- Email: [email protected]
Evolutionary Multi-Objective Optimization of an Activated Sludge Process
10th International Symposium on Process Systems Engineering - PSE2009 Rita Maria de Brito Alves, Claudio Augusto Oller do Nascimento and Evaristo Chalbaud Biscaia Jr. (Editors) © 2009 Elsevier B.V. All rights reserved.
747
Evolutionary Multi-Objective Optimization of an Activated Sludge Process Rosana Kazuko Tomitaa, Song Won Parka a
Department of Chemical Engineering - USP Av. Prof. Luciano Gualberto, n. 380, trav. 3, CEP 05508-900, São Paulo - SP, [email protected]
Abstract Many real problems involve multiple objectives that are often conflicting and non comparable. Rare are the problems in which the objectives to be achieved (maximized or minimized) are single. Such problems are not adequately solved by using single objective function optimization techniques. Non comparable solutions are associated to the multi-objective optimization problems that represent the tradeoff among objectives. These solutions are named Pareto optimal solutions. In this work, a general review about different methods that deal with multi-objective optimization problems is presented. Special attention is given to evolutionary methods, detailing the NSGA II algorithm. This algorithm was implemented in Matlab to deal with crossover and mutation applied directly to real variables and constrained optimization problems. This method of evolutionary multi-objective optimization was applied in an activated sludge wastewater treatment system, in which effluent quality and operational cost are optimized. For this purpose, set-points of nitrate and dissolved oxygen concentrations were used as decision variables. The Pareto curves obtained at the end of 50 generations were analyzed. Finally, for illustration, one point in the Pareto curve was chosen to demonstrate the optimal set of values. In this study, the results obtained show the ability of the algorithm to achieve Pareto optimal solutions. Keywords: NSGA II, Evolutionary Multi-Objective Optimization, Activated Sludge.
1. Introduction Many real problems involve multiple objectives that are often conflicting and non comparable, and their solutions are associated to the multi-objective optimization problems, that represent the trade-off among the objectives, named Pareto optimal solutions. Most techniques to solve these problems comprise two phases: a multiobjective phase, which is more mathematical, and a subjective phase, namely decisionmakers problem. There are many methods to solve multi-objective problems. Very recently, the heuristic methods, such as those based on genetic algorithms, show increased application. Evolutionary algorithms are becoming an alternative to the classical optimization methods, because they permit to work with great search spaces, can generate the best compromise solution among several objectives using an unique run of optimization algorithm, besides not needing extra information, as function derivative. The evolutionary algorithms reproduce the natural evolution principles to drive the search towards an optimal solution. Such algorithms refer to a class of stochastic optimization methods that simulate the natural evolution process. According to Deb [1], the ability of an evolutionary algorithm to find multiple optimal solutions in one single simulation
748
R.K. Tomita and S.W. Park
run makes an evolutionary algorithm unique in solving multi-objective optimization problems. The first implementation of a multi-objective evolutionary algorithm occurred in the mid 1980s with the J. Schaffer studies, but only in the mid-1990s that the evolutionary multi-objective optimization take shape, with the development of several important algorithms [2]. Among them, the multiobjective genetic algorithm (MOGA), developed by Fonseca and Fleming [3]; non-dominated sorting genetic algorithm (NSGA), by Srinivas and Deb [4]; and the niched Pareto genetic algorithm (NPGA) elaborated by Horn et al. [5]. Also Deb et al. [6] proposed the elitist non dominated sorting genetic algorithm (NSGA II) so as to alleviate three principal criticisms of the NSGA: computational complexity, non-elitist approach and the need of specifying a sharing parameter.
2. Evolutionary Multi-Objective Method The elitist non dominated sorting genetic algorithm, denoted as NSGA II, incorporates the concept of elitism and use an explicit diversity preserving mechanism to make it more powerful than the previous NSGA algorithm. The NSGA II was proposed to alleviate three difficulties presented in NSGA: computational complexity, non-elitist approach, and the need for specifying a sharing parameter. The flowchart for NSGA II used here is shown in Figure 1. The blocks on right of this figure indicate our Matlab functions implemented for this work.
SBX – Simulated Binary Crossover Create
two
xi( 2,t 1)
>
solutions
0,5 1
E qi xi(1,t )
1
xi(1,t 1)
E qi xi( 2,t )
>
0,5 1 E qi xi(1,t ) 1 E qi xi( 2,t )
@ after two parental solutions
xi(1,t )
e
@
and
xi( 2,t )
. This
operator simulates recombination operator in one point for binary variables: first it generates the random number
E qi
0 d u i d 1 . The parameter
2(1 u i ) 1 (K 1) c
E qi
2u i 1 (K 1) c
when u i d 0.5
and
when 0.5 u i d 1 is calculated so that its probability area is the
number value u i . The Kc is the recombination distribution index. The more elaborated version of this SBX was implemented in the present paper.
Polinomial Mutation Apply
the
distribution
yi(1,t 1)
xi(1,t 1)
and G i
1 >2(1 u i )@
of
the
polynomial
probability
G i ' max . The parameter is estimated as G i 1 (K m 1)
to
create 1 (K m 1)
2ui
a
solution
when u i d 0.5
when 0.5 u i d 1
where K m is the index for the mutation probability distribution and
' max is the maximum
allowed perturbation. We implemented the more elaborated version of this parameter estimation
Evolutionary Multi-Objective Optimization of an Activated Sludge Process
NSGA.m 1. Initialize parameters
inputdados.m
2. Generation =0 3. random nº generator
4. Create initial population
5. Evaluate objective function and constraints. Classify solutions in fronts
realinit.m
func.m rankcon.m indcmp3.m
6. Sharing of P0
share.m sortf.m
7. Selection
nselect.m
8. Crossover 9. Mutation 10. Evaluate objective function and constraints of Q (offspring 11. Global population (P+Q) and selection of fronts that remain in the next generation 12. generation = generation+1
Not
warmup_random.m advance_random.m
13. generation < max generation
Yes End
Figure 1: Flowchart for NSGA II
realcross.m real_mutate.m
func.m
keepalive.m rankcon.m indcmp3.m share.m sort.m gsort
Generation of offspring population
749
750
R.K. Tomita and S.W. Park
3. Evolutionary Multi-objective Optimization of the Activated Sludge Process The wastewater treatment system dynamics was modeled in Simulink, see Sotomayor et al. [7-9] for details. The multi objective functions to be minimized are the Kjeldahl nitrogen EQNK , the nitrate EQNO and the energy required for aeration AE . ° EQNK ° ° min ® EQ NO ° ° ° AE ¯
1 ³ S NK,e t Qe t dt T 1 ³ S NO,e t Qe t dt T 1 3 2 ³ ¦ 0,4032 K LA i 7,8408 K LA i dt T i 2
(1)
with the restraints SS max SS t 0 DQOmax DQO t 0 N total ,max N total t 0 DBOmax DBO t 0
dX B, H dt dX B, A
F X B, H entra X B, H r1 r2 r4 V F X B, A entra X B, A r3 r5 V
dt dX S F X S entra X S 1 f p r4 r5 r7 dt V dX P F X P entra X P f P r4 r5 dt V dX ND F X ND entra X ND iXB iXP f P r4 r5 r8 V dt dSS F 1 SS entra SS r1 r2 r7 dt V YH
dS ND dt
F S ND entra S ND r8 r6 V
dS NH dt
§ F S NH entra S NH iXB r1 r2 ¨¨ iXB 1 V YA ©
dS NO dt
F S NO entra S NO 1 YH r2 1 r3 V YA 2,86YH
dS O dt dS ALK dt
· ¸r3 r6 ¸ ¹
F SO entra SO 1 YH r1 4,57 YA r3 K LA SO sat SO V YH YA F S ALK entra S ALK iXB r1 r2 r3 1 YH r2 1 r3 1 r6 V 14 40,4YH 7YA 14
(2)
751
Evolutionary Multi-Objective Optimization of an Activated Sludge Process
For the details of this activated sludge ASM1 model with the explanation of the reaction rates r1 ,..., r8 , see Tomita et al. [10]. The parameters applied for this problem were: the size of the population 100, number of iterations 50, recombination probability 0.9, mutation probability 0.1, recombination distribution index 1, mutation distribution sp NOmin
0 g .m 3 ,
index
1,
OD2sp,max
6 g.m 3 , OD3sp,min
3 g .m 3 ,
sp NOmax
0 g.m 3 , OD3sp,max
OD2sp,min
0 g.m 3 ,
6 g.m 3 For the control of this
activated sludge wastewater treatment, see Sotomayor et al. [11]. The simple and partial results of the evolutionary multi-objective optimization are shown in Figure 2.
Aeration energy (kWh/h)
Nitrate concentration (g/h) 200
50
180
45
160 40
140 120
35
100 30
80 60
25
40 20 15 20
20 25
30 35 Kjeldahl niitrogen(g/h)
40
45
0 20
50
(a)
25
30 35 40 Kjeldahl nitrogen (g/h)
45
50
(b)
Figure 2. (a) nitrate concentration versus Kjeldhl nitrogen (b) aeration energy Kjeldhl nitrogen
In the ASM1 activated sludge model the process is highly nonlinear due to the kinetics similar to Monod. In Figure 3-b, it is observed that the convergence of the evolutionary optimization is not finished. However, these results are good enough to illustrate the methods of multi-objective optimization. In our complete work [12] the sensitivity analysis of the optimization results was performed due to parameters of the NSGA II algorithm, but this is not the main scope of the present paper.
4. Concluding remarks The main contribution of the paper is the Matlab implementation of the NSGA II with more efficient and flexible programming for chemical engineering problems. Here, it can be integrated with a dynamic process model. This is illustrated with an ASM1 model implemented in Simulink. The usual implementation of NSGA II considers the maximum iteration as one of the stopping criteria, with no guarantee of optimization convergence. More efficient convergence criteria for chemical process applications have to be developed. Another issue for further development is the presentation of the clear decision-maker criteria for the upper layer of the multi-objective applications. The evolutionary algorithm also deserves more developments. Finally, after this experience,
752
R.K. Tomita and S.W. Park
the authors prefer to work, in the future, with differential evolutionary studies rather than adaptive and co-evolutionary algorithms.
References 1. Deb K (2001). Multi-objective optimization using evolutionary algorithms. John Wiley & Sons: New York 2. Coello CAC (2003). Guest editorial: special issue on evolutionary multiobjective optimization. IEEE Transactions on Evolutionary Computation 7(2) 97-99. 3. Fonseca CM and Fleming PJ (1993). Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. In: Kaufmann M (ed) Proceedings of the 1993 International Conference On Genetic Algorithms. San Mateo, pp. 416-423. 4. Srinivas N and Deb K (1994). Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation 2(3): 221-248. 5. Horn J, Nafpliotis N and Goldberg DE (1993). Multiobjective optimization usingthe niched pareto genetic algorithm. Urbana-Champaign, Illinois: Genetic Algorithms Laboratory, University of Illinois. (IlliGAL Report, 930051993). 6. Deb K, Pratat A, Sameer A and Meyarivan T (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): 182-197. 7. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2003). Multivariable identification of an activated sludge process with subspace-based algorithms. Control Engineering Practice, v. 11, n. 8, p. 961-969 8. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2002). Software sensor for on-line estimation of the microbial activity in activated sludge systems. ISA Transactions v. 41, p. 127-143. 9. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2001). A Simulation Benchmark to Evaluate the Performance of Advanced Control Techniques in Biological Wastewater Treatment Plants. Brazilian Journal of Chemical Engineering, v. 18, n. 01, p. 81-101. 10. Tomita RK, Park SW and Sotomayor OAZ (2002). Analysis of activated sludge process using multivariate statistical tools – a PCA approach. Chemical Engineering Journal 90: 283-290. 11. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2002).A reference model for evaluating control strategies in activated sludge wastewater treatment plants. Revue Des Sciences de L'eau, v. 15, n. 2, p. 543-556. 12. Rosana Kazuko Tomita (2004). Multi-objective Optimization of Wastewater Treatment Systems.PhD Thesis. University of São Paulo. 207p. Brazil. (in portuguese) Otimização Multiobjetivo de Sistemas de Tratamento de Efluentes . 2004. 207 f. Tese de Doutorado em Engenharia Química - Universidade de São Paulo, Orientador: Song Won Park.
747
Evolutionary Multi-Objective Optimization of an Activated Sludge Process Rosana Kazuko Tomitaa, Song Won Parka a
Department of Chemical Engineering - USP Av. Prof. Luciano Gualberto, n. 380, trav. 3, CEP 05508-900, São Paulo - SP, [email protected]
Abstract Many real problems involve multiple objectives that are often conflicting and non comparable. Rare are the problems in which the objectives to be achieved (maximized or minimized) are single. Such problems are not adequately solved by using single objective function optimization techniques. Non comparable solutions are associated to the multi-objective optimization problems that represent the tradeoff among objectives. These solutions are named Pareto optimal solutions. In this work, a general review about different methods that deal with multi-objective optimization problems is presented. Special attention is given to evolutionary methods, detailing the NSGA II algorithm. This algorithm was implemented in Matlab to deal with crossover and mutation applied directly to real variables and constrained optimization problems. This method of evolutionary multi-objective optimization was applied in an activated sludge wastewater treatment system, in which effluent quality and operational cost are optimized. For this purpose, set-points of nitrate and dissolved oxygen concentrations were used as decision variables. The Pareto curves obtained at the end of 50 generations were analyzed. Finally, for illustration, one point in the Pareto curve was chosen to demonstrate the optimal set of values. In this study, the results obtained show the ability of the algorithm to achieve Pareto optimal solutions. Keywords: NSGA II, Evolutionary Multi-Objective Optimization, Activated Sludge.
1. Introduction Many real problems involve multiple objectives that are often conflicting and non comparable, and their solutions are associated to the multi-objective optimization problems, that represent the trade-off among the objectives, named Pareto optimal solutions. Most techniques to solve these problems comprise two phases: a multiobjective phase, which is more mathematical, and a subjective phase, namely decisionmakers problem. There are many methods to solve multi-objective problems. Very recently, the heuristic methods, such as those based on genetic algorithms, show increased application. Evolutionary algorithms are becoming an alternative to the classical optimization methods, because they permit to work with great search spaces, can generate the best compromise solution among several objectives using an unique run of optimization algorithm, besides not needing extra information, as function derivative. The evolutionary algorithms reproduce the natural evolution principles to drive the search towards an optimal solution. Such algorithms refer to a class of stochastic optimization methods that simulate the natural evolution process. According to Deb [1], the ability of an evolutionary algorithm to find multiple optimal solutions in one single simulation
748
R.K. Tomita and S.W. Park
run makes an evolutionary algorithm unique in solving multi-objective optimization problems. The first implementation of a multi-objective evolutionary algorithm occurred in the mid 1980s with the J. Schaffer studies, but only in the mid-1990s that the evolutionary multi-objective optimization take shape, with the development of several important algorithms [2]. Among them, the multiobjective genetic algorithm (MOGA), developed by Fonseca and Fleming [3]; non-dominated sorting genetic algorithm (NSGA), by Srinivas and Deb [4]; and the niched Pareto genetic algorithm (NPGA) elaborated by Horn et al. [5]. Also Deb et al. [6] proposed the elitist non dominated sorting genetic algorithm (NSGA II) so as to alleviate three principal criticisms of the NSGA: computational complexity, non-elitist approach and the need of specifying a sharing parameter.
2. Evolutionary Multi-Objective Method The elitist non dominated sorting genetic algorithm, denoted as NSGA II, incorporates the concept of elitism and use an explicit diversity preserving mechanism to make it more powerful than the previous NSGA algorithm. The NSGA II was proposed to alleviate three difficulties presented in NSGA: computational complexity, non-elitist approach, and the need for specifying a sharing parameter. The flowchart for NSGA II used here is shown in Figure 1. The blocks on right of this figure indicate our Matlab functions implemented for this work.
SBX – Simulated Binary Crossover Create
two
xi( 2,t 1)
>
solutions
0,5 1
E qi xi(1,t )
1
xi(1,t 1)
E qi xi( 2,t )
>
0,5 1 E qi xi(1,t ) 1 E qi xi( 2,t )
@ after two parental solutions
xi(1,t )
e
@
and
xi( 2,t )
. This
operator simulates recombination operator in one point for binary variables: first it generates the random number
E qi
0 d u i d 1 . The parameter
2(1 u i ) 1 (K 1) c
E qi
2u i 1 (K 1) c
when u i d 0.5
and
when 0.5 u i d 1 is calculated so that its probability area is the
number value u i . The Kc is the recombination distribution index. The more elaborated version of this SBX was implemented in the present paper.
Polinomial Mutation Apply
the
distribution
yi(1,t 1)
xi(1,t 1)
and G i
1 >2(1 u i )@
of
the
polynomial
probability
G i ' max . The parameter is estimated as G i 1 (K m 1)
to
create 1 (K m 1)
2ui
a
solution
when u i d 0.5
when 0.5 u i d 1
where K m is the index for the mutation probability distribution and
' max is the maximum
allowed perturbation. We implemented the more elaborated version of this parameter estimation
Evolutionary Multi-Objective Optimization of an Activated Sludge Process
NSGA.m 1. Initialize parameters
inputdados.m
2. Generation =0 3. random nº generator
4. Create initial population
5. Evaluate objective function and constraints. Classify solutions in fronts
realinit.m
func.m rankcon.m indcmp3.m
6. Sharing of P0
share.m sortf.m
7. Selection
nselect.m
8. Crossover 9. Mutation 10. Evaluate objective function and constraints of Q (offspring 11. Global population (P+Q) and selection of fronts that remain in the next generation 12. generation = generation+1
Not
warmup_random.m advance_random.m
13. generation < max generation
Yes End
Figure 1: Flowchart for NSGA II
realcross.m real_mutate.m
func.m
keepalive.m rankcon.m indcmp3.m share.m sort.m gsort
Generation of offspring population
749
750
R.K. Tomita and S.W. Park
3. Evolutionary Multi-objective Optimization of the Activated Sludge Process The wastewater treatment system dynamics was modeled in Simulink, see Sotomayor et al. [7-9] for details. The multi objective functions to be minimized are the Kjeldahl nitrogen EQNK , the nitrate EQNO and the energy required for aeration AE . ° EQNK ° ° min ® EQ NO ° ° ° AE ¯
1 ³ S NK,e t Qe t dt T 1 ³ S NO,e t Qe t dt T 1 3 2 ³ ¦ 0,4032 K LA i 7,8408 K LA i dt T i 2
(1)
with the restraints SS max SS t 0 DQOmax DQO t 0 N total ,max N total t 0 DBOmax DBO t 0
dX B, H dt dX B, A
F X B, H entra X B, H r1 r2 r4 V F X B, A entra X B, A r3 r5 V
dt dX S F X S entra X S 1 f p r4 r5 r7 dt V dX P F X P entra X P f P r4 r5 dt V dX ND F X ND entra X ND iXB iXP f P r4 r5 r8 V dt dSS F 1 SS entra SS r1 r2 r7 dt V YH
dS ND dt
F S ND entra S ND r8 r6 V
dS NH dt
§ F S NH entra S NH iXB r1 r2 ¨¨ iXB 1 V YA ©
dS NO dt
F S NO entra S NO 1 YH r2 1 r3 V YA 2,86YH
dS O dt dS ALK dt
· ¸r3 r6 ¸ ¹
F SO entra SO 1 YH r1 4,57 YA r3 K LA SO sat SO V YH YA F S ALK entra S ALK iXB r1 r2 r3 1 YH r2 1 r3 1 r6 V 14 40,4YH 7YA 14
(2)
751
Evolutionary Multi-Objective Optimization of an Activated Sludge Process
For the details of this activated sludge ASM1 model with the explanation of the reaction rates r1 ,..., r8 , see Tomita et al. [10]. The parameters applied for this problem were: the size of the population 100, number of iterations 50, recombination probability 0.9, mutation probability 0.1, recombination distribution index 1, mutation distribution sp NOmin
0 g .m 3 ,
index
1,
OD2sp,max
6 g.m 3 , OD3sp,min
3 g .m 3 ,
sp NOmax
0 g.m 3 , OD3sp,max
OD2sp,min
0 g.m 3 ,
6 g.m 3 For the control of this
activated sludge wastewater treatment, see Sotomayor et al. [11]. The simple and partial results of the evolutionary multi-objective optimization are shown in Figure 2.
Aeration energy (kWh/h)
Nitrate concentration (g/h) 200
50
180
45
160 40
140 120
35
100 30
80 60
25
40 20 15 20
20 25
30 35 Kjeldahl niitrogen(g/h)
40
45
0 20
50
(a)
25
30 35 40 Kjeldahl nitrogen (g/h)
45
50
(b)
Figure 2. (a) nitrate concentration versus Kjeldhl nitrogen (b) aeration energy Kjeldhl nitrogen
In the ASM1 activated sludge model the process is highly nonlinear due to the kinetics similar to Monod. In Figure 3-b, it is observed that the convergence of the evolutionary optimization is not finished. However, these results are good enough to illustrate the methods of multi-objective optimization. In our complete work [12] the sensitivity analysis of the optimization results was performed due to parameters of the NSGA II algorithm, but this is not the main scope of the present paper.
4. Concluding remarks The main contribution of the paper is the Matlab implementation of the NSGA II with more efficient and flexible programming for chemical engineering problems. Here, it can be integrated with a dynamic process model. This is illustrated with an ASM1 model implemented in Simulink. The usual implementation of NSGA II considers the maximum iteration as one of the stopping criteria, with no guarantee of optimization convergence. More efficient convergence criteria for chemical process applications have to be developed. Another issue for further development is the presentation of the clear decision-maker criteria for the upper layer of the multi-objective applications. The evolutionary algorithm also deserves more developments. Finally, after this experience,
752
R.K. Tomita and S.W. Park
the authors prefer to work, in the future, with differential evolutionary studies rather than adaptive and co-evolutionary algorithms.
References 1. Deb K (2001). Multi-objective optimization using evolutionary algorithms. John Wiley & Sons: New York 2. Coello CAC (2003). Guest editorial: special issue on evolutionary multiobjective optimization. IEEE Transactions on Evolutionary Computation 7(2) 97-99. 3. Fonseca CM and Fleming PJ (1993). Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. In: Kaufmann M (ed) Proceedings of the 1993 International Conference On Genetic Algorithms. San Mateo, pp. 416-423. 4. Srinivas N and Deb K (1994). Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation 2(3): 221-248. 5. Horn J, Nafpliotis N and Goldberg DE (1993). Multiobjective optimization usingthe niched pareto genetic algorithm. Urbana-Champaign, Illinois: Genetic Algorithms Laboratory, University of Illinois. (IlliGAL Report, 930051993). 6. Deb K, Pratat A, Sameer A and Meyarivan T (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): 182-197. 7. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2003). Multivariable identification of an activated sludge process with subspace-based algorithms. Control Engineering Practice, v. 11, n. 8, p. 961-969 8. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2002). Software sensor for on-line estimation of the microbial activity in activated sludge systems. ISA Transactions v. 41, p. 127-143. 9. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2001). A Simulation Benchmark to Evaluate the Performance of Advanced Control Techniques in Biological Wastewater Treatment Plants. Brazilian Journal of Chemical Engineering, v. 18, n. 01, p. 81-101. 10. Tomita RK, Park SW and Sotomayor OAZ (2002). Analysis of activated sludge process using multivariate statistical tools – a PCA approach. Chemical Engineering Journal 90: 283-290. 11. Sotomayor, O.A.Z.; Park, S.W.; Garcia, C (2002).A reference model for evaluating control strategies in activated sludge wastewater treatment plants. Revue Des Sciences de L'eau, v. 15, n. 2, p. 543-556. 12. Rosana Kazuko Tomita (2004). Multi-objective Optimization of Wastewater Treatment Systems.PhD Thesis. University of São Paulo. 207p. Brazil. (in portuguese) Otimização Multiobjetivo de Sistemas de Tratamento de Efluentes . 2004. 207 f. Tese de Doutorado em Engenharia Química - Universidade de São Paulo, Orientador: Song Won Park.