A Simple and Efficient Initialization Strategy for Optimizing Water-Using Network Designs

Transcription

1 Ind. Eng. Chem. Res. 2007, 46, A Simple and Efficient Initialization Strategy for Optimizg Water-Usg Network Designs Bao-Hong Li* Department of Chemical Engeerg, Dalian Nationalities UniVersity, Dalian , Cha Chuei-T Chang Department of Chemical Engeerg, National Cheng Kung UniVersity, Taan, Taiwan 70101, Republic of Cha An efficient itialization strategy is developed this work to solve the NLP and MINLP models for synthesizg water-usg networks with multiple ants. A fairly good estimate of the optimal solution can be generated for any given problem accordg to a simple calculation procedure. Two published examples are adopted to demonstrate the effectiveness of the proposed approach. The optimum solutions obtaed this study are at least as good as those reported the literature, while the computation time required to achieve convergence is significantly less. Introduction * To whom correspondence should be addressed, Tel: Fax: libh@dlnu.edu.cn. Water is consumed very large quantity the chemical and petrochemical dustries. In recent years, the severe shortages resources and strict governmental regulations on dustrial effluents have created strong centives for water conservation. Takama et al. 1 first developed a mathematical program for optimal water allocation a petroleum refery based on a superstructure, which all possible reuse and regeneration opporties are embedded. Later, the so-called pch method was proposed by Wang and Smith 2,3 to design water-usg and wastewater-treatment networks separately. These pioneerg works stimulated a series of enthusiastic research activities. 4-6 Notice that, to avoid handlg the complex teractions between a water-usg network and wastewater treatment network, most studies only focused on the design issues concerng either one of these two subsystems. It has been well recognized that the pch method is very helpful for understandg the trsic nature of a water-usg network. In addition, the necessary theorem proposed by Bagajewicz and Savelski 7,8 is another useful tool for this purpose. In general, the sgle-ant water networks can be optimized routely with available methods. 2,9 As for the multiant case, the pch method is really not applicable. The existg design approaches can be classified to two types. One of them can be regarded as a sequential procedure; 5,10 that is, the design of a water-usg network is decomposed to a series of simpler steps and an acceptable solution can be obtaed gradually. The other is referred to as the simultaneous method, which a tree search strategy 11 or a superstructurebased mathematical program 12,13 is adopted to identify the optimal networks. Obviously, the solutions obtaed with the sequential approach are usually suboptimal. On the other hand, the drawbacks of simultaneous optimization strategy are maly due to the computation difficulties encountered locatg the global optimum. Notice that the itial guesses adopted solvg an NLP or MINLP model always exerts a profound impact on the convergence process. Other than Chang and Li, 13 the itialization issues synthesizg the optimal water networks have often been neglected the past. In this paper, a simple and efficient strategy has been developed to generate near-feasible itial guesses for solvg the NLP or MINLP problems needed for water network synthesis. The remag paper is organized as follows. The problem statement and superstructure are provided the next section. The mathematical programmg model is then formulated, and the method to generate reliable itial guesses is explaed detail. At the end of this paper, the effectiveness of our proposed model and itialization method are demonstrated with two published examples. The solutions obtaed with the proposed approach are at least as good as those reported the literature, while a significant amount of computation time is saved the convergence process. Problem Statement and Superstructure Accordg to Bagajewicz et al., 11 the synthesis problem of a water-usg network can be described as follows: Given a set of water-usg s which multiple ants are volved, it is desired to determe a network of terconnections of water streams among the s so that the overall fresh water consumption is mimized while the s receive water of adequate quality. To facilitate model formulation to solve the above problem, it is necessary to build a superstructure to corporate all possible flow configurations. A substructure of that suggested by Chang and Li 13 is adopted the present work (see Figure 1). Followg is its construction procedure: (1) Place a mixg node at the let of every water-usg. (2) Place a mixg node before the wastewater sk. (3) Place a splittg node after the source; the split branches from this node are connected to all the mixg nodes established step 1. (4) Place a splittg node at the exit of every water-usg. The split branches from every such node are connected to all the mixg nodes stalled steps 1 and 2 except the one before itself; that is, self-recycle is not allowed. It should be noted that the self-recycle structure around every water-usg is forbidden this superstructure because such practice cannot promote water conservation /ie070702y CCC: $ American Chemical Society Published on Web 11/08/2007

2 8782 Ind. Eng. Chem. Res., Vol. 46, No. 25, 2007 where f u WW is the flow rate of wastewater generated by waterusg u. (5) Upper let and outlet concentration limits, c e C c out e Cout u U,k K (8) Figure 1. Generalized superstructure for water-usg network. Mathematical Programmg Model To describe the mathematical programmg model accurately, let us first troduce the defitions of two sets, i.e.: U ) {u u is the label of a water-usg ; u ) 1, 2,..., }; K ) {k k is the label of a ant that affects the water quality; k ) 1, 2,..., N K }. The design objective can be expressed as where f FW T is the total consumption rate of the water-usg network. It is subjected to the followg constrats: (1) At the source splittg node, the flow balance can be written as where f u FW is the consumption rate of water-usg u. (2) At the mixg node before water-usg u, the water balance equation is where f u1,u represents the branch flow rate from water-usg u1 to u and f u is the total flow rate after mixg node of u. In addition, the mass balances of ants should be where c f u c f u ) f u FW + ) M f T FW f FW FW T ) f u u)1 and c out are the concentrations of ant k at the let of u (after the mixg node) and the outlet of u1 (before the splittg node) respectively. (3) Around a water-usg, the water and ant balances are where F u L is the rate of water loss u and M is the mass load of ant k to be removed u. (4) At the splittg node after each water-usg, the flowrate balance is (1) (2) f u1,u u,u1 U (3) f c out u1,u u1,k f u ) f u out + F u L f u c + M ) f out out u c f u out ) N u u,u1 U,k K (4) u U (5) u U,k K (6) f + f WW u,u1 u u,u1 U (7) where C and Cout denote the imum allowable concentrations of ant k at let and outlet of u respectively. (6) Structural constrats. To manipulate structural complexity of the network, the imum number of branch streams enterg a mixg node or leavg a splittg node can be specified with the followg constrats. n + n FW u1,u u e NM u n + n WW u,u1 u e NS u u,u1 U (9) where, n u,u1 and n u1,u are bary variables used to signify respectively whether the branches from u to u1 and vice versa exist; n u FW and n u WW are bary variables associated with the branch streams from source to u and from u to sk, respectively; NM u and NS u represent the imum branch numbers allowed to enter the mixg node and to leave the splittg node of u, respectively. To relate these bary variables to the correspondg flow rates, the followg constrats are needed: f u,u1 e n u,u1 F u,u1 f u FW e n u FW FW u f u WW e n u WW WW u u,u1 U (10) u,u1 U (11) u U (12) u U (13) where F u,u1,fw u, and WW u denote the upper bounds of f u,u1, f FW u, and f WW u, respectively. Obviously, the flow rate will be forced to zero if the correspondg bary variables are zero. Notice also that, this study, the above-mentioned imum flow rates are not just arbitrarily set to a sufficiently large value. 6 They are carefully chosen based on the design objective and also specifications of the water-usg s. This practice is adopted to cut down search space and computation time. More details will be provided the next section. Accordg to Yang et al., 12 it is possible to obta an optimal solution that contas a number of water streams with very low flow rates. To avoid this pitfall network synthesis, the followg constrats must also be troduced: m f u,u1 g n u,u1 F u,u1 f u FW g n u FW FW u m f u WW g n u WW WW u m u,u1 U (14) u U (15) u U (16) where F m u,u1,fw m u, and WW m u represent the lower bounds of f u,u1, f FW u, and f WW u, respectively.

3 Ind. Eng. Chem. Res., Vol. 46, No. 25, Notice that the above formulation is essentially a MINLP problem. However, if the structural constrats are excluded, the remag model forms a NLP. Initialization Strategy Although the search space of the aforementioned NLP or MINLP problem is clearly defed, most solution procedures still call for a good itial guess. This is due to the fact that the startg pot usually exerts a profound fluence on the convergence process. Thus, a reliable itialization method has been developed this work to satisfy this need. Let us assume that the given data of the network synthesis problem clude the followg: the mass load of each ant every water-usg, the rate of water loss each, and the upper bounds of the correspondg let and outlet concentrations. Based on these data, the followg procedure can be followed to produce an itial guess: Step 1. Determe the imum consumption rate of each and assign it as the itial guess to f u FW. Assume that wastewater reuse is not allowed and only one ant (say k) is considered, then the mimum consumption rate of u can be determed if the imum outlet concentration of ant k is reached, i.e., G FW ) M Cout u U,k K (17) here Cout is the upper bound of outlet concentration, G FW is the mimum flow rate needed to satisfy the mass load of ant k u. For a water-usg with multiple ants, the ant k with imum value of G FW is the key ant of u. Sce the design objective is to mimize the total usage, the imum consumption rate of u, denoted by FW u, should be FW u ) G FW u U,k K (18) k Obviously, it is possible that there exist more than one key ant for a and the concentrations of the nonkey ants at the outlet do not reach their upper bounds when is supplied at the rate of FW u. So the itial value of f u FW can be set to (f u FW ).l ) FW u u U (19) Here (f FW u ).l represents the itial value of f FW u and the same notation is adopted throughout this paper. Step 2. Set the itial guesses of c out and f u,u1. The necessary conditions of optimality 8 imply that the outlet concentration of at least one ant reaches its imum. So the itial guess of c out is set to be Cout temporally, i.e., (c out ).l ) Cout (20) For an NLP or MINLP model, nonzero and reasonable itial values should be assigned to the flow rates of the terconnectg streams the superstructure as much as possible. To mata feasibility, the flow rate between any two s the network should be set to zero on the basis of eq 19. However, it is desirable to provide a small nonzero flow rate for each f u,u1 to avoid zero derivatives. i.e., where R is a small adjustable positive parameter. Step 3. Estimate the itial guesses of f FW T, f u, f out u, f WW u, and c. With the values of (f FW u ).l, (c out ).l, and (f u,u1 ).l, the aforementioned variables can be itialized accordg to the followg equality constrats: ( (c ).l ) Step 4. Modify the itial guess of (c out ).l. Usually, not all outlet concentrations reach their imum; the value of every (c out ).l can be corrected accordg to the followg equation: At this pot, the itialization process for the NLP model is completed. For the MINLP model, the followg steps should also be carried out: Step 5. Initialize the bary variables. Step 6. Set the upper bounds of remag flow rates, i.e., F u,u1 and WW u. Substitute eq 5 to eq 6 to elimate the variable f u and then express f out u as a function of the other variables, that is, out In this equation, if c (f u,u1 ).l )R (21) (f FW T ).l ) (f FW u ).l (22) u)1 (f u ).l ) (f ).l + (f u1,u u ).l u, u1 U (23) u1)1 u1*u (f u out ).l ) (f u ).l - F u L and c u U (24) N u (f WW u ).l ) (f out u ).l - (f u,u1 ).l u, u1 U (25) u1)1 u1*u u1)1 u1*u (f ).l(c out u1,u u1,k).l)/(f u ).l u, u1 U, k K (26) (c out ).l ) ((f u ).l(c ).l + M )/(f out u ).l u U,k K (27) (n u FW ).l ) 1 (n u WW ).l ) 1 u U (28) (n u,u1 ).l ) 1 u,u1 U and u * u1 (29) L c f out u ) F u + M c out - c u U,k K (30) simultaneously reach their imum, the correspondg flow rate will be the limitg flow rate of u when only ant k is considered, i.e., F lim ) F L uc + M Cout - C u U,k K (31)

4 8784 Ind. Eng. Chem. Res., Vol. 46, No. 25, 2007 Table 1. Stream Data of a Water-Usg Network 2 no. limitg flow rate C Cout hydrocarbon distillation 45 H 2S salt 0 35 hydrocarbon H 2S salt hydrocarbon desalter 56 H 2S Salt Table 2. Initial Guess Generated for Example 1 flow rate 2 hydrodesulfurization no. f u,u1 c out c hydrocarbon H 2S salt F 1,2) 0.1 hydrocarbon F 3,2 ) H 2S salt F 1,3) 0.1 hydrocarbon F 2,3 ) H 2S salt total where F lim is the limitg flow rate of u for ant k. The actual limitg flow rate of u should be where Fout lim u represents the limitg flow rate of u. Sce F u,u1 and WW u represent the imum flow rates of the branch streams connectg the splittg node of u to the mixg nodes before u1 and wastewater sk respectively, they should equal to the correspondg limitg flowrate of u, i.e., F u,u1 ) Fout lim u Fout u lim ) k K F lim WW u ) Fout u lim u U (32) u,u1 U (33) Illustrations Two published examples are adopted here to illustrate the effectiveness of proposed itialization strategy. All models were solved by GAMS modules 14 (version 22.4) on an IBM notebook (model A22e). Example 1. The WUN design problem presented here was origally studied by Wang and Smith 2 and later redesigned by Bagajewicz et al. 11 and Wang et al. 10 The given stream data are presented Table 1. The design objective is to synthesize a network with mimum usage. The reported global optimum, which was identified with a tree-search method, calls for a consumption rate of t/h. The aforementioned NLP model (without the structure constrats) was formulated the present example. The itial guesses were generated with the proposed method described the previous section and the value of R was taken to be 0.1 t/h (see Table 2). It can be observed that these values are nearly feasible while the feasible ones are given boldfaced numbers. The module CONOPT3 14 was then used to solve the NLP problem. Notice also that, other than the stream, Figure 2. Optimum network for example 1. Table 3. Stream Data of Another Water-Usg Network 11 no. load (kg/h) C Cout A B C A B ,000 C A B C A B C A B C A B C A B C A B C A B C A B C Table 4. Comparison the Optimum Network Obtaed by Different Methods method objective value computg time (s) no. of water reuse streams m flow rate of reuse streams tree search <) NLP model <) MINLP model <) the flow rates of all put streams of 1 (i.e., distillation) were set to zero itially. This is due to the fact that the imum allowable let concentrations of all ants this should be zero and it is impossible to accept reused waters here. The optimum network is found 1 s with almost identical objective value, i.e., t/h. This solution is presented Figure 2. In addition, the same solution can also be found with 1 s with the other NLP solvers GAMS, i.e., MINOS or SNOPT. Comparg with the tree search method 11 and the procedure-based method, 10 our proposed method is simpler and more efficient. Example 2. A more complex water-usg system is considered this example. The stream data presented Table 3 are

5 Ind. Eng. Chem. Res., Vol. 46, No. 25, Figure 3. Optimum network obtaed with MINLP model for example 2. adopted from Bagajewicz et al. 11 The reported objective value is t/h, and there are ne wastewater reuse streams the correspondg network. In our study, this WUN synthesis problem was solved with the proposed NLP and MINLP models separately to compare the effects of structural constrats. Sce s 3 and 5 only accept pure waters, their put streams from other s the superstructure were all removed. In addition, the structural constrats were not imposed upon the streams comg from the source and also wastewater streams dischargg to the sk, that is, n FW u and n WW u do not appear on the left of eqs 9 and 10, respectively, and eqs 12, 13, 15, and 16 are not cluded the MINLP model. This practice is due to the fact that, a realistic chemical plant, the facilities to troduce and discharge wastewater are usually available for every water-usg. The NLP model was formulated without any structural constrat, and the value of R was chosen to be 2 t/h for generatg the itial guesses. This model was then solved with MINOS, and the resultg mimum consumption rate was found to be t/h. Notice that this result is slightly better than that reported the literature. It should also be noted that the required computation time the present case is much shorter (see Table 4). Fally, notice that additional suboptimal solutions can be found if R takes values other than 2 t/h; for example, the objective value is t/h if R equals 1 t/h. As for the MINLP-based synthesis approach, the value of R and the mimum flow rates of reuse streams, i.e., F m u,u1, were all set to be 1 t/h. The upper bounds NM u and NS u were all chosen to be 3 except that associated with the splittg node of 5; i.e., NS 5 ) 5. The reason for this choice is that the limitg throughput and also the imum outlet concentrations of 5 are all quite moderate when compared with those of other s the network, and thus, its wastewater has the highest potential to be reused. The NLP solver of MINOS and MIP solver of CPLEX are called the MINLP solver of DICOPT. Under the aforementioned specifications, an optimal network can be identified. The key features of this solution can be found Table 4. A more detailed version can be found Figure 3 and Table 5. It can be observed from Table 4 that the number of reuse branches is reduced from 14 the NLP solution to 10 the MINLP solution, while the crease the objective value the latter case is 0.5%. More importantly, Table 5. Optimum Network Obtaed by MINLP Method for Example 2 process no. f u,u1 mimum flow rate c out c F 3,1)8.000 A F 5,1) B , F 10,1)1.776 C A F 10,2) B C A B C A F 5,4) B C A B C A F 10,6) B C A F 5,7) B C A F 5,8) B C A F 6,9) B C A F 5,10) B C total notice the computation time for solvg the more difficult MINLP model is still very short, i.e., less than 4 s. Conclusions A simple and efficient itialization strategy is developed this work to solve the NLP and MINLP models for synthesizg the optimal water-usg networks with multiple ants. The NLP model is suitable for the relatively small-scale problems. Sce the structural constrats are corporated maly for the purpose of simplifyg network configuration,

6 8786 Ind. Eng. Chem. Res., Vol. 46, No. 25, 2007 the MINLP model can be used to optimize larger water-usg systems. Two examples from the literature are used to justify the proposed approach; the obtaed optimum solutions are at least as good as the reported optimums but with significantly less computation time. Nomenclature Sets, Parameters, and Variables R)parameter used to itialize wastewater reuse stream N K ) the number of elements set K ) the number of elements set U c ) concentration of ant C ) given parameter of imum concentration at let of a Cout ) given parameter of imum concentration at outlet of a f ) flow rate of a water stream, t/h F ) given parameter of flow rate parameter FW ) given parameter of flow rate of G ) parameter of needed mimum flow rate of if no water reuse is allowed K ) the set consists of ant M ) parameter of mass load, kg/h n ) bary variable to represent whether a water stream exists NM ) parameter to represent the number of branches enterg a mixg node NS ) parameter to represent the number of branches leavg a splittg node U ) the set consists of water-usg s WW ) given parameter of flow rate of wastewater Superscripts FW ) stream ) let L ) loss lim ) limitg ) imum m ) mimum out ) outlet WW ) wastewater Subscripts k ) the label of a ant T ) total amounts u,u1 ) the label of a water-usg Acknowledgment Fancial support provided by Research Fund of Dalian Nationalities University under Grant is gratefully acknowledged by the first author. Literature Cited (1) Takama, T.; Kuriyama, T.; Shiroko. K.; Umeda. T. Optimal water allocation a petroleum refery. Comput. Chem. Eng. 1980, 4, 251. (2) Wang, Y. P.; Smith, R. Wastewater mimization. Chem. Eng. Sci. 1994, 49, 981. (3) Wang, Y. P.; Smith, R. Design of distributed effluent treatment systems. Chem. Eng. Sci. 1994, 49, (4) Bagajewicz, M. A review of recent design procedures for water networks referies and process plants. Comput. Chem. Eng. 2000, 24 (9), (5) Li, B. H.; Fei, W. Y. Design water-usg network with multiple ants by step by step lear programmg. J. Chem. Ind. Eng. 2005, 56 (2), 285. (In Chese). (6) Zheng, X. S.; Feng, X.; Shen, R. J.; Seider, W. D. Design of optimal water-usg networks with ternal water mas. Ind. Eng. Chem. Res. 2006, 45 (25), (7) Savelski, M. J.; Bagajewicz, M. J. On the optimality conditions of water utilization systems process plants with sgle ants. Chem. Eng. Sci. 2000, 55, (8) Savelski, M. J.; Bagajewicz, M. J. On the necessary conditions of optimality of water utilization systems process plants with multiple ants. Chem. Eng. Sci. 2003, 58, (9) Savelski, M. J.; Bagajewicz, M. J. Algorithmic procedure to design water utilization systems featurg a sgle ant process plants. Chem. Eng. Sci. 2001, 56, (10) Wang, B.; Feng, X.; Zhang, Z. X. A design methodology for multiple-ant water networks with sgle ternal water ma. Comput. Chem. Eng. 2003, 27, 903. (11) Bagajewicz, M. J.; Rivas, M.; Savelski, M. J. A new approach to the design of water utilization systems with multiple ants process plants. Annual AIChE Meetg 1999, Dallas, TX, (12) Yang, Y. H.; Lou, H. H.; Huang, Y. L. Synthesis of an optimal wastewater reuse network. Waste Manage. 2000, 20, 311. (13) Chang, C. T.; Li, B. H. Improved optimization strategies for generatg practical water-usage and -treatment network structures. Ind. Eng. Chem. Res. 2005, 44, (14) GAMS Development Corp. GAMS: The solver manuals. GAMS Development Corp.: Washgton DC, ReceiVed for review May 16, 2007 ReVised manuscript received September 2, 2007 Accepted September 11, 2007 IE070702Y