Ind. Eng. Chem. Res. 2001, 40, 3895-3901
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Leaching of Manganiferous Ores by Glucose in a Sulfuric Acid Solution: Kinetic Modeling and Related Statistical Analysis F. Veglio` ,*,† M. Trifoni,‡ and L. Toro‡ Dipartimento di Ingegneria Chimica e di Processo, Facolta` di Ingegneria, Universita` degli Studi di Genova, via Opera Pia, 15, 16145 Genova, Albaro, Italy, and Dipartimento di Chimica, Facolta` di SMFN, Universita` degli Studi “La Sapienza”, P.le A. Moro, 5-00185 Roma, Italy
A kinetic study of a manganiferous ore leaching in acid media by glucose as the reducing agent is here reported. The shrinking-core model opportunely modified was used to describe the kinetics of the manganese dissolution. Considering the complex network of chemical reaction involved in this process, a variable activation energy term was introduced in the kinetic model to take into consideration the different chemical reaction evolution. A nonlinear regression analysis was performed in order to estimate the modeling parameters, and their related statistical analysis was carried out as well. The proposed model was verified at low and high ore content (10 and 200 g/L, respectively). This comparison showed a quite good agreement between the experimental data and the theoretical prevision, confirming the possible application of the estimated kinetic parameters in a wide range of ore content. 1. Introduction Leaching is one of the central unit operations in the hydrometallurgical processes. A careful kinetic study concerning the controlling reaction step(s), the factors kinetically influencing the metal extraction yield and the estimation of the modeling parameters becomes very important for an efficient design process of hydrometallurgical operations. Several kinetic models relating the metal extraction yield to the leaching time are shown in the literature.1-5 As is known, one of the most useful mathematical models used to represent the kinetics of noncatalytic heterogeneous reactions is the shrinking-core model (SCM).6 In the present work the acid leaching of manganese dioxide ores by using glucose as the reducing agent was kinetically studied by using the SCM opportunely modified. The overall chemical reaction which takes place during the leaching treatment is reported in the following:
C6H12O6 + 12MnO2 + 12H2SO4 f 6CO2 + 12MnSO4 + 18H2O (1) As is shown elsewhere,7 a complex network of reactions may be supposed by considering the carbohydrates behavior in an acid medium. In fact, although the stoichiometry reported in eq 1 can well represent the overall dissolution process, several steps of carbohydrates chemical degradation may be considered during the leaching treatment time. For this reason it should be necessary to monitor all of the components produced during the process to give a complete modeling * Corresponding author. E-mail:
[email protected] or veglio@ ing.univaq.it. † Universita ` degli Studi di Genova. ‡ Universita ` degli Studi “La Sapienza”.
description of the manganese dissolution during the leaching time. A previous study8 showed that there is a relationship between the stoichiometry of the reaction (1) and the reagent concentrations: from the analysis of those experimental results, the presence of intermediate partially oxidized products slowing the reaction rate (because of worse reductants than glucose) was supposed and experimentally evaluated.8 The existence of intermediate products in the reaction medium constitutes one of the uncertainties in the collection and interpretation of the leaching data.9 For this reason a kinetic model able to describe a complex set of chemical reactions was considered of interest without needing a mathematical description for each intermediate reactive step. The aim of the present work was the evaluation of a suitable mathematical model to describe the reductive acid leaching of manganese dioxide by glucose considering only the global chemical reaction (1) and using a standard mathematical modeling description (SCM) relating the activation energy with the manganese conversion dissolution. The use of this model was successfully employed in the pyrrhotite bioleaching process and may be useful in the kinetic modeling of a number of leaching/bioleaching processes.10 2. Materials and Methods 2.1. Ore. The manganiferous ore used in the experimental tests comes from an Italian mine (Latium, Italy). The sampling of its size fractions employed in the tests was performed according to Gy’s rules to obtain representative samples for analysis. Table 1 shows the overall chemical composition of the investigated ore as metal oxides. 2.2. Experimental Procedures. Each batch test was performed in an Erlenmeyer flask with a screw plug placed in a thermostatic Dubnoff shaker. Ore, sulfuric acid, and glucose were placed in the shaken flask
10.1021/ie0004040 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/04/2001
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Table 1. Elemental Composition of the Investigated Ore (LOI ) Loss on Ignition)
Table 3. Experimental Treatments of the 23 Full-Factorial Designa
component
% w/w
component
% w/w
run
H2SO4 (M)
glucose (g/L)
particle size (µm)
MnO2 Fe2O3 Al2O3 SiO2 CaO MgO BaO ZnO CuO
37.90 2.43 12.2 25.71 2.67 0.30 0.59 0.03 0.01
NiO SO3 P2O5 CO2 H2O Na2O SO2 K2O LOI
0.09 0.90 0.89 0.45 0.86 3.05 0.90 1.73 10.23
1 2 3 4 5 6 7 8
1.25 2.00 1.25 2.00 1.25 2.00 1.25 2.00
20 20 30 30 20 20 30 30
75-106 75-106 75-106 75-106 210-250 210-250 210-250 210-250
Table 2. Experimental Treatments of the 3IV4-1 Fractional Factorial Designa run
H2SO4 (M)
glucose (g/L)
temp (°C)
particle size (µm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
0.5 1.5 1.0 1.5 1.0 0.5 1.0 0.5 1.5 1.5 1.0 0.5 1.0 0.5 1.5 0.5 1.5 1.0 1.0 0.5 1.5 0.5 1.5 1.0 1.5 1.0 0.5
10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30 10 20 30
30 30 30 50 50 50 70 70 70 30 30 30 50 50 50 70 70 70 30 30 30 50 50 50 70 70 70
75-106 75-106 75-106 75-106 75-106 75-106 75-106 75-106 75-106 150-180 150-180 150-180 150-180 150-180 150-180 150-180 150-180 150-180 210-250 210-250 210-250 210-250 210-250 210-250 210-250 210-250 210-250
a
Ore concentration 10 g/L, mixing condition 200 rpm.
according to the selected experimental conditions. Two sets of experimental runs differing in the ore content were performed (10 and 200 g/L). Mixing conditions at 200 rpm were found suitable to reduce the external film mass-transfer influence in the kinetic study.6 2.2.1. Experimental Runs with Low Ore Content. The experimental tests performed by using a low ore content (10 g/L) were arranged by a 3IV4-1 fractional factorial design.11 The four factors taken into consideration were temperature, sulfuric acid concentration, glucose concentration, and particle size. Table 2 shows the experimental plan selected according to the following defining contrast:11 I, ABCD, A2B2C2D2. 2.2.2. Experimental Runs with High Ore Content. The experimental tests performed by using a high ore content (200 g/L) were arranged by a 23 full-factorial design.11 The three factors investigated were sulfuric acid concentration, glucose concentration, and particle size at two levels. Temperature was selected at 50 °C. Table 3 shows the investigated experimental plan. 2.3. Chemical Analysis. Samples for analysis were collected at different times and diluted in a HNO3 solution (pH ) 2). The manganese concentration during the leaching time was detected by a Varian atomic absorption spectrophotometer.
a Ore concentration 200 g/L, temperature 50 °C, mixing condition 200 rpm.
3. Results 3.1. Mathematical Modeling. One of the most employed kinetic models suitable for noncatalytic heterogeneous reactions is the SCM.6 As is shown elsewhere,7,12 a large temperature effect on the manganese extraction yield and a direct proportionality between the leaching time and particles sizes were experimentally observed. In these cases a chemical reaction controlling step may be supposed for the leaching process. The kinetic model equation can then be used to relate the shrinking-core radius of the reacting particle (rc) with the metal dissolution conversion (X), and as reported in the literature,6 if the reagent concentrations are constant, the standard formulation of the SCM can be expressed as follows:
t ) 1 - (1 - X)1/3 τ
(2)
where X is the mineral dissolution conversion, t is the leaching time, and τ is the leaching time necessary to obtain the complete mineral dissolution (X ) 1). To find a suitable mathematical model able to describe manganese extraction yield vs time in the tested experimental conditions, the following aspects were taken into consideration: (i) The reagent concentration (sulfuric acid and glucose) does not stay constant during the leaching treatment.8,13 (ii) The reaction order could not be of the first order. (iii) The reaction rate decreases during the leaching time due to the change of the chemical reaction stoichiometry.7 In 1975 Brittan proposed a model named the “variable activation energy model” for the leaching of copper ore by hydrochloric acid.14 In this model the presence of the extracted metal concentration in the kinetic exponential term was suggested to take into consideration different mineralogical forms (and then different reaction sites) present in the investigated ore. Taking into consideration the available literature and the considerations on the chemical reactions involved in the manganese dissolution, the following kinetic model is proposed to describe the behavior of manganese dioxide leaching in acid media by a carbohydrate as the reducing agent. Taking into consideration the investigated experimental conditions,8,12 the chemical reaction was supposed to be the limiting kinetic step of the process:
[(
)]
b1Xb2 EA 1 1 dX C′ exp + ) dt Rp R T T′ RT
(
)
×
(CA0 - CAsX)na(CG0 - CGsX)ng(1 - X)2/3 (3)
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where X ) manganese conversion, C′ ) constant (µm/ M(na+ng)‚min), Rp ) particle size (µm), EA ) activation energy (kJ/mol), T′ ) reference temperature (K), b1 and b2 ) parameters relating the conversion to the activation energy (b1, kJ/mol; b2, dimensionless), CA0 ) initial acid concentration (M), CG0 ) initial glucose concentration (M), na ) reaction order with respect to the acid, ng ) reaction order with respect to the glucose, CAs ) stoichiometric sulfuric acid requirement (see eq 1), CGs ) stoichiometric glucose requirement (see eq 1). When b1 ) 0, the standard form of the SCM is obtained; for b2 ) 1, a linear relationship of the activation energy is postulated with respect to the metal extraction or b2 can be a further parameter which can be found in the nonlinear regression analysis. The last two terms can be evaluated considering the stoichiometry of reaction (1) and the manganese content in the investigated ore: CAs ) [ore concentration (g/L)][ore manganese content (%)] (4) 100AWMn CGs ) [ore concentration (g/L)][ore manganese content (%)] × 100AWMn 1 12
1 1
n
∑
(6) 1
n i)1 Ti where n is the investigated experimental condition number (n ) 27) and Ti is the selected temperature for each condition. Equation 3 was numerically solved (for both parameter estimation and simulation purposes) by the RungeKutta method. The model parameters C′, EA, b1, b2, na, and ng were fitted by a nonlinear regression analysis (Microsoft Excel 97 solver program) minimizing the function
Φ)
∑i ∑k [Xi(tk) - Xcali(tk)]2
manganese extraction yield (%) run
10 min
20 min
30 min
40 min
50 min
60 min
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
9 44 76 17 42 72 14 45 87 10 28 55 11 30 67 9 41 72 7 22 63 5 34 64 11 34 60
18 68 87 33 66 81 32 67 92 20 54 68 21 50 78 18 62 81 14 36 73 12 57 75 22 55 72
26 80 90 48 78 85 42 76 94 30 65 76 32 63 85 25 72 86 20 44 83 19 69 83 32 68 80
35 83 93 55 80 90 51 79 96 38 68 79 42 67 87 33 75 89 27 53 85 25 75 86 39 71 85
44 84 94 62 81 91 57 81 97 45 70 82 48 69 88 42 76 91 34 62 87 30 77 88 45 73 86
50 84 94 69 83 91 65 82 98 55 71 82 55 70 89 50 77 91 40 70 88 35 80 89 52 75 86
a
Ore concentration 10 g/L, mixing condition 200 rpm.
(5)
where AWMn is the manganese atomic weight. In the investigated case, CAs ) 0.04359 (mol/L) and CGs ) 0.00363 (mol/L). The reference temperature T′ introduced in the model was necessary in order to reduce the correlation between the activation energy and the preexponential constant in the Arrhenius term.15 T′ was calculated as the inverse of the average of all of the investigated temperature reciprocals:
T′ )
Table 4. Manganese Extraction Yield (%) vs Timea
(7)
where i is the run index, k is the time index in the ith run, and X is the manganese conversion. The minimization procedures can be selected in the program: a direct search method was selected to find the suitable model parameters. Although this mathematical model was formulated including six parameters which make it more flexible with respect to the experimental data, it is founded on
physical consideration of the possible chemical reactions involved in the leaching process. In this manner the kinetic model shown here allows one to reproduce the higher reaction rate observed in the first leaching times with respect to the low one noticed during the reaction course. The presence in the investigated ore of (1) different accessibility of the mineralogical forms present in the investigated ore, (2) different reactivity of the intermediate products obtained from the degradation reagents, (3) amorphous phases in the ore structure more reactive than the crystalline ones (XRD analysis, data not reported here, showed a not completely crystalline ore structure), (4) the decrease of the total surface area of the reaction interface as the core of the unreacted mineral shrinks could be the possible physical causes of the chemical reaction rate slowing down during the leaching time.9 3.2. Leaching Tests by Low Ore Concentration. Table 4 shows the manganese extraction yield (%) during the leaching time for the tests carried out at low relatively ore concentration (see Table 2). These experimental results were used for parameter estimation purposes. Figure 1 shows the flowchart of the nonlinear regression analysis. The initial parameter values reported in Figure 1 were selected considering previous works7,12 and preliminary minimization procedures: different initial guess values were also considered to verify the possibility of finding a local minimum (i.e., C′ ) 1; EA ) 10 000; b1 ) 5000; b2 ) 1; na ) 1; ng ) 1). In any case the minimization procedure was performed with constraints considering the physical meaning of the parameters (i.e., b1g 1; ng e 1; na e 1). The estimated parameters (see eqs 3 and 7) and their confidence interval limits are reported in Table 5. The parameter errors were estimated by a 95% confidence limit ((1.96σ). Each standard deviation parameter was calculated by using the principle diagonal
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[
]
The Jacobian matrix can be expressed according to the following equation:
J)
( ) ( ) ( ) ( ) ( ) ( ) ∂f ∂p1 ∂f ∂p1 ∂f ∂p1
1
∂f ∂p2
i
∂f ∂p2
N
∂f ∂p2
( ) ( ) ( )
1
∂f · · · ∂p j
i
∂f · · · ∂p j
N
∂f · · · ∂p j
( ) ( ) ( )
1
∂f · · · ∂p M
1
i
∂f · · · ∂p M
i
N
∂f · · · ∂p M
N
(10)
where i is the experimental run index (i ) 1, n; n ) 27), k is the leaching time index in the ith run (k ) 1, s; s ) 7), j is the parameter index (j ) 1, M; M ) 6), and N ) ns. In the case under study:
f)
dt ) X(ti) ∫0t dX dt i
[]
C′ EA b p) 1 b2 na ng
Each term in the Jacobian matrix (eq 10) was numerically calculated by the conversion incremental ratios obtained by varying each parameter value of 10% with respect to the estimated ones reported in Table 5. The experimental error variance was considered to be equal to the residual variance16 and calculated in the following manner:
S02 )
Table 5. Estimated Parameters
a
parameter
estimated value 12 ( 1 55 ( 1 13 ( 2 3.6 ( 0.4 0.42 ( 0.05 0.28 ( 0.04
Ckq )
Ore concentration 10 g/L, confidence limit 95%, T′ ) 321.8 K.
terms in the variance-covariance matrix. This matrix was obtained by the following equation:16
V ) (JTJ)-1S02
(8)
where J is the Jacobian matrix (N × M) and S02 is the experimental error variance. The Jacobian matrix and the experimental error variance are calculated as follows:
if fi ) ηi(xi,p)
(11)
The obtained variance-covariance matrix (see eq 8) allows one to calculate the correlation matrix of the estimated parameters. The generic element of this matrix is given by the following relation:16
Figure 1. Flowchart of the regression analysis.
C′ (µm/M(na+ng)‚min) EA (kJ/mol) b1 (kJ/mol) b2 na ng
Φ N-M
(9)
where xi is the variable array in the ith run and p is the parameter array.
σkq
xσk2σq2
(12)
where k and q are the indexes of different parameters and σ is the generic element of the variance-covariance matrix. Table 6 shows the variance-covariance matrix and the correlation matrix of the estimated parameters. Figure 2 shows a comparison between some experimental manganese extraction yields and the related calculated ones by using the parameters reported in Table 5. As shown in this figure, there is a satisfactory agreement between the experimental data and those calculated by eq 4. The scatter diagram of the experimental manganese extraction yields vs the calculated ones is plotted in Figure 3. These results show a quite good assumption of the experimental error normal distribution. Finally, it is necessary to mention that the stoichiometry reported in eq 1 is substantially correct because, although other possible chemical reactions can be present (i.e., between other oxides and acid), this contribution can be considered negligible:8 in other
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Figure 2. Manganese extraction yields vs time: a comparison between experimental data and calculated values (ore concentration 10 g/L, runs 10-18).
Figure 3. Scatter diagram: calculated and experimental manganese extraction yields (ore concentration 10 g/L, runs 1-27).
words, the process is very selective with respect to the manganese dissolution.
Figure 4. Comparison between experimental data (ore concentration 200 g/L) and calculated values obtained by using the low ore concentration kinetic parameters.
3.3. Leaching Tests by High Ore Concentration. The aim of this part of the work was to investigate the possible application of the estimated parameters (model shown in eq 3) for a wide range of ore concentrations. Table 7 shows the manganese extraction yield (%) during the leaching time for the tests carried out at relatively high ore concentration (see Table 3). These experimental data were simulated by using the parameters fitted by the experimental results obtained in the low ore concentration tests. A comparison between the experimental results collected at high ore concentration and the simulated data (obtained by low ore concentration fitted parameters) is shown in Figure 4. As is possible to observe, although there is a quite good agreement between simulated and experimental results, these last ones show systematically low manganese extraction values (average value of 3-5%). This is probably due to the major influence of the mixing condition on the kinetic dissolution in the used experimental system for large ore concentration. In fact, some further experimental tests at different stirring conditions (data here not reported) showed a little positive effect of the agitation on the slurry: this effect was also
Table 6. Variance-Covariance Matrix (above the Principle Diagonal) and Correlation Matrix (under the Principle Diagonal) of the Estimated Parameters (See Table 5) C′ EA na ng b1 b2
C′
EA
na
ng
b1
b2
0.342 -0.328 -0.219 0.936 0.126 0.210
-135.528 498.274 × 103 -0.04 -0.386 0.314 0.088
-3.338 × 10-3 -0.741 6.760 × 10-4 -0.220 0.027 0.037
1.255 × 10-2 -6.254 1.310 × 10-4 5.260 × 10-4 0.176 0.378
64.887 195.525 × 103 0.619 3.547 775.505 × 103 0.855
2.822 × 10-2 14.349 2.240 × 10-4 1.991 × 10-3 172.757 5.265 × 10-2
Table 7. Manganese Extraction Yield (%) vs Timea manganese extraction yield (%) run
5 min
10 min
15 min
20 min
25 min
30 min
35 min
40 min
45 min
50 min
55 min
60 min
1 2 3 4 5 6 7 8
30 34 34 40 18 19 19 22
45 55 51 63 28 34 32 39
58 65 60 68 35 45 41 50
62 68 65 74 42 50 46 55
67 73 70 76 48 55 51 60
70 76 73 80 55 60 56 65
72 77 74 80 57 63 59 67
74 78 77 82 60 65 61 69
75 79 78 83 62 68 64 71
77 80 80 84 63 69 66 72
80 81 81 85 64 70 67 73
81 83 82 86 65 71 69 74
a
Ore concentration 200 g/L, mixing condition 200 rpm.
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Figure 5. Scatter diagram: experimental manganese extraction yield vs the calculated ones estimated by the “constant activation energy” model (see eq 13).
of minor importance with respect to the chemical reaction as controlling the rate-limiting step as reported elsewhere in other works.12 3.4. Comparison between “Variable Activation Energy” and “Constant Activation Energy” Models. To show the improved fitting obtained by adding the conversion to the activation energy in the exponential term (see eq 3), the experimental data collected at low ore concentration were simulated by the classical Arrhenius term (selecting b1 as 0). The kinetic model used in this case is shown below:
[ (
)]
EA 1 dX C′ 1 ) exp × dt Rp R T T′ (CA0 - CAsX)na(CG0 - CGsX)ng(1 - X)2/3 (13) As for eq 3, this mathematical expression was numerically solved by the Runge-Kutta method. The model parameters C′, EA, na, and ng were fitted by a nonlinear regression analysis by using the low ore concentration estimated parameters (see Table 5) as starting values. A comparison between the calculated manganese extraction yields and the experimental ones is shown in Figure 5. It is possible to note the worse agreement between estimated and experimental data with respect to that obtained by using eq 3. Anyways an acceptable fitting can be noticed up to 45-50% conversion, but beyond this value the scatter notably increases. This aspect is confirmed by the simulation plot of Figure 6 where the trend of the EA + b1Xb2 term vs conversion is shown. This term, in fact, stays quite constant up to about X ) 0.5, but beyond this limit, its value swiftly augments. No good results were found also by using b2 ) 1. These considerations may highlight the increase of the activation energy during the chemical reaction. Obviously also, if a different manganese composition is not uniform along the radial direction of the ore particle, there could be an apparent dependence of the activation energy on the conversion X. In the studied case, the dependence of the activation energy on the conversion was considered to be most probable also by considering some previous experimental results reported elsewhere,7 in which the kinetic degradation of
Figure 6. Simulation plot: “variable activation energy” term (Var Ea ) EA + b1Xb2) vs the conversion (X).
glucose and sucrose during the time was taken into consideration: in any case further work is in progress performing tests with pure MnO2 and using different possible intermediate reaction products (gluconic and glucaric acids, etc.) to elucidate this process. Whatever is this interpretation of the leaching phenomena, the proposed model was considered suitable from the engineering point of view to describe the experimental data and to perform simulation for optimization purposes. 4. Conclusions In the present work a single physical model for the manganiferous ore leaching is proposed. The kinetic model takes into consideration an Arrhenius term decreasing with the conversion for a selected temperature. This “variable activation term” succeeds in describing the diminishing metal extraction yield during the treatment time for experimental tests performed by using a selected size fraction. The experimental results obtained in the runs performed at low ore concentration were used to obtain the parameters modeled by nonlinear regression analysis. The same parameters were used in order to simulate the experimental data collected for the high ore concentration tests. A satisfactory agreement between the experimental and the calculated results have been obtained, and a simple description of a complex stoichiometry and kinetic process has been successfully proposed. A comparison between the scatter diagrams obtained by the “constant activation energy” model and the “variable activation energy” model shows the improved agreement between estimated and experimental data reached by using the second kinetic model. Acknowledgment The authors thank Dr. Francesca Beolchini and Marcello Centofanti for their precious contribution in the experimental work. This work was carried out by the financial support of CNR. Literature Cited (1) Ma, K.; Ek, C. Rate Processes and Mathematical Modelling of the Acid Leaching of a Manganese Carbonate Ore. Hydrometallurgy 1991, 27, 125.
Ind. Eng. Chem. Res., Vol. 40, No. 18, 2001 3901 (2) Crundwell, F. K.; Bryson, A. W. The Modelling of Particulate Leaching Reactors: the Population Balance Approach. Hydrometallurgy 1992, 29, 275. (3) Ma, K.; Ek, C. Engineering Application of the Acid Leaching Kinetics of a Manganese Carbonate Ore. Hydrometallurgy 1992, 28, 223. (4) Crundwell, F. K. Progress in the Mathematical Modelling of Leaching Reactors. Hydrometallurgy 1995, 39, 321. (5) Momade, F. W. Y.; Momade, Zs. G. A Study of the Kinetics of Reductive Leaching of Manganese Oxide Ore in Aqueous Methanol-Sulphuric Acid Medium. Hydrometallurgy 1999, 54, 25. (6) Levenspiel, O. Chemical Reactions Engineering; John Wiley & Sons: New York, 1972. (7) Veglio`, F.; Toro, L. Reductive Leaching of a Concentrate Manganese Dioxide Ore in Acid Solution: Stoichiometry and Preliminary Kinetic Analysis. Int. J. Miner. Process. 1994, 40, 257. (8) Veglio`, F.; Volpe, I.; Trifoni, M.; Toro, L. Surface Response Methodology and Preliminary Process Analysis in the Study of Manganiferous Ores by Using Whey or Lactose in Sulfuric Acid Solutions. Ind. Eng. Chem. Res. 2000, in press. (9) Prosser, A. P. Review of Uncertainty in the Collection and Interpretation of Leaching Data. Hydrometallurgy 1996, 41, 119.
(10) Veglio`, F.; Beolchini, F.; Nardini, A.; Toro, L. Bioleaching of a Pyrrhotite Ore by a Sulfoxidans Strain: Kinetic Analysis. Chem. Eng. Sci. 2000, 55, 783. (11) Montgomery, D. C. Design and Analysis of Experiments; John Wiley & Sons: New York, 1991. (12) Veglio`, F.; Toro, L. Fractional Factorial Experiments in the Development of Manganese Dioxide Leaching by Sucrose in Sulphuric Acid Solutions. Hydrometallurgy 1994, 36, 215. (13) Trifoni, M.; Veglio`, F.; Taglieri, G.; Toro, L. Acid Leaching Process by Using Glucose as Reducing Agent: a Comparison Among the Efficiency of Different Kinds of Manganiferous Ores. Miner. Eng. 2000, 13 (2), 217. (14) Brittan, M. I. Variable Activation Energy Model for Leaching Kinetics. Int. J. Miner. Process. 1975, 2, 321. (15) Dovı`, V. G.; Reverberi, A. P.; Acevedo-Duarte, L. New Procedure for Optimal Design of Sequential Experiments in Kinetic Models. Ind. Eng. Chem. Res. 1994, 33, 62. (16) Himmelblau, D. M. Process Analysis by Statistical Methods; John Wiley & Sons: New York, 1970.
Received for review April 12, 2000 Accepted April 13, 2001 IE0004040
