i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/he Modeling, synthesis and optimization of heat exchanger networks. Application to fuel processing systems for PEM fuel cells Diego G. Oliva, Javier A. Francesconi, Miguel C. Mussati, Pio A. Aguirre* INGAR Instituto de Desarrollo y Diseño (CONICET-UTN), Avellaneda 3657 (S3002GJC) Santa Fe, Argentina article info abstract Article history: The development of biofuels has gained much attention in recent years. Thermodynamic Received 20 October 2010 analyses to obtain energy from biofuels using fuel cells were addressed in previous works Received in revised form for a variety of processes. In those processes, the determination of the best conditions to 14 April 2011 achieve high efficiency values in the conversion of chemical energy into electrical power is Accepted 15 April 2011 a critical issue from the net global energy efficiency point of view. In this regard, a main Available online 23 May 2011 aspect is to address the energy integration of the whole process. In a previous paper, the authors dealt with energy integration studies for glycerin- and ethanol-based processors Keywords: coupled to PEM fuel cells resorting on the “multi-stream heat exchanger” feature provided HEN synthesis by the simulation tool HYSYS. In that work, the aim was to maximize the energy recovery Optimization from the process streams that renders the maximum achievable net global efficiency. In SYNHEAT model this paper, the aim is to synthesize and design the optimal heat exchangers network (i.e. Glycerin determination of the process configuration and units sizes) while maintaining the net Ethanol global efficiency of the whole system at its achievable value. PEM fuel cell Three modifications to the original SYNHEAT model developed in 1990 by Yee and Grossmann for synthesizing heat exchanger networks are proposed in this work aiming at a better problem description, and consequently searching for best problem solutions. First, a modification in computing the minimum approach temperature difference is proposed. Second, the called “operation line method” is coupled to the SYNHEAT model to built-up the network superstructure to be optimized. Finally, the SYNHEAT model’s hypothesis of constant cp value for modeling heat exchange between process streams is improved by considering enthalpy variable instead of temperature variable, which is convenient when latent heat is transferred. The model variables number involved in the heat exchanger network synthesis problems solved has been reduced to less than a half by applying the operation line method. The proposed methodology and modifications made are of general application and not just for the specific cases addressed in this work. Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. * Corresponding author. Tel.: þ54 342 4555229; fax: þ54 342 4553439. E-mail addresses: doliva@santafe-conicet.gov.ar (D.G. Oliva), javierf@santafe-conicet.gov.ar (J.A. Francesconi), mmussati@santafeconicet.gov.ar (M.C. Mussati), paguir@santafe-conicet.gov.ar (P.A. Aguirre). 0360-3199/$ e see front matter Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2011.04.097 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 1. Introduction The development of biofuels has gained much attention in recent years. Previous works have addressed the thermodynamic analysis to obtain energy from biofuels using fuel cells in a wide variety of processes [1e16]. The system analyzed in this work is that one previously described by the authors [17,18], where the focus was mainly put on process efficiency issues. Briefly, the selected process involved the following main components: steam reforming (SR) reactor, water gas shift (WGS) reactors, carbon monoxide preferential oxidation (COPrOx) reactor, proton exchange membrane (PEM) fuel cell, combustor, pumps, compressor, and expanders. The fuel fed to the process was alternatively glycerin or ethanol. The “composite curve” approach was used to perform the energy integration of all process streams, but the heat exchangers network was not synthesized. Two major methodologies have been proposed to synthesize heat exchangers networks (HENs): sequential and simultaneous approaches. Among the formers, the pinch design method [19] is one of the most widely known and applied. In this method, the minimum utility demand, the minimum number of exchange units, and the HEN’s minimum capital cost are obtained sequentially by using heuristic rules. Among the latter ones, methodologies based on mathematical programming techniques are included. Furman and Sahinidis [20] performed a complete review and classification of different methods and procedures for HEN synthesis. Yee and Grossmann [21] developed a basic framework for HEN synthesis using a staged-superstructure formulated as a Mixed Integer Non Linear Programming (MINLP) model aimed at simultaneously minimizing utility and capital costs. Several extensions of that framework were proposed addressing flexibility [22,23], incorporating detailed design aspects of the exchange units [24e27], and focusing on global optimality issues [28,29]. In this work, the HEN 9099 synthesis for PEM fuel cell integrated to biofuel reforming processes is obtained using the original SYNHEAT model (Yee and Grossman [21]) and a modified or extended version of it. Those modifications are intended to consider situations left out of consideration in the original model. This paper is organized as follows. The problem definition is presented in section 2. The original SYNHEAT model and the proposed modifications are presented and discussed in section 3. The resulting heat exchanger network for glycerin and ethanol-fed processes are described in section 4 and 5, respectively. Finally, conclusions are drawn in section 6. 2. Problem definition The determination of the best operation conditions to reach high efficiency values in converting chemical energy into electrical power from glycerin and ethanol by integrated fuel processor-fuel cell systems was addressed in previous work [17] using an energy integration tool provided by the simulation software HYSYS. More specifically, a simulation study was performed based on the “multi-stream heat exchanger” tool to obtain the energy process integration that maximizes the global net process efficiency for a wide operation range of the main process variables. However, it just only allows identifying the minimal hot and cold utilities and estimating the total heat exchange area. Now, the interest is to obtain the optimal configuration of heat exchangers for this process, i.e. optimal HEN synthesis, as well as their individual area, based on optimal criteria. A generic process scheme is depicted in Fig. 1. The input stream named “fuel” can be either glycerin or ethanol. Operation temperatures are detailed in Table 1 and Table 2 for a glycerin- and ethanol-based processor, respectively. Process cold streams (CS) are defined as those that “move” from a given energy level to a higher one (i.e. they must be heated); while hot streams (HS) are defined as those that Fig. 1 e Flow sheet of the fuel processor system coupled to PEMFC. 9100 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 1 e Streams specification using glycerin as a fuel in the processor system. Number in Fig. 1 #4 #5 #6 #7a #7b #11 #3a #3b #3c #3d #10 #9 Type Stream TIN ( C) TOUT ( C) HS HS HS HS HS HS CS CS CS CS CS CS CU HU H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 25 807 705.00 542.14 247.36 471.31 87.51 806.55 39.00 132.93 133.93 269.66 25.00 80.00 28 807 500.00 150.00 237.00 87.51 80.00 272.82 133.93 133.93 269.66 705.00 300.00 500.00 “move” from a given energy level to a lower one (i.e. they must be cooled). Sensible heat is transferred to a process stream that changes its energy level modifying its temperature; while latent heat is transferred to a stream that changes its energy level at a constant temperature. The problem addressed in this paper can be stated as follows. Given (i) a set of cold and hot streams of the investigated process; (ii) their thermal levels incoming to/outgoing from each process unit; (iii) availability of hot and cold duties for tasks that require heating and cooling, respectively; (iv) a cost model including cost of cooling water and steam, heat exchangers area, and heat exchangers installation, synthesize and design the HEN for the process aiming at minimizing the total annual cost (TAC). Heat capacity flow rate (W  C 1) Heat transfer coefficient (W  C 1 m 2) 0.700 0.665 0.656 0.712 4.109 1.766 1.129 289.230 1.940 0.656 0.003 1.681 87.77 10.22 87.15 31.54 57.21 34.41 97.43 10.22 190.01 190.01 31.77 9.78 194.02 13.23 more clearly the modifications proposed in this paper. Symbols are defined in the Nomenclature section. Overall energy balance for a hot stream i and a cold stream j:
TINi XX  _ i $cpi ¼ TOUTi $m qi;j;k þ qCU;i i˛HS (1) j˛CS (2) k˛ST j˛CS  TOUTj  X X _ j $cpj ¼ TINj $m qi;j;k þ qHU;j k˛ST i˛HS Energy balance for stream i and j in stage k:
Ti;k
Tj;k X  _ i $cpi ¼ qi;j;k Ti;kþ1 $m i˛HS; k˛ST (3) X  _ j $cpj ¼ qi;j;k Tj;kþ1 $m j˛CS; k˛ST (4) j˛CS i˛CS Inlet temperature assignment for stream i and j: 3. Models for HENs synthesis and design 3.1. Original SYNHEAT model TINj ¼ Tj;kþ1 TINi ¼ Ti;1 Yee and Grossmann (1990) proposed a MINLP model for synthesizing and designing heat exchanger networks, known as the SYNHEAT model, which is presented below to explain j˛CS (5) i˛HS (6) Temperature feasibility for stream i and j: Ti;k  Ti;kþ1 i˛HS; k˛ST (7) Table 2 e Streams specification using ethanol as a fuel in the processor system. Number in Fig. 1 #4 #5 #7a #7b #11 #3a #3b #3c #3d #10 #9 Type Stream TIN ( C) TOUT ( C) HS HS HS HS HS CS CS CS CS CS CS CU HU H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 C6 20 810.84 709.00 538.92 405.67 94.98 810.84 41.76 98.25 98.25 126.50 25.00 80.00 25 810.84 500.00 150.00 94.98 80.00 286.64 98.25 99.25 126.50 709.00 300.00 500.00 Heat capacity flow rate (W  C 1) Heat transfer coefficient (W  C 1 m 2) 0.6268 0.5966 0.6223 2.3012 1.6787 0.9482 111.9641 10.8001 0.5482 0.0178 1.5921 87.77 10.22 87.15 31.54 34.41 97.43 10.22 190.01 190.01 31.77 9.78 194.02 13.23 9101 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 j˛CS; k˛ST Tj;k  Tj;kþ1 (8) Temperature feasibility for hot stream i around cold utility: i˛HS TOUTi  Ti;kþ1 (9) Temperature feasibility for cold stream j around hot utility: j˛CS TOUTj  Tj;1 (10) Heating and cooling duties:  _ i $cpi ¼ qCU;i i˛HS TOUTi $m  _ j $cpj ¼ qHU;j j˛CS Tj;1 $m
Ti;kþ1  TOUTj (11) (12) Upper bound constraints for heat exchange: qi;j;k Qmax $yi;j;k  0 i˛HS; j˛CS; k˛ST (13) qCU;i Qmax $yCU;i  0 i˛HS (14) qHU;j Qmax $yHU;j  0 j˛CS (15) Minimal temperature difference for heat exchangers:  Tj;k þ DTmax i;j $ 1 DTi;j;k  Ti;k DTi;j;kþ1  Ti;kþ1 DTCU;i  Ti;kþ1 DTHU;j  TOUT;HU yi;j;k  Tj;kþ1 þ DTmax i;j $ 1  TOUT;CU þ DTmax CU;i $ 1  Tj;1 þ DTmax HU;j $ 1  (16) yi;j;k  yCU;i yHU;j (17)  (18)  (19) Logarithmic mean temperature differences:    1
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $ DTi;j;k þ DTi;j;kþ1 þ $ DTi;j;k DTi;j;kþ1  0 6 3 LMTDi;j;k LMTDCU;i LMTDHU;j   1
TINCU $ DTCU;i þ TOUTHu 6  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 2 þ $ DTCU;i $ TOUTHu TINCU  0 3 ð21Þ    1 $ DTHU;j þ TINi TOUTj 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 0 þ $ DTHU;j $ TINi TOUTj 3 (23) (24) (25) þ j˛CS CFi;CU $yCU;i þ X j˛CS i˛HS CFj;HU $yHU;j þ TINi ; TINj TOUTi ; TOUTj TINi ; TOUTj TOUTi  (27) CAi;j i˛HS j˛CS k˛ST X


b b b X CAj;HU $ areaj;HU HU CAi;CU $ areai;CU CU þ  areai;j;k þ i˛HS As illustration example, two isothermal streams are supposed given. The temperature of the cold stream is assumed to be higher than the hot one. Then, they cannot be matched for energy integration as the upper bound constraints [16] and [17] of the original model cannot be fulfilled, and the model results infeasible. The reason is that the parameter DTmax defined by eq [27]. does not consider this situation: Then, a modification to the DTmax definition is proposed: i˛HS j˛CS k˛ST XXX Modification of the parameter DTmax  DTmax ¼ max 0; TINj i;j Total annual cost (objective function): X X XXX TAC ¼ CFi;j $yi;j;k CCU$qCU;i þ CHU$qHU;j þ X Three modifications to the original SYNHEAT model are proposed for a better description or representation of the HEN synthesis problem, and consequently for searching for better solutions. The resulting model is hereafter referred as the “modified model”. First, a modification of the parameter DTmax is proposed for allowing the exploration of potential configurations not considered in the original model. Second, the theoretical method called ‘operation line method’ (OLM) [30,31] is applied. Unless this approach was initially developed for synthesizing power systems, it can be applicable to a wide range of synthesis problems. It allows reducing the problem size and finding configurations not included in the original superstructure of the SYNHEAT model. This model modification leads to local optimal solutions closer to the global optimum. Finally, the original model assumes constant cp value. This limitation can be improved by replacing enthalpy variable by temperature variable in some model constraints and adding new ones for relating these variables. A discretization of the heat exchangers is also proposed, computing in each discretized exchange point both the cp value and the logarithmic mean temperature difference. In doing so, more accurate enthalpytemperature curves can be approximated. Indeed, such discretization and domain change allow for a more accurate heat exchange representation of an isothermal stream. More precisely, streams with latent heat are fragmented into three pieces (liquid, phase change and vapor). In the original model, the phase change is represented by assigning a fictitious and arbitrarily big cp value and a fictitious and arbitrarily small ∆T value (normally 1  C) (e.g. stream #3b in Table 1). Such threepiece fragmentation strategy is also performed in the modified model, but the consideration of the enthalpy variable (instead of temperature variable) avoids assigning fictitious cp y ∆T values; it is only necessary to assign the amount of latent heat corresponding to those operation conditions, keeping thus the isothermal characteristic of that piece. Following, a more detailed discussion on the proposed modifications is presented. 3.2.1. qHU;j ¼0 UHU;j $LMTDHU;j i˛HS Modifications to the original SYNHEAT model ð22Þ Heat exchanger area requirement: qi;j;k areai;j;k ¼0 Ui;j $LMTDi;j;k qCU;i ¼0 areaCU;i UCU;i $LMTDCU;i areaHU;j (20) 3.2. j˛CS (26) if TOUTi TOUTj  DTmin then : DTmax ¼ DTmin þ TOUTj TOUTi i;j else DTmax ¼0 i;j (28) 9102 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 where DTmin is the fixed lower bound for variables DTi;j;k and DTi;j;kþ1 . With this definition, a given stream (with sensible or latent heat) that exchanges heat in superior stages does not lead to model infeasibilities in the following stages or “forces” the model to avoid them. To clarify, consider the two isothermal streams mentioned above and a DTmin design value of 5  C. The cold stream is at temperature level of 100  C and the hot stream at 90  C. Note that in this example binary variables (yi;j;k ) have to be zero because the heat exchange between these streams is not feasible. The DTmax value computed by eq. (27) is 10  C. With this value, the right-hand side of eq. (16) or (17) results in 0  C for all stages, which is lower than the design DTmin . Then, eq. (16) and (17) lead to infeasibilities and, consequently, do not allow solving the model. If the proposed modification is used instead (eq. (28)), the computed DTmax value is 15  C. Then, the right-hand side of eq. (16) or (17) gives 5  C fulfilling all model constraints. This parameter modification is used throughout this work. TAC ¼ C$area þ CUtilities $qu (33) where: qu ¼ X qCU;i þ i˛HS X qHU;j (34) j˛CS In the optimum, small perturbations of the solution (e.g. an area increase) produce a zero cost variation (Karush-KuhnTucker -KKT- optimal conditions): d TAC ¼ 0 (35) Taking into account that: X XX qi;j;k (36) Constant ¼ q þ qu (37) q¼ k˛ST i˛HS j˛CS and then 3.2.2. Operation line method OLM The operation line method OLM was initially developed for maximizing the production of power systems with a concomitant maximum energy recovering by process streams integration [30,31]. In this paper, the OLM approach is only applied for obtaining the energy integration strategy among hot and cold streams at temperature levels T and t, respectively, since the power generation is not targeted. The functionality T¼f(t) is analyzed to design an initial HEN superstructure different from the original one proposed by SYNHEAT model. The first step consists of plotting the operation line and the streams involved in the process, corresponding the ordinate axis to hot stream temperature (T) and the abscise axis to cold ones (t). The operation line is linear with a slope of 1 and a DTInitial shift over the T-axis, i.e. T ¼ t þ DTInitial . The parameter DTInitial is computed as follows: DTInitial ¼ C U$CUtilities (29)

b b P CAi;CU $ areai;CU CU þ CAj;HU $ areaj;HU HU i˛HS j˛CS   P
P
C¼ areaj;HU areai;CU þ P CUtilities ¼ P CCU$qCU;i þ i˛HS P qCU;i þ U¼ UCU;i þ P UHU;j j˛CS N P CHU$qHU;j qHU;j (31) j˛CS i˛HS P P j˛CS i˛HS (30) j˛CS i˛HS (32) where C and CUtilities are weighted average values of the area cost and utilities costs, respectively; U is the average value for the global heat transfer coefficient. N is the total number of process streams. C , CUtilities and U calculation considers that all process streams exchange heat exclusively with process utilities. Briefly, DTInitial value is obtained from the optimum conditions derived for the operation line problem and its relation with an economic optimum. The approximate TAC is computed as follows: d area ¼ d qu 1 U$DTInitial (38) By algebraic manipulation, the following expression can be obtained in the optimum: d TAC ¼ C$d area þ CUtilities $d qu ¼ C$d qu þ CUtilities $d qu ¼ 0 U$DTInitial (39) From eq. (39), the relation expressed by eq. (29) is thus obtained: DTInitial ¼ C U$CUtilities (40) Although this methodology was conceived for power systems, it is useful in designing the initial HEN superstructure. Indeed, it allows: (a) reducing the number of analyzed heat exchangers when compared to the original SYNHEAT superstructure model, (b) finding configurations not considered by the original SYNHEAT superstructure model. Here, the first steps of the OLM approach are used to propose an initial HEN superstructure according to the problem cost parameters. Specifically, the exchange regions in the space (T, t) determine such superstructure, which is usually different from that proposed in the original SYNHEAT model. Following, the OLM approach is briefly described for clarity. As explained, the inlet and outlet temperature of hot streams (T) and the inlet and outlet temperature of cold streams (t) are represented in the (T, t) plane, as shown in Fig. 2. Parallel straight lines to the abscise axis are plotted from each hot stream extreme T; analogously, parallels to the ordinate axis are drawn from each cold stream extreme t. In doing so, rectangular regions are formed in the plane Tet. Afterwards, the operation line is plotted as explained above. Each region delimited by a rectangular region and the operation line is called “stage”. Each stage has equipment units that arise as a result of intersections between cold and hot temperatures. These stages determine the initial superstructure, which usually results in more stages but in fewer bifurcations (units) in a stage when comparing to the original SYNHEAT model. That is advantageous from 9103 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 3 e Cost and data of process streams and utilities in example 1. H1 Direction of routes for stages Stream TIN ( C) TOUT ( C) Heat capacity flow rate (kW  C 1) Heat transfer coefficient (kW  C 1 m 2) Cost ($ kW 1 year 1) CU HU H1 H2 C1 C2 C3 C4 25 325 180 240 40 120 40 80 40 325 75 60 230 260 130 190 e e 30 40 20 15 25 20 1.0 2.0 2.0 2.0 1.5 1.5 2.0 2.0 20 120 e e e e e e Stage 1 H2 Hot stream temperature (°C) T Stage 2 Stage 3 Stage 4 Stage 5 on ati lin e er Op C1 C2 t Initial Cold stream temperature (°C) Fig. 2 e OLM scheme. a computational effort point of view. In this problem type, the obtained superstructure leads to local solutions close to the global optimum. Now, it may occur that a pair of streams (i.e. a cold stream with a hot stream) having appropriate temperature levels for heat exchanging has not been included as a heat exchange option at any stage, and they appear in the obtained solution requiring utilities. Such pairs of streams are hereafter called “orphan” pairs. In that case, the next step is to add the orphan pair (represented as a rectangular region) as an additional integration option in those stages of the previous scheme that Fig. 3 e Flowchart of OLM. limit with the region corresponding to that pair. Afterwards, the model is solved again. If any orphan pair still remains, it is included in adjacent stages, and the model is solved once again. These steps are iteratively repeated until no orphan streams pair appears in the model solution. The procedure is schematized in Fig. 3. Following, the OLM approach is applied for illustration to two case studies selected from literature. 3.2.2.1. OLM approach. Example 1. This example corresponds to a process taken from Björk and Westerlund [32]. Table 3 lists the process streams and the duties unitary costs for the TAC minimization problem. Fixed cost of process units is $ 8000, the area’s cost coefficient is 50 $ m 2, and the scale factor is b ¼ 0:85. Fig. 4 depicts the operation line scheme for this case. Table 4 shows and compares the superstructure derived from the OLM approach and that proposed by the original model. The rectangular regions limiting with the operation line, i.e. the stages of the superstructure, are included. The structure obtained from the original model has only 4 stages while that one derived from the OLM approach has 8 stages. The DTInitial parameter value computed by eq. (40) is 0.47  C. The TAC value obtained by the original SYNHEAT model is $ 145,277.83. Based on a global optimization technique, Bergamini et al. [33] reported a TAC value of $ 140,367.07 with 1% Fig. 4 e Representation of OLM in example 1. 9104 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 4 e Stages depending on adopted initial superstructure. Example 1. Stage Streams involved SYNHEAT superstructure 1 2 3 4 5 6 7 8 H1 H2 C1 C2 C3 C4 H1 H2 C1 C2 C3 C4 H1 H2 C1 C2 C3 C4 H1 H2 C1 C2 C3 C4 e e e e Number of heat exchange options by stage OLM superstructure SYNHEAT superstructure OLM superstructure H2 C2 H2 C1 C2 H2 C1 C2 C4 H1 H2 C1 C2 C4 H1 H2 C1 C2 C3 C4 H1 H2 C1 C3 C4 H1 H2 C1 C3 H2 C1 C3 8 8 8 8 e e e e 1 2 3 6 8 6 4 2 tolerance, while the computed TAC value using OLM is $ 140,349.25, which is within that tolerance. Fig. 5 represents the HEN resulting from the OLM approach. The structure obtained by Bergamini et al. and this one are equal, but small differences exist in the calculation of heat exchanger areas. The heat exchanger “2” matching streams H2 and C2 exhibits a DT value equal to DTInitial ¼ 0.47  C. Both the original and modified models resulted in 32 heat exchangers. The model based on the OLM approach was implemented in General Algebraic Modelling System GAMS, which a general-purpose computational tool for modeling, simulation and optimization [34]. The codes XPRESS [35], CONOPT 3 [36] and DICOPT [37] were used to solve the mixed integer programming (MIP), non linear programming (NLP) and mixed integer non linear programming (MINLP) problems, respectively. The scaling technique of problem variables was adopted to gain robustness in both the original and modified models. 3.2.2.2. OLM approach. Example 2. This example is proposed for illustrating the existence of HEN configurations not contemplated by the original SYNHEAT model. Fig. 6 depicts the operation line scheme for this process. Table 5 lists process streams and utility costs. The fixed cost of units is $ 5500, the area’s cost coefficient is 150 $ m 2, and the scale factor is b ¼ 1. Table 6 shows and compares the superstructure derived from the OLM and from the original SYNHEAT model. The computed DTInitial value is 4.92  C. The TAC value obtained by the original SYNHEAT model and the OLM approach is $ 145,139.23 and $ 135,611.40, respectively. Fig. 7 represents the HEN obtained by the OLM approach. Note the original SYNHEAT model is unable to obtain this configuration as the heat exchangers “2” and “3” cannot be represented with it. Certainly, consider in Fig. 7 hot streams H1 and H2, cold streams C1 and C2, and the only two stages that the original SYNHEAT model can propose for this case. By Fig. 5 e Heat exchanger network of example 1 using OLM. 9105 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 exchangers and 2 stages. The heat exchanger “1” exhibits a DT value equal to 8.48  C. This value is close to the DTInitial value computed by the OLM approach. The model was solved using the same computational tools and strategies as done in example 1. 3.2.3. Fig. 6 e Representation of the OLM in example 2. inspecting hot streams, it can be observed that H1 exchanges with C1 in heat exchanger “1” at stage 1, and with C2 in heat exchanger “2” at stage 2. While H2 exchanges with C1 in heat exchanger “3” at stage 1, and with C2 in heat exchanger “4” at stage 2. By inspecting now cold streams, it can be observed that C1 exchanges with H2 in heat exchanger “3” at stage 2, and with H1 in heat exchanger “1” at stage 1. While C2 exchanges with H2 in heat exchanger “4” at stage 2, and with H1 in heat exchanger “2” at stage 1. Then, by inspecting hot streams, heat exchanger “2” should be at stage 2; while inspecting cold streams, heat exchanger “2” should be at stage 1, which is contradictory. Analogously, it can be concluded that the existence of heat exchanger “3” is also infeasible. So, the original SYNHEAT superstructure is unable to represent the configuration obtained with the superstructure generated using the OLM approach. The OLM approach required 10 heat exchangers and 5 stages while the original SYNHEAT model computed 8 heat Streams discretization In the original SYNHEAT model the cp values are assumed to be constant, Here, that hypothesis is refined by discretizing the process streams. With this discretization and by including enthalpy as new variable, it is possible to represent the process streams exchanging latent heat without assuming small DT value for computing energy flows (eq. (1) to (4)), e.g. stream #3b in Table 1. The reformulation of the original problem requires modifying some model constraints and adding new ones. More _ specifically, the occurrences of the factor ðm,cp,TÞ in the streams energy balances are replaced by the (new) enthalpy variable H. In addition, eq. (5) to (10) related to temperature assignment and feasibility have to be replaced with the enthalpy assignment constraints. Then, constraints (1) to (12) of the original model are modified as follows: 3.2.3.1. Modification to constraints of the original model. Overall energy balance for each hot stream i and a cold stream j:
HINi  XX HOUTi ¼ qi;j;k þ qCU;i i˛HS (41) k˛ST j˛CS  HOUTj  X X HINj ¼ qi;j;k þ qHU;j j˛CS (42) k˛ST i˛HS Energy balance for stream i and j in stage k:
Hi;k
Hj;k  X Hi;kþ1 ¼ qi;j;k i˛HS; k˛ST (43)  X qi;j;k Hj;kþ1 ¼ j˛CS; k˛ST (44) j˛CS i˛CS Table 5 e Cost and data of process streams and utilities in example 2. Stream CU HU H1 H2 C1 C2 TIN ( C) TOUT ( C) Heat capacity flow rate (kW  C 1) Heat transfer coefficient (kW  C 1 m 2) 350 680 650 590 410 350 500 680 470 370 650 500 e e 10 20 15 13 1.00 5.00 1.00 1.00 1.00 1.00 Cost ($ kW 1 year 1) 80 15 e e e e Table 6 e Stages depending on the adopted initial superstructure. Example 2. Stage 1 2 3 4 5 Streams involved Number of heat exchange options by stage SYNHEAT superstructure OLM superstructure SYNHEAT superstructure OLM superstructure H1 H2 C1 C2 H1 H2 C1 C2 e e e H1 C1 H1 H2 C1 H1 H2 C1 C2 H2 C1 C2 H2 C2 4 4 e e e 1 2 4 2 1 9106 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Fig. 7 e Heat exchanger network of example 2 using OLM. Table 7 e Cost and data of process streams and utilities as an example of stream discretization. Stream TIN ( C) TOUT ( C) Mass flow rate (kg sec 1) Heat transfer coefficient (kW  C 1 m 2) 280 1020 920 880 305 400 285 1020 300 400 900 800 e e 3 6 4 6 1.00 2.50 1.80 2.00 1.80 2.00 CU HU H1 H2 C1 C2 Inlet enthalpy assignment for stream i and j: HINj ¼ Hj;kþ1 HINi ¼ Hi;1 j˛CS 1 year 1) 10 100 e e e e Enthalpy feasibility for hot stream i around cold utility: (45) i˛HS Cost ($ kW (46) Enthalpy feasibility for stream i and j in stage k: HOUTi  Hi;kþ1 i˛HS (49) Enthalpy feasibility for cold stream j around hot utility: HOUTj  Hj;1 j˛HS (50) Heating and cooling utilities: Hi;k  Hi;kþ1 Hj;k  Hj;kþ1 i˛HS; k˛ST j˛CS; k˛ST (47) (48)
H  i;kþ1 HOUTj  HOUTi ¼ qCU;i i˛HS  Hj;1 ¼ qHU;j j˛CS Fig. 8 e Comparison between areas obtained with and without stream discretization. (51) (52) 9107 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 2 Tj;k 2 Tj;k 6 Tj;k 6 1000 1000 6 Aj $ þBj $ þCj $ 6 1000 2 3 _ j6 1000$m 4 6 Hj;k ¼ T j;k 6 MW 6 Ej 1000 6 þFj Gj 6 Dj $ 4 Tj;k 4 1000 3 3 7 7 þ7 7 7 7 7 7 7 7 5 j˛CS (54) Fig. 9 e Discretized-linearized ratio profiles of heat exchangers 1, 2 and 3. _i 1000$m MW 2 2 Ti;k Ti;k 6 6 Ti;k 1000 1000 6 Ai $ þCi $ þBi $ 6 2 3 1000 6 4 6 6 Ti;k 6 Ei 1000 6 þFi Gi 6 Di $ 4 Ti;k 4 1000 7 7 þ7 7 7 7 7 7 7 7 5  Tj;k ¼ TINj þ TOUTj TINj  Hj;k $ HOUTj HINj HINj i˛HS (55) j˛CS (56) approximates the temperature of a hot stream i or cold stream j at the inlet and outlet of a stage k, but not at its interior. Then, it is proposed to discretize the temperature domain inside a stage k adopting an arbitrary number of discretization elements Elk according to the required precision. The energy fraction Phi;j;k that a hot stream i exchanges with a cold stream j depending on the number of bifurcations within a stage k, is expressed as follows: Phi;j;k ¼ i˛HS (53) Hi;k qi;j;k Hi;kþ1 (57) Analogously, the energy fraction Pci;j;k that a cold stream j exchanges with a hot stream i depending on the number of bifurcations within stage k, is expressed as follows: Pci;j;k ¼ 3 Hj;k qi;j;k Hj;kþ1 (58) Then, the interior enthalpy flow Hhi;j;k;e of the eth element of a hot stream i exchanging heat with a cold stream j in a bifurcation inside a stage k is calculated as: Hhi;j;k;e ¼ Phi;j;k $Hi;k Stage 14 Stage 13 Stage 12 Stage 10 qi;j;k $ðe Elk 1Þ (59) Stage 3 H6 800 H1 Stage 1 Stage 2 H2 600 H4 Stage 4 Stage 5 Stage 6 Stage 7 400 H3 Stage 9 200 Stage 11 H5 Hot stream temperature (°C) 1000 3  Hi;k HINi TINi $ HOUTi HINi 3.2.3.3. Model discretization. Correlations (53) and (54) only 3.2.3.2. New model constraints added. As mentioned, additional constraints have to be added for relating temperature with enthalpy. More specifically, it is needed to relate eq. (41) to (52) of the discretized model with the constraints not modified of the original model (eq. (13) to (26)). For doing so, the constraints (53)e(54) or (55)e(56) are added depending on the case. The former two correlations are used to obtain a more approximate T-H profile, where “A” to “G” are correlation coefficients. The latter two ones are linear relationships, and used for describing isothermal streams, i.e. with latent heat. Hi;k ¼
Ti;k ¼ TINi þ TOUTi C1 C2 0 C3 C4 C5 -200 -200 0 200 C6 400 600 800 Cold stream temperature (°C) Fig. 10 e Representation of OLM in the glycerin processor system. 1000 9108 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 8 e Number of heat exchangers by stage depending on adopted initial superstructure. Glycerin processor system. Streams involved (Glycerin-based processors) Stage 1 2 3 4 5 6 7 8 9 10 11 12 13 14 SYNHEAT superstructure H1 H2 H1 H2 H1 H2 H1 H2 H1 H2 H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 H3 H4 H5 H6 C1 C2 C3 C4 C5 C6 e e e e e e e e OLM superstructure qi;j;k $ðe Elk 1Þ SYNHEAT superstructure OLM superstructure 36 36 36 36 36 36 e e e e e e e e 216 1 2 3 6 4 6 9 12 8 12 8 4 4 2 81 H6 C4 H1 H6 C4 H1 H2 H6 C4 H1 H2 H6 C4 C6 H2 H6 C4 C6 H2 H4 H6 C4 C6 H2 H4 H6 C4 C5 C6 H2 H4 H6 C2 C3 C5 C6 H2 H4 C2 C3 C5 C6 H2 H3 H4 C2 C3 C5 C6 H2 H4 C2 C3 C5 C6 H4 C2 C3 C5 C6 H4 C1 C2 C5 C6 H5 C1 C5 Total while the interior enthalpy flow Hci;j;k;e of the eth element of a cold stream j exchanging heat with a hot stream i in a bifurcation inside a stage k is calculated as: Hci;j;k;e ¼ Pci;j;k $Hj;k Number of heat exchange options by stage (60) Thus, the temperature Thi;j;k;e of the eth element of a hot stream i exchanging heat with a cold stream j in a bifurcation inside stage k, is computed as follows: Hhi;j;k;e ¼ 2 _ i6 Phi;j;k;horc $1000$m 6Ai $ Thi;j;k;e 4 MW 1000 Thi;j;k;e 1000 þCi $ 3 3 Thi;j;k;e 1000 þDi $ 4 Thi;j;k;e 1000 þBi $ 2 2 4 Ei þFi Thi;j;k;e 1000 3 7 Gi 7 5 (61) If it is a fragmentation piece of a hot stream i that exchanges latent heat, its temperature is computed as: Fig. 11 e Heat exchanger network of the glycerin processor system obtained using OLM plus stream discretization. Comparison between areas obtained with and without stream discretization. 9109 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 9 e Total heat exchanger network area obtained according to the adopted superstructure. Glycerin processor system. Type of initial superstructure adopted Discretization of streams Number of heat exchangers in final structure obtained Total area of heat exchangers in final structure (dm2) Number of stages in final solution No No Yes 12 12 12 229.3 116.9 114.2 4 7 7 SYNHEAT OLM approach OLM approach (64) Tci;j;k;e ¼ Tj;k Table 10 e Heat exchanger network costs obtained according to the adopted superstructure. Glycerin processor system. Investment cost for heat exchangers Annual hot utility cost Annual cold utility cost Total cost Original SYNHEAT OLM approach $ 1168.06 $ 844.06 DTi;j;k;e  Thi;j;k;e $e $ 5216.29 $ 6384.35 $e $ 5216.29 $ 6060.35 th Analogously, the temperature Tci;j;k;e of the e element of a cold stream j exchanging heat with hot stream i in a bifurcation inside stage k, is computed as: 2 Tci;j;k;e 1000 þBj $ 2 Tci;j;k;e 1000 þDj $ 4 4 _ j6 Pci;j;k $1000$m 6Aj $ Tci;j;k;e Hci;j;k;e ¼ 4 MW 1000 3 2 Ej þFj Tci;j;k;e 1000 3 7 Gj 7 5  Tci;j;k;e þ DTmax i;j;k $ 1 yi;j;k  (65) 3.2.3.4. Model discretization. Example 3. This example is (62) Thi;j;k;e ¼ Ti;k Tci;j;k;e 1000 þCj $ 3 Finally, the discrete temperature differences DTi;j;k;e between a hot stream i and a cold stream j in an element e inside each stage k is computed by eq. (65), differing with the original model as it only computes DTi;j;k at the inlet and outlet of a stage k. (63) If it is a fragmentation piece of a cold stream j that exchange latent heat, its temperature is computed as: intended to illustrate the differences in results between the original SYNHEAT model and the discretized model. Two cold streams and two hot streams containing CH4 at atmospheric pressure are considered. Stream specifications, and cost of hot and cold utilities are listed in Table 7. Note that only stream mass flow rates are given. The energy (enthalpy) flow rates are calculated by (61) and (63). Results are depicted and compared in Fig. 8. It can be observed that the heat exchangers area estimates obtained from both models are clearly different. Certainly, the total area for heat exchangers “1”, “2” and “3” computed by the original and modified models is 962.67 and 513.88 m2, respectively, differing more than 87%. Fig. 9 plots the ratio between DT computed by a non linear correlation (i.e. variable cp) using the proposed discretized model and by a linear correlation (i.e. constant cp) using the SYNHEAT model (which was discretized to perform this comparison only). The ratio curves correspond to heat exchangers 1, 2 and 3. It can be observed that the computed area values depend on the correlation type used. Fig. 12 e Representation of OLM in the ethanol processor system and orphan streams after solving the first proposed superstructure. 9110 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 11 e Number of heat exchangers by stage depending on the adopted initial superstructure. Ethanol processor system. Stage Streams involved (Ethanol-based processors) SYNHEAT superstructure 1 2 3 4 5 6 7 8 9 10 11 12 H1 H1 H1 H1 H1 H1 H2 H2 H2 H2 H2 H2 H3 H3 H3 H3 H3 H3 H4 H4 H4 H4 H4 H4 Number of heat exchange options by stage OLM superstructure SYNHEAT superstructure OLM superstructure H5 C4 H1 H5 C4 H1 H2 H5 C4 H1 H2 H5 C4 C6 H2 H5 C4 C6 H2 H3 H5 C4 C6 H2 H3 H5 C4 C5 C6 H2 H3 C4 C5 C6 H3 C4 C5 C6 H3 C2 C3 H3 C1 C2 H4 C1 Total 30 30 30 30 30 30 e e e e e e 180 1 2 3 6 4 6 9 6 3 2 2 1 45 H5 C1 C2 C3 C4 C5 C6 H5 C1 C2 C3 C4 C5 C6 H5 C1 C2 C3 C4 C5 C6 H5 C1 C2 C3 C4 C5 C6 H5 C1 C2 C3 C4 C5 C6 H5 C1 C2 C3 C4 C5 C6 e e e e e e 4. HEN synthesis for glycerin-based processors The heat exchangers network of a PEM fuel cell system fed with a water/glycerin mixture is synthesized. Fig. 1 shows a flow sheet for this process. The temperature of the water/ glycerin mixture increases from 39 to 705  C, transferring both latent and sensible heats in this range. As explained, the original SYNHEAT model does not deal rigorously with streams exchanging latent heat. Then, the use of streams discretization is a useful tool in this case. On the other hand, the resulting number of heat exchangers obtained by adopting the initial superstructure proposed by the original model is relatively large; whereas if the initial superstructure derived from the OLM approach is used, such amount can be reduced down to less than a half in some cases. That is relevant due to when the model is discretized the number of “interior” variables increases according to the number of heat exchangers present in the superstructure. It is aimed at minimizing the total annual cost (TAC) while maintaining the global net energy efficiency. A fixed cost per unit of 1.00 $, an area cost coefficient of 379.50 $ m 2, and a scale factor b of 0.65 are adopted. Table 1 lists the specifications of the process streams to be integrated. It can be observed that the heat exchangers cost is negligible compared to utilities cost. The investment cost includes fixed costs and costs related to the heat exchangers size. Fig. 10 depicts the operation line and the resulting stages for this process. Applying the methodology described in section 3.2.2, no orphan pair is obtained. Then, this is the initial superstructure to be optimized to find the solution for the HEN synthesis problem (differing from the ethanol-based study case addressed in the next section). This initial OLMbased superstructure resulted in 81 heat exchange options distributed in 14 stages, while the initial superstructure from the original SYNHEAT model resulted in 216 units distributed in 6 stages (Table 8). Fig. 11 represents the final solution for the HEN synthesis problem obtained with the OLM approach, where the legends “Area 1” and “Area 2” for each heat exchanger correspond to area values calculated using and not using stream discretization, respectively. It is observed that the computed Table 12 e Orphan pair added in limiting regions (after second solution). Stage Streams involved (Ethanol-based processors) OLM superstructure after add the first orphan pair 1 2 3 4 5 6 7 8 9 10 11 12 H5 C4 H1 H5 C4 H1 H2 H5 C4 H1 H2 H5 C4 C6 H2 H5 C4 C6 H2 H3 H5 C4 C6 H2 H3 H5 C4 C5 C6 þ (H2 C3) H2 H3 C4 C5 C6 þ (H2 C3) H3 C4 C5 C6 þ (H2 C3) H3 C2 C3 þ (H2 C3) H3 C1 C2 þ (H2 C3) H4 C1 Second orphan pair H5 C3 H5 C3 Total Number of heat exchange options by stage OLM superstructure after add the first orphan pair OLM superstructure after add the second orphan pair 1 2 3 6 4 6 10 7 4 3 3 1 50 1 2 3 6 4 6 11 8 4 3 3 1 52 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 9111 Fig. 13 e Representation of OLM in the ethanol processor system and orphan streams after solving the second proposed superstructure. differences in those area values are negligible. Twenty elements per stream were adopted for discretization. The total area value computed by the OLM approach resulted in 116.90 dm2 with 12 heat exchangers allocated in 7 stages. When using the original SYNHEAT superstructure, the final solution resulted in a total area value of 229.30 dm2 with 12 heat exchangers allocated in 4 stages (see Table 9). The area cost for heat exchangers using the OLM superstructure is 28% less than that obtained using the original model at the same required auxiliary energy (Table 10). Finally, it is required cold utility for cooling only one hot stream for the considered cost scenario. As the system efficiency depends on the required utilities, this result maintains the maximum efficiency of the whole system since the Fig. 14 e Heat exchanger network of the ethanol processor system obtained using OLM plus stream discretization. Comparison between areas obtained with and without stream discretization. 9112 i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Table 13 e Total heat exchanger network area obtained according to the adopted superstructure. Ethanol processor system. Type of initial superstructure adopted SYNHEAT OLM approach OLM approach Discretization of streams Number of heat exchangers in final structure obtained Total area of heat exchangers in final structure (dm2) Number of stages in final solution No No Yes 11 11 11 119.8 102.6 102.3 2 9 9 resulting cooling utility demand is equal to the value computed by HYSYS simulator for operating the system at its maximum efficiency [17]. 5. HEN synthesis for ethanol-based processors The heat exchangers network of a PEM fuel cell system fed with a water/ethanol mixture is synthesized. The aim is the same as the one pursued in the previous section for glycerinbased systems and the same fixed cost per unit, area cost coefficient and scale factor values are adopted. Table 2 lists the specifications of the process streams to be integrated. Fig. 12 depicts the operation line and the resulting stages for this process. The number of heat exchangers computed by the original SYNHEAT superstructure and the OLM-based superstructure are compared in Table 11. Note that in Fig. 12 the hot stream H2 and the cold stream C3 have not been integrated, “appearing” in the solution demanding utilities (i.e. the rectangular region formed by these streams is not adjacent to the operation line curve). Consequently, this orphan pair is not listed in Table 11. However, this orphan pair is able to exchange heat as the temperature levels of the streams involved are appropriate. Following the methodology described, this orphan pair is then added to the neighboring rectangular regions (i.e. the orphan pair is included in stages 7, 8, 9, 10 and 11) (Fig. 12). The problem is solved again obtaining another orphan pair formed by streams H5 and C3 (Fig. 13). As before, it is added to the neighboring rectangular regions (Fig. 13 and Table 12), and the problem is solved once again. As no orphan pair is obtained, this is the initial superstructure to be optimized to find the solution for the HEN synthesis problem. This initial OLM-based superstructure resulted in 52 heat exchange options distributed in 12 stages (Table 12), while the initial superstructure from the original SYNHEAT model resulted in 180 options distributed in 6 stages (Table 11). Fig. 14 represents the final solution for the HEN synthesis problem obtained with the OLM approach, where the legends “Area 1” and “Area 2” have the same meaning as in Fig. 11. It is observed that the differences in those area values calculated with and without discretization are also negligible. The total area of heat exchangers obtained with the OLM approach is 102.6 dm2 with 11 heat exchangers allocated in 9 stages. When using the original SYNHEAT superstructure, the final solution resulted in a total area value of 119.8 dm2 with 11 heat exchangers allocated in 2 stages. In contrast to the glycerinbased system, total areas obtained are similar with both methods (Table 13). Similarly to the glycerin-base processor, only cold utility for cooling one hot stream is needed for the considered cost scenario. As the system efficiency depends on the required utilities, this result maintains the maximum efficiency of the whole system since the resulting cooling utility demand is equal to the value computed by HYSYS simulator for operating the system at its maximum efficiency [17,18]. 6. Conclusions Optimal heat exchanger networks for glycerin- and ethanolbased processors coupled to PEM fuel cells were synthesized based on modifications made to the original SYNHEAT model. One main modification consisted on developing a novel iterative methodology for energy integration based on the Operation Line Method. Note that the proposed approach is of general application and not just for specific cases addressed in this work. The discretization of streams temperature domain and the consideration of variable enthalpy instead of temperature allow for a more rigorous analysis of the heat exchangers synthesis problems when streams with isothermal phase changes are involved. Applying domains discretization, the computed differences in the estimated total HEN area become more significant when non linear approximations are used instead of linear ones. To avoid infeasibilities caused by numerical values that certain model variables can take during the optimization run, a modification to the model parameter DTmax is proposed. For the glycerin processor system, the OLM-based initial superstructure computes 81 heat exchangers distributed in 14 stages, while the original SYNHEAT superstructure computes 216 units distributed in 6 stages. For the ethanolbased system, the first one computes 52 heat exchangers in 12 stages, while the second one 180 heat exchangers in 6 stages. Then, the number of variables involved in some discussed problems has been reduced to less than a half. In addition, in the glycerin processor system, the total area of heat exchangers obtained with the OLM was 116.90 dm2 with 12 heat exchangers allocated in 7 stages, while it was 229.30 dm2 when using the original superstructure with 12 heat exchangers allocated in 4 stages. Note that the total area obtained with the OLM is smaller than that obtained with the original superstructure proposed in SYNHEAT model. For both glycerin and ethanol-based systems, differences derived from streams discretization are negligible. i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 6 ( 2 0 1 1 ) 9 0 9 8 e9 1 1 4 Acknowledgments The authors acknowledge the financial support from CONICET (Consejo Nacional de Investigaciones Cientı́ficas y Técnicas) and ANCYPT (Agencia Nacional de Promoción Cientı́fica y Técnica) from Argentina. TINj TOUTi TOUTj Tci;j;k;e Thi;j;k;e DTi;j;k Nomenclature DTi;j;k;e area CA CUtilities C CCU CHU CS CF cp Elk Hi,k Hj,k HINi HINj HOUTi HOUTj HEN Hci;j;k;e DTInitial DTmax DTMIN U U yi,j,k ycu,i heat transfer area area cost coefficient average value between CCU and CHU average value of the area cost cost of cold utility cost of hot utility cold stream fixed charge for exchangers specific heat capacity amount of elements in stage k enthalpy of hot stream i at the hot end of stage k enthalpy of cold stream j at the hot end of stage k inlet enthalpy for hot stream i inlet enthalpy for cold stream j outlet enthalpy for hot stream i outlet enthalpy for cold stream j heat exchanger network interior enthalpy value of the eth element of the cold stream “j” exchanging heat with the hot stream “i” inside the stage k interior enthalpy value of the eth element of the hot Hhi;j;k;e stream “i” exchanging heat with the cold stream “j” inside the stage k HS hot stream LMTDi,j,k log-mean temperature difference _ m mass flow rate MW molecular weight MINLP mixed integer non linear programming N total number of process streams NOK total number of stages Pci;j;k energy ratio exchanged between the cold stream “j” and the hot stream “i” depending the number of branches within the stage k energy ratio exchanged between the hot stream “i” Phi;j;k and the cold stream “j” depending the number of branches within the stage k q heat exchanged by process streams heat exchanged between hot process stream i and qijk cold process stream j in stage k heat exchanged between cold utility and hot stream i qcu,i heat exchanged between hot utility and cold stream j qhu,i qu heat exchanged by utilities upper bound for heat exchange Qmax ST set of stages in superstructure TAC total annual cost temperature of hot stream i at the hot end of stage k Ti,k temperature of cold stream j at the hot end of stage k Tj,k inlet temperature of hot stream i TINi yhu,j 9113 inlet temperature of cold stream j outlet temperature of hot stream i outlet temperature of cold stream i temperature of a cold stream “j” in heat exchange with hot stream “i” in the eth element inside the stage “k” temperature of a hot stream “i” in heat exchange with cold stream “j” in the eth element inside the stage “k” temperature approach difference for match (i,j) at temperature location k discrete temperature differences between hot stream “i” and cold stream “j” in the interior of stage “k” at the element “e” parameter of the operation line method upper bound for temperature difference minimum approach temperature difference overall heat transfer coefficient average value of the overall heat transfer coefficient binary variables for match (i,j) in stage k binary variables for match between cold utility and hot stream i binary variables for match between hot utility and cold stream j Greek letter b exponent for area in cost equation Subscripts CU cold utility e numbering of interior elements in stage k horc hot or cold tag HU hot utility i hot process stream j cold process stream k index for stage (1,.,NOK) and temperature location (1,.,NOKþ1) references [1] Casas Y, Arteaga LE, Morales M, Rosa E, Peralta LM, Dewulf J. 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