Multiobjective Optimization of a Semibatch Epoxy ... - ACS Publications

Ind. Eng. Chem. Res. 2004, 43, 6055-6063

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Multiobjective Optimization of a Semibatch Epoxy Polymerization Process Using the Elitist Genetic Algorithm Kishalay Mitra,† Saptarshi Majumdar,*,‡ and Sasanka Raha‡ Engineering and Industrial Services, Tata Consultancy Services, 54B Hadapsar Industrial Estate, Pune 411013, India, and Tata Research Development and Design Centre, 54B Hadapsar Industrial Estate, Pune 411013, India

Multiobjective Pareto optimal solutions for a semibatch isothermal epoxy polymerization process are obtained by adapting the binary-coded nondominated sorting genetic algorithm II (NSGA II). The number-average molecular weight and polydispersity index are taken as two objectives, where the first one is maximized and the second one is minimized. The decision variables are addition profiles of various reagents, e.g., the amount of addition for monomer, sodium hydroxide, and epichlorohydrin at different times, whereas the solution of all species balance equations is treated as a constraint. Because the number-average molecular weight and polydispersity index are sometimes not sufficient to describe the desired species growth, additional objectives such as the maximization of preferential formation of lower oligomers with the minimization of NaOH addition are also studied. For all practical purposes, semibatch-mode operations are preferred even if for the fulfillment of certain objectives batch-mode operations are theoretically competitive. A well-validated model taking care of all physicochemical aspects of a reaction mechanism is a prerequisite for this kind of study. Process simulation and optimization with close proximity to the available experimental conditions is definitely a distinguishable feature of this work that can direct the results toward actual plant realizations. Introduction The Taffy process is the most popular industrial process for preparing epoxy polymers, where Bisphenol A (AA0, monomer) and epichlorohydrin (EP), in excess, are reacted in the presence of sodium hydroxide (NaOH).1 This reaction mixture leads to polymer formation of a glycidyl ether end group (building block) at both ends. The advancement process is the other route where pulverized NaOH is added in steps to the reaction mixture of AA0 and EP dissolved in a solvent.2 Actually, alkali controls oligomeric impurities in the advancement process.1,3 However, the role of the addition of other reagents (AA0 and EP) in polymer formation is not wellestablished. In this work, the investigation carried out is to identify whether the reactor performance can be improved further, from a batch reactor benchmark condition, if all of the reagents are added to the reaction mixture in a semibatch mode with various types of objectives defined on prevailing process requirements. There are very few experimental and theoretical studies of the epoxy polymerization process in the open literature. Raha and Gupta4 used a species balance and equation of moment’s approach to study the process. They gave special importance to building the entire modeling framework in order to study the effect of kinetic parameters and reagents on the performance of the reaction process in the batch mode. Raha et al.5,6 extended this work for semibatch-mode operation, where instead of the addition of all three reagents only at the * To whom correspondence should be addressed. Tel.: 9120-4042406. Fax: 91-20-4042399. E-mail: [email protected]. † Engineering and Industrial Services, Tata Consultancy Services. ‡ Tata Research Development and Design Centre.

beginning of the reaction process, NaOH is added in a semibatch manner. After these studies, a very obviously perceived need was to vary all three reagents in semibatch mode to investigate the improvement, if any, in the polymerization process performance. In the first case study presented in this work, the idea is to obtain optimal profiles or histories (reagent addition at intermediate times) of all three reagents for which the final product attains the user-specified objectives [minimization of the polydispersity index (PDI) and maximization of number-average molecular weight (Mn) simultaneously]. Now the final product obtained by this optimization procedure is a mixture of various species, out of which the operator may be interested in the most important one. In the epoxy process, one of the user requirements is to have maximum lower oligomers. Because the objective of Mn maximization does not necessarily attain maximization of a desired species in the final product unless specified as a separate objective, the concentration of the most desired species is set as another objective to be maximized with the minimization of chain growth of that species. This species distribution is obtained without violating some obvious requirements on the product quality. This is the second case study of this work. Several other issues have been considered in the second case study like the basic stoichiometric requirements of reagents, and restrictions have been posed to the total additions of reagents so that conditions can remain close to the process conditions of the available experimental data. In the third case study, the constraint of total NaOH additions as presented in case study 2 is moved from the category of constraint to objective. So, this becomes a three-objective optimization problem, where the first two objectives are identical to those of case 2 and the additional objective becomes a minimization of the total addition of NaOH.

10.1021/ie034153h CCC: $27.50 © 2004 American Chemical Society Published on Web 07/30/2004

6056 Ind. Eng. Chem. Res., Vol. 43, No. 19, 2004

This is done to investigate whether moving the total NaOH addition from constraint to objective helps in getting better operating points that provide a product of the same quality (as that found in case 2) with less NaOH addition. Multiobjective optimization problems lead to a set of optimal solutions, known as Pareto-optimal (PO) solutions, as opposed to the single solution provided by any single-objective optimization task. Mainly, two approaches exist in the open literature to find a set of PO solutions: (i) methods based on single-objective optimization techniques7,8 and (ii) methods based on multiobjective evolutionary algorithms (MOEAs).9 It is well established that the second approach is preferred over the first simply because of the time and effort saved in simultaneous computation of PO solutions.9 Most of these MOEAs are population-based approaches, whereas single-objective conventional methods are single-point marching techniques. MOEAs are capable of generating the whole PO set under a single simulation run as opposed to the minimum of as many simulations as there are points in the final PO set required in the case of the other approach. Broadly, these MOEAs are based on the working principle of nondominated sorting and Pareto ranking. In the former approach, all potential solutions in a population are checked for their dominance over other solutions. A solution is said to strictly dominate another solution if the previous one is found to be superior to the other in terms of both objectives. Some popular methods that use nondominated sorting are the nondominated sorting genetic algorithm (NSGA),9 elitist nondominated sorting genetic algorithm (NSGA II),9 etc. In the approach of Pareto ranking, the selection of individuals is based on the Pareto optimality of the individuals. However, there are several ways of performing this Pareto ranking, and on the basis of these ways, there are several well-known approaches under the category of MOEAs. Some of these popular approaches are the multiobjective genetic algorithm (MOGA),10 niched Pareto genetic algorithm (NPGA),11 Pareto archived evolutionary strategy (PAES),12 and strength Pareto evolutionary algorithm (SPEA).13 Deb9 describes all of these methods and shows clearly the edge that one method can have over others using various test functions. Though the multiobjective optimization problem described in this work could have been solved by any of the methods mentioned above, the method chosen by the authors for their work is NSGA II. Polymerization processes are generally good test beds for various optimization methods because of their inherent complex behavior, and MOEAs are relatively newly developed methods in the field of multiobjective optimization. Thus, some applications of MOEAs on polymerization as well as other complex chemical engineering problems have become increasingly popular these days.14-22 The performance of epoxy polymerization with the different addition patterns for all the reactants is not the only important feature of this present effort. A sincere effort has been made to give a totally new dimension to the quality objective of epoxy polymerization by defining new objective functions, new constraints through which realistic conclusions can be built. Mn and PDI were treated as the sole quality objectives in epoxy polymerization studies to date. Introduction of the maximization of a desired oligomer has never been addressed anywhere in epoxy polymerization literature.

Table 1. Reaction Scheme for Epoxy Polymerization

A detailed analysis has been done to understand the dynamics of the epoxy process with specific growth of a desired polymer species under several relevant process constraints. This study, with the maximization of the desired lower oligomers close to the reported experimental conditions, adds a totally new dimension to this research work. Formulation and Optimization Aspects Model. The complete kinetic scheme for the epoxy polymerization system is given in Table 1. Raha and Gupta4 and Raha et al.6 have modeled a batch and semibatch polymerization process, respectively, using the same scheme and subsequently validated the results of the model with experimental data. During the modeling of the process, mass and moment balance equations are written for 10 different species considered for modeling. These equations lead to a system of ordinary differential equations-initial value problems (ODEIVP) that is solved by a well-tested RK type explicit technique.23 The kinetic parameters appearing in these equations are estimated using the genetic algorithm (GA). The above-mentioned 48 ODEs can easily be written in condensed form as follows:

dxi/dt ) fi(x,U); i ) 1, 2, ..., 48

(1)

where x and U are vectors of the state and control variables (amount of intermediate additions for different reagents at different times). The control variable vector consists of three histories, namely, the history for NaOH

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addition [U1(t)], history for EP addition [U2(t)], and history for monomer addition [U3(t)]. The various molecular species including the monomer considered for the modeling exercise, all derived ODEs, details on modeling, solution procedure, and parameter estimation can be found in work by Raha and Gupta4 and Raha et al.5 Optimization Problem Definition. For the first case study, the objective function, I, is a vector consisting of two objectives: maximization of the numberaverage chain length (Mn), i.e., I1, and minimization of PDI, i.e., I2. Because a polymer of higher Mn and lower PDI is generally a desired one, attaining the abovementioned objectives simultaneously is going to provide the high-quality polymer. One simulation runs from zero (initial condition) to tsim ()7 in this case) hours. Each of the three profiles [U1(t), U2(t), and U3(t)] are, therefore, discretized into seven equally spaced points [21 (NDigit) altogether, u1, u2, ..., u21, i.e., u1-u7 belong to U1(t), u8-u14 belong to U2(t), and u15-u21 belong to U3(t)] at seven different times (time ) 0, 1, 2, 3, 4, 5, and 6 h), and each of these points is treated as a decision variable, thereby leading to 21 decision variables for the problem. Each of these decision variables has their respective upper and lower bounds. The optimization of all three profiles means finding the optimal values of these 21 parameters for which the stated objectives are met without violating the constraints (simultaneous solution of all mass balance equations). The optimization problem can be expressed in mathematical form as follows:

Objective 1: I1 ) max Mn ui

experimental conditions given by Batzer and Zahir.24,25 To make the results more reliable, fundamental stoichiometry is maintained throughout; i.e., the total additions of reagents are restricted to being around the amount of the reagents added for the batch experimental (benchmark) case. To achieve that, restrictions are set on the total amount of reagents. This optimization problem can be expressed in mathematical form as follows: n Objective 1: I1 ) max λEE 0

ui

Objective 2: I2 ) min ui

n λEE 1 n λEE 0

Decision variables: uLi e ui e uU i , i ) 1, 21 Constraints: mass balance equations PDI < PDIB Mn > Mn,B 14

ui ∑ i)8 7

Objective 2: I2 ) min PDI

> 3.0

ui ∑ i)1

ui

Decision variables: uLi e ui e uU i , i ) 1, 21 Constraints: mass balance equations

(2)

where PDI, Mn, and u1, ..., u21 are the polydispersity index, number-average molecular weight, 21 decision variables, respectively, and superscripts L and U denote the lower and upper bounds of the above-mentioned variables. Normally Mn and PDI are the obvious choices in the case of most of the polymer systems. However, sometimes having more information than the average properties such as Mn and PDI seems important, especially for systems such as epoxy, where a species balance approach is adopted for modeling and experimentally validated with experimental data under some specified process conditions. In the case of epoxy, EEn is one of the species that has glycidyl ether groups at both ends, and lower oligomers of this species are preferred for industrial practices. So, another set of optimization objectives is derived from this requirement in the second case study. This can be achieved by maximizing the EEn population with a decrease in its chain propagation. Maximization of the zeroth moment of EEn gives the idea of increasing the EEn population in the reaction mixture, and minimization of the ratio of the first and zeroth moment of EEn can give the idea of chain propagation. This is to be achieved by maintaining the basic stoichiometry (EP to be added more than three times as much as NaOH in the reaction mixture) of the

7

ui < 1.0 ∑ i)1 14

ui < 3.0 ∑ i)8 21

∑ ui < 1.0

(3)

i)15

n λEE is the concentration of the most desired species 0 n is with a glycidyl ether group at both ends, and λEE 1 the first moment of that species. PDIB and Mn,B are the PDI and Mn values for the benchmark case (defined in the next section). In addition to case study 2, another objective to minimize the total amount of NaOH additions is added in case study 3; as in a more alkaline situation, hydrolysis of EP can be of high probability and needs to be tackled effectively. Also, as stated earlier, the ratio of EP to NaOH has a strong impact over the product composition as well as the overall productivity of this process. In this formulation, the constraint of a ratio of the total addition of EP and NaOH greater than the


benchmark addition is omitted. So, the third formulation can be written in the following form: n Objective 1: I1 ) max λEE 0

ui


n λEE 1 n λEE 0

7


ui ∑ i)1

Decision variables: uLi e ui e uU i , i ) 1, 21 Constraints: mass balance equations PDI < PDIB Mn > Mn,B Figure 1. Multiobjective Pareto plot for Mn vs PDI.

7

ui < 1.0 ∑ i)1

Table 2. Process Performance Analysis for Different Addition Patterns of Reagents with Respect to Benchmark Data

14

ui < 3.0 ∑ i)8 21

∑ ui < 1.0 i)15

(4)

Here all of the symbols mean the same as they do in the formulation of case study 2. All optimization exercises are carried out for a reaction time of 7 h. The binary version of NSGA II, proposed by Deb et al., is adapted for all three multiobjective studies presented above. The description on the binary-coded NSGA II is omitted here for the sake of brevity and can be found readily in the open literature.9,26 Results and Discussion Experimental data for the epoxy processes are rarely available. As for authors’ information, no experimental or plant data are available for PDI and Mn for epoxy processes. Raha and Gupta4 validated their momentbased model with the monomer concentration data of Batzer and Zahir.24,25 This is quite reasonable because, after some hours of polymerization reactions, if one matches the monomer concentration profile with the experimental data very closely, that itself proves the validity of the proposed mechanism as well as the entire modeling exercise. Raha et al.6 have done this parameter estimation in their study with a simple GA. Because the search space for the parameter estimation can also have multimodality, handling the same with a better method for finding global optima like GAs is found to be justified. Incidentally, the parameters found by a simple GA provided a marginally better fitting of the model predictions with industrial data as compared to a similar study executed by Raha and Gupta4 using the Box complex method. These parameters were used here to simulate various scenarios. In the experiment conducted by Batzer and Zahir,24,25 the values used for one time initial addition of reagents are 0.4 kmol/m3 for NaOH, 1.280 kmol/m3 for EP, and

benchmark case set 1 (used in plots) set 2 (used in plots) set 3 (used in plots) set 4

Mn

PDI

633.2 6606.7 814.7 666.1 625.6

1.61 1.94 1.61 1.57 1.53

0.4 kmol/m3 for AA0. If the model is simulated using these sets of values, it provides a reasonable Mn (633.2) and PDI (1.61) of industrial relevance. Authors consider this to be benchmark performance data. Multiobjective optimization with NSGA II for case study 1 converged to a set of PO solutions as shown in Figure 1. This Pareto front is a combination of equally competitive solutions. If one moves along the Pareto front, one objective improves at the cost of the other. An operator can choose one solution out of this set of solutions by either using his/her intuition or using some higher level information. Regarding incorporation of preferences among Pareto optimal solutions, one can refer to the work of Coello27 and Cvetkovic and Parmee.28 Out of various PO solutions in the obtained PO front, each one having three different addition profiles for three different reagents, three specific points are picked up to analyze the epoxy system under various conditions. These points (sets 1-3) along with the benchmark point are presented in Table 2. Figure 2 shows the distribution of polymer species concentrations for these three different sets at the end of 7 h (reaction time). From this figure, it is clear that the concentration distribution for the higher PDI set (set 1) is quite dispersed compared to that with lower PDI sets (sets 2 and 3), as expected. For PDI less than 1.7, only a few polymer species show up in the concentration distribution at the end of 7 h of reaction time, making the polydispersity less. Mn and PDI increase quite steadily with time but their dynamics are obviously different for different addition patterns. Figures 3-5 show the optimized concentration profiles of NaOH, EP, and AA0 for all three data sets mentioned above during the time of reaction, respectively. The amounts of EP and AA0 are


Figure 2. Variation of the concentrations of different polymer species for set 1 (V2), set 2 (V3), and set 3 (V4) of Table 2.

Figure 4. Plot of the EP concentration vs time for set 1 (V2), set 2 (V3), and set 3 (V4) data of Table 2.

Figure 3. Plot of the NaOH concentration vs time for set 1 (V2), set 2 (V3), and set 3 (V4) data of Table 2.

much higher in the case of PDI ) 1.94 (set 1 in Table 2). This has happened because the enhanced kinetics of the chain growth steps (steps 4 and 5 in Table 1) require more of EP and AA0. When the same two chain growth steps are utilized, this particular case has grown to a huge Mn. NaOH, as a catalyst, was consumed very effectively in this case. The other two sets with PDI ) 1.61 and 1.57 (sets 2 and 3 in Table 2) are also undoubtedly better than the benchmark performance with respect to Mn and PDI. Semibatch operations are thus recommended for better operation and control of epoxy polymerization processes if Mn and PDI are considered as objectives. For set 2 in Table 2 (PDI ) 1.61 and Mn ) 814.73), the hourly change in the concentration of different polymer species is analyzed. This is presented in Figure 6. The trend observed here clearly indicates that some specific polymer species, not all of them, are growing as compared to others, and ultimately after 5 h of reaction, only those species hold the sole contribution toward the species distribution in the final product. PDI

Figure 5. Plot of the AA0 concentration vs time for set 1 (V2), set 2 (V3), and set 3 (V4) data of Table 2.

in this case is definitely low with respect to many other cases, and this is in tune with less diversity in the products. A higher Mn with a lower or equal PDI indicates the superiority of this set over the benchmark case. Keeping the quality of the products the same, one can achieve as much as 30% more Mn in this case. From numerous simulations carried out with all PO solutions and other cases and their analyses, one can make out some points regarding the process mechanism. A high amount of NaOH addition is required in the first phase of the reaction for better initiation of polymerization. EP should be added in a large amount at a later stage because it actively participates in chain growth. Also, more EP results in more BFn, EFn, and FFn kinds of polymer species in the final polymer product. It is experienced by the authors that if a relatively smaller amount of EP is added (especially in the later part of the reaction process), one can expect more growth of ABn and BBn types of polymer species. AA0 also follows the


Figure 6. Hourwise (after the first hour, after the third hour, and after the fifth hour) variation of polymer species concentrations.

Figure 8. Species distribution for an optimized addition pattern (for the minimum value in the Figure 7 plot).

EEn n n Figure 7. Multiobjective Pareto plot for λEE and λEE 1 /λ0 0 .

Figure 9. Species distribution for the benchmark case.

same trend as NaOH because initially the requirements for AA0 are high compared to the later part. Figure 7 shows the optimal Pareto front coming out of case study 2. Though all points are equally competitive in the Pareto front, the lowest value of the front, where chain propagation is touching a minimum, has been taken for further discussion. Figure 8 shows the species distribution for that case for important species over a long span of the reaction time. One can find that the desired species, EEn, dominates the population at about 15 h. Actually, the optimization was done for 7 h only in order to have a clear comparison with the benchmark case, and this case has more Mn and less PDI compared to the benchmark data at the end of 7 h. However, unfortunately, neither the benchmark case nor this present case has a stable product composition at the end of 7 h (Figure 9 is the species distribution of the benchmark case). A lot of unstable species remain at the end of 7 h, which will be dissociated further before giving a stable product. That is why an extended time simulation was performed for this case as well as for the benchmark case to stabilize the product composition.

It is also clear from Figures 8 and 9 that the benchmark case takes more time to stabilize than the optimized case. However, because basic stoichiometry for the batch process was maintained even in a semibatch mode, if the total amount of the three reagents for the optimized semibatch mode are added all together at the beginning of the process, the stability of the product is brought a little ahead of the proposed semibatch mode. However, that does not substantiate the superiority of batch-mode operations over semibatch operations because dumping all reagents together may pose bigger problems in terms of mass transport and interfacial reactions (between NaOH and EP) that have not been considered in the present modeling framework. Even removal of heat (to make the process isothermal like it was originally) will become a problem. So, the theoretical superiority of the batch process for these new optimization objectives (based on the present modeling scheme) may not have a practical appeal to process applications. This clearly suggests that the semibatch optimized addition patterns can give a practical improvement (feasible as well) to the epoxy process. So, from these two optimization


Figure 12. Optimized addition patterns for NaOH, EP, and AA0 additions. EEn EEn n Figure 10. Multiobjective Pareto surface for λEE 1 /λ0 , λ0 , and NaOH additions.

Figure 11. Species distribution for an optimized addition pattern (for the minimum NaOH addition pattern in Figure 10).

studies, it can be said that, although objective functions with Mn and PDI are good enough to give gross ideas about the process, better objectives can be thought of that are related to specific species growth. Also, from a Mn-PDI study, the stability of the product will be difficult to guess. One has to analyze the species distribution to have a clear view of the stable product composition. Figure 10 shows the PO solutions for case study 3. This set of three objectives study clearly opens a wide range of choices to the operator to operate a system based on practical and feasible considerations. A point is taken from the PO solution set that is very close to 0.4 kgmol/m3 of the total NaOH addition, and Figure 11 is the species distribution for this point under an extended time of reaction. With the additional objective of NaOH minimization, performancewise better results cannot be achieved under the same duration of reaction time (7 h). However, if more hours of reaction are allowed, as shown in Figure 11, unless productivity suffers by a great extent, a stabilized product can be achieved with a much smaller amount of the total NaOH addition (0.4 kgmol/m3 as compared to 0.9 kgmol/m3 in

most of the cases of case study 2). Figure 12 shows the three-dimensional plot of the optimized total addition of all three reagents at the end of 7 h. The requirements of EP are high in most of the cases with wide variations of NaOH, though AA0 is found to vary within a very tight band. AA0 triggers the polymerization process initially with a higher demand of EP for more production of the desired species with glycidyl ether groups at both ends (EEn). With more additions, NaOH, being a catalyst, forms unstable species in a faster fashion and desired species are formed by those unstable species getting depleted immediately. Following this finding, Figure 11 (with minimum NaOH additions) indicates a lower population of intermediate unstable species that results in a relatively slow growth of the desired species population at the end of the extended reaction time. The NSGA II parameters used to obtain the PO solutions for the present study are as follows: population size ) 250 (500 for case study 2 and 3); number of decision variables ) 21; string length for each decision variable ) 5; maximum number of generations ) 50; crossover probability ) 0.9; mutation probability ) 0.001. It took 24 generations to converge to the Pareto front by NSGA II starting with all random solutions in the given search space (for case study 1). The effect of varying several GA parameters on the Pareto solution was studied next. If the number of populations is decreased from 250 to 100, as the population number decreases, convergence of the initially generated inferior fronts to the Pareto set front was relatively faster (requirement of a smaller number of generations for convergence), but the spread of the Pareto set obtained was relatively poor. The effect of a change of the substring length (from 5 to 9) did not provide any improvement or deterioration on the final Pareto front. However, the total simulation time for the case with string length 9 was more because the number of bits computation was more in that case. The effect of varying the value of the crossover probability, from 0.9 to 0.7, again led to no significant changes in the optimal solutions though the number of generations taken to converge to the final Pareto was more (in the case of 0.7) than earlier (in the case of 0.9). It even took a larger number of generations to get the same spread as earlier. The effect of a change in the value of the mutation probability, from 0.001 to 0.01, neither affects the spread and number of points nor imparts any improvement in the Pareto set. Once the Pareto front is achieved, any


increase in the maximum number of generations successfully maintains the diversity in the Pareto front from generation to generation. Conclusion A well-validated large set of moment-based ODEs has been utilized for multiobjective optimization of the epoxy polymerization process using NSGA II. The primary aim was to extract the addition patterns of the reagents for maximum Mn with minimum PDI simultaneously. Growth of specific species along with the formation of lower oligomers is a set of better objectives because Mn and PDI can only reflect the average properties of the epoxy process. A three-objective optimization with an additional objective of minimizing the total NaOH additions produces a set of choices for the operator, from which he/she can choose the appropriate operating conditions using proper engineering judgment without sacrificing the product quality. Different species distributions and their evolutions with time have been studied. Trends among the profiles of different reagents as well as their relations with the reaction mechanisms are established. Utility of NSGA II for such multidimensional nonconvex processes can also be understood from this study. It has also become clear that a semibatch mode of operations for epoxy processes is supposed to be superior to one-time addition processes (batch mode) for all practical purposes. The mass-transfer effect can be included in this modeling framework to have a better physical picture. Acknowledgment Authors acknowledge the support of TRDDC and TCS management in the course of this work. Nomenclature AA0 ) Bisphenol A (monomer) EP ) epichlorohydrin I ) vector of objective functions Ki ) reaction rate constant Mn ) number-average molecular weight Mw ) weight-average molecular weight NaOH ) sodium hydroxide U ) vector of control variables, U1(t), U2(t), U3(t) (i) uPop,k ) value of the control variable at the end of the kth time interval for the ith chromosome max min uPop,j , uPop,j ) upper and lower bounds on the control variable at the end of the kth time interval PDI ) polydispersity index (Mw/Mn) tSim ) simulation time (h) λjk ) kth moment of the jth species, k ) 0, 1, 2 x ) vector of state variables, xi Subscripts/Superscripts max ) maximum value min ) minimum value

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Received for review September 29, 2003 Revised manuscript received June 13, 2004 Accepted June 23, 2004 IE034153H

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