Computers and Chemical Engineering

Transcription

1 Computers and Chemcal Engneerng 32 (2008) Contents lsts avalable at ScenceDrect Computers and Chemcal Engneerng ournal homepage: Synthess of non-sothermal heat ntegrated water networks n chemcal processes Mloš Bogata a, Mguel J. Bagaewcz b, a Faculty of Chemstry and Chemcal Engneerng, Unversty of Marbor, Smetanova 17, Marbor SI-2000, Slovena b School of Chemcal, Bologcal and Materals Engneerng, Unversty of Oklahoma, Norman, OK, 73019, USA artcle nfo abstract Artcle hstory: Receved 1 August 2007 Receved n revsed form 18 Aprl 2008 Accepted 12 May 2008 Avalable onlne 27 May 2008 Ths paper presents a new approach for the smultaneous synthess and optmzaton of heat ntegrated water networks. A new superstructure for heat exchanger network (HEN) synthess s proposed. The procedure s based on mxed nteger non-lnear mathematcal programmng (MINLP). Four relevant examples are presented to llustrate varous aspects of the proposed approach Elsever Ltd. All rghts reserved. Keywords: Water networks Waste water mnmzaton Heat ntegraton Superstructure Process synthess 1. Introducton Water and energy are among the most mportant commodtes used n the process ndustres. For example, water s used n petrochemcal plants and refneres for strppng and lqud lqud extracton; n ron and steel ndustres prmarly as a coolant, n food and agrcultural ndustres n a varety of washng operatons. In certan stuatons sgnfcant amounts of water need to be heated up or cooled down to meet process operatng condtons. As a consequence, large energy consumpton n the form of coolng and heatng utltes s needed. In such cases, when both the qualty and temperature of water are mportant, water and energy management need to be consdered smultaneously. Dfferent methods, rooted n conceptual desgn or mathematcal programmng, have been developed for water mnmzaton as well as for the heat exchanger network (HEN) synthess problem. The reader s referred to Bagaewcz (2000) for a comprehensve revew of technologes developed to solve the water mnmzaton problem and to Furman and Sahnds (2002) for a revew of the HEN synthess technologes. The most wdely used technology n HEN synthess feld s the well-known Pnch Technology (Lnnhoff, Townsend, Boland, Correspondng author. Tel.: ; fax: E-mal address: bagaewcz@ou.edu (M.J. Bagaewcz). & Hewtt, 1982). However, desgns usng the pnch methodology were shown to be n many cases non-optmal, manly due to ts sequental nature (mnmze energy frst, followed by strct unt number mnmzaton), although some mprovements have been noted (Supertargetng). To overcome the drawbacks of the pnch method dfferent approaches usng mathematcal programmng were presented over the last two decades. Of these, one can classfy them as transportaton transshpment orented and superstructure orented. One of the latest models on the transportaton transshpment type s the one proposed by Barbaro and Bagaewcz (2005), whch s lnear and allows non-sothermal mxng as well as multple matches between two streams. Among the superstructure-based models, the most popular method s a stage-wse superstructure approach (Yee & Grossmann, 1990). Smplcty of pnch methodology and some smlartes between water mnmzaton and energy mnmzaton problem nduced a development of conceptual desgn approaches n the feld of water mnmzaton (Maoz, Brouckaert, & Buckley, 2006; Wang & Smth, 1994). The conceptual approach s mostly useful for the sngle-contamnant case, wth very lmted applcablty to multcontamnant cases. Multple contamnant water problems requre more elaborate methods (Karuppah & Grossmann, 2006; Savelsk, Rvas, & Bagaewcz, 1999). Despte all the enablng technologes the nfluence of heat ntegraton on the soluton of water allocaton plannng (WAP) has been scarcely addressed n past years. Savelsk and Bagaewcz (1997) frst studed the problem pontng out the exstence of a trade off. A /$ see front matter 2008 Elsever Ltd. All rghts reserved. do: /.compchemeng

2 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Nomenclature Indces hot process stream cold process stream k temperature locaton and stage ndex l contamnant mx mxer unt n, m, fw, ww stream p process unt r regeneraton unt s spltter unt T C,,k T CU T HU q,,k q CU q HU temperature approach for match (,) n cold end of stage k (K) temperature approach for match of cold utlty and hot stream (K) temperature approach for match of hot utlty and cold stream (K) heat exchanged between streams and n stage k heat exchanged between hot stream and cold utlty (kw) heat exchanged between cold stream and hot utlty (kw) Sets CP HP M P R S ST cold process streams hot process streams mxer unts process unts regeneraton unt spltter unts stages Parameters c l IN,cl maxp OUT maxp maxmal nlet/outlet concentraton of contamnant l for process p (mg/kg) C HE fxed costs for heat exchangers ($) C A area cost coeffcents ($/m 2 ) C CON fxed costs for HEN connectons ($) C HU hot utlty costs ($/(kw a)) C CU cold utlty costs ($/(kw a)) C FW fresh water costs ($/t) C R costs of waste water regeneraton ($/t) C R,v regeneraton unt capacty cost ($ h/t) Cp heat capacty (kj/kg K) EMAT exchanger mnmum approach temperature f a /1 tme fracton of operaton h stream heat transfer coeffcent (kw/(m 2 K)) Lp l mass load of contamnant l n process p (kg/s) N h number of hours per year (h/a) NOK number of stages T FWS temperature of fresh water source (K) T WWS temperature of waste water snk (K) U overall heat transfer coeffcent (kw/(m 2 K)) upper bound for temperature dfference (K) upper bound for heat exchange (kw) /1 exponent for regeneraton unt capacty ˇl/1 exponent for area cost l /1 removal effcency Contnuous varables CF H,IN,CF C,IN nlet heat capacty flow rate of hot stream, and cold} stream (kw/k) CF H,k,CFH,,k,CFH,k CF C heat capacty flow rate of hot (kw/k),k,cfc,,k,cfc,k stream, and cold stream F mass flow rate (kg/s) T IN,T IN nlet temperature of hot stream, and cold stream }(K) T,T,k T,T temperature of hot stream, and cold stream (K),k T H,,k temperature approach for match (,) n hot end of stage k (K) Bnary varables z,,k exstence of heat exchanger for match (,)nstagek Z CU exstence of heat exchanger for match (cold utlty, ) Z HU exstence of heat exchanger for match (hot utlty, ) y H,,k,yC,,k exstence of splts of hot and cold streams n HEN Subscrpts/superscrpts CU cold utlty FW fresh water FWS fresh water source HU hot utlty HEN heat exchanger network IN nlet LO lower OUT outlet UP upper WW waste water WWS waste water snk WN water network graphcal procedure was ntroduced by Savulescu and Smth (1998) attemptng to solve the energy effcent WAP problem. The method they used s sequental and was recently extended to consder water and heat mnmzaton smultaneously (Savulescu, Km, & Smth, 2005a; Savulescu, Km, & Smth, 2005b). However, the approach s lmted to a sngle-contamnant case. In turn, Bagaewcz, Rodera, & Savelsk (2002) solved the problem usng mathematcal programmng. Wth mnor modfcatons ther approach can be extended to handle the mult-contamnant case. The model s, nonetheless, sequental. An mportant realzaton about all these systems s that, n the absence of regeneraton, systems are pnched at the lowest (nlet) temperature. In addton, what makes the desgn challengng s that mxng of streams s a part of the desgn, especally f mxng of streams s used to acheve target temperatures, and therefore avod the use of heat exchangers or utltes. It has been shown that clever mxng can reduce the number of exchangers n the system (Bagaewcz et al., 2002; Savulescu, Sorn, & Smth, 2002). Ths paper ntroduces a new approach for smultaneous synthess of energy effcent water networks, portons of whch have been advanced, wthout the ncluson of mportant detals, by Bogata and Bagaewcz (2007). The model s MINLP and the man feature of the formulaton s mxng and splttng of streams wthn the HEN superstructure, thus enablng drect heat exchange n order to reduce the number of heat exchangers as well as to reduce the complexty of heat ntegrated process structure.

3 3132 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Problem statement Gven s a set of water usng/water dsposng processes, whch requre water of adequate qualty and temperature. The obectve s to determne the optmal network of water stream nterconnectons among the processes and the correspondng heat exchanger network by smultaneously optmzng annual operatng costs (fresh water, regeneraton, and utlty costs), and captal costs (heat exchanger costs). The followng assumptons are used: The level of contamnant s low enough that the total flow rate can be consdered constant, processes operate sothermally or non-sothermally (fxed temperature change), water s present only n a lqud phase, fresh water s free of contamnants, streams have constant heat capacty (Cp = kj/(kg K)), heat transfer coeffcents are constant, heat exchangers are countercurrent, only one hot and one cold utlty are avalable, no heat losses are consdered for process streams. 3. Superstructure In ths secton three superstructures relevant to ths work are presented. Frstly, a smple superstructure for the synthess of water networks, smlar to the one presented by Alva-Argáez, Kokosss, and Smth (1998), s depcted n Fg. 1a. A set of water usng process unts (P) s nterconnected usng a set of mxer unts (M) and a set of spltter unts (S). A sngle regeneraton unt (R), f embedded n the WN superstructure (Fg. 1b), accepts contamnated water from all the water usng process unts, selectvely removes contamnants, and dstrbutes treated water back to the water usng process unts. No self-recyclng was consdered n ths work. Dashed lnes are the only ones allowed to exchange heat n heat exchangers. We dscuss ths choce n detal later. For the heat exchange, the stage-wse superstructure (Yee & Grossmann, 1990) for HEN synthess was modfed to account for drect heat exchange (mxng of streams) as well as the regular heat exchangers for ndrect heat transfer. Fg. 2 presents a 2-hot 2-cold streams representaton of the modfed stage-wse superstructure. In ths superstructure, each hot/cold stream can potentally mx wth other hot/cold streams n each stage of the superstructure. We omt the detaled descrpton of the basc features of the org- Fg. 1. (a) Smple WN superstructure for process water network synthess. (b) WN superstructure for process water network synthess wth embedded regeneraton unt. nal model because they are well-known and concentrate on the addtons: the mxng of streams at each stage. In Fg. 2 ndex k refers to crossngs of temperature locatons n the drecton of flow, rather than stages. For ths reason k values concde wth stage numbers n the case of hot streams, but not n the case of cold streams. Therefore, CF C s the heat capacty flow,k rate of cold stream whch crossed the temperature locaton k. On the other hand CF H s the heat capacty flow rate of hot stream,k whch crossed the temperature locaton k. Ths reasonng apples to all the varables. After hot stream wth heat capacty flow rate CF H, enters stage,k k, t can splt and send water to mx wth stream * (CF H, ), or,k Fg. 2. Modfed HEN superstructure.

4 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Fg. 3. Four possble types of splttng/mxng. receve water from stream * (CF H ). The same s allowed for cold,,k streams. Ths mxng of streams can be forbdden when certan contamnants are not to be mxed. Fg. 3 shows all possble mxng/splttng patterns that one can consder. Because we expect that ether mxng or splttng wll take place among streams, but not both, any of these structures s approprate. In ths work Type A was chosen. Fnally, n Fg. 4 a combned superstructure, consstng of a water network (WN) superstructure (Fg. 1) and a heat exchanger network superstructure (Fg. 2) s presented. Only certan streams (dashed lnes n Fg. 4 correspondng to dashed lnes n Fg. 1) are consdered to take part n heat ntegraton, that s, fresh water streams, waste water streams, and streams connectng mxer unts wth water usng process unts and mxer unt wth regeneraton unts. Arguments supportng selecton of these streams, and problems accompanyng ths decson are presented n the next secton. Also, as dscussed above, some streams are not allowed to mx wthn the HEN superstructure (see Fg. 4). These streams are the streams nterconnectng mxer unts wth process unts, and stream nterconnectng mxer unts wth regeneraton unts Stream labelng To determne whether a certan process stream s cold or hot (assumng no phase change takes place as t s n our case) one needs to know n advance ts supply and target temperatures. If the temperature of the fresh water s assumed to be equal or lower than Fg. 4. Superstructure for smultaneous WN and HEN synthess. the lowest operatng temperature of process unts, each of the fresh water streams can be consdered as a cold stream. In addton, f the temperature of waste water streams dscharged to the envronment s set to be equal or lower than the lowest operatng temperature of process unts, each of the waste water streams can be consdered as a hot stream. Unfortunately, such unambguous decsons are mpossble to make for the rest of the streams n the water WN superstructure. Water networks can have several alternatve solutons featurng the same, mnmum fresh water ntake, but wth dfferent connectons (Bagaewcz & Savelsk, 2001). For llustraton, consder the example presented n Fg. 5. Its uppermost part shows a generalzed WN superstructure consstng of two process unts, each operatng at dfferent temperatures. Assume that T P1 > T P2 and T FW <(T P1, T P2 ). Addtonally, consder the two arbtrary selected solutons among all feasble solutons wth equal fresh water ntake shown n lower part of the fgure. As apparent n the fgure, under certan crcumstances, the complcatng stream connectng the mxer Fg. 5. Effect of alternatve solutons on thermodynamc propertes of streams.

5 3134 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) unt and the process unt (dashed lne) can be consdered, from the perspectve of the HEN, ether cold or hot. Clearly, we are confronted wth two ssues. Frst, how many and whch streams n the WN superstructure should be consdered to take part of heat ntegraton? In addton, on what bass should the desgner decde whether the complcatng streams are to be treated as hot or cold? To make sure that the model captures the true optmal soluton all, the streams n the WN superstructure should be consdered for heat ntegraton. Ths would most lkely ncrease the computatonal effort needed. The reason for ths s that the total number of streams n WN superstructure (Fg. 1a) equals N 2 +3N, where N s the number of process unts. In addton, snce all the complcated streams (n general, all but fresh water and dscharge water streams) can potentally be ether cold or hot, the number of streams ncreases to 2(N 2 +2N). What s more, addtonal logc would need to be mplemented to treat these complcated streams exclusvely as hot or cold. All of ths would n turn make the model combnatorally very demandng. Also, t would ncrease the undesrable effect of non-convextes due to enlarged number of blnear terms. Therefore, the decson to select only the above dscussed streams (dashed lnes n Fg. 1) can be ustfed by the reduced sze of the model. In addton, these streams represent the mnmal number of streams that gve feasble heat ntegrated structures and at the same tme allow maxmal explotaton of drect heat transfer possbltes. 4. Mathematcal model 4.1. Water network model Water usng process unts (p P) and regeneraton unt (r R) To model the water usng process unt p, accordng to representaton n Fg. 6a, we need to formulate the overall mass balance (Eq. (1)), contamnant mass balance ((Eq. (2)), and temperature nlet outlet relaton (Eq. (3)). Process unt s consdered to have a sngle nlet stream and sngle outlet stream. F n = F m n p IN,m p OUT (1) F n c l n + Ll p = F mc l m p P, n p IN,m p OUT (2) T n = T m n p IN,m p OUT (3) The temperature of stream n enterng the process unt p should be equal to the operatng temperature of the process (T p ): T n = T p p P, n p IN (4) When a water usng process unt p s consdered not to operate sothermally, ts nlet and outlet temperatures are not equal, and therefore Eq. (3) becomes: T n + T p = T m p P, n p IN,m p OUT (5) where T p R s a fxed temperature change. The constrants on maxmum process nlet and outlet concentratons are: c l n cl IN maxp p P, n p IN (6) c l m cl OUT maxp p P, m p OUT (7) As n the case of process unts, a regeneraton unt r s also regarded as havng a sngle nlet and a sngle outlet stream (Fg. 6b). The outlet concentraton of each contamnant s reduced (wth respect to the nlet concentraton) accordng to ts removal effcency l. The followng equatons model the regeneraton unt r: F n = F m n r IN,m r OUT (8) F n c l n (1 l ) = F m c l m n rin,m r OUT (9) T n = T m n r IN,m r OUT (10) In the case of a non-sothermal operaton of regeneraton unt r, assumng a fxed temperature change, Eq. (10) s replaced by: T n + T r = T m r R, n r IN,m r OUT (11) A mxer unt (mx M) s shown n Fg. 7a, havng multple nlet streams ((n, fw) mx IN ) and a sngle outlet stream (m mx OUT ). The ndex fw s assocated wth a sngle fresh water stream, and ndex n wth all the other streams (comng from spltter unts (s S)). The overall mass balance s gven by Eq. (12), the ndvdual contamnant balances by Eq. (13), and the energy balance by Eq. (14). In turn, a spltter unt (s S) s depcted n Fg. 7b, consstng of a sngle nlet stream (n s IN ) and multple outlet streams, namely streams m s OUT lnked to mxer unts (mx M), and a sngle waste water dscharge stream ww s OUT. Spltter unts are modeled usng Eqs. ((15) (19)) Mxer unts (mx M) F m = F n + F fw m mx OUT, (n, fw) mx IN (12) n F m cm l = F n cn l m mx OUT,n mx IN (13) n F m T m = F n T n + F fw T fw m mx OUT, (n, fw) mx IN (14) n Spltter unts (s S) F n = F m + F ww (m, ww) s OUT,n s IN (15) m c l n = cl m m s OUT,n s IN (16) Fg. 6. (a) Schematc representaton of process unt. (b) Schematc representaton of regeneraton unt. Fg. 7. (a) Schematc representaton of mxer unt. (b) Schematc representaton of spltter unt.

6 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) c l n = cl ww ww s OUT,n s IN (17) T n = T m m s OUT,n s IN (18) T n = T ww ww s OUT,n s IN (19) Note that a mxer unt and a spltter unt lnked to a regeneraton unt (see Fg. 1b) dffer from the rest of the mxer and spltter unts. The former does not have the fresh water feed whle the latter does not have the waste water dscharge stream. The formulaton of ths mxer unt s straghtforward: only the varables assocated wth fresh water stream need to be excluded from Eqs. (12) and (14).For the spltter unt, the flow rate of the dscharged waste water stream (F ww ) s excluded from the formulaton of the overall mass balance (Eq. (15)), and Eqs. (17) and (19) become redundant HEN model As descrbed above, for ths model, we extend the MINLP formulaton of Yee and Grossmann (1990) by addng mxng at each stage. The frst man dfference between the orgnal stage-wse formulaton and the proposed one s addtonal contnuous varables needed to model the non-sothermal mxng n each stage. Note that even when nlet heat capacty flow rates (CF H,IN,CF C,IN ) and nlet/outlet temperatures are fxed the new energy balance ntroduces several addtonal blnear terms addng to the model non-convexty. Also, the overall heat balance of each stream s not explctly stated because there s splttng and mxng takng place. The overall heat balance s, nonetheless, satsfed Assgnment of superstructure nlet temperatures and nlet heat capacty flow rates T H,IN = T H,1 HP (20) T C,IN = T C,NOK+1 CP (21) CF H,IN = CF H,1 HP (22) CF C,IN = CF C,NOK+1 CP (23) Stage heat balance for hot and cold streams (T H,k T H,k+1 )CFH,k = q,,k k ST, HP (24) CP (T C,k T C,k+1 )CFC,k+1 = q,,k k ST, CP (25) HP Cold and hot utlty load q CU = CF H H (T,NOK+1,NOK+1 T H,OUT ) HP (26) q HU = CF C,1 C,OUT (T T C ) CP (27), Stage heat capacty flow rate balances for hot and cold streams CF H,k + CF H,,k CF H,,k = CFH,k k ST NOK + 1, HP /= (28) CF C,k + CF C,,k CF C,,k = CFC,k k ST NOK + 1, CP /= (29) Stage enthalpy balances for hot and cold streams CF H,k T H,k + CF H,,k T H,k CF H,,k T H,k = CFH,k T H,k k ST NOK + 1, HP /= (30) CF C,k T C,k + CF C,,k T C,k CF C,,k T C,k = CFC,k T C,k k ST NOK + 1, CP /= (31) Note that the Eqs. (28) (31) are not restrcted to k ST. Snce, for example, mxng of hot streams s possble even after the crossng of last temperature locaton (k = 3 n Fg. 2), the equatons should hold also for k + 1 (NOK + 1). The same s true cold streams Overall stage heat capacty flow rate balance CF H,k = CF H HP, k ST NOK + 1 (32),k CF C,k = CF C CP, k ST NOK + 1 (33),k Bounds on splts (non-sothermal mxng) CF H,,k CFH HP /=,k ST NOK + 1 (34),k CF C,,k CFC,k CP /=,k ST NOK + 1 (35) Stage nlet and outlet heat capacty flow rate relaton CF H,k+1 = CFH,k HP, k ST (36) CF C,k = CFH,k+1 CP, k ST (37) Logcal constrants needed to determne the exstence of a heat exchange match (,) n stage k q,,k z,,k 0 HP, CP, k ST (38) q CU CU z CU 0 HP (39) q HU HU z HU 0 CP (40) Logcal constrants actvatng temperature dfferences n stage k T H,,k T H,k T C,k + (1 z,,k) HP, CP, k ST (41) T C,,k T H,k+1 T C,k+1 + (1 z,,k) HP, CP, k ST (42) T CU T H,NOK+1 T OUT CU + (1 z CU ) HP (43) T HU T OUT HU T C,1 + (1 z HU ) CP (44) The above four constrants can be expressed as nequaltes because the costs of heat exchangers decrease wth the ncrease n the temperature dfferences. And snce the obectve functon s to be mnmzed, the temperatures wll be drven to take the hghest possble value. The role of bnares n these constrants s to ensure postve drvng forces Lower bounds on temperature dfferences T H,,k EMAT HP, CP,k ST (45) T C,,k EMAT HP, CP, k ST (46) T CU EMAT HP (47) T HU EMAT CP (48)

7 3136 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Feasblty of temperatures T H,NOK+1 T H,OUT HP (49) T H,k T H,k+1 HP, k ST (50) T C,1 T C,OUT CP (51) T C,k+1 T C,k CP, k ST (52) T H,k max{0,th,in } HP (53) T C,k mn{c, T C,IN } CP, (c R + c T C,IN ) (54) Monotonc decrease of temperature at each successve stage for each stream s not requested and expected to take place n ths formulaton. However, monotonc decrease wthn each stage s enforced through Eqs. (49) (52). Also, note that the vald upper bound on temperatures for hot streams, whch are allowed to mx, s the nlet temperature of the hottest hot stream (Eq. (53)). Lkewse, the vald lower bound on temperatures for the cold streams, whch are also allowed to mx, s the nlet temperature of the coldest stream (Eq. (54)).In turn, forbdden stream splttng and mxng can be acheved by the followng set of constrants: CF H,,k = 0 HP /=, k ST NOK + 1 (55) CF H,,k = 0 HP /=,k ST NOK + 1 (56) CF C,,k = 0 HP /=, k ST NOK + 1 (57) CF C,,k = 0 HP /=,k ST NOK + 1 (58) The constrants represented by Eqs. (55) (58) (.e. fxng the values of the varables to zero) are the smplest way to enforce forbdden mxng/splttng of streams n the HEN superstructure. However, ths formulaton gves the desgner no control over the number of splts and ther heat capacty flow rates. For example, from strctly practcal reasons, one may want to mpose some lower and upper bounds on the heat capacty flow rates of splts, essentally settng bounds on water flow rates of splts. Furthermore, to obtan a less complex HEN topology t s benefcal to lmt the number of splts n each stage. Ths can be done through the followng constrants: CF H,,k CFH,LO y H,,k (59) CF H,,k CFH,UP y H,,k (60) y H,,k + yh,,k 1 (61) (CF H,LO,CF H,UP ) R + y H {0, 1},,k HP /=,k ST NOK + 1 CF C,,k CFC,LO y C,,k (62) CF C,,k CFC,UP y C,,k (63) y C,,k + yc,,k 1 (64) (CF C,LO,CF C,UP ) R + y C {0, 1},,k HP /=,k ST NOK + 1 Usng the above formulaton the followng effects on the HEN topology s acheved. Bounds on splttng flow rates: Eqs. (59) and (60) state that f the bnary varable y H, equals one n stage k, then splts CFH,k, of hot,k stream can take some value between lower and upper bounds (CF H,LO, CF H,UP ). Otherwse, splts are forced to zero. Lmtaton on the number of splt streams: If a sngle splt CF H,,k exsts n stage k, then the all the splts CF H n ths stage are,,k forced to zero nequalty constrant on bnary varable (Eq. (61)). The same s true for splts of cold streams (CF C,,k, CFC ), Eqs.,,k ((62) (64)). Also note that all of the types of mxng depcted n Fg. 3 are represented by Eqs. (59) (64) because cross-mxng s excluded Constrants connectng the WN and the HEN Fresh water streams The temperature of fresh water source (T FWS ) s assumed to be fxed, therefore, the nlet temperatures of fresh water streams can be treated as parameters. T C,IN = T FWS CP (65) On the other hand, ther heat capacty flow rates (CF C,IN ) are varables whose values are to be determned by solvng the combned WN HEN model. CF C,IN = CF C CP (66),NOK+1 T C,OUT = T fw fw mx IN,mx M, CP (67) CF C,1 = F fwcp n mx IN,mx M, CP (68) Streams connectng mxer unts (mx M) to process unts (p P), and mxer unt (mx M) to regeneraton unt (r R) For the streams consdered as cold streams the varables n HEN superstructure and WN superstructure are related through equatons: T C,IN = T m m mx OUT,mx M, CP (69) CF C,IN = F m Cp m mx OUT,mx M, CP (70) T C,OUT = T n n p IN,n r IN,p P,r R, CP (71) CF C,1 = F ncp n p IN,n r IN,p P,r R, CP (72) In contrast, for the streams consdered as hot streams the followng equatons are used: T H,IN = T m m mx OUT,mx M, HP (73) CF H,IN = F m Cp m mx OUT,mx M, HP (74) T H,OUT = T n n p IN,n r IN,p P,r R, HP (75) CF H,NOK+1 = F ncp n p IN,n r IN,p P,r R, CP (76) Regardless of whether these streams are hot or cold, equatons to make the concentratons of contamnants and flow rates of streams leavng the mxer unts (streams enterng the HEN superstructure) equal to the ones n the streams feedng the process unts or the regeneraton unt (streams leavng the HEN superstructure) are needed. They are the followng: F m = F n m mx OUT,n p IN,n r IN mx M, p P, r R (77) c l m = cl n m mx OUT,n p IN,n r IN mx M, p P, r R (78)

8 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Waste water dscharge streams T H,IN = T ww ww s OUT,s S, HP (79) CF H,IN = F ww Cp ww s OUT,s S, HP (80) The outlet temperature of all the dscharge waste water streams s assumed to be equal to the temperature of waste water snk (T WWS ): T H,OUT = T WWS HP (81) 4.4. Obectve functon The annualzed cost of the HEN, comprsng annual utlty costs and nvestment costs s: Ca HEN = C HE, z,,k + C HE z CU HP CPk ST HP + C HE z HU + f a CU q CU + C HU q HU CP HPC CP + C A, Aˇ,,k + C A Aˇ CU + C A Aˇ HU (82) HP CPk ST HP CP + C CON y H,,k + y C,,k HP HP k ST CP CP k ST /= /= where f a s the tme fracton of operaton n a year. The areas and heat transfer coeffcents, n turn, are gven by the followng standard relatons: q,,k A,,k = HP, CP, k ST (83) U, ln T,,k A CU = q CU U,CU ln T,CU HP (84) q A HU = HU U,HU ln T,HU CP (85) 1 = U, h h HP, CP (86) 1 = U,CU h h CU HP (87) 1 = U,HU h h HU CP (88) Fnally, the logarthmc mean temperature dfference ( ln T) s approxmated accordng to Chen (1987): ln T,,k = ( T H,,k T C,,k ( T H,,k + T C,,k ) 2 ) 1/3 HP, CP, k ST (89) ln T,CU = ( T CU (T OUT T IN CU ) ( T CU + (T OUT 2 ) T IN CU )) 1/3 HP (90) ( ) 1/3 ( T ln T,HU = T HU (T IN OUT HU + (T IN HU T OUT )) HU T ) 2 CP (91) The last term of the obectve functon represents the fxed costs assocated to splts of hot and cold streams. One can ustfy ths by the captal costs of addtonal equpment (valves, regulaton, etc.) needed to operate the HEN. The annualzed WN costs are gven by: C WN a = N h f a CFW F fw + C R F n + CR,v F n fw fw mx IN,n r IN (92) where N h s the number of hours n a year. Whle the frst term of the WN obectve functon (Eq. (92)) corresponds to operatng costs of WN, namely to fresh water and regeneraton costs, the last term corresponds to varable costs assocated wth regeneraton unt capacty. 5. Soluton procedure To overcome the ambguty caused by the need of labelng streams as hot or cold we propose the followng strategy: Step 1: The WN model was solved usng a local NLP solver. However, pror to solvng the WN model a mnor reformulaton s needed. Indeed, to meet the target process operatng temperatures, postve slack varables ς CU mx and ςhu mx representng external heatng and coolng are added nto the formulaton of the mxer unt energy balance. Then, Eq. (14) becomes: F m T m + ς HU mx ςcu mx = n mx n F n T n mx M,m mx OUT (93) The WN obectve functon (Eq. (92)) s reformulated ncludng the slack varables and addng a small prce () to them. ( ) Ca WN = N h f a C FW F fw + C R F n + C R,v F n fw + (ςmx HU + ςcu mx ) (94) mx The above obectve functon mnmzes the water related cost, but equally mportant, t tends to maxmze the drect heat transfer n mxer unts by mnmzng the values of the slack varables. Also mportant s that due to small prces assgned to the slack varables the possblty of both havng postve value n a partcular mxer unt s excluded. In addton, both varables can take the value of zero. Labelng of the complcated streams s then performed accordng to the values of the slack varables n the soluton of the reformulated WN model. Frst, f varable ςmx HU takes a postve value then the stream connectng a mxer unt wth a process unt s consdered to be cold, because addtonal heatng s needed to acheve ts target temperature. On the other hand, f the value of ςmx CU s greater than zero the stream s consdered a hot stream. When both the slack varables are zero, the stream s not consdered to take part n heat ntegraton, snce ts target temperature s met by drect heat transfer n the correspondng mxer unt. Step 2: Solve the combned WN HEN model. Accordng to our experences the values of contnuous varables (flow rates,

9 3138 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Table 1 Cost and operatng parameters for WN and HEN Parameter Parameter Parameter C CON ($) C HU ($/(kw a)) 260 T IN HU ( C) 126 C HE ($) C CU ($/(kw a)) 150 T OUT HU ( C) 126 C A ($/m 2 ) 860 C FW ($/t) 2.5 T IN CU ( C) 15 ˇ/ C R ($/t) 0.95 T OUT CU ( C) 20 / C R,v ($ h/t) h CP,HP,CU (kw/(m 2 K))=1,h HU (kw/(m 2 K)) = 5, CF (C,H),LO (kw/k) = Examples temperatures, etc.) obtaned n the soluton of the reformulated WN model serve as a good ntal guess for the assgnment of the HEN superstructure nlet condtons. Four examples are gven n ths secton to llustrate the proposed approach. In all of the examples the same cost factors, nlet/outlet utlty temperatures were used (Table 1). The lower bound on splts are 1 t/h and the tme fracton of operaton n a year, f a = 0.95, was assumed n all cases. Also, n all of the examples the fresh water temperature s 20 C, and the dscharge temperature of waste water s 30 C. Fnally, the examples were mplemented n GAMS (Brooke, Kendrck, Meeraus, & Raman, 1998) and solved usng DICOPT (Vswanathan & Grossmann, 1990) wth CPLEX as a MIP solver and SNOPT (Gll, Murray, & Saunders, 2002) as a NLP solver on a PC machne (3.2 GHz, 1 GB RAM). To reduce the problems caused by nfeasble NLP sub-problems, the combned model was reformulated usng the nteger-nfeasble path strategy (Soršak & Kravana, 2002). The reformulaton was performed only on the bnary varables assgned to determne the heat exchange matches (z,,k, z CU, z HU ). Table 2 Data for example 1 Process L l (kg/h) cmax IN (mg/kg) cout max (mg/kg)/ Tp ( C) P P P P Table 3 Data for example 2 Process L l (kg/h) cmax IN (mg/kg) cout max (mg/kg) Tp ( C) P P P P Example 1 The frst example s the one orgnally proposed by Savulescu and Smth (1998); a sngle-contamnant case comprsng four water usng process unts. The data s presented n Table 2. To be able to compare results wth solutons obtaned by Savulescu et al. (2005b) and Bagaewcz et al. (2002) the heat exchanger network was desgned exclusvely usng fresh water and dscharge waste water streams wthout consderng the process to process streams. The network obtaned usng the proposed approach wth annual operatng costs of M$/a and HEN captal cost of M$ s depcted n Fg. 8. The correspondng HEN conssts of three heat exchangers featurng a total area of m 2, and a sngle heater (132.2 m 2 ). Fresh water ntake equals 324 t/h. The network presented n Fg. 8 s dentcal to the one reported n Bagaewcz et al. (2002). The soluton obtaned by Savulescu et al. (2005b) s depcted n Fg. 9; the authors used a systematc Fg. 8. Heat ntegrated water network for example 1.

10 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Fg. 9. Soluton of example 1 by Savulescu et al. (2005b). conceptual approach amng to smplfy the network topology by consderng non-sothermal mxng. Because not enough nformaton was gven n the orgnal paper, the areas of the heat exchanger of Fg. 9 were calculated usng the data of Table 1. Clearly, the fresh water ntake s dentcal as n the above presented soluton (see Fg. 8), but, the HEN s dfferent. Note that the total area of the three heat exchangers s approxmately 7% larger than the one obtaned usng our approach. Also, besdes the 13% larger hot utlty consumpton, 485 kw of cold utlty s needed. Summarzng, the soluton obtaned usng our proposed approach s found to be superor to the one reported by Savulescu Fg. 10. Heat ntegrated water network for example 2.

11 3140 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Table 4 Data for example 3 Process L l (kg/h) cmax IN (mg/kg) cout max (mg/kg) Tp ( C) C 1 C 2 C 3 C 1 C 2 C 3 C 1 C 2 C 3 P P P P et al. (2005b) when captal as well as annual operatng costs are consdered Example 2 The second example (Table 3) s the same as the frst one, except that, the contamnant loads are scaled down by a factor of 3.6. As n the prevous example, the heat exchanger network was desgned to handle fresh water and dscharge waste water streams, excludng process to process streams. The network presented n Fg. 8 s also a feasble soluton of example 2, f water flow rates, heat dutes, and areas of heat exchange unts are scaled down by the same factor used to reduce contamnant loads. The correspondng total area of all the heat exchange unts (ncludng the heater) s m 2 and the correspondng captal costs, and annual operatng costs would be M$ and M$/a, respectvely. Nevertheless, such a soluton s suboptmal. The optmal network has the same annual operatng costs (2.131 M$/a), but lower HEN captal cost (0.281 M$) and s depcted n Fg. 10. The correspondng HEN conssts of only two heat exchangers wth a total area of m 2, and a sngle heater (36.7 m 2 ). The fresh water ntake s 90 t/h. Note that the nterconnectons between water usng process unts (water reuse), comparng the solutons presented n Fgs. 8 and 10, have not changed due to the contamnant load scaledown. However, the topology of HEN has changed. The reason s Table 5 Data for example 4 (water usng process unts) Process L l (kg/h) cmax IN (mg/kg) cout max (mg/kg) Tp ( C) C 1 C 2 C 3 C 1 C 2 C 3 C 1 C 2 C 3 IN OUT P P Table 6 Data for example 4 (regeneraton unt) Regeneraton unt (%) T r ( C) C 1 C 2 C 3 IN OUT R that due to the economy of scale, t s more cost effectve to have smaller number of heat exchangers wth 25% larger area. We conclude that smultaneous cost-drven synthess of heat ntegrated networks s mportant for obtanng economcally attractve solutons and that solutons solely based on mnmzng a weghted sum of fresh water consumpton and utlty usage gnorng captal nvestment, or sequental procedures are not necessarly the best Example 3 The thrd example (Table 4) s an extenson of the second example to consder multple contamnants (three). The number of process unts and ther operatng temperatures are the same as n example 2. Also, the mass load of contamnant C 1, and ts nlet outlet concentraton constrants are the same as n example 2. The soluton of example 3 s depcted n Fg. 11. Asnexample 2, no coolng s needed the problem s pnched at the fresh water temperature (20 C). However, the topology of the network dffers from the one presented n Fg. 10. Frst, the presence of addtonal contamnants altered the nterconnectons among the water usng Fg. 11. Soluton of mult-contamnant example.

12 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) Fg. 12. Soluton of mult-contamnant example wth regeneraton. Table 7 Summary of results for the four examples Example F FW (t/h) C FW (k$/a) C R (k$/a) C R,v (k$) C HEN (k$) a C HU (k$/a) C CU (k$/a) No. of splts (HEN) a Incl. costs of splts (C CON). process unts. Second, the heat exchangers are placed on streams nterconnectng mxer unts wth process unts, as well as on fresh and dscharge water streams. The captal cost of the HEN (whch has a total area of m 2 ) s M$ and the fresh water consumpton s t/h resultng s an annual operatng costs of M$/a Example 4 Fnally, the fourth example s a small scale mult-contamnant case ncludng the possblty of waste water regeneraton. Note that n ths example the water usng process unts and regeneraton unt operate non-sothermally. A fxed temperature change, regardless of the water flow rate through the unts s assumed. The data for operatng condtons of water usng process unts s gven n Table 5. The operatng condtons of regeneraton unt are presented n Table 6. The soluton of example 4 s presented n Fg. 12. Unlke n the frst three examples, heatng of 1602 kw and coolng of 1409 kw n total s needed. Heat exchanger network conssts of fve heat exchange unts (one heater, two coolers, and two heat exchangers) wth total area of m 2, and captal costs of M$. The total annual operatng cost of the resultng network s M$/a, of whch approxmately 16% corresponds to costs of water regeneraton, 22% to annual utlty costs, and 62% to annual fresh water costs. Varable cost assocated wth regeneraton unt capacty s approxmately M$. The results for the four examples presented are summarzed n Tables 7 and 8. All four water networks operate at mnmal fresh water ntake. One explanaton for ths s that an ncrease n fresh Table 8 Problem szes and computatonal tme Example No. of contnuous varables No. of bnary varables Total CPU tme (s) water ntake causes the ncrease n utlty consumpton. Also, n all the examples, the contrbuton of annual fresh water costs outweghs the contrbutons of other annual and captal costs. Snce the obectve n all cases was mnmzaton of annualzed costs ths outcome s n fact not surprsng. Ths may, however, change when some revenue s assocated wth the operaton of water usng process unts. In such case, mnmal fresh water consumpton may not be the optmal one. 7. Conclusons We presented a mathematcal programmng model to smultaneously synthesze process water networks and ther correspondng HENs. A modfed HEN superstructure s proposed to allow non-sothermal stream mxng of process streams. The combned model conssts of NLP formulaton of WN superstructure and MINLP formulaton of embedded HEN superstructure. Snce the maorty of equatons/constrants n the combned model are non-lnear and non-convex more than one optmal soluton may exst. For ths reason, an effcent ntalzaton s needed to obtan globally or at least very good locally opt-

13 3142 M. Bogata, M.J. Bagaewcz / Computers and Chemcal Engneerng 32 (2008) mal solutons. The soluton strategy presented n the paper s effcent enough to overcome the problems assocated wth ntalzaton. In addton t helps to consderably reduce the szes of the problems. Four examples have been presented, clearly showng that the proposed method can be used n synthess of snge- and multcontamnant heat ntegrated water networks. The desgns obtaned show farly low topologcal complexty, whch s from ndustral applcaton pont of vew hghly desrable. Acknowledgements Mloš Bogata would lke to acknowledge fnancal support from the Scentfc and Educatonal Foundaton of the Republc of Slovena under grant 2005/ Mloš Bogata also expresses grattude to Prof. Dr. Peter Glavč and Prof. Dr. Zdravko Kravana for ther advce and support. References Alva-Argáez, A., Kokosss, A. C., & Smth, R. (1998). Wastewater mnmsaton of ndustral systems usng an ntegrated approach. Computers and Chemcal Engneerng, 22(Suppl.), S741 S744. Bagaewcz, M. J. (2000). A revew of recent desgn procedures for water networks n refneres and process plants. Computers and Chemcal Engneerng, 24, Bagaewcz, M. J., & Savelsk, M. (2001). On the use of lnear models for the desgn of water utlzaton systems n process plants wth sngle contamnant. Chemcal Engneerng Research and Desgn (Part A), 79, Bagaewcz, M. J., Rodera, H., & Savelsk, M. (2002). Energy effcent water utlzaton systems n process plants. Computers and Chemcal Engneerng, 26, Barbaro, A., & Bagaewcz, M. J. (2005). New rgorous one-step MILP formulaton for heat exchanger network synthess. Computers and Chemcal Engneerng, 29, Bogata, M., & Bagaewcz, M. (2007). Desgn of non-sothermal process water networks. In V. Plesu & P. S. Agach (Eds.), 17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 (pp ). Elsever. Brooke, A., Kendrck, D., Meeraus, D., & Raman, R. (1998). GAMS A user gude. Washngton D.C.: GAMS Development Corporaton. Chen, J. J. J. (1987). Letter to the edtors: Comments on mprovement on a replacement for the logarthmc mean. Chemcal Engneerng Scence, 42, Furman, K. C., & Sahnds, N. V. (2002). A crtcal revew and annotated bblography for heat exchanger network synthess n the 20th century. Industral and Engneerng Chemstry Research, 41, Gll, P. E., Murray, W., & Saunders, M. A. (2002). SNOPT: An SQP algorthm for largescale constraned optmzaton. SIAM Journal on Optmzaton, 12, Karuppah, R., & Grossmann, I. E. (2006). Global optmzaton for the synthess of ntegrated water systems n chemcal processes. Computers and Chemcal Engneerng, 30, Lnnhoff, B., Townsend, D. W., Boland, D., & Hewtt, G. F. (1982). User gude on process ntegraton for the effcent use of energy. UK: Insttute of Chemcal Engneers. Maoz, T., Brouckaert, C. J., & Buckley, C. A. (2006). A graphcal procedure for wastewater mnmsaton n batch processes. Journal of Envronmental Management, 78, Savelsk, M. J., & Bagaewcz, M. J. (1997). Desgn and retroft of water utlzaton systems n refneres and process plants. Los Angeles: Amercan Insttute of Chemcal Engneerng Annual Meetng, paper 188g. Savelsk, M., Rvas, M., & Bagaewcz, M. (1999). A new approach to the desgn of water utlzaton systems wth multple contamnants n process plant. Dallas, TX: Annual Amercan Insttute of Chemcal Engneerng meetng. Savulescu, L. E., & Smth, R. (1998). Smultaneous energy and water mnmsaton. Mam Beach: Amercan Insttute of Chemcal Engneerng Annual Meetng, paper 13c. Savulescu, L. E., Sorn, M., & Smth, R. (2002). Drect and ndrect heat transfer n water network systems. Appled Thermal Engneerng, 22, Savulescu, L., Km, J.-K., & Smth, R. (2005a). Studes on smultaneous energy and water mnmsaton I. Chemcal Engneerng Scence, 60, Savulescu, L., Km, J.-K., & Smth, R. (2005b). Studes on smultaneous energy and water mnmsaton II. Chemcal Engneerng Scence, 60, Soršak, A., & Kravana, Z. (2002). Smultaneous MINLP synthess of heat exchanger networks comprsng dfferent exchanger types. Computers and Chemcal Engneerng, 26, Vswanathan, J., & Grossmann, I. E. (1990). A combned penalty functon and outer approxmaton method for MINLP optmzaton. Computers and Chemcal Engneerng, 14, Wang, Y. P., & Smth, R. (1994). Wastewater mnmzaton. Chemcal Engneerng Scence, 49, Yee, T. F., & Grossmann, I. E. (1990). Smultaneous optmzaton models for heat ntegraton II. Computers and Chemcal Engneerng, 14,