Kinetic Modeling and Parameter Estimation for Competing Reactions

Article pubs.acs.org/IECR

Kinetic Modeling and Parameter Estimation for Competing Reactions in Copper Removal Process from Zinc Sulfate Solution Bin Zhang, Chunhua Yang,* Hongqiu Zhu, Yonggang Li, and Weihua Gui School of Information Science and Engineering, Central South University, Changsha 410083, China ABSTRACT: In zinc hydrometallurgy, an advanced copper removal process purifies zinc sulfate solution through a series of chemical reactions with recycled underflow by using zinc powder in zinc hydrometallurgy. This paper focuses on the kinetic modeling of the competitive-consecutive reaction system in the copper removal process, and proposes an adaptive parameter optimal selection strategy for different industrial conditions. In the system model, copper cementation, one of the removal reactions, is described by a surface controlled pseudo-first-order rate equation; cuprous oxide precipitation, the other removal reaction, is described by a shrinking core model of a noncatalytic fluid−solid reaction. Because there are several kinetic parameters in the system model, parameter estimation plays an essential role. Because of the complexity and variation in the practical removal process, the kinetic parameters are usually sensitive to alterations in the process conditions. This work solves the parameter estimation problem using an optimal selection strategy. In the strategy, the industrial conditions are classified adaptively according to the system model performance, then the kinetic parameters are selected optimally by evolutionary and particle swarm optimization algorithms for different industrial conditions. Three different representative industrial data sets are used to test the effectiveness and flexibility of the proposed modeling and parameter optimal selection approach in various situations. Finally, the kinetic model is applied to the soft measurement of the practical copper removal process with underflow, and the results demonstrate that the model effectively captures the trends of the removal reactions.

1. INTRODUCTION Zinc hydrometallurgy utilizing atmospheric direct leaching is an advanced technological method that is both more economically and environmentally friendly for the recovery of zinc from zinc concentrates.1−4 The zinc hydrometallurgy involves five processesgrinding, leaching, purification, electrowinning, and castingas shown in Figure 1. In these processes, trace impurities (e.g., Cu, Co, Ni, Cd) in the leaching solution will lower the current efficiency and reduce the quality of the zinc ingot.5−8 Adequately purifying the zinc sulfate solution is an essential step before electrowinning. Copper is commonly removed in the first stage of purification by the use of zinc powder because copper has a more negative oxidation potential than the other impurities. The effectiveness of the copper removal process determines the stability and quality of the subsequent removal processes. The conventional unit operation for copper removal is cementation of copper ions with a noble metal in a series of agitator tanks.9−12 In the zinc purification process, the copper ions are normally precipitated in accordance with the following reaction: CuSO4 + Zn → ZnSO4 + Cu↓ (1)

oxide also enables the chloride brought from direct leaching to be removed from the zinc sulfate solution.14 Therefore, the copper in this process is deposited in two ways: copper ions are precipitated to metallic copper by zinc powder, in accordance with the cementation reaction 1, and a portion of the metallic copper continuously reacts with ionic copper and produces cuprous oxide according to reaction 2: CuSO4 + Cu + H 2O → Cu 2O↓ + H 2SO4

Thus, the overall reaction to produce cuprous oxide is 2CuSO4 + Zn + H 2O → ZnSO4 + Cu 2O↓ + H 2SO4 (3)

It is observed that 1 mol of zinc powder is consumed in precipitating 1 mol of copper ions as metallic copper in reaction 1, while only 1/2 mol of zinc powder is needed in reaction 2 to precipitate 1 mol of copper ions as cuprous oxide. Moreover, in the advanced copper removal process, the unreacted zinc powder is recycled and used again in the copper removal reactors. Recycling the zinc powder reduces zinc consumption. Meanwhile, the cuprous oxide can be used as a reagent in a separate chloride removal process, where chloride is precipitated as cuprous chloride according to reaction 4.15 2Cl− + Cu 2O + 2H+ → 2CuCl + H 2O

The cementation occurs because zinc has a more positive oxidation potential than copper. It is observed that 1 mol of zinc powder reduces 1 mol of copper ions for an ideal stoichiometric conversion. Actually in the practical process, more zinc powder is needed because pieces of the powder are unreacted and left in residues. In the copper removal process of zinc hydrometallurgy utilizing atmospheric direct leaching, part of the copper residue is returned to the first reactor where most copper ions are precipitated as cuprous oxide.13 The precipitation of cuprous © 2013 American Chemical Society

(2)

(4)

Compared with the conventional process, the advanced copper removal process has some new characteristics. First, the mechanisms are more complex for stoichiometric conversion. Received: Revised: Accepted: Published: 17074

May 21, 2013 November 6, 2013 November 6, 2013 November 6, 2013 dx.doi.org/10.1021/ie401619h | Ind. Eng. Chem. Res. 2013, 52, 17074−17086

Industrial & Engineering Chemistry Research

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Figure 1. Flowchart of zinc hydrometallurgy utilizing atmospheric direct leaching.

Figure 2. The copper removal process with recycle underflow.

copper cementation by zinc powder from solution.20 The reaction was modeled as a first order kinetic equation with copper concentration, temperature, and stirring speed using reaction data. Ahmed et al. investigated the effects of pH, temperature, zinc amount, copper concentration, and shaking time on copper cementation.21 These previous works provide inspiration and motivation for the kinetic modeling of the copper removal process in this paper. Note that previous research focused on the kinetics of the copper cementation reaction. To the best of the authors’ knowledge, few studies have reported kinetic models of the competitive-consecutive reactions in a copper removal process.16−21 In this paper, a competitive-consecutive reactions model is proposed for the copper removal process with recycle underflow by combining a cementation model and a shrinking core model of noncatalytic heterogeneous reactions. Generally, the precision of a kinetic model depends significantly on parameter estimation. Various parameter estimation algorithms have been proposed for process modeling, such as nonlinear programming method,22 influence function,23 mean-squared-error,24 artificial neural networks,25 gradient-based optimization,26 trial-and-error,27 and approximate maximum likelihood.28 Most of these parameter estimation strategies are designed for laboratory scale experiments. The estimated kinetic parameters cannot be applied directly to the studied copper removal process. Theoretically, the chemical reactions are influenced by the presence of several species in the reaction environment and other various factors (temperatures, pH values, stirring speeds, etc.). The species and the factors are constantly changing in an actual process, so the reaction conditions vary frequently. By using conventional parameter optimization methods, the kinetic model poorly

The advanced process consists of two solid−liquid reactions, including copper cementation (eq 1) and cuprous oxide precipitation (eq 2). On one hand, both of the major reactions consume ionic copper; on the other hand, the metallic copper produced in the former reaction is treated as the reactant in the later reaction. These reactions comprise a typical competitiveconsecutive reaction system. Second, in practice, random disturbances in the advanced process will result in much larger changes than in the conventional process. The outlet copper concentration and the mass ratio of the metallic copper and cuprous oxide remain stable only when the copper cementation (eq 1) and cuprous oxide precipitation (eq 2) are balanced with relatively constant reaction rates. If a slight variation occurs in the reaction environment, the provisional balance would be broken, which easily leads to an immediate fluctuation in the industrial process. Third, the difficulty of process operation is increased. The fluctuation mentioned above is hard to eliminate manually under most operating conditions in a continuous process. Sometimes it produces an unstable production for a long time. For better operation and stable production, an essential task is to analyze the kinetics of the copper removal reactions and model the process under complex industrial conditions. A number of kinetic models for hydrometallurgical copper removal processes have been reported in the literature.16−21 Dönmez et al. studied the kinetics of copper cementation from sulfate solutions onto aluminum.18 Stole−Hansen modeled the cementation reactor as a dynamic continuous stirred tank reactor (CSTR) and defined a stoichiometric efficiency factor to correct the model. The precipitation reaction rate was reported to follow first order kinetics and form a diffusion boundary layer.19 Demirkiran et al. studied the kinetics of 17075

dx.doi.org/10.1021/ie401619h | Ind. Eng. Chem. Res. 2013, 52, 17074−17086


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describes the reaction changes when the reaction conditions are variable. Moreover, since some of the chemical species and factors are undetectable or unknown, it is difficult to identify manually the reaction conditions in the constantly changing industrial situation. Hence, to build a precise kinetic model on industrial scale, an adaptive parameter optimal selection strategy for multiconditions (APOSS-MC) is proposed in this paper, where the reaction conditions are classified adaptively according to the kinetic model performance, and the kinetic parameters are estimated separately for each condition. The remaining parts of this paper are organized as follows. In section 2, the copper removal process with recycle underflow is described in detail, and the characteristics of the process are analyzed. Section 3 establishes the kinetic models for copper cementation and cuprous oxide precipiation, which constitute the competitive-consecutive reactions model for the advanced copper removal process. Section 4 introduces the proposed parameter optimal selection strategy, which consists of three steps: condition labeling, condition balancing, and optimal selection of condition parameters. The performance of the proposed kinetic model and parameters optimal selection strategy is verified and discussed in section 5. The industrial applications of the model are then shown in section 6. Finally, conclusions are provided in section 7.

(3) Inadequateness of measurements: The measurements of copper concentration, as well as the component analysis of the solid particles in the underflow, are incomplete. In the practical process, all of the metal concentrations in the inlet solution are detected, while only the copper concentration is detected from the purified solution (from the second tank) and clean solution (overflow), as shown in Table 1. The other metal Table 1. The Available Material Measurements in the Copper Removal Processa

a

materials

Zn2+

Cu2+

Fe2+

Co2+

Ni2+

Cd2+

Cu

Zn

inlet solution purified solution clean solution precipitate

∗ − − −

∗ ∗ ∗ −

∗ − − −

∗ − − −

∗ − − −

∗ − − −

− − − ∗

− − − ∗

(∗) Measured; (−) unmeasured.

concentrations are ignored due to their relatively low content. In the precipitate, only the amount of copper and zinc elements are determined. However, in the practical process, these metal ions could also react with zinc powder, which may decrease the copper removal efficiency especially when the ferrous concentration exceeds the index limitation. If better measurements of these components in the actual process are available, it will be easier to control the process precisely. Therefore, it is essential to build a precise kinetic model that considers the variations in industrial conditions for the copper removal operation.

2. PROCESS ANALYSIS The copper removal process with recycled underflow is a long and complex process consisting of serially connected units that include impurity precipitation and solid−liquid separation. A simplified schematic diagram of the process is shown in Figure 2. The leaching solution contains various impurities, among which the concentration of copper is the highest. The inlet valve is controlled by the distributed control system (DCS) and feeds the solution into two CSTRs for impurity precipitation. Operators set the amount of zinc powder continuously added to each reactor by a weight belt. After the precipitation reaction is completed (approximately 50 min), the purified solution is sent to the thickener to separate clean solution and the precipitate. The clean solution at the top of the thickener (overflow) flows to the next zinc purification removal stage, while part of the precipitate (underflow) is pumped into the first reactor and the remaining precipitate is sent to the copper recovery process. The purpose of process control is to maintain the outlet copper concentration within the desired range. However, because of uncertainties in the practical process, the conditions of the copper removal reactions are complex and variable. These variations in the reaction conditions are difficult for even experienced operators to determine. Thus, the outlet copper concentration is not as stable as expected. Sometimes, it will exceed the limits of the production indices when the industrial conditions change. Generally, the uncertainties in the practical process come from the following sources: (1) Concentration instability: The diversity of zinc concentrate sources leads to a variety of copper concentrations in solutions, which causes difficulties in obtaining the copper concentration mass balance. (2) Undetectable species in the reactions: Since the zinc sulfate solution is at a high temperature and is seriously corrosive, the online element analyzer does not always work stably in the field. Thus, changes in the reactants and the occurrence of side reactions are difficult to observe in the copper removal reactors.

3. MODELING OF THE REACTION KINETICS In this section, a semiempirical kinetic model is constructed to explain the mineralogical changes that occurr in the copper removal reactions. The model provides numerical verification for the chemical competing reactions mechanism. First, independent kinetic models are built to describe the copper cementation and cuprous oxide precipitation reactions. Next, the competing reaction system is constructed according to the monomodels to simulate the multireactions in the improved copper removal process. Finally, the parameters used in the models (such as variable coefficients and undetectable parameters) are identified based on the parameter optimization algorithm for multiconditions. 3.1. Copper Cementation Model. Copper cementation occurs in the early stage of the copper removal process and is presented in eq 1. The copper cementation takes place on the surface of zinc particles in solution. In recent research, the cementation reaction is assumed to be a surface-controlled first order reaction;29 the reaction can be modeled as K S r1 = −KC = − D A CCu2+ (5) V 2+ where r1 is the reaction rate, CCu is the copper concentration in the reactor, KD is the mass transfer coefficient and V is the volume of the reaction solution. In addition, SA is the reaction surface area for the copper cementation reaction. The reaction surface area depends on the amount of zinc powder added from zinc bunkers and the zinc powder contained in the underflow. In the industrial process, the purification effectiveness depends on the particle size of zinc powder. Zinc powders with small particle size have a higher specific surface area than powders with larger particle size. Hence, the particle size of zinc 17076



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Figure 3. The shrinking core model. n

powder determines the surface area available for the copper cementation reaction. To measure the total surface area of the zinc powder used in the practical process, the particle size of zinc powder should be sieved when the powder source is altered. The number of size intervals is denoted as Nscreen, the powder weight of screen size i as mi, with the weight fraction of zinc particles of screen size i calculated by eq 6:

S=

∑

mi = mi /M

i=1

(6)

N

(7)

1 Ntotalαiπdi 3 6

(8)

Assuming particles of different weight fraction have the same density ρ, the surface of zinc powder is then calculated with its weight by eq 9: Nscreen

W = ρVA = ρ

∑ i=1

1 Ntotalαiπdi 3 6

(9)

According to the above equations, the total surface area of zinc particles is represented using the weight and density of zinc powder and the mass ratios of the powder after being segregated by different sizes of screen holes: SA =

n

ρ ∑i = 1 αidi 3

6KD(gZn + γZng under ) ∑i = 1 αidi 2 n

Vρ ∑i = 1 αidi 3

r2,1 = −4πR S2V −1k G(CCu2+g − CCu2+s)

CCu2+

(12)

(13)

where r2,1 is the reaction rate, RS is the particle radius, kG is the mass transfer coefficient, CCu2+g is the copper concentration in the bulk phase, CCu2+s is the copper concentration at the outer surface of the copper particle, and V is the volume of the particles. In the second step, the fluid reactants (CuSO4 and H2O) diffuse through the solid ash layer (cuprous oxide). The reaction rate can be presented as

n

6 ∑i = 1 αidi 2

(11)

3.2. Cuprous Oxide Precipitation Kinetics. Copper ions will continue to react with metallic copper in the copper cementation process and form cuprous oxide in the copper removal reactors, as described in eq 2. The cuprous oxide precipitation commonly exists as an unexpected side reaction in a variety of chemical processes, such as the copper plating process. To the best of the authors’ knowledge, there are few kinetic studies of cuprous oxide precipitation. Hence, the universal model for noncatalytic fluid solid reactions is applied to the cuprous oxide producing process, and the shrinking core model (SCM) is chosen to describe the cuprous oxide particle forming process.30,31 The schematic diagram of the shrinking core is shown in Figure 3. Cuprous oxide precipitates on the surface of the metallic copper particles, which are assumed to be spherical, and reacts with the zinc sulfate solution isothermally. According to the SCM theory, the cuprous oxide precipitation consists of three steps: diffusion through the liquid film surrounding the solid particles, diffusion through the solid ash layer, and chemical reaction at the surface of the unreacted core. In the first step, the reaction rate is calculated as

where Ntotal is the total number of zinc particles used in the tests, and di is the average diameter of the particle of screen size i (the average value of the upper limit and lower limit of the interval size). Similarly, the total volume of zinc particles of screen size i is calculated as follows: VAi =

(gZn + γZng under )

n

r1 = −

mi . where M = ∑i =screen 1 Supposing the particles in the zinc powder are spherical, then the total surface area of the particles of screen size i is calculated by eq 7: SAi = Ntotalαiπdi 2

n

ρ ∑i = 1 αidi 3

where gZn denotes the weight of zinc powder added from the zinc bunkers, and gunder denotes the weight of zinc powder from the solid particles in the underflow. Moreover, γZn is the weight proportion of residual zinc powder in the solid returned from the underflow. Finally, the copper cementation model can be expressed as

Nscreen

αi ≈ mi /

6 ∑i = 1 αidi 2

W (10)

In the improved copper removal process, the surface area of the residual zinc powder particles in the returning underflow should also be considered. Thus, the surface area of copper cementation in this process, S, consists of two parts, 17077


Industrial & Engineering Chemistry Research ⎛ dC 2 + ⎞ r2,2 = −4πR S2V −1De⎜ Cu ⎟ ⎝ dR ⎠ R = R

C

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particles in the former reaction are reactants in the latter reaction, while both reactions consume copper ions in the solution. These reactions comprise a typical competitiveconsecutive reaction system:

(14)

where r2,2 is the reaction rate, De is the effective diffusivity of copper sulfate through the cuprous oxide layer, and RC is the distance from the center of the sphere to the reaction surface. In the third step, the precipitation occurs at the surface of the unreacted copper core, and the reaction is assumed to be a firstorder reaction. Then, the reaction rate is expressed as r2,3 = −4πR C 2V −1kCCu2+C

k1

A + B → C↓ + D k2

A + C + E → R↓ + H

n

k1 =

−1

(16)

=

WCu 2O WCu 2O + WCu 3

(R S − R C )ρCu O 3

3

R C ρCu + (R S − R C )ρCu O 2

≈1−

RC

3

2R S3 − R C3

(17)

where WCu2O and WCu are the weights of cuprous oxide and metallic copper in the precipitate, respectively, and ρCu2O and ρCu are the densities of cuprous oxide and metallic copper, respectively. Accordingly, the radius of the unreacted core can be expressed as RC −

3

2 − 2θ R S = ηR S 2−θ

⎡1 (η − 1)R S η2 ⎤ k 2 = 4πR S2V −1⎢ + + ⎥ De k⎦ ⎣ kG

(23)

rCu2+

(24)

n ⎧ + γZng under ) ∑i = 1 αidi 2 ⎪ 6KD(g Zn = −⎨ n ⎪ ρ ∑i = 1 αidi 3 ⎩

⎫ ⎡1 (η − 1)R S η 2 ⎤⎪ ⎬V −1CCu2+ + 4πRS 2⎢ + + ⎥⎪ De k ⎦⎭ ⎣ kG (25)

(18)

where rCu is the rate of the change of the copper concentration, KD, kG, De, and k are the unknown kinetic parameters that need to be estimated. According to the flow sheet of the copper removal process, the CSTR model of the two reactors can be described by the following differential equations:32 2+

where η = [(2 − 2θ)/(2 − θ)]−1/3. In the practical process, the solid content remains stable in the reactors. The radius of the solid particles can be calculated by the solid weight in the reactors. Substituting eq 18 into eq 16, the overall reaction rate is ⎡1 (η − 1)R S η2 ⎤ r2 = −4πR S2V −1CCu2+g ⎢ + + ⎥ De k⎦ ⎣ kG

(22)

These equations describe the behavior of the copper removal reactions for kinetic conversion. Among these different chemical species, the copper concentration is the most important variable. This variable is the quality index in the traditional process, and the rate of change of the copper concentration is described according to eq 25.

2

3

n

Vρ ∑i = 1 αidi 3

⎧ rA = −r1 − r2 ⎪ r = −r 1 ⎪B ⎪ rC = −r1 + r2 ⎪ r ⎨ rD = r1 ⎪ r = −r 2 ⎪E ⎪ rR = r2 ⎪ ⎩ rH = r2

In the improved copper removal process, θ, the quality ratio of cuprous oxide to the whole precipitate, is examined regularly. Because this process requires sufficient contact between the metallic copper particles and the copper sulfate solution, θ depends on RC and RS from the formation of cuprous oxide surrounding the metallic copper particles:

3

6KD(gZn + γZng under ) ∑i = 1 αidi 2

On the basis of the principle of independence of coexisting chemical reactions, the competitive-consecutive reaction system of this copper removal process is described by the following equations:

⎡1 R (R − R C ) R 2 ⎤ r2 = −4πR S V CCu2+g ⎢ + S S + S2 ⎥ R CDe RC k ⎦ ⎣ kG

θ=

(21)

where A, B, C, D, E, R and H denote CuSO4, Zn, Cu, ZnSO4, H2O, Cu2O, and H2SO4, respectively. Moreover, k1 and k2 denote the rate constants of reactions 20 and 21, respectively. These rate constants are functions of the process parameters (such as volume and solid content).

(15)

where r2,3 is the reaction rate, k is the reaction rate constant, and CCu2+C is the copper concentration at the reaction surface. In principle, if the step with the slowest reaction rate is treated as the controlling regime, the rate at this step is considered as the reaction rate of the process. However, singlestep rate-determining processes are rare among practical industrial chemical processes. The cuprous oxide precipitation is generally influenced simultaneously by all three steps. Therefore, the overall reaction rate should be the sum of the three rates: 2

(20)

0 • VCCu = QCCu 2+ 2 + (t ) − (1 + q(t ))QCCu 2 +,1(t ) − VrCu 2 + ,1 ,1

(19)

0 • VCCu = (1 + q(t ))QCCu 2+ 2 + (t ) ,2 ,2

3.3. The Kinetics of Competing Reactions System. The copper cementation and cuprous oxide precipitation occur simultaneously in the same reactors. The deposited copper

− (1 + q(t ))QCCu2+,2(t ) − VrCu2+ (26) 17078



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Figure 4. The multicondition parameter optimization framework. • where CCu 2+ , ,i i = 1,2 is the rate of change of copper 0 concentration in the ith reactor, CCu 2+ , i = 1,2 is the inlet ,i copper concentration of the ith reactor, Q and q are the flow rates of zinc sulfate solution and the returned underflow, respectively.

4.1. Condition Labeling. The chemical reaction rate is influenced by various factors. In the laboratory, the kinetic parameters are usually determined for different reaction conditions which are classified according to these factors. Accordingly, condition labeling and classification are important for increasing the precision of the kinetic model. If the industrial conditions can be labeled and classified according to the model performance, the condition classification will be more reasonable, and the model will respond better to different conditions. Usually, the condition labeling is performed manually according to operators’ experience. Clearly, manual conditional labeling is labor intensive, and sometimes inaccurate. Hence, an automatic condition labeling method based on model performance is proposed in this section. Substituting eq 25 into eq 26, the outlet copper concentration can be obtained by numerical integration of the differential equations. For simplicity, the outlet copper concentration is denoted as a function of time, industrial parameters, and unknown parameters:

4. PARAMETER OPTIMIZATION FOR MULTIPLE CONDITIONS As described in section 2.2, due to the uncertainties in the copper removal process, a mathematical model with fixed kinetic parameters cannot describe the process precisely under all working conditions. However, if the conditions of the copper removal process do not change, a mathematical model using a fixed set of kinetic parameters is still accurate. Thus, to improve the precision of the kinetic model, a parameter optimization algorithm for multiple conditions is proposed in this section. The framework of parameter optimization for multiple conditions is shown in Figure 4, where the optimization process consists of three stages (condition labeling, condition balancing, and parameter optimization). In the first stage, the industrial data are divided into several groups according to the model performance, and each group is considered as an industrial condition. In the second stage, artificial data are added to the groups with smaller sizes to solve the size imbalances among the conditions. In the last stage, the kinetic parameters are optimized for each group using the particle swarm optimization algorithm (PSO) and the evolutionary algorithm (EA).

CCu 2+ = MODEL(t , pK , pUnk )

(27)

where CCu2+ is the vector of m measured copper concentrations, MODEL is a m−component vector-valued function of the copper concentration, t is time, pK is a vector of independent variables (C0Cu2+,1(t), C0Cu2+,2(t), Q, q, γZn, gZn, gunder αi, di, ρ, V, RS and η), and pUnk is a vector of unknown parameters (KD, kG, De, and k). 17079



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Step 1: Obtain the sizes of all the labeled conditions (m1, ..., m2, ..., mCN) and the number of conditions CN, and set the ratio threshold vratio. CN Step 2: Calculate the size ratio, ri = ri = mi /∑ j = 1 mj , (i = 1, 2, ..., CN) for each condition. Step 3: If the size ratio of a condition is smaller than vratio, the condition is considered as a minority condition. Step 4: Create the artificial data according to eq 25 with the data in the minority condition; the balanced conditions are denoted as {S1*,S2*, ... SCN * }. 4.3. Parameter Optimization. After the conditions are labeled and balanced, the unknown parameters can then be estimated separately for each condition. The PSO algorithm is widely applied to estimate parameters for nonlinear models.34,35 However, this algorithm can become easily trapped in a local optimum. Thus, we use the evolutionary algorithm to improve the PSO algorithm in the parameter optimization process. In this process, the unknown parameters in the model are treated as the position of particle, and the ranges of all the parameters are set first. The evolutionary algorithm mutates some of the particles during the iteration. After several iterations, the best values can be found. In the EA-PSO algorithm, the velocity and position of particles are updated as

During the labeling process, industrial data are used to assess the kinetic model. The data with smaller relative errors in the kinetic model are treated as one set of conditions. Next, the remaining data are treated in the same way to find the next set of conditions. In this way, all of the industrial data are labeled according to the performance of the kinetic model. A detailed procedure for the proposed condition labeling method is expressed as follows. Step 1: Select the industrial data randomly from the copper removal process, and form a sample set S0 = {si|i = 1, ..., m0}. Set the precision threshold and the size threshold to be vprecision and vsize, respectively. Step 2: Estimate the parameters P0 = {KD0, kG0, De0, k0} of the kinetic model from S0, and denote the model with the estimated parameters P0 as M(P0). Step 3: Calculate the relative errors for all the data from S0: ei0 = |(yi0 − ci)/ci| ,

i = 1, 2, ..., m0

(28)

where, y0i is the ith output of the model, M(P0), and ci is the corresponding measured copper concentration of the ith sample. Step 4: Judge whether the errors of the model M(P0) are less than the precision threshold vprecision, and select samples with the errors smaller than vprecision from S0 to form a new data set, S1, which is labeled as the first condition: S1 = {si|i = 1, ..., m1, e0i < vprecision, 0 < m1 < m0}. Step 5: If the number of remaining data is greater than vsize × m0, repeat Steps 2 to 4 to find the next condition. Step 6: If the number of remaining data is less than or equal to vsize × m0, take the remaining data set SCN as the last condition. Then the entire industrial data set can be divided into CN conditions {S1, S2, ..., SCN}. 4.2. Condition Balancing. The industrial data are divided into several condition classes according to the performance of the kinetic model. Samples in the same class have similar kinetic features. However, the sample sets for the different conditions will usually be of unequal size. This is because the samples under some extreme conditions (e.g., an overabundance or shortage of zinc powder, high or low copper content in solution, etc.) are rare, whereas, the samples under the regular conditions (proper zinc powder and temperate copper content in solution) are abundant. The samples in the minority conditions are insufficient to estimate the unknown parameters precisely. Furthermore, the size imbalances among the conditions will hinder the online condition identification of the industrial conditions. To address these concerns, the industrial samples should be balanced before estimating the parameters for each condition. In this section, the synthetic minority oversampling technique (SMOTE) is introduced to enrich samples in the minority conditions. The SMOTE algorithm creates artificial data according to the feature similarities between existing minority samples.33 For the minority condition data set Smin, we first find the K-nearest neighbors XI,K = {xi,1, xi,2, ..., xi,K} for each sample xi ∈ Smin, where XI,K are defined as K samples from S0, whose Euclidean distances from xi are of the smallest magnitude. Next, the artificial samples are obtained as x new. j = x i + (x i , j − x i) × b ,

j = 1, 2, ..., K

vi(k + 1) = wvi(k) + c1R1(pIbesti(k) − xi(k)) + c 2R 2(pGbest(k) − xi(k))

xi(k + 1) = xi(k) + vi(k + 1)

(30) (31)

where k is the current iteration number; vi and xi are the velocity and position of the ith particle, respectively; pIbest is the best position found by the particle itself; pGbest is the global optimum found by the whole swarm; w is an inertial weight; c1 and c2 are constants; R1 and R2 are random numbers within the range [0,1]. The procedure of the EA-PSO algorithm is expressed as follows: Step 1: Set the size of population Npopul and the threshold of fitness value Tfitness. Initialize the particles in the population and produce the initial position and initial velocity of each particle according to the dimensions of the solution space. Xi(0) = {xij(0)}, (i = 1, ..., Npopul; j = 1, ..., m) Vi(0) = {vij(0)}, (i = 1, ..., Npopul; j = 1, ..., m) Meanwhile, the optimal Xi(0) of each parameter is recorded. Step 2: Let k = 0. Calculate and sort the current fitness values of the particles using the fitness function, and find the position of the particle with the lowest fitness value (pGbest(0)). Step 3: Calculate the new velocities and new positions of the particles according to eq 30 and eq 31, respectively. Step 4: Calculate the new fitness value FV for all the particles, and update pIbest(k) and pGbest(k). Step 5: Sort the particles according to the fitness value (in ascending order), and compare FV with Tfitness for every particle. If FVi > Tfitness, then mutate the particle by the evolutionary algorithm, k = k + 1. Step 6: If the iteration number exceeds the maximum number of iterations, or pGbest(k) remains constant several times in succession, then the PSO search is terminated and the global optimum has been found. Otherwise, the algorithm returns to Step 2 and updates the particle velocities and positions using eq 30 and eq 31, respectively.

(29)

where b ∈ [0,1] is a random number. Then, the proposed condition balancing method consists of four steps: 17080



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Figure 5. Online application of the kinetic model of the copper removal process.

Table 2. The Ranges of Input Parameters in the Three Data Sets data set

inlet copper concentration (g/L)

inlet flow rate (m3/h)

underflow rate (m3/h)

solid content in underflow (kg/L)

zinc powder (kg/h)

pH value

temperature (°C)

1 2 3

0.9−1.0 0.7−1.2 0.8−1.2

295−305 180−260 200−250

29−31 28−30 29−30

1.4−1.5 1.4−1.5 1.4−1.5

120−145 60−156 70−150

3.9−4.2 3.9−4.2 3.9−4.2

62−66 62−66 62−66

are optimally selected for each condition in the proposed parameter optimal selection strategy, where vprecision, vsize, and vratio are set as 0.1, 15%, and 15%, respectively. After optimal parameter selection, the system model is used to calculate the outlet copper concentration. The number of testing samples is 150, consisting of 50 samples from each data set. To evaluate the performance of the kinetic model, four indices are used in this paper, including mean absolute error (MAS), root-meansquare error (RMSE), maximum absolute error (MAXABS), and accuracy probability (AP). Among these indices, AP considers the requirements of the zinc purification process and measures the similarity between the calculated and measured value. This index is in the form of

In this procedure, different kinetic parameters are optimized for each of the different conditions, and the kinetic model for the ith condition can be expressed as MODEL(t,pK,pconi). When the kinetic model is applied online, the condition should be identified using the preprocessed industrial data. Next, the corresponding kinetic parameters are selected according to the selected condition. With the selected kinetic parameters, the kinetic model can be used to predict concentration or an additive amount calculation. The procedure for online application of the model is shown in Figure 5.

5. RESULTS AND DISCUSSION To evaluate the proposed kinetic modeling method and parameter optimal selection process (KM&APOSS-MC), we considered three actual industrial situations. Additionally, we compared our new multicondition model with a singlecondition model using a set of fixed parameters optimized by PSO algorithm. All of the simulations are conducted in the MATLAB7.1 environment running on a Pentium 4, 2.66 GHZ CPU with 512 RAM. The simulation data were sampled from a practical copper removal process and were divided into three different representative data sets. In the first data set, both the copper concentration and the flow rate of the inlet solution were high and relatively stable. The outlet copper concentration in this data set was maintained within a narrow range of 0.3 ± 0.05 (g/L3), providing a stable situation. In the second data set, the copper content of the inlet solution fluctuated significantly, and zinc powder was not added continuously because of a machine fault by the belt weigher. In the second set, the outlet copper concentration fluctuated significantly, ranging from 0.1 to 0.51 (kg/m3). This situation is called the fluctuant situation. During the third data set, the industrial conditions also fluctuated, but few process faults occurred. The outlet copper concentration varied from 0.1 to 0.52 with an obvious trend. This situation was considered a macroscopic trend. The ranges of the input parameters for the three data sets are shown in Table 2. The number of training samples is 300; these are selected randomly from the three data sets. The training samples are divided among several conditions, and the kinetic parameters

A C = P(Ye(i) ∈ [Ym(i) − ς , Ym(i) + ς])

(32)

with Ye(i) = Yc(i) − Ym(i)

(33)

⎧ 0.02 0 < Ym(i) ≤ 0.15 ⎪ ⎪ ς = ⎨ 0.03 0.15 < Ym(i) ≤ 0.45 ⎪ ⎪ 0.04 0.45 ≤ Y (i) ⎩ m

(34)

where Ym(i) is the measured value, Yc(i) is the calculated value, Ye(i) is the error between the measured and calculated value, and ζ is the precision threshold of the accepted region for different numerical ranges, which is defined according to the requirements of the practical process. The outputs of KM&APOSS-MC and the single-condition kinetic model optimized by PSO algorithm (KM&PSO) are reported for the stable situation in Figure 6. It is noted that the calculated values of these two methods agree well with the measured values when the situation is stable. Although the outputs of KM&PSO are closer to the measured values compared with KM&APOSS-MC (as shown in Figure 7), the average errors of the proposed method are smaller than the kinetic model with fixed parameters (as shown in Figure 8). When the situation is fluctuant, poor agreement is observed at the 5th, 26th, and 46th samples for KM&PSO, whereas KM&APOSS-MC captures the dynamics of the outlet copper 17081



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concentration exactly at these points, as shown in Figure 9. And the errors of the proposed method are mostly lower than that of KM&PSO (Figures 10 and 11).

Figure 6. Calculations from KM&APOSS-MC and KM&PSO in the stable situation.

Figure 9. Calculations from KM&APOSS-MC and KM&PSO in the fluctuant situation.

Figure 7. Comparisons of KM&APOSS-MC and KM&PSO to the calculated values in the stable situation.

Figure 10. Comparisons of KM&APOSS-MC and KM&PSO to the calculated values in the fluctuant situation.

In the third situation, KM&PSO captures the macroscopic trend of the copper concentration, while when the concentration reaches a peak or hits a low point, obvious discrepancies are observed as shown in Figure 12. However, KM&APOSSMC performs better in these points, and the calculations are close to the measured concentrations. Figure 13 and Figure 14 also prove that the errors of the proposed method are lower than that of KM&PSO. From Table 3, we see that the kinetic model with fixed parameters works well when the copper concentration is relatively constant. However, the calculation results of our multicondition model are superior to the fixed parameter one when considering absolute error categories: MAE, RMSE, and MAXABS in the situation with fluctuations. Especially, the former model hardly satisfies the requirement of the zinc purification production (production index: AP > 0.85) if the copper concentrations fluctuate frequently. Generally, the

Figure 8. Errors of KM&APOSS-MC and KM&PSO in the stable situation.

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Figure 11. Errors of KM&APOSS-MC and KM&PSO in the fluctuant situation.

Figure 14. Errors of KM&APOSS-MC and KM&PSO in the situation with macroscopic trend.

Table 3. Results Analysis for KM&APOSS-MC and KM&PSO under Three Situations situations stable situation

fluctuant situation macroscopic trend

methods KM&APOSSMC KM&PSO KM&APOSSMC KM&PSO KM&APOSSMC KM&PSO

MAE (g/L)

RMSE

MAXABS (g/L)

AP

0.010

0.019

0.036

0.96

0.013 0.021

0.020 0.023

0.038 0.078

0.91 0.85

0.029 0.017

0.026 0.014

0.078 0.041

0.80 0.90

0.021

0.019

0.058

0.86

industrial situation consists of several reaction conditions, each of which has different kinetic parameters. Thus, the model with only one set of fixed parameters cannot describe the industrial situation with multiconditions exactly. Table 4 shows the conditions classified in the proposed parameter optimal selection method for each industrial

Figure 12. Calculations from KM&APOSS-MC and KM&PSO in the situation with macroscopic trend.

Table 4. The Numbers of Industrial Data for Each Condition in KM&APOSS-MC situations

condition 1

condition 2

condition 3

condition 4

stable situation fluctuant situation macroscopic trend

41 18 20

9 14 8

0 16 2

0 12 10

situation. In the proposed method, all the industrial data are classified into four conditions. In the stable situation, only nine samples differ from the others, which have less influence on the precision of the kinetic model with fixed parameters. When the situation fluctuates frequently, more samples belong to different reaction conditions. So the multicondition kinetic model describes the changes of the copper concentration precisely in the last two situations. The optimal values of the kinetic parameters and their 95% confidence intervals for the different conditions are shown in Table 5. Copper ions are precipitated at the fastest rate in condition 3 and the slowest rate in condition 4. The removal reaction rates are neither too fast nor too slow in the other conditions. Accordingly, the kinetic parameters vary greatly in

Figure 13. Comparisons of KM&APOSS-MC and KM&PSO to the calculated values in the situation with macroscopic trend.

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Table 5. Parameters Estimation and Their 95% Confidence Interval conditions

KD × 10−5

condition condition condition condition

4.97 5.13 5.38 4.61

1 2 3 4

± ± ± ±

kG × 10−5

0.08 0.11 0.13 0.14

3.21 3.42 3.61 3.09

± ± ± ±

0.09 0.12 0.14 0.10

De × 10−7 3.17 3.27 3.44 3.01

± ± ± ±

0.13 0.10 0.12 0.09

k × 10−5 4.52 4.71 5.02 4.32

± ± ± ±

0.11 0.13 0.18 0.08

the second and third situation, while the parameters keep relatively constant in the stable situation.

6. INDUSTRIAL APPLICATION Supported by these successive experiments, the proposed system model and optimal parameter selection strategy are Table 6. The Computation Time Required for the APOSSMC Method

off-line online

condition labeling

condition balancing

parameter optimization

403 s

45 s

51 s

condition classification

model calculation

45 s

7s

applied to an actual zinc purification and copper removal process located at the largest lead−zinc smelting factory in China. The proposed model and strategy are used for the soft measurement of the outlet copper concentration and provide essential monitoring and control of this process. The soft measurement system based on the proposed model was developed in the visual C++ language on the Windows 2000 platform. Access 2007 technology stored the original industrial data sampled by the DCS system. In the soft measurement system, the databases include two parts: the sample base and kinetic parameter base. The former base consists of 1000 representative industrial samples of 10 parameters, containing inlet copper concentrations, inlet flow rate, amounts of zinc powder added in the first and second reactors, solid content and flow rate of the returning underflow, quality ratio of cuprous oxide to the precipitate, pH value, temperature, and the corresponding industrial condition label. The latter base consists of the industrial condition label and four optimized kinetic parameters (KD, kG, De, and k). During the process of soft measurement of the outlet copper concentration, the input data are classified to determine the

Figure 16. Predicted error from the kinetic model after running for 50 days.

condition label with the former database. Next, the corresponding kinetic parameters are determined according to the label by traversing the latter database. Lastly, the outlet copper concentration is calculated by the kinetic model from the determined parameters. In this section, 500 groups of the estimated and measured concentrations, collected over almost 45 days after the system was put into operation, are selected to verify the efficiency of the kinetic model. In the application, the computation time of the proposed method is divided into two parts: the required time of off-line computation and online computation, as shown in Table 6. Before the kinetic model calculates the copper concentration online, the kinetic parameters should be optimized for the different conditions of that the industrial data are also labeled and balanced off-line. It requires about 8 min for off-line

Figure 15. Predicted outlet copper concentration from the kinetic model after 50 days. 17084



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hydrometallurgy plant in China. The running results show the proposed model works stably and reliably.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.:/Fax: +86-731-88836876. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 61273185 and 61174133), the National Science Foundation for Distinguished Young Scholars of China (Grant No. 61025015), and the National Key Technology R&D Program of China (Grant No. 2012BAF03B05).

■

Figure 17. Diagram of the distribution of prediction errors from the kinetic model.

REFERENCES

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computation. When the model was applied online, it only cost 53 s for the model to identify the conditions to which the current sample belongs and calculate the copper concentration. Figure 15 compares the results for the estimated and measured outlet copper concentrations. The proposed kinetic model accurately captures the trend of the copper concentration. From Figure 16 and Figure 17, almost 90% of the estimation errors are lower than 0.02 g/L. These results prove that the proposed system model and parameter selection strategy adapt to changes in the copper concentration with stable estimation accuracy.

7. CONCLUSIONS In this study, a kinetic system model of coexisting competitiveconsecutive reactions and parameter optimal selection method are proposed for a copper removal process with recycle underflow. The system model is accomplished by combining a cementation kinetic model and shrinking core model for copper cementation and cuprous oxide precipitation. To increase the model precision for applying in the actual process, the model parameters are estimated for different industrial conditions. This procedure consists of condition labeling, condition balancing, and parameter optimization, where the samples are classified adaptively into different conditions according to the model performances, and used to estimate the model parameters for each condition. The kinetic model with the proposed APOSS-MC is tested for three different situations in the practical process, including stable, fluctuant, and tendency situations, and compared with the model with one set of fixed parameters. Numerical results show good agreement between the model calculations and measured values in each situation, while the calculations in the concentration peaks enhance that the kinetic model performs more precisely by using the proposed parameter optimal selection method. And the AP index also proves that the kinetic model with the proposed APOSS-MC reaches the requirements of the industrial production. The proposed model, with the parameter optimal selection method, is being used for the soft measurement of copper concentration in an actual copper removal process of zinc 17085



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