On the Degeneracy of the Water/Wastewater Allocation Problem in Process Plants

Transcription

1 4340 Ind. Eng. Chem. Res. 2010, 49, On the Degeneacy of the Wate/Wasteate Allocation Poblem in Pocess Plants Déboa C. Faia and Migel J. Bagajeicz* UniVesity of Oklahoma, 100 E. Boyd, Noman, Oklahoma Seveal methodologies to design ate systems in pocess plants ae based on minimizing feshate consmption. The objective is the appopiate one hen ate is scace and costs ae not a big isse. It has also been sed as a sbstitte fo cost in the belief that ate and final teatment costs ovehelm othe fixed capital and opeating costs of egeneation pocesses. Among gaphical and algoithmic methods, the popla Pinch Technology -based pocede has ealy poponents and contempoay advocates ho conside and defend it as a good method to povide insights into the ight anse. In this pape, e sho that the minimm feshate poblem has sometimes a lage nmbe of altenative degeneate soltions, something that gaphical and algoithmic pocedes like the afoementioned pinch technology can hadly identify systematically. The fact that the degeneate soltions ae many makes it ose. We povide a mathematical pogamming method to identify these soltions and to point ot those that make moe economical sense. We also povide means to identify sboptimal soltions that ae vey close to the optimm one. We illstate these degeneacies in seveal cases of single and mltiple contaminants. Intodction Wate allocation poblems (WAP) stated to be of inteest in the ealy eighties and have been extensively stdied theeafte. Bagajeicz 1 pesents a evie of the most popla method pesented ntil In geneal, these appoaches can be divided in to big classes: mathematical pogamming-based methods and methods based on gaphical, heistic, o algoithmic pocedes. In addition, even thogh the seminal pape by Takama et al. 2 advocates cost minimization, most of the methods poposed late, 3 13 focs on minimizing feshate consmption. Althogh the eason has hadly been made explicit, e believe the assmption is made in analogy to pinch technology-based methods fo heat exchanges, that is, that most cost-effective soltions old feate minimm consmption. In othe ods, the ndelying assmption is that feshate costs ae the dominant costs. While mathematical pogamming-based methods have also been sed to solve the minimm feshate consmption poblem, they ae the appopiate tool fo a vaiety of cost objectives, to handle fobidden matches, 14,18,19 to add the cost of egeneation nits, 17,18 and fo maximizing pofitability Impotant effots to solve the WAP to global optimality ee pesented. 17,26,27 The poponents and cent advocates of the se of simplified pocedes as Pinch Technology applied to ate poblems, 28 o simplified models like soces and sinks, claim that these methods povide qick insights into the ight anse. They ae sccessfl in single component cases to coectly pedict minimm feshate consmption and can even be extended to the case of total ate systems, hee patial egeneation and ecycle fom end-of-pipe teatment is added (see a discssion in Faia and Bagajeicz 29 ). In the case of mlticomponent systems, they ae not igoos, bt seveal heistics that easonably obtain minimm feshate consmption have been poposed. These methods, hoeve, enconte some difficlties hen the objective is cost. Despite the sccesses in pedicting feshate consmption and above and beyond the difficlties in obtaining good soltions hen minimm cost is soght, the isse of obtaining the coesponding netok has been consideed as seconday, despite indications that * To hom coespondence shold be addessed. seveal altenative configations feating the same feshate consmption (degeneate soltions), might exist. Bagajeicz et al. 30 pesented an algoithmic pocede that combines mathematical pogamming and necessay optimality conditions. 31 They applied thei methodology to a lage mltiple contaminant example and, in addition to hat they conside the optimm soltion, fo sboptimm ones ee pesented. Pta and Ammindin 19 appoached the existence of hat they call class of good soltions. These soltions ae diffeent design options that find the same optimm (o nea optimm soltions), bt sho diffeent pespectives concening cost, layot (complexity) o efficiency of the egeneation pocesses. They find the class of good soltions by fixing the maximm nmbe of connection to an opeation o existence of egeneation-ecycling, and then minimizing the feshate consmption. They find fo good soltions and compae them ith thee othes fond by pevios oks Althogh thei point of analyzing diffeent options is valid, thei pocede is not able to effectively geneate a consideable nmbe of altenatives. Popleski et al. 35 also analyze altenative soltions fo the case of netoks fomed by ate-sing nits that have fixed flo ates and ae modeled as LP/MILP poblems. In eality, thee is no systematic ay knon so fa to pedict befoehand ho many altenative soltions a poblem has. In addition, it is nclea if sch degeneacy exists fo the case of minimizing total costs. In this pape, an atomatic method to find altenative soltions is poposed. The seach fo altenative soltions is done in a matte in hich a ne netok configation (connections among feshate soce, ate sing nits, egeneation pocesses and sink) is sccessively fond ith espect to a cetain objective fnction o a peset conditions (minimm feshate consmptions fo example) that ill chaacteize degeneate soltions fo the given condition. At each ne seach, the netoks peviosly fond ae exclded fom the set of feasible soltions. In a poblem ith high degeneacy, the optimm soltion (objective fnction vale) may be epeated fo many of the fond stctes and the altenative soltions povide a moe flexible scope in the decision making pocess. On the othe hand, hen the poblem is not highly degeneate, the altenative soltions can povide nonoptimm soltions in /ie901574g 2010 Ameican Chemical Society Pblished on Web 04/09/2010

2 Ind. Eng. Chem. Res., Vol. 49, No. 9, Fige 1. Scheme of a complete ate system. Fige 2. Scheme of the spestcte sed to descibe the ate system. hich othe citeia (mch loe investment costs, easie opeability, etc.) can be impoved. The pape is oganized as follos: e fist pesent the geneal mathematical model fo the WAP, a complete integated ate system, as pesented by Faia and Bagajeicz. 29 Next, a method to identify degeneate and nea-optimm soltions is pesented, and then seveal examples ae pesented. Mathematical Model The ate allocation poblem aims at optimizing a given indstial ate system sing one o moe objectives thogh the ese, ecycle and/o egeneation of sed ate. Fige 1 shos a conventional, seqentially odeed, nonintegated CWS. A complete ate system (CWS) as descibed by Faia and Bagajeicz 29 as the one involving the thee typical sbsystems in pocess plants: ate peteatment, ate-sing and asteate teatment bt adding all possible ecycles among nits of any sbsystem. Ths, the optimization of these systems can be stated as follos: GiVen sets of ate sing nits, feshate soce and potential egeneation pocesses (ate peteatment and/o asteate teatment nits), one ants to obtain a ate/ asteate netok that globally optimize a chosen objective fnction. The spestcte sed to model this poblem is pesented in Fige 2. All connections ae alloed among the sbsystems and ithin the sbsystem. Note that althogh the connections ithin the sbsystems ae not shon, all connection ae alloed inside the boxes epesenting the sbsystems. A geneal nonline model to solve the ate allocation poblem is given by the folloing set of eqations: Wate Balance at the Wate-Using Units. No ate losses ae consideed hee. FWU, + * FUU *, + * FRU, ) FUS,s + s FUU,* + FUR, (1) hee FWU, is the flo ate fom feshate soce to nit, FUU *, is the flo ate beteen nit * and, FRU, is the flo ate fom egeneation pocess to nit, FUS,s is the flo ate fom nit to sink s, and FUR *, is the flo ate fom nit to egeneation pocess. Wate Balance at the Regeneation Pocesses. FWR, + FUR, + * * FRR *, ) FRU, + FRR,* + FRS,s (2) s hee FWR, is the flo ate fom feshate soce to the egeneation pocess, FRR *, is the flo ate fom egeneation pocess * to egeneation pocess, and FRS,s is the flo ate fom egeneation pocess to sink s. In fact, e assme hee that the set of egeneation pocesses existing in the system is fomed by the set of ate peteatments and the set of asteate teatments. If one ants to diffeentiate beteen these to categoies of egeneation pocesses, to sbsets fo the egeneation pocesses set can be easily ceated and diffeent constaints applied to each sbset. Contaminant Balance at the Wate-Using Units. (CW,c FW, ) + (FUU *,,c C ot *,c ) + * (FRU,,c CR,c (FUS,s,c C,c s ot ) + M,c ) * ot ) + (FUU,*,c C ot,c ) + (FUR,,c C ot,c ), c (3) hee CW,c is the concentation of contaminant c in the feshate soce, M,c is the mass load of contaminant c extacted in nit,c ot,c is the otlet concentation of contaminant c in nit, and CR ot,c is the otlet concentation of the not teated contaminant c in egeneation. Maximm Inlet Concentation at the Wate-Using Units. It is common to impose limits to the maximm inlet and otlet concentations of ate-sing nits.

3 4342 Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 (CW,c FW, ) + (FUU *,,c C ot *,c ) + * (FRU,,c CR ot,c ) e C,c in,max ( FUW, + FUU *, + * FRU, ), c (4) hee C in,max,c is the maximm alloed concentation of contaminant c at the inlet of nit. Maximm Otlet Concentation at the Wate-Using Units. ot C *,c ot,max e C,c, c (5) hee C ot,max,c is the maximm alloed concentation of contaminant c at the otlet of nit. Flo ate togh the egeneation pocesses. FR ) FWR, + FUR, + FRR *, (6) * hee FR is the flo ate thogh the egeneation pocess. Contaminant Balance at the Regeneation Pocesses. To types of egeneation pocesses ae consideed in this model: one ith a pedefined and fixed otlet concentation and anothe ith a pedefined and constant ate of emoval. FR,c CR in,c ) (FWR, CW,c ) + (FUR, C ot,c ) + (FRR *, CR *,c * ot ), c (7) CR,c ) CR,c (1 - XCR,c )(1 - RR,c ) + CRF,c XCR,c, c (8) hee CR,c in is the concentation of contaminant c at the inlet of egeneation pocess and XCR,c is a binay paamete that indicates if contaminant c is teated by egeneation pocess as a pocess ith fixed otlet concentation (XCR,c ) 1) o as a pocess ith knon ate of emoval (XCR,c ) 0). CRF,c ot is the concentation of the teated contaminant hen it is knon and constant, and RR,c is a given ate of emoval of contaminant c in egeneation pocess. Maximm inlet concentation of the egeneation pocesses. Uppe limits on the inlet of egeneation pocesses ae also imposed. CR in,c in,max e CR,c, c (9) hee CR in,max,c is the maximm concentation of contaminant c alloed at the inlet of egeneation pocess. Maximm Alloed Dischage Concentation. Dischage limits ae nomally imposed in all kind of sinks. When the sink is a ive fo example, the limits ae elated to envionmental eglations. In cases in hich the sinks ae pocesses that only consme ate, these limits ae elated to technical specifications as in the case of ate-sing nits. (FUS,s,c C,c C s,c ot ) + dischage,max ( (FRS,s,c CR ot,c ) e FUS,s + FRS,s ) s, c (10) hee C dischage,max s,c is the maximm alloed concentation at sink s. Minimm Flo Rates. It is ell-knon that many soltions of the ate poblem may inclde small flo ates that ae impactical. To avoid these e se the folloing constaints: FWU, g FWU min, YWU,, (11) FWR, g FWR min, YWR,, (12) FUU,* g FUU min,* YUU,*, * (13) FUS,s g FUS min,s YUS,s, s (14) FUR, g FUR min, YUR,, (15) FRU, g FRU min, YRU,, (16) FRR,* g FRR min,* YRR,*, * (17) FRS,s g FRS min,s YRS,s, s (18) hich ses a set of binay vaiables (YWU,, YWR,, YUU,*, YUS,s, YUR,, YRU,, YRR,*, and YRS,s ) that ae eqal to one hen the coesponding flo ate is diffeent fom zeo and zeo otheise. Maximm Flo Rates. To ense that the connections do not spass maximm vales, e se the folloing constaints: FWU, e FWU max, YWU,, (19) FWR, e FWR max, YWR,, (20) FUU,* e FUU max,* YUU,*, * (21) FUS,s e FUS max,s YUS,s, s (22) FUR, e FUR max, YUR,, (23) FRU, e FRU max, YRU,, (24) FRR,* e FRR max,* YRR,*, * (25) FRS,s e FRS max,s YRS,s, s (26) Objective Fnctions. Minimm feshate consmption: min ( Minimm total annal cost: max[op( R ( FWU, + FWR, ) (27) FWU,m + FWR, ) + OPN FR ) - af FCI] (28) hee OPN ae the opeational cost of the egeneation pocesses, OP is the hos of opeation pe yea. The last tem is the annalized capital cost, hee FCI is the fixed capital cost and af is any facto that annalizes the capital cost (sally 1/N, hee N is the nmbe of yeas of depeciation). The fixed capital of investment is calclated sing the sm of the piping costs and the ne egeneation nits costs as follos: FCI ) YWU (, CCWU, + YUR, CCUR, + ** ( YUU,* CCUU,* + YUS,s CCUS,s ) + s YWR, CCWR, + YRR,* CCRR,* + ** YRU CCRU + YRS,s CCRS,s + CCR (FR ) ) 0.7 (29) s hich ses a set of capital cost paametes to assign cost to the connections (CCWU,, CCWR,, CCUU,*, CCUS,s,

4 Table 1. Limiting data of example 1 pocess mass load (kg/h) C in,max (ppm) C ot,max (ppm) CCUR,, CCRU,, CCRR,*, and CCRS,s ) and to the egeneation pocesses (CCR ). Degeneacy and Sb-Optimal Soltions With the exception of Bagajeicz et al., 30 ho pesents a pocede to identify optimm and sboptimm soltion sing the necessay conditions of optimality, 31 and Pta and Ammindin, 19 ho pesent an appoach to geneate hat they call class of good soltions, e kno of no othe ok that has pesented a methodology to find degeneate and sboptimm soltions of ate allocation poblems. Pta and Ammindin 19 poposed a to-step appoach to find the mltiple soltions. In the fist step the stcte of the netok is defined sing an MILP model, and then a NLP model is sed to find the conditions fo the fond stcte. They claim this stategy endes a global optimm, bt they offe no poof of this assetion. Becase of the to step stategy poposed, e dobt it is. The class of good soltions is fond fixing the piping connections, hich can be elated to the nmbe of ate ese steams, maximm nmbe of connection to an opeation o existence of egeneation-ecycling, and minimizing the feshate consmption. Even if degeneacy and sboptimm soltions can be fond sing this pocede, thee can still exist othe altenative soltions fo the same piping netok. To amelioate the above poblems, e popose an atomatic method to find a significant highe nmbe of options, if not all of them. The seach fo altenative soltions is done in a matte in hich a ne netok configation (connections among feshate soce, ate sing nits, egeneation pocesses and sink) is sccessively fond ith espect to a cetain objective fnction. At each ne seach the pevios fond netok ae Ind. Eng. Chem. Res., Vol. 49, No. 9, exclded fom the feasible soltion. In a poblem ith high degeneacy, the optimm soltion (objective fnction vale) ill be epeated fo many of the fond stctes and the altenative soltions povide a moe flexible scope in the decision making pocess. On the othe hand, hen the poblem is not highly degeneated, the altenative soltions can povide non optimm soltions in hich pesent othe advantages sch as mch loe investment costs, easie opeability, etc. The altenative soltions ae fond as follos: Step 1. Rn the model pesented in section 3. Step 2. Fobid the netoks peviosly fond. Step 3. Go back to step 1. To fobid the netoks, the folloing intege ct 36 is added to the model (i,j) { (,),(,*),(,), (,s),(,),(,*),(,s) } NYIJn,i,jYIJi,j + (1 - NYIJn,i,j)(1 - YIJ i,j ) e CARD(NYIJ) - 1 n < n fond (30) hee n coesponds to the nth netok peviosly fond, n fond is the nmbe of netoks peviosly fond and NYIJ n,i,j ae the vales of the binay vaiables obtained in n n, hich define the configation of each netok. In tn, CARD(NYIJ) is the cadinality of the set of binay vaiables NYIJ. Ths, the netok exclsion constaints fobid combinations of possible connections fond all pevios iteations. The left-hand side of the eqation is sed to accont fo existing (fist tem) and nonexisting connections (second tem) in the nth soltion. In othe ods, all the peviosly fond combinations ill have the smmation eqal to CARD(NYIJ) and theefoe cannot be epeated. Ths, to geneate a ne netok, at least one of the connections needs to be inclded o exclded. It is essential that the fist soltion fond is a global optimm becase all degeneate soltions oght to have the same objective. A local optimize cold be sed aftead becase one old be looking fo feasible soltions having the same Fige 3. Hnded fist soltions fo minimm feshate consmption of the ate-sing sbsystem single contaminant example fom Wang and Smith. 3

5 4344 Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Table 2. Limiting Data of Example 1 pocess contaminant mass load (kg/h) C in,max (ppm) C ot,max (ppm) 1 A B A B Table 3. Altenative Netok Configations fo the Wate-Using Sbsystem of the Mltiple Contaminants Example fom Wang and Smith 3 a nit 1 EOP netok 1 54 ton/h feshate t/h nit 1 21 t/h 19 t/h 35 t/h netok ton/h feshate 40 t/h t/h nit 1 40 t/h t/h netok ton/h feshate t/h t/h nit t/h t/h netok ton/h feshate t/h nit t/h t/h t/h netok ton/h feshate t/h nit t/h t/h a A minimm flo ate of 1 t/h as sed. objective. We hoeve believe that this may not gaantee that all degeneate soltion ill be fond becase the local optimize may fail. Reslts Reslts have shoed that fo some poblems pesent a significant nmbe of degeneate soltions egading minimm feshate consmption. On the othe hand, thee ae poblems in hich degeneacy is not pesent o is vey small. We stat ith a single contaminant case and then analyze mltiple contaminant cases. Example 1. This example coesponds to the ate-sing sbsystem example pesented by Wang and Smith, 3 hich has fo ate-sing nits. The data fo this poblem is shon in Table 1. We solve the poblem minimizing feshate consmption to global optimality to find the 100 fist netoks. We se a minimm flo ate of 1 t/h fo all connections. Hee, the minimm flo ate is not only elated to pactical isses, bt also to avoid the existence of combinations of netoks that, in eality, have zeo flo ate thogh the connections. We sed the global optimization appoach pesented by Faia and Bagajeicz 37,38 ith a 1% toleance gap. Fige 3 illstates the feshate consmption and the nmbe of connections fo the fist 100 soltions, as they ee obtained (not soted); the fist 96 feate the minimm consmption of 90 t/h (odinate on the left) and the last fo exhibit a slightly highe vale. The nmbe of connections is also shon (odinate on the ight). All soltions ee obtained minimizing feshate adding the coesponding connections exclsion constaint (30). All the 100 soltions ee fond sing an Intel Xeon 2.67 GHz and 2.5 GB of RAM in 1 h (all clock time). To detemine hat the ight netok is, one needs to add cost. This can be done by Making an assessment of the cost of each netok afte they ae fond, a stategy that may ok ell if the nmbe of netoks is small. Solving the poblem again, fixing the flo ate to its minimm and minimizing capital cost, o cost of egeneation, o both. We note that only in the case hee the efflent fom the end-of-pipe teatment is not ecycled and totally disposed of, the cost of egeneation is popotional to the cost of feshate and theefoe teatment costs cannot be sed as an economical objective. 25 We point ot hee the coe of o claim, that pinchtechnology-based methods as ell as othe gaphical and algoithmic pocedes ae in pinciple incapable of pefoming the above poposed soting and theefoe they fail to povide pope insights beyond identifying the vale of minimm consmption, something that mathematical pogamming can also easily detemine. Example 2. This is the case of ate-sing sbsystem optimization pesented by Wang and Smith, 3 hich involves to ate-sing nits and to contaminants and minimizes Fige 4. Feasible netoks fo the total ate system of the mltiple contaminants fom Wang and Smith. 3

6 Ind. Eng. Chem. Res., Vol. 49, No. 9, Table 4. Altenative Soltions at Minimm Consmptions fo the Total Wate System of the Mltiple Contaminants fom Wang and Smith 3 a nit 1 EOP netok 1 40 t/h feshate 40 t/h nit 1 40 t/h 85 t/h EOP 45 t/h netok 2 40 t/h feshate 40 t/h nit t/h t/h t/h EOP 20 t/h netok 3 40 t/h feshate 40 t/h nit 1 40 t/h t/h EOP t/h netok 4 41 t/h Feshate 40 t/h 1 t/h Unit 1 39 t/h 1 t/h Unit 2 99 t/h EOP 59 t/h netok 5 41 t/h feshate 40 t/h 1 t/h nit 1 40 t/h t/h EOP t/h netok 6 41 t/h feshate 40 t/h 1 t/h nit 1 40 t/h 40 t/h 100 t/h EOP 59 t/h *. a A minimm flo ate of 1 ton/h as sed. Table 5. Wate Using Units Limiting Data of Example 3 ate nits contaminant mass load (kg/h) C in,max (ppm) C ot,max (ppm) (U1) steam stipping HC H 2S SS (U2) HDS-1 HC H 2S SS (U3) desalte HC H 2S SS (U4) VDU HC H 2S SS (U5) HDS-2 HC H 2S SS feshate consmption. Table 2 pesents the limiting data of this poblem. The minimm feshate consmption of this netok ithot ese is ton/h. As no egeneation pocess is sed in this example, only to cases ae analyzed: No ecycle of the end-of-pipe teatment (optimization of ate-sing sbsystem); The efflent steam fom the end-of-pipe teatment can be esed by the ate-sing nits (total ate system). We discssed the advantages and disadvantages of these systems ealie. 29 Fo the end-of-pipe teatment, an otlet concentation of 10 ppm fo both contaminants is assmed, hich ae in ageement ith the maximm alloed to disposal. Fo the fist case (no ecycle of end-of-pipe teatment alloed) the minimm feshate consmption can be edced to 54 t/h, hich is appoximately 15% less than the cent consmption obtained hen no ate ese is consideed. When altenative soltions ae investigated, it indicates the existence of a niqe soltion (no-degeneacy) at 54 t/h, that is, no degeneacy. The next possible soltion identified hen the fist is exclded Table 6. Regeneation Pocesses Data of Example 3 egeneation pocess contaminant emoval atio (%) OPN VRC (R1) steam stipping HC ,800 H 2S 99.9 SS 0 (R2) biological teatment HC ,600 H 2S 90 SS 98 (R3) API sepaato HC ,800 H 2S 0 SS 50 Table 7. Distances fo Example 3 d i,j WU 1 WU 2 WU 3 WU 4 WU 5 RG 1 RG 2 RG 3 dischage FW WU WU WU WU WU RG RG RG feates t/h, hich is the netok ithot ese and is not degeneate eithe. If fo some eason (cost fo example, as it as exploed in Faia and Bagajeicz 25 ) one old ant to exploe highe consmptions, 3 possible netoks consming ton/h ae fond. Note that if one ants to minimize nmbe of connection, the optimm netok is netok 5, hich is a netok in seies and has the lagest consmption. All these netoks ae pesented in Table 3. Next, e analyze the case in hich the ecycle of the efflent steam fom the end-of-pipe teatment is alloed. In sch case, the minimm feshate consmption can be fthe edced to 40 ton/h feshate consmption netok. This is 26% loe than the pevios case (and 36.8% loe than the consmption ithot ese). Eleven feasible altenative netoks ee fond in this case, in hich the fist thee soltions obtained consme 40 t/h of feshate and the next thee 41 t/h. The eleven feasible soltions ae smmaized in Fige 4. The 3 soltions at minimm consmption and the sbseqent 3 slightly highe ae pesented in Table 4. Qite clealy, in this case, the netoks se a vey small flo ate in some connections and fo this eason ill not be even consideed. Othes, like netok 3, exhibit independent cycles, hich ae sally avoided. The fact that pinch technology o othe gaphical and algoithmic pocedes ae not designed to look fo cost, take into accont fobidden connections, and have difficlties handling mlticomponent cases, is knon. Theefoe, nless these methods become able to conside fobidden combination of connections, the above execise cannot be made sing methods othe than mathematical pogamming. When costs ae consideed, the statement also applies as e shall see belo. We no pesent examples ith mch lage degeneacy and discss the isse of costs. Example 3. Pta and Ammindin 19 analyzed a lage efiney poblem, hich as oiginally pesented by Ko and Smith 6 and late also investigated by Gnaatman et al. 16 and Alva-Agaez et al. 18 This is a total ate system poblem that has five ate-sing nits, thee egeneation pocesses and consides thee contaminants. Pta and Ammindin 19 shoed

7 4346 Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Fige 5. Hnded minimm consmption (58 t/h) altenative netok configations of efiney example fom Ko and Smith. 6 Fige 6. Wate-sing nit flo ates: Hnded altenative netok configations at minimm consmption (58 t/h) fo the efiney example fom Ko and Smith. 6 fo altenative soltions fo this poblem and compaed them ith the eslts peviosly obtained by othes. Ganaatman et al. 16 and Alva-Agaez et al. 18 solved fo total annalized cost, inclding piping cost. Tables 5-7 sho the data sed in this example. The dischage limits of this system ae 20 ppm fo HC, 5 ppm fo H 2 S and 100 ppm fo sspended solids (SS). The feshate cost is $0.2/t and the system opeates 8600 h pe yea. A 10% ate of discont is assmed. We sed a minimm flo ate thogh the connection of 5 t/h and a maximm thogh the connection and pocesses of 200 t/h. Using the distances of Table 7 and assming a velocity of 1 m/s, the piping costs ae given by: FIJC i,j ) 124.6d i,j i {W, U, R}, j {W, U, R, S} (31) VIJC i,j ) 1.001d i,j i {W, U, R}, j {W, U, R, S} (32) The best knon soltion minimizing TAC is $616,824, given by Alva-Agaez et al. 18

8 Ind. Eng. Chem. Res., Vol. 49, No. 9, Fige 7. Regeneation pocesses flo ates: Hnded altenative netok configations at minimm consmption (58 t/h) fo the efiney example fom Ko and Smith. 6 Fige 8. Netok ith minimm TAC (and minimm opeating cost). Fige 9. Netok ith minimm capital cost. In o pocede, e identified the minimm consmption (58 t/h) by solving the poblem ithot costs. We also solved fo minimm total annal cost (ithot fixing the feshate flo ate) to global optimality sing Baon and specifying 1% toleance. The n took 7 h, 5 min and endeed a netok feating a minimm total annal cost of $574,155, hich happens to feate the peviosly identified minimm consmption of 58 t/h. To analyze the degeneacy of this poblem at the minimm consmption, e fix the consmption at its minimm (58 t/h) and look fo feasibe soltions p to 100 soltions. We do this by sing a minimm cost objective fnction, and a 99% gap fo the global method pesenetd by Faia and Bagajeicz. 37,38 This is diffeent fom hat as done in examples 1 and 2. Hee e ae having the explicit ppose of saving comptational time. Indeed, if one ns minimizing feshate and fobids peviosly

9 4348 Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Fige 10. Netok ith minimm nmbe of connections. Fige 11. Hnded minimm consmption (58 t/h) altenative netok configations of efiney example fom Ko and Smith 6 inclding the pactical isses pointed ot by Pta and Ammindin. 19 Fige 12. Netok ith minimm TAC of example 3 consideing pactical isses. fod netoks, the comptational time is highe. In addition, e can identify loe costs netoks ealie. Finally, one cold ty to n only once in ode to identify the netok ith loest cost. Sch a n takes mch longe than o altenative (7 h vs 1 h and 40 min to find 100 feasible netoks). We claim that the minimm cost netok that one old identify if one ns to 0% gap feates a set of connections that is eventally identified late, as long as all degeneate soltions ae exploed and one does not stop ealie. That said, e obtain the same connections, bt not necessaily the same flos. We eiteate that e ae not gaanteeing that e obtain the global soltion feating minimm cost. We ae simply obtaining altenative netoks feating the same feshate consmption. The eslts fo the netoks identified ae pesented in Fige 5. We pesent them in a inceasing cost ode of total annalized cost, hich is not necessaily the ode they ae fond. We also

10 Ind. Eng. Chem. Res., Vol. 49, No. 9, Fige 13. Costs: Fifty altenative netok configations at minimm TAC fo the modified example 4 fom Kappiah and Gossmann. 17 Fige 14. Regeneation pocesses flo ates: Fifty altenative netok configations at minimm TAC fo the modified example 4 fom Kappiah and Gossmann. 17 add opeating cost, annalized capital cost and nmbe of connections fo completeness. The oveall nning time of o methodology is 201 CPUs to identify the minimm feshate consmption, and 2525 CPUsto find the hnded degeneate soltions. We point ot that the fist 20 ae fast and then, becase of the netok exclsion constaints, the nning time pe n inceases fo some of them. Note that the loest TAC fond among these 100 soltions is $572,767, hich is loe than the one fond by BARON sing 1% global optimality toleance. It is oth pointing ot that this best soltion fond among the 100 options as the fifth netok, hich took 76 CPUs. The flos thogh the atesing nits and egeneation pocesses coesponding to these soltions ae pesented in Figes 6 and 7, espectively. We note that this pocede does not gaantee global optimality of costs, even thogh e can see it identifies fo altenative soltions ith objective fnction loe than the optimm one fond sing BARON sing 1% toleance. Additionally, if one ants to look at diffeent citeia (alays opeating at minimm consmption), one can chose the netok ith minimm TAC, o minimm opeating cost, minimm capital cost o smalle complexity (hee identified as the nmbe of connections). Table 8 compaes these options (the bold nmbes ae the ones coesponding to the minimm vale of

11 4350 Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010 Table 8. Netoks ith Best Citeia citeia TAC ($/yea) opeating cost ($/yea) capital cost ($) nmbe of connections TAC 572, ,073 1,006, opeating cost 572, ,073 1,006, capital cost 1,062, , , nmbe of connections 1,141,479 1,012,617 1,288, Table 9. Wate Using Units Limiting Data of Example 4 pocess contaminant mass load (kg/h) C in,max (ppm) F max (t/h) 1 A B C A B 1 50 C A B 1 50 C A B 2 50 C A B 1 25 C 0 25 the optimization). Note that the netok ith minimm TAC has also the minimm opeating cost. Hoeve, among the 100 fond soltions thee ae othe 14 netoks that have the same opeating costs. Figes 8-10 sho the thee netoks pesented in Table 8. The eslts pesented so fa do not consideed stctal constaints o pactical consideations othe than the ones given by the inpt data. Pta and Ammindin 19 discss some of these isses. Thei concens ee egading the folloing to pactical isses: 1. The API sepaato shold be placed in the psteam of biological teatment de to incease in pefomance (highe inlet concentation) and to gaantee that oil is not sent to the biological teatment; 2. To avoid nlimited accmlation of cetain contaminants, ecycling aond egeneation nits shold not be alloed. In othe ods, the egeneation pocess cannot send teated ate back to the nits that sent asteate to it. This ill accmlate contaminants that ae not teated. Applying these citeia, they eliminate 2 of the 4 altenative soltions fond by thei pocede. In o case, e apply o pocede inclding these pactical isses. To conside the fist one, e jst added a maximm inlet concentation of 66 ppm of HC to the biological teatment. This vale coesponds to the maximm vale above hich the biological teatment is not able to bing the concentation don to the HC envionmental limit (20 ppm). Fo the second isse, e add a constaint to fobid all diect ecycles, ese ecycling and egeneation ecycling. This constaint is the folloing: YIJ i,j + YJI j,i e 1 (i, j) {(, *), (, ), (, ), (, *)} (33) The minimm feshate consmption obtained sing this modified poblem is also 58 t/h. As befoe, e fix the consmption and find the fist 100 altenative soltions. The costs ae pesented in Fige 11 and the minimm TAC fond among the 100 soltion is $592,573. This optimm netok is pesented in Fige 12. We notice that incopoating this constaint foced the netok to avoid a diect ecycle to the same egeneation pocess, bt it fond a ecycle thogh anothe nit. In eality additional Table 10. Regeneation Pocesses Data of Example 4 pocess contaminant emoval atio (%) OPN CCR 1 A ,800 B 0 C 0 2 A ,500 B 0 C 95 3 A ,600 B 95 C 0 constaints cold be added to avoid ecycles to any nits thogh as many pocesses as one ants. Instead of tying to fobid ecycles, becase of the fea that some contaminants/impities/ inets may accmlate, a bette modeling anse is to add stict inlet limitations to all nits, inclding egeneation pocesses, fo all componds that ae picked p in the ate nits and that ae not sbject to emoval. This ill ok becase hen all contaminants that can accmlate ae inclded, a steady state soltion ith a ecycle that does not emove them is infeasible. Example 4. In this example e ant to find the fist 50 soltions minimizing TAC ithot fixing the feshate consmption at its minimm. Fo this example e se the total ate system pesented as example 4 by Kappiah and Gossmann. 17 The data fo this example is pesented in Tables 9 and 10. Additionally, the dischage limit of all the contaminants is 10 ppm. Kappiah and Gossmann 17 solved the poblem as an NLP poblem. Hoeve, setting aside the fact that the NLP model fo this poblem endes a soltion ith npactical small flo ates, to consistently fobid netoks e need to impose a minimm alloed flo ate thogh the connection so the connection only exists if thee is a flo ate diffeent than zeo. Ths, o poblem becomes an MINLP. The optimm soltion (ithin 1% toleance) of this MINLP poblem feates a cost of $1,033, hen the minimm flo ate thogh connections is set as 1 t/h in CPU sec. In tn, Baon fond a minimm TAC of $1,036,384 in 287 s sing a 1% toleance. The loest TAC fond sing o poposed pocede is $1,033,832, hich is also slightly loe than both 1% toleance global soltion fond by Faia and Bagajeicz 36 and Baon. The eason fo this is that the netok ith TAC of $1,033,832 is not the fist netok fond ith 1% toleance. In eality it is fond in afte fobidden the 4 fist netoks fond sing 1% toleance. At this exact cost thee ae othe 7 altenatives and the 50th lagest TAC is $1,035,288. Note that this high degeneacy in TAC can be attibted to the absence of connection costs. In this poblem the only vaiables that accont fo the TAC ae the feshate consmption and flo ates thogh egeneation pocesses. Figes 13 and 14 sho the costs and egeneation flo ates fo the 50 loest TAC soltion obtained fo this poblem. Becase this pocede as done to find the global soltions evey time a netok is fobidden, it takes mch longe than the pevios one (20 h). Hoeve, hen e n this poblem ith 99% gap ith the ppose of only finding feasible

12 netoks, e ae able to identify 500 netoks in 12 h and 30 min, the fist 50 altenative netoks fond in 25 min. Conclsions In this aticle e pointed ot the fact that minimm feshate soltions of ate management poblems in pocess plants exhibits sometimes a lage degeneacy. We age that even fo single contaminant cases, pinch-technology based methods as ell as othe algoithmic ones cannot povide the insights they claim they can povide and ae nable to deal effectively ith the identification of all the degeneate soltions, not even sho hethe the degeneacy is small o lage. We sho that degeneacy not only shos in minimm feshate consmption poblems, bt also in cases hee total cost is minimized. We aged that ONLY mathematical pogamming-based methods can pefom this task and e pesented a methodology to identify as many degeneate and sboptimal soltions as one desies. Finally, e believe that the degeneacy shold disappea ith moe detailed modeling. Fo example, once maximm inlet concentations ae imposed fo egeneation pocesses, o thei efficiencies ae fnction of inlet conditions (concentation, flo ates, etc), some of the altenatives may not be feasible anymoe. Hoeve, given the above eslts, e still expect sboptimal soltions close to the global optimm. Acknoledgment Déboa Faia acknoledges sppot fom the CAPES/ Flbight Pogam (Bazil). Liteate Cited (1) Bagajeicz, M. J. A evie of ecent design pocedes fo ate netoks in efineies and pocess plants. Compt. Chem. Eng. 2000, 24, (2) Takama, N.; Kiyama, T.; Shioko, K.; Umeda, T. Optimal Wate Allocation in a Petolem Refiney. Compt. Chem. Eng. 1980, 4, (3) Wang, Y. P.; Smith, R. Wasteate minimisation. Chem. Eng. Sci. 1994, 40 (7), (4) Wang, Y. P.; Smith, R. Wasteate minimisation ith floate constaint. Tans. IChE 1995, 73-A8, (5) Olesen, S. G.; Polley, G. T. A simple methodology fo the design of ate netoks dandling single contaminants. Tans. IChE 1997, 75A. (6) Ko, W.-C. J.; Smith, R. 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A ne appoach fo global optimization of MINLP poblems ith bilinea and concave nivaiate tems. AIChE J. 2010, in pess. ReceiVed fo evie Octobe 8, 2009 ReVised manscipt eceived Febay 16, 2010 Accepted Mach 11, 2010 IE901574G