseudo rate constant that depends on various parameters governing the flotation process, and may vary with time. There has been a great deal of discussion over the actual order of the flotation process ŽArbiter, 1951; Bogdanov et al., 1954; De Bruyn and Modi, 1956; Klassen and Mocrousov, 1963; Tomlinson and Fleming, 1965; Somasundaran and Lin, 1973., but the first-order kinetics introduced by Garcia-Zuniga Ž1970. and Schuhmann Ž1942. has been

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used most extensively. It is based on the assumptions that the rate of the particle–bubble collision process is first-order with respect to the number of particles and that the bubble concentration remains constant ŽSutherland, 1948.. Batch flotation test data in the literature support the first-order rate equation under reasonable operating conditions ŽImauzimi and Inoue, 1963; Tomlinson and Fleming, 1965; Harris and Chakravarti, 1970; Jameson et al., 1977; Dowling et al., 1985; Rastogi and Aplan, 1985.. Jowett and Safvi Ž1960. also reported observing first-order kinetics in continuous flotation tests. Solution of the first-order rate equation results in the classical first-order flotation model. The corresponding equation, along with relations for other models are given in Table 1. The classical first-order equation needs to be modified, however, for representing the flotation rate data for a wide range of conditions. Kapur and Mehrotra Ž1974. classified these modifications in a species–phase matrix. The multiple phase concept arises from the fact that flotation occurs at least in two phases, namely pulp and froth; a review of multi-phase models was presented by Harris and Rimmer Ž1966. and Harris Ž1978.. Models with distributed species arise because the actual feed to a flotation cell might differ in particle size, shape, surface properties, etc. The species distribution could be assumed to be discrete ŽMorris, 1952; Kelsall and Asquith, 1981; Jowett, 1974; Huber-Panu et al., 1976; Thorne et al., 1976. or continuous. Extending multiphase models to include a distribution of species results in expressions that cannot be integrated in closed form. If the material to be floated consists of particles whose k’s could be expressed by a continuous distribution function, f Ž k ., the recovery at time t will be equal to: `

R Ž t . s R` 1 y

H0

f Ž k . exp Ž ykt . d k

Ž 2.

Here, R` is the ultimate recovery at long times 2 ŽImauzimi and Inoue, 1963.. Various distribution functions have been proposed by different investigators to account for the variability in the k’s. These include as gamma ŽImauzimi and Inoue, 1963; Loveday, 1966., rectangular ŽHuber-Panu et al., 1976; Klimpel, 1980., triangular ŽHarris and Chakravarti, 1970., and sinusoidal ŽDiao et al., 1992.. There is disagreement in the literature as to which function is better suited to represent the actual flotation rate distribution, especially for a wide range of flotation conditions ŽJowett, 1974; Harris and Cuadros-Paz, 1978; Feteris et al., 1987; Wanangamudi and Rao, 1986; Diao et al., 1992.. In a comprehensive analysis of 13 commonly used models, Dowling et al. Ž1985. concluded that there is no single model that is sufficient to represent flotation rate data and the ‘‘best’’ model may be different for various flotation conditions. Another complication arises because model parameters cannot be compared among different first-order models since they correspond to a different property of the distribution function for each model. For example, the rate parameter for the classical first-order

2 The parameter R` is not an arbitrary parameter. One can write Eq. Ž2. by considering two independent distribution functions, f 1Ž k . and f 2 Ž k . with respective fractions of F0 and 1y F0 where f 2 Ž k . is a dirac-delta function situated at k s 0.

N Normal Distribution

3 e

2

Ž m

k

0 2m - k - `

'

m 2p

9 y1

.

2

0F kF 2m

erf

mty 3

y erf

2

2e

m ty

18

m2 t 2

2

ž' / ž' /

mtq 3

1

ey 2

1

Ž

s

ky m

m

m2

.

s2

2

0F kF 2m0 2m- k- `

'

s2

s2

ž /ž /

m2

s 2p

G

Ž1 .r ŽŽ1 q Kt . n .

ŽŽ krK . ny 1 e Žy k r K . .r Ž k G Ž n ..

mk s2

G Gamma Distribution

s2

e

Ž1 y Ž2 Ktr p .e yK t .r Ž1 q Ž2 Ktr p .. 2 .

Žp r2 K .Sin Žp kr2 K . @0 F k F K 0 @K- k- `

S Sinusoidal Distribution

y1

Ž1 q e y K t y 2e K t r2 .r ŽŽ Ktr2 . 2 .

4 krK 2 @0 F k - Ž K .r2 4 Ž K y k .rK 2 @ Ž K .r2 - k F K 0@ K - k - `

T Triangular Distribution

m2

0 @0 F k - m y Ž w .r2 Ž1 .r Ž w . @ m y Ž w .r2 F k F m q Ž w .r2 0 @ m q Ž w .r2 - k - ` 0 @0 F k - Ž2 m y w .r2 Ž4 Ž k y m q Ž wr2 ..rw 2 @ Ž2 m y w .r2 F k - m Ž4 Ž2 m y k ..rw 2 @ m - k F Ž2 m q w .r2 0 @ Ž2 m y w .r2 - k - ` 0 @0 F k - m y Ž2 w .r3 Žp .r Ž2 w .Sin ŽŽp Ž k y m ..r 2 w q Žp r3 . @ m y Ž2 w .r Ž3 . F k F m q Ž2 wr6 . 0 @ m q Ž2 w .r Ž6 . - k - `

Ž1 y e yK t .rKt

0 @0F k - K ` @ks K 0 @K- k- ` 1rK @0 - k F K 0 @K- k- `

C Single-valued Distribution ŽClassical.

k

fŽk. Not available

ey K t

fŽk.

R Rectangular Distribution

Three-parameter Form 1 y R Ž t .rR `

Two-parameter Form

First-order models with

Table 1 First-order models with distribution of flotation rate constants Ž k ’s . analyzed in this work

1 y R Ž t .rR `

erf

ž

s

m

/

m2 s2

y erf

s ty

s

m

'2

2e m t

2

s 2t2

'2

ž / ž /

s tq

m

mqs 2t

Že y m tq Ž2 w t .r Ž3. y Ž2 wt .r Žp . e y m tŽ2 w t .r Ž6. .r Ž1 q Ž2 wtr p . 2 .

Že Ž w t .r 2 q e Ž w t .r 2 y 2 .r Že y m t Ž wtr2 . 2 .

Že y m tq Ž w t .r 2 y e y m tŽ w t .r 2 .rwt

Not available

148 M. Polat, S. Chanderr Int. J. Miner. Process. 58 (2000) 145–166

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model represents some kind of mean flotation rate while the rate parameter for the first-order model with a rectangular distribution of k’s corresponds to the flotation rate of the fastest floating component. The evaluation of flotation kinetics using an appropriate first-order model can be carried out in several ways. Most often, the total flotation recovery is used in the estimation of k distributions. Since the ‘‘true’’ distribution of k’s is a function of both the size and hydrophobicity of the particles, such estimations using pre-selected functions are not satisfactory for most cases. The effect of particle size and hydrophobicity may be distinguished by employing narrow, non-overlapping size fractions of the flotation products for kinetic analysis. Fractionation of the froth products into size and hydrophobicity intervals is yet another, but more involved way of estimating the distribution of k’s. A systematic evaluation of these methods on the basis of their accuracy and reliability in estimating the ‘‘true’’ distribution of k’s is a necessary step in advancing the current state of macro-scale flotation kinetic modeling. The purpose of this paper is two fold. Ž1. To redefine the first order flotation kinetics models with a distribution of flotation rates such that all the models can be represented by the same three parameters, namely a mean k Ž m ., the standard deviation or the width of the distribution around the mean Ž s , or w ., and the ultimate recovery of floatable material Ž R` . ŽTable 1.. This would ensure that the width of the k distribution is independent of its location as well as providing a common ground for comparison of all the models. Ž2. To develop a physically representative and reliable kinetics analysis procedure using the modified models to estimate the ‘‘true’’ distribution of k’s. The analysis in this article is limited to coal flotation. 3

2. Experimental and statistical analysis procedures 2.1. Materials and sample preparation Three coal samples were used in this investigation: As-mined raw coals from the Upper Freeport and Pittsburgh seams were chosen as typical of those processed in a flotation plant. These are designated as UF and PR, respectively. A third sample, which was the product from a jigging plant was chosen since it was believed to have a narrow distribution of k’s in narrow size intervals due to a very small ash content. This sample was designated as PC. Selected physical properties of these coals are given in Table 2. The float-sink analysis of these coals is given in Table 3. The as-received coal samples were crushed down to y3.35 mm and stored under an argon atmosphere in aluminum-laminated plastic bags of 15 kg capacity. The 15 kg samples were further divided into samples of 500 g as needed and stored in smaller bags 3 For coals, the hydrophobicity of particles is related to their maceral and mineral matter content. Hence, specific gravity fractionation may be used to fractionate the particles into hydrophobicity intervals since both different macerals and mineral matter are known to have distinctly different densities.

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Table 2 Description and some properties of the coal samples used in the study Sample

ID

Upper Freeport, Raw Pittsburgh, Raw Pittsburgh, Clean

UFyR PR PC

Rank

Weight Ž%.

lv bituminous hvA bituminous hvA bituminous

H 2O

VM

FC

Ash

1.6 3.1 2.6

23.1 22.8 36.9

49.9 37.2 57.1

36.0 40.0 6.0

under the same conditions. Prior to each flotation test these samples were wet-ground to y600 mm nominal size using a rod mill. 2.2. Flotation kinetics Flotation tests were carried out in a 5-l cell using a Wemco laboratory flotation machine at a solids concentration of 10%. The flotation cell was equipped with paddles for automatic froth removal based on an earlier design by Miller ŽMiller, 1980; Luttrell and Yoon, 1983. which minimized the effect of froth height on the flotation kinetics. Make-up water was added as necessary to maintain a constant pulp level. Dodecane and methyl isobutyl carbinol ŽMIBC. were employed as flotation reagents. 2.3. Fractionation of froth products Froth products were collected at preset time intervals and were wet screened at 37 mm to prevent caking during drying. Both the q37 mm and y37 mm fractions of each froth product were filtered and dried overnight at 808C. They were stored separately for the subsequent size or size-specific gravity fractionation. 2.3.1. Size fractionation The q37 mm fraction obtained from the wet screening step was dry-screened at screen sizes of 300, 150, 75 and 37 mm. An overall y37 mm material was obtained for each froth product by combining the screen undersize from the wet and dry screening steps. The size-fractionated flotation products were weighed and analyzed for ash to obtain the flotation recovery, R i Ž t ., in each time-size fraction.

Table 3 Float-sink analysis of the coal samples used in the study Specific gravity

F @1.4 1.4=1.6 1.6=2.0 S @2.0

Weight Ž%.

Ash Ž%.

UF-R

PR

PCU

UF-R

PR

PCU

47.0 11.8 5.9 35.3

43.8 15.4 8.3 33.5

93.1 4.0 1.3 1.6

10.2 23.1 49.0 81.4

6.8 20.3 47.2 85.6

4.5 21.8 47.4 61.2

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2.3.2. Size-specific graÕity fractionation The q37 and y37 mm fractions were classified separately at specific gravities of 1.3, 1.5, 1.7 and 1.9. The specific gravity fractions of the q37 mm material were further classified at the screen sizes given above. A centrifuge was used for specific gravity fractionation of the y37 mm material. Additional details of the procedure for fractionation of froth products into various size-specific gravity fractions are given in an article by Zhou et al. Ž1993.. The size-specific gravity fractionated froth products were weighed and analyzed for ash to obtain a flotation recovery, R i j Ž t ., for each time-size-specific gravity fraction. It was seen early in the test work that some size-specific gravity fractions contained rather small amounts of material based on the 500 g feed used in a flotation test. Hence, a given flotation test was repeated between 7 to 10 times under identical conditions until a sufficient amount of material was obtained. The reproducibility of these tests was fairly good ŽPolat et al., 1993.. 2.4. Statistical analysis of the kinetics data The recovery vs. time data for the total, size fractionated and size-specific gravity fractionated materials were subjected to a curve fitting procedure using various first-order models. Each curve fitting procedure resulted in the best estimates of the parameters of a given model. The curve fitting errors for the total material, E l , and for the size fractionated, Ei , and size-specific gravity fractionated, Ei j , materials were calculated using the equations given below: q

El s Ý ts1 q

Ei s Ý ts1 q

Ei j s Ý ts1

R Ž t . y RX Ž t .

2

;

qyp R i Ž t . y RXi Ž t .

2

qyp R i j Ž t . y RXi j Ž t . qyp

; 2

Ž 3.

RX Ž t ., RXi Ž t . and RXi j Ž t . are the model-predicted recovery values in these equations. The parameter q is the number of data points and p is the number of model parameters. 2.5. Microscopy The size fractionated froth products from various time increments were analyzed using oil immersion microscopy to determine the amount of mineral matter locked with carbonaceous material. Cylindrical pellets molded by mixing the coal particles with an epoxy resin were prepared to create a surface for reflected-light microscopic analysis. The microscope was coupled to a automatic counter with a stage translator. A scale of locking was established on the basis of the percentage of the particles’ projected surface

152

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area occupied by the ash material. Frequency plots showing the percent of particles in each locking interval were obtained for groups of 500 particles. 2.6. Film flotation These studies were carried out to determine the hydrophobicity distribution of the size fractionated samples obtained either from the flotation feed or from froth products at different time intervals. Film flotation is based on successive fractionation of a given sample, preferably of a narrow size fraction, by aqueous methanol solutions of varying surface tension ŽHornsby and Leja, 1984; Fuerstenau and Williams, 1987; Fuerstenau et al., 1988.. A sample amount of about 10 to 30 mg was used in a given test. A conical vessel filled with the water–alcohol mixture of desired surface tension and with a liquid surface area of about 175 cm2 was utilized in the tests. The particles were dropped at the air–water interface one by one from a height of 1 cm using an adjustable vibratory feeder. Special care was taken so that the particles would not fall on top of each other. The details of the film flotation techniques are given in an article by Fuerstenau et al. Ž1988..

3. Results and discussion This study was carried out in two phases. In the first phase of the work ŽSection 3.1., the flotation rate data were subjected to a curve fitting procedure to compare the modified forms of the models within themselves and with their original forms. The predictions of the k distributions by these models were cross-checked using independent analysis methods of microscopy and film flotation. In the second phase of the work ŽSection 3.2., the rate data were evaluated by different methods of flotation kinetics analysis using the modified models. For the sake of brevity, only the results for the selected models will be presented in this paper. However, they have been observed to be true for all the other models tested as well ŽPolat, 1995.. 3.1. Modification and eÕaluation of first-order models with distribution of k’s 3.1.1. Kinetic flotation tests The two raw coal samples UF and PC were floated at dodecane and MIBC concentrations of 0.3 kgrT and the rate data were analyzed using standard curve fitting procedures. The distributions of flotation rate constants obtained by fitting Models R2 and R3 as a function of particle size are given in Fig. 1a for the UF and Fig. 1b for the PC coal samples. The corresponding curve fitting errors for all the models tested are given in Fig. 2. The following observations could be made. Ž1. For a given data set, both the magnitudes of the model parameters and the curve fitting errors were virtually the same for all the modified three-parameter models. On the other hand, they varied considerably for the two-parameter models.

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Fig. 1. Flotation rate constant distributions for Ža. upper Freeport, UF, and Žb. Pittsburgh, PC coals estimated by models R2 and R3. The frequency at a given flotation rate was multiplied by the weight fraction of that material in the feed to represent the relative contribution.

Ž2. The modified models predicted narrow distributions of k’s for all size fractions for the PC sample whereas the distributions were wider for the UF sample. The distributions approached a single-valued function, corresponding to Model C, for several size fractions of the sample PC. Ž3. Fine size fractions, especially the y37 mm fraction, generally gave relatively large curve fitting errors; most probably because flotation of fines deviates from

Fig. 2. Curve fitting errors generated by various two- and three parameter models for the Upper Freeport ŽUF. and Pittsburgh ŽPC. coal samples.

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first-order kinetics. Nevertheless, the curve fitting errors were significantly smaller for the modified 3-parameter models, especially for the sample PC. 3.1.2. Verification of the distributions of k’s predicted by three parameter models The k distributions predicted by model R3 were quite different from those predicted by model R2, for various size fractions of the two coal samples, as can be seen from the results in Fig. 1. Also, for model R3 the predicted distributions were broad for the coal sample UF, whereas they were narrow for the sample PC. These results are consistent with the fact that the former sample contained a considerable amount of locked-high specific gravity material, which can be inferred from the data in Tables 2 and 3. In comparison, the sample PC was made up of mostly free coal particles. In order to evaluate the accuracy of the distributions predicted by R2 and R3, two independent experimental procedures, namely optical microscopy and film flotation, were employed. These tests were aimed, respectively, at determining the locking behavior and estimating the hydrophobicity distribution and the results are discussed in the paragraphs that follow. Microscopy. The 300 = 600 mm size fraction of the froth products of UF coal and the 75 = 150 mm size fraction of the PC coal were examined under a microscope. Only the froth samples from the time intervals of 0–20, 20–40 and 60–120 s were employed since the froth products obtained at longer times did not have sufficient material for analysis. The results of the microscopic evaluation are given in Fig. 3a and 3b for these two size fractions. The k distributions predicted by Model R3 are also included as insets in each figure for comparison. Fig. 3a shows that the amount of locked ash in the flotation product increases significantly as flotation time increases for the UF sample. While the majority of

Fig. 3. Degree of locking as determined by microscopy for selected froth products of Ža. the 300=600 mm size fraction from the Upper Freeport coal sample ŽUF. and Žb. the 75=150 mm size fraction from the Pittsburgh coal sample ŽPC.. U : The percent ash stands for the estimated amount of locked ash in a given particle by microscopy.

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155

particles floating in the initial 20 s were low in ash, those obtained at longer times contained notable amounts of locked particles. Since the kinetics analysis is based on flotation recovery vs. time data, the presence of locked particle may be considered as confirmation of a broad distribution of k’s as predicted by Model R3. The flotation products of the 75 = 150 mm size fraction of the PC sample, on the other hand, predominantly consisted of practically ash-free coal particles at all the time intervals analyzed. This type of behavior suggests that material floating at different times had similar hydrophobicities, which is consistent with a narrow distribution of k’s, as predicted by Model R3. Film flotation. To further confirm the presence of particles with different hydophobicities, the two size fractions discussed above were subjected to film flotation. The tests were carried out on the feed material, and the 0–20 s and 20–40 s froth products. The results are presented in Fig. 4a and 4b for the UF and the PC samples, respectively. The topmost graph in each figure displays the cumulative hydrophobic fraction vs. surface tension plots. The three bar graphs at the bottom were obtained by differentiating the cumulative distributions. They represent the hydrophobicity distribution of particles in various materials. The particles of the UF feed material displayed a wide distribution of hydrophobicities as can be seen from the results in Fig. 4a. The froth product for the 0–20 s time interval consisted of particles of very high hydrophobicity with a narrow hydrophobicity distribution. The froth product for the 20–40 s time interval had a wider distribution that tailed off at high surface tensions, indicating the presence of locked particles. When the

Fig. 4. Partition curves obtained by film flotation for the feed materials of Ža. the Upper Freeport ŽUF. and Žb. Pittsburgh ŽPC. coals and selected froth products.

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hydrophobicity of the froth product changes with flotation time, a broad distribution of k’s can be expected when kinetic models are used to fit the data. The behavior of the coal sample PC was quite different as can be seen from Fig. 4b. The hydrophobicity distribution of the feed material was relatively narrow with no material sinking at high surface tensions. The shapes of the distributions for 0–20 and 20–40 s froth products were almost identical, suggesting that the material coming out of the flotation cell did not change in hydrophobicity with increasing time of flotation. As a consequence one would expect a narrow distribution of k’s. 3.1.3. Robustness of two- Õs. three-parameter models The preceding discussion established that the modified three-parameter, first-order models represented the flotation rate data much better than their two-parameter counterparts both in terms of the physical meaning of the model parameters and the curve fitting errors. In this section, the robustness of the model, was evaluated using the kinetics data from Fig. 1. A model was considered robust if the model parameters are independent of small changes in the actual data. It was calculated using the procedure outlined by Mazumdar Ž1994. after Meyers Ž1990.. It involves calculation of predictive sum of square error ŽPRESS. residuals. The details of the procedure are summarized as follows. Ža. Assume a set of rate data consisting of recoveries RŽ t1 ., RŽ t 2 ., . . . , RŽ t q . at times t1 , t 2 , . . . , tq . Žb. Delete the first pair Ž RŽ t 1 ., t 1 . from the data set and calculate the model parameters by curve fitting on the remaining pairs and obtain a predicted recovery, RU Ž t 1 . at t 1 , from the model parameters. Žc. Repeat the process q times for all data pairs. Žd. Calculate the average PRESS residual from the formula: q

Ý E PRESS s

R k Ž t k . y RUk Ž t k .

2

ks1

q

Ž 4.

For a robust model, the PRESS residuals are small. The average PRESS residuals for the flotation of the intermediate size fractions of the UF and PC coal samples are given in Fig. 5 for models R2 and R3. It is apparent from the significantly smaller PRESS residuals that R3 is a much more robust model than R2 for all size fractions. This was observed to be true for the other modified models as well, with the exception of G3. Dowling et al. Ž1985. also reported ‘‘gross dilution of parameter confidence limits’’ for this model. For a more detailed discussion, refer to Polat Ž1995.. 3.2. DeÕelopment of a reliable kinetics analysis method using modified models The UF and PR coal samples were floated in the absence and presence Ž0.65 kgrT. of dodecane in the second phase of the work. The MIBC concentration was 0.65 kgrT in all cases. The flotation rate data were evaluated using the modified three-parameter models by the application of three different methods to determine the best approach to

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Fig. 5. Average PRESS residuals for a. Upper Freeport ŽUF. and b. Pittsburgh ŽPC. coal samples for the twoand three-parameter first-order models with rectangular distribution of flotation rate constants.

kinetics analysis. Method I is based on fitting the model to the total amount of material floated as a function of time, RŽ t ., and calculating an overall flotation rate distribution, f I Ž k . from the model parameters. Method II is based on fractionating the flotation products into individual size fractions and obtaining the flotation rate data, R i Ž t . for each size fraction. In Method III, flotation products are fractionated into both size and specific gravity intervals to obtain flotation rate data, R i j Ž t ., for individual size and specific gravity intervals. The k distributions for each size and size-specific gravity interval, f i Ž k . and f i j Ž k ., can be obtained by fitting the model to R i Ž t . and R i j Ž t . data, respectively. Overall flotation rate distributions, f II Ž k . and f III Ž k . can then be calculated from the f i Ž k . and f i j Ž k . values by taking a weighed average using the equations: m

m

f II Ž k . s Ý f i Ž k . g i ;

f III Ž k . s Ý

is1

n

Ý fi j Ž k . g i j

Ž 5.

is1 js1

where m and n are the number of size and size-specific gravity intervals. the respective frequency of these intervals in the overall flotation feed are given by g i and g i j . Overall errors, E II , for Method II and, EIII , for Method III can be calculated by taking the weighed average of the errors for the size and size-specific gravity fractions: m

E II s Ý Ei g i ; is1

m

EIII s Ý

n

Ý Ei j g i j

Ž 6.

is1 js1

3.2.1. Analysis using total flotation products, Method I The total flotation recovery that was used in the kinetic analysis by Method I was obtained by analyzing the flotation products for weight and ash prior to any fractionation. A summary of the model parameters for the total flotation recovery is given in Table 4 for the five kinetics models. The table shows that the increase in the ultimate recovery by addition of the collector is minimal for the both coals Žabout 3% in both

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Table 4 Kinetics parameters generated by the various first-order models for the two coal samples Collector

UF-R

PR

None

0.65 kgrT

None

0.65 kgrT

C

RA mC

86.8 1.86

91.5 3.12

89.2 2.93

91.5 6.60

R3

RA mR w

89.3 2.10 3.61

93.0 3.69 6.03

90.0 3.25 4.36

92.8 7.78 15.6

S3

RA mS w

88.5 2.08 3.11

92.8 3.61 5.42

90.5 3.37 5.04

92.3 7.68 11.5

N3

RA mN sN

87.7 1.95 0.65

92.3 3.34 1.11

90.5 3.13 1.04

91.9 6.86 2.29

G3

RA mG sG

88.4 2.16 1.28

93.0 3.81 2.16

90.0 3.29 1.43

92.9 7.62 5.70

cases.. Also, there is not an appreciable difference in the ultimate recoveries predicted by the various models. The standard deviations or the widths of the distributions suggest that the flotation rates are widely distributed for all cases. This is expected for these coals due to their broad size distributions and high ash contents, which indicates a broad k distribution in the total feed. It should be noted that the mean flotation rates generated by the five models, in their modified forms, are very similar for a given test. Hence, a single model could be chosen to discuss the results for varying experimental conditions. Model N3 is selected for the purpose. It can be seen that the mean flotation rate varied appreciably for both coals upon addition of the collector. It increased from 1.95 miny1 to 3.34 miny1 for the UF Žabout a 71% increase. and from 3.13 miny1 to 6.86 miny1 for the PC sample Žabout 120% increase.. It is clear that addition of the collector promoted the flotation of Pittsburgh coal sample considerably more than that of the Upper Freeport sample. However, it was also seen that addition of oil promoted the flotation of ash-forming minerals as it increased the mean flotation rate, and this was more significant for the Pittsburgh coal. However, the reasons for this behavior were not clear from the kinetics analysis of the total flotation products. 3.2.2. Analysis of flotation products in indiÕidual size fractions, Method II The kinetic behavior of the size-fractionated flotation products was analyzed using the modified first-order models. It was observed that the kinetics of the individual size fractions varied substantially both with respect to ultimate recovery and the mean flotation rate. The width of the flotation rate distribution also varied for different size fractions. An example of the kind of flotation rate distributions predicted by various

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159

Fig. 6. Predicted flotation rate distributions for various size fractions of the Upper Freeport coal sample in the absence of collector. Note that the frequency at a given flotation rate has been multiplied by the frequency of that size fraction in the feed material for estimation of the overall distribution. Various curves correspond to rectangular, sinusoidal, normal and gamma distributions and can be easily recognized from their shape.

models is given in Fig. 6 for the Upper Freeport sample in the absence of collector for the six size fractions. It can be seen that, despite the differences in the shape of the distribution functions, the mean flotation rate and the width of the distributions do not vary substantially for different models. The same is also true for the ultimate recovery values that showed negligible variations from one model to another. Hence, the parameters for the flotation kinetics of individual size fractions were evaluated using Model N3 and the results are given in Table 5 for the six size fractions, and for the total material. The ultimate recoveries of the middle and fine size fractions varied slightly whereas the ultimate recovery of the coarser size fractions increased significantly, suggesting that the coarser size fractions benefited more from the addition of the collector. This could be seen clearly from an analysis of the mean flotation rates. It was shown above that in Method I the mean flotation recovery increased by about 71% for the Upper Freeport sample and by about 120% for the Pittsburgh sample upon addition of the collector. The relative increase in the mean flotation rate for various size fractions is also included in Table 5 for the both coal samples. Upon addition of the collector, the Pittsburgh coal sample benefits more and the mean k’s of the coarser size fractions increase substantially more than those of the finer fractions. An analysis of the ash units after short flotation times showed that the large increase in the flotation rate for the coarse size fractions was accompanied by a large increase in the amount of ash-forming minerals reporting to the froth in these size fractions, most probably in the form of locked particles. 3.2.3. Analysis of flotation products in indiÕidual size-specific graÕity fractions, Method III The kinetics analysis was carried out on the size-specific gravity fractionated flotation products. The parameters predicted by various models were observed to be quite similar

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Table 5 Kinetics parameters predicted by N3 for individual size fractions Collector

% increase in m N

Upper Freeport

Pittsburgh

None

0.65 kgrT

None

0.65 kgrT

UF-R

PR

q600 m

RA mN sN

45.8 2.59 0.86

66.0 5.45 1.82

56.5 1.53 0.51

70.7 6.94 2.31

110

354

300=600

RA mN sN

81.8 2.82 0.94

90.0 5.54 1.79

84.7 3.48 1.16

90.0 9.90 3.30

97

184

150=600

RA mN sN

90.6 2.49 0.83

95.6 4.43 1.45

93.0 4.25 1.20

94.7 10.7 3.43

78

152

75=150

RA mN sN

95.0 2.02 0.67

97.3 3.33 0.70

94.7 3.43 1.10

95.9 8.62 2.78

65

151

37=75

RA mN sN

97.0 1.62 0.52

97.9 2.68 0.73

94.8 2.55 0.82

96.3 5.47 1.59

65

115

y37 mm

RA mN sN

92.9 0.94 0.31

92.4 1.48 0.49

88.1 1.60 0.40

87.0 2.81 0.94

57

76

Total

RA mN sN

87.7 1.95 0.65

92.3 3.34 1.11

90.5 3.13 1.04

91.9 6.86 2.29

71

120

for a given size-specific gravity fraction. Also, the width of the flotation rate distributions approached zero and Models R3, S3, N3 and G3 approximated Model C for the majority of the size-specific gravity intervals ŽPolat et al., 1993; Chander and Polat, 1994; Polat, 1995.. This supports the underlying assumption of the classical first-order model that the particles would have single-valued k’s if the analysis were carried out on fractions that are sufficiently narrow with respect to size and specific gravity. Therefore, Model C has been used in the kinetics analysis of the data in this part of the study. It was also observed that the final combustible matter recoveries obtained for the individual size-specific gravity fractions did not differ appreciably for the two coal samples when the collector was added. Apart from some scatter, the ultimate recoveries in the absence of oily collector were very close to those obtained in the presence of the collector. Hence, further discussion of the flotation kinetics in various size-specific gravity intervals will be limited to the flotation rate distributions. The flotation rate distributions for the UF and PR samples are given in Fig. 7. For ease of comparison, the results are presented as bar graphs for each size-specific gravity interval. The open and hatched bar graphs correspond, respectively to the floatation rate in the absence and presence of the oily collector. In absence of the collector, the

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161

Fig. 7. First-order flotation rate constants for the Upper Freeport ŽUF. and Pittsburgh ŽPR. coal samples for various size-specific gravity fractions. The open and closed bars correspond to flotation rates in the absence and presence of collector, respectively.

flotation rate decreased with increase in specific gravity for both the coals. With decrease in particle size the flotation rate first increase and then decreased indicating an optimum size for flotation. With increase in specific gravity, the particle size for maximum rate of flotation decreased to lower values. In general, addition of oily collector had a pronounced effect on the flotation kinetics of both the PR and UF samples. The increase in flotation rate was more for the raw Pittsburgh seam coal, however. For this coal, the increase in flotation rates was more even for the material with higher specific gravity. Since the material in higher specific gravity intervals, and for large size intervals to some extent, is expected to be composed mostly of locked particles, it is clear that collector addition favored the flotation of locked particles more than the free coal particles. On the other hand, the oil droplets attached to ‘‘free’’ coal particles may have minimal effect on their k’s since the hydrophobicity is already high for these particles. 3.2.4. Error analysis The overall curve fitting errors for the various models were determined for Methods I, II and III using Eqs. Ž3. and Ž6.. It was observed that the magnitudes and the general trends observed with respect to errors were similar among the four tests. Therefore, only the results with the Upper Freeport coal sample in the absence of collector are given in this paper, see Fig. 8. It can be seen that Method I generated relatively small errors when the models were modified using the 3-parameters. This also shows that when the modified form of these models was used the difference in the curve fitting power becomes small. On the other

162

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Fig. 8. Sum of square errors calculated by Methods I, II and III for the Upper Freeport coal sample in the absence of collector.

hand, the errors by Method III are quite large compared to Method I. This was mostly due to large errors observed in the ‘‘tails’’ fractions of the size-specific gravity spectrum. Since the amount of material in these fractions was very small in some cases, the large errors were most likely due to experimental or statistical reasons rather than modeling. The errors generated by Method II were the smallest for all the models that were tested. The errors in the individual size fractions that were used in calculating the overall error for Method II are given in Fig. 9. The following observations could be made: 1. The magnitude of the error did not differ from one model to another, especially for the intermediate size fractions. 2. Very small errors were obtained for the intermediate size fractions.

Fig. 9. Sum of square errors obtained with the five first-order models for individual size fractions from the coal samples in the absence and presence of collector.

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Fig. 10. Overall flotation rate distribution for models R3, S3, G3, and N3 calculated by Ža. Method I and Žb. Method II. For clarity in presentation to avoid overlapping curves the origin in the ordinate has been shifted downwards for each curve. See text for details.

3. The errors in the q600 mm and y37 mm size fractions were large. The large errors for the q600 mm size fraction could be due to the very large sizes of these particles. It was seen that some q600 mm particles were quite large and did not float even though they had sufficient hydrophobicity. The large errors in the case of y37 mm size fraction are most likely due to deviation from the first-order kinetics due to water carry-over. It was seen that the recovery in the y37 mm fraction was proportional to the water recovery from the flotation cell. 3.2.5. Total flotation rate distributions by Methods I and II The flotation rate distributions calculated by Method I are given in Fig. 10a for the UF coal in the presence of 0.65 kgrT of collector using Models R3, S3, G3, and N3. The origin of the frequency scale is shifted vertically to separate the distributions and avoid overlap of curves. It can be seen that the distributions obtained with Method I are quite different for the four models, both with respect to their shape and the spread along the flotation rate axis. The corresponding distributions calculated by Method II, using f II Ž k . from Eq. Ž5., are given in Fig. 10b. It can be seen that the general shape of the distribution of flotation rates obtained for the 3-parameter rectangular, sinusoidal, gamma and normal models, is quite similar. In other words, if one draws smooth lines through the calculated data in Fig. 10b, all the lines will have a similar shape for each of the model. Such a smooth line may be considered as a ‘‘true’’ distribution of flotation rate constants. The calculation of an overall distribution of flotation rate constants is possible for Method III also. However, due to large curve fitting errors observed for Method III, reliability of such a procedure was questionable and it was omitted in this study. 4. Conclusions The following conclusions can be made on the basis of this investigation. Ž1. A substantial improvement was made when three-parameter models were used to fit flotation kinetics data to the commonly used first-order models that incorporate a

164

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distribution of flotation rate constants. The parameters and the curve fitting errors for all the models that were tested became virtually independent of the chosen distribution function. The curve fitting errors seem to be more sensitive to the location of the distribution function rather than to its shape. These results imply that the choice of a correct model is not very critical when the modified 3-parameter first-order models are used to describe the kinetics of flotation. All models converge to give the same final result. Ž2. The curve fitting errors were much smaller and the robustness of the model as determined by PRESS analysis was much better for the three-parameter models compared to their two-parameter counterparts. Ž3. Detailed information can be obtained on the kinetics of flotation by gradual fractionation of the flotation products and conducting kinetics analysis for material in different size-specific gravity factions. Ž4. Three methods for estimation of ‘‘true’’ flotation rate distributions were compared in this article: In Method I, the total amount of material floated as a function of time was used to determine model parameters. In Method II, the froth products were fractionated into several size fractions and in Method III, the froth products were fractionated into different size-specific gravity fractions. No useful information about the distribution of flotation rate constants could be obtained by Method I. Method II can be used to determine the overall flotation rate distributions from weighted averages of the distributions of the individual size fractions. The overall distributions were less sensitive to the model selected for analysis and the smallest amount of curve fitting errors were obtained with this method. Method III gave relatively large errors, most likely due to combined contributions from both experimental and statistical errors. The experimental errors were large, possibly resulting from successive fractionation of froth products and from the diminishingly small amounts of material in certain size-specific gravity intervals. Method III also required large amounts of time and effort.

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