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Computers and Chemical Engineering 32 (2008) 1861–1876

Synthesis of cost-optimal heat exchanger networks using differential evolution夽 Krishna M. Yerramsetty 1 , C.V.S. Murty ∗ Chemical Engineering Division, Indian Institute of Chemical Technology, Hyderabad 500007, India Received 9 October 2006; received in revised form 4 September 2007; accepted 16 October 2007 Available online 23 October 2007

Abstract Heat exchanger network synthesis (HENS) has been one of the most-studied problems in process synthesis. Nevertheless, the complexity of the HENS problem provides enough scope for the development of novel algorithms involving the application of specialized optimization techniques. Evolutionary algorithms (EA) have emerged as viable alternatives to traditional methods for optimizing functions of both continuous and discrete variables. Differential evolution (DE) is one such evolutionary algorithm that promises simple, fast and robust optimization. The present study illustrates the application of this novel technique for the synthesis of heat exchanger networks. The HENS model proposed here considers stream splitting, does away with the simplifying assumption of isothermal mixing of the split streams and has the capability to handle compulsory and forbidden matching of streams. The DE-based model (DEM) does not rely on the decomposition of the problem into subproblems but employs a simultaneous method of approach to optimize the structure of the network of heat exchangers, the heat loads of these exchangers, the split stream heat flows and the minimum approach temperature. The proposed model has been applied to some case studies available in the literature and the results of these studies are very encouraging. The present work represents thus a step forward in the search for robust and efficient global optimization algorithms for the solution of the HENS problem. © 2007 Elsevier Ltd. All rights reserved. Keywords: Differential evolution; Heat exchanger network synthesis; Global optimization; Stream splitting; Energy integration

1. Introduction Continued focus on research in areas such as heat exchange network synthesis (HENS) is essential for sustaining energy conservation efforts, which are indispensable in the present-day context of fast depleting fossil fuel resources and skyrocketing energy prices. HENS has been a hot topic for research in the 1980s and 90s and considerable progress has been achieved over the years in the development of robust methods of design and efficient operation of heat exchanger networks. There have been some excellent reviews of the work carried out in this area (Furman & Sahinidis, 2002; Gundersen & Naess, 1988).

夽

IICT Communication Number 70307. Corresponding author. Tel.: +91 40 27193852; fax: +91 40 27193626. E-mail addresses: [email protected], [email protected] (C.V.S. Murty). 1 School of Chemical Engineering, Oklahoma State University, Stillwater, OK 74078, United States. ∗

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2007.10.005

In general terms, the objective of HENS is to find out the structure of a heat exchanger network, which facilitates the task of the cooling of a given set of hot streams and the heating of a given set of cold streams to the desired levels with a minimum of investment and operating costs. Basically, there are two types of approach for solving the HENS problem: (i) sequential methods and (ii) simultaneous methods. The sequential methods attempt to reduce the computational complexity of the problem by decomposing the main problem into subproblems, which are then solved sequentially. The simultaneous methods solve the problem without any decomposition. The sequential methods seldom lead to globally optimal solutions. Optimization methods form the backbone of the HENS models. For a specified number of hot and cold streams, there are a large number of possibilities of network structure. HENS attempts to find the optimum among all the network configurations from the standpoint of minimum utility consumption, minimum number of units and minimum cost, etc. Although HENS has been one of the most-studied problems in process synthesis, even small HENS problems have not been

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Nomenclature a, b and c three individuals selected for mutation area area of an exchanger (m2 ) B0 , . . ., B8 constants in the objective function (Eq. (7)) CC capital cost of a network of exchangers Colstr cold stream exchanging heat in an exchanger CP heat capacity flowrate of a hot or cold stream (W K−1 ) CR crossover factor dt Chen’s approximation for LMTD (K) fc heat capacity flowrate vector of a split cold stream (W K−1 ) fc heat capacity flowrate of a split cold stream (W K−1 ) fh heat capacity flowrate vector of a split hot stream (W K−1 ) fh heat capacity flowrate of a split hot stream (W K−1 ) f( ) objective function F scaling factor Fc heat capacity flowrate of a cold stream (W K−1 ) Fh heat capacity flowrate of a hot stream (W K−1 ) Hotstr hot stream exchanging heat in an exchanger LEN maximum number of exchangers (except utility exchangers) that can be placed on all the cold streams LMTD log mean temperature difference (K) m mutation vector Max Q the maximum limit for the heat load of an exchanger (W) n noisy vector n number of decision variables; noisy element NC number of cold streams NH number of hot streams NP total number of heat exchanger networks in a population OC operating cost of a network of exchangers Q heat load vector (W) Q heat load of an exchanger (W) rand(a, b) random number in the specified range a to b S string vector S string element SP maximum number of splits allowed per stream ST total number of stages in the network t trial vector t trial element ti inlet temperature of a cold stream in any heat exchanger (K) tin specified inlet temperature of a cold stream (K) to outlet temperature of a cold stream in any heat exchanger (K) tout specified outlet temperature of a cold stream (K) Ti inlet temperature of a hot stream in any heat exchanger (K) Tin specified inlet temperature of a hot stream (K)

To

outlet temperature of a hot stream in any heat exchanger (K) Tout specified outlet temperature of a hot stream (K) TC total annualized cost of a heat exchanger network t temperature difference between the hot and cold streams at one end of an exchanger (K) tmin minimum approach temperature preset for the network in the program (K) Tmin optimized value of the minimum approach temperature for the network (K) U overall heat transfer coefficient (W m−2 K−1 ) Ucost unit cost of a utility Viol( ) number of violations in a heat exchanger network X heat load vector or split stream heat capacity flowrate vector X heat load or the split stream heat capacity flowrate y target vector y target element ymax upper limit of the target vector y ymin lower limit of the target vector y Superscripts i hot stream number i, CU cooler placed on hot stream i i, s hot stream i in stage s j cold stream number j, HU1 heater placed on the outlet end of cold stream j j, HU2 heater placed on the inlet end of cold stream j j, s cold stream j in stage s k exchanger number in the network m count of the decision variable Subscripts CU cold utility HU hot utility i hot stream number j cold stream number p number denoting an individual heat exchanger network in a population

solved to global optimality to date. In fact, even finding feasible solutions using simultaneous synthesis methods has been troublesome. The complexity of the HENS problem provides, therefore, enough scope for the development of specialized optimization algorithms (Furman & Sahinidis, 2002). What makes the HENS problem particularly hard to solve is its combinatorial nature. Until recently, most of the published work employed traditional optimization methods. However, as newer and more efficient optimization algorithms became available, these found application for synthesizing large heat exchanger networks. Examples for some of these are: Genetic Algorithm (GA) (Androulakis & Venkatasubramanian, 1991; Lewin, 1998; Lewin, Wang, & Shalev, 1998; Ravagnani, Silva, Arroyo, & Constantino, 2005; Wang, Yao, Yuan, Yu, & Shi, 1997; Wang, Qian, Huang, & Yao, 1999), Simulated Anneal-

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

ing techniques (Athier, Floquet, Pibouleau, & Domenech, 1996, 1997a, 1997b, 1998; Dolan, Cummings, & LeVan, 1987; Nielsen, Hansen, & Joergensen, 1996), combination of Genetic Algorithm and simulated annealing (Yu, Fang, Yao, & Yuan, 2000), Tabu search (Lin & Miller, 2004), etc. Of all these, Genetic Algorithm has been the most popular. GA belongs to a new generation type of methods called Evolutionary Algorithms (EA) (Goldberg, 1989). One can use EA for problems that are difficult to solve with traditional optimization techniques, including problems that are not well defined or are difficult to model mathematically. In recent times, Genetic Algorithm has become a viable alternative to the traditional optimization methods, as demonstrated in several studies. The method of differential evolution (DE), which also belongs to the EA class, is a relative latecomer. Price and Storn have first proposed it in 1997. Its popularity has been catching up, of late. It is fast in numerical optimization and is more likely to find the true optimum (Price & Storn, 1997). Differential evolution has been used for optimization purposes in a few modeling studies. In a recent publication (Kiranmai, Jyothirmai, & Murty, 2005), the differential evolution method has been used for the modeling of a fixed-film bio-reactor. It has been shown in this work that DE scores better than GA in terms of computational speed for the same level of accuracy. The main objective of the present study is to formulate a DE-based model (DEM) for solving the HENS problem and then applying it to some of the HENS problems available in the literature for evaluating the performance of DE relative to the other methods. Typically, DE is suitable for optimizing problems with continuous variables. Broad HENS algorithms, such as the one dealt with here, have continuous variables in the form of heat exchanger duties and heat capacity flows of split streams and discrete (integer) variables arising out of the matching of the hot and cold streams. DE has been suitably modified in order to handle this kind of situations involving both continuous and discrete variables. Minimum approach temperature Tmin is a crucial parameter, which affects the cost of the heat exchanger network. Tmin has a direct relationship with energy costs and an inverse relationship with fixed costs (exchanger surface area). The attendant energy-area trade-off implies that, for any problem there is an optimal value for the minimum approach temperature in a cost-optimal network. In the present study, Tmin is optimized implicitly during the course of the synthesis process. Yee, Grossmann, and Kravanja (1990) and Yee and Grossmann (1990) made the simplifying assumption of isothermal mixing to avoid non-linear constraints but they also stated that a limitation of their model was that the isothermal mixing assumption might lead to an overestimation of the cost of the exchanger area for the network structure. The present model handles stream splitting and also does away with the assumption of isothermal mixing for obvious reasons. Both the hot and cold streams can be split any number of times, although optimal designs require only few splits. The model does not consider more than one exchanger per split stream and also does not take into account stream bypassing. Restrictions are allowed on the network structure in the form of forbidden/compulsory matches

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of streams, which might become necessary for practical and safety considerations. The solution algorithm for the synthesis problem does not rely on the decomposition of the problem into subproblems but, instead, tries to optimize simultaneously for the best structure of the heat exchangers, the minimum approach temperature, the heat loads of these exchangers and the heat capacity flows of split streams. 2. Problem statement In a process plant, there are hot streams and cold streams and these are cooled and heated respectively to their target temperatures using utilities such as cooling water, steam, etc. The load on the utilities, and thereby the cost, can be reduced to a great extent using a proper system of heat exchangers to exchange heat between the hot and cold streams. The basic objective of the HENS problem is to synthesize a network of heat exchangers, which facilitate the desired heat exchange, while keeping the investment (exchanger area) and operating costs (utilities consumption) to a minimum. The synthesis of cost-optimal heat exchanger networks is a combinatorial problem. Even for a two-hot stream and two-cold stream problem, the number of possible network structures is quite large and the complexity of the problem increases rapidly, as the number of streams increases. Differential evolution (DE) is employed in the present work to minimize the objective function, i.e. the total annualized cost (TC) of the network of heat exchangers, which comprises the capital cost (CC) and the operating cost (OC) of the network. 2.1. Capital cost Capital cost is the cost of setting up of the heat exchangers in the network. The usual form of the capital cost is CC = B0 + B1 × (area)B2

(1)

where B0 , B1 and B2 are positive real values. The area of an exchanger k, areak , is a function of the heat load of the exchanger, Qk , the overall heat transfer coefficient, U and the log mean temperature difference between the hot stream and the cold stream in the exchanger, LMTDk : areak =

Qk U × LMTDk

(2)

Temperatures of the hot and cold streams at either end of the exchanger should be known to calculate LMTD from: LMTDk =

t1k − t2k ln(t1k /t2k )

(3)

This function cannot be used directly, because of the logarithmic term in the denominator which leads to mathematical difficulties when t1 = t2 . For this reason, an approximation dtk , for LMTD, called the Chen’s approximation is used:

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dt = k

t1k

× t2k

×

0.33

t1k + t2k 2

where Qi,CU and Qj,HU are the heat loads for the cold and hot utilities, respectively, and UcostCU and UcostHU are the unit costs of the cold and hot utilities, respectively. The total annualized cost of the network TC is a sum of the capital costs and the operating costs of the network:

(4)

B2 B5 NH Qk Qi,CU B0 + B 1 × + B + B × + TC = 3 4 U k × dt k U i,CU × dt i,CU i=1 k=1 B8 B8

NC Qj,HU1 Qj,HU2 + 2 × B 6 + B7 × + U j,HU1 × dt j,HU1 U j,HU2 × dt j,HU2

LEN

(7)

j=1

NH

Qi,CU × UcostCU +

i=1

i=1

×

Qi,CU U i,CU × dt i,CU

×

B5 +

Qj,HU1 U j,HU1 × dt j,HU1

NC j=1

B8

2 × B6 + B7

Qj,HU2 + U j,HU2 × dt j,HU2

(Qj,HU1 + Qj,HU2 ) × UcostHU

j=1

This approximation for LMTD slightly underestimates the actual value and hence overestimates the area and the cost of the exchanger, thus making the real cost of the network slightly lower than the cost predicted using the Chen’s approximation. The capital cost can be then expressed for the heat exchangers and the hot and cold utilities in the network as B2 LEN NH Qk CC = B0 + B 1 × + B3 + B 4 U k × dt k k=1

NC

2.3. Constraints of the problem Synthesizing a heat exchanger network is a complex task, since many constraints need to be satisfied in the process. The HEN synthesized by DEM must comply with the thermodynamic constraints on heat loads for the network to be feasible. These constraints are placed on the amount of heat that a particular stream can exchange. Each hot and cold stream must satisfy the following: (a) 0 ≤

B8

LEN

Qk ≤ Fhi × (T ini − T outi )

k=1

subject to Hotstrk = i (5)

LEN

(8)

The parameters Q in the second and third terms on the right hand side of the above equation represent utility heat loads and LEN is the maximum number of exchangers that can be placed between the hot and cold streams. One feature of Eq. (1) is worth highlighting. The coefficient B2 in the equation is usually a number between 0.5 and 1. If B2 is a fraction, and B0 is relatively a small value, then one exchanger with a large area has less capital cost than two smaller exchangers, collectively having the same area. Thus, a globally optimal solution for this problem would usually have relatively less number of units. On the other hand, if B2 were 1, the opposite would be true and the algorithm would not penalize smaller exchangers. Consequently, the globally optimal solution in such a situation would usually have a higher number of exchangers.

where Tout and Tin are the specified outlet and inlet temperatures of the hot stream, tout and tin are the specified outlet and inlet temperatures of the cold stream, Hotstr and Colstr are the hot and cold streams associated with a heat exchanger. There would be NH + NC such constraints for each network. Apart from these, every exchanger must also satisfy the following constraints on the temperatures of its hot and cold streams.

2.2. Operating cost

(c) T ik − tok ≥ tmin

The other component of the total cost is the operating cost. It is a simple function of the utility loads of the network: OC =

NH

Qi,CU × UcostCU +

i=1

× UcostHU

NC

(Qj,HU1 + Qj,HU2 )

j=1

(6)

(b) 0 ≤

Qk + Qj,HU1

≤ Fcj × (toutj − tinj )

k=1

subject to Colstrk = j

(d) T ok − tik ≥ tmin

(9)

(10)

where tmin is the minimum approach temperature preset for the network in the program. These two constraints need to be checked for all the LEN number of exchangers. Thus, each network needs to be checked for NH + NC + 2 × LEN number of violations.

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

3. Method of differential evolution The method of differential evolution consists mainly of four steps: initialization, mutation, recombination and selection. In the first step, a random population of potential solutions is created within the multi-dimensional search space. These then undergo the subsequent steps, which, together may be termed as the processes of evolution, to yield the desired optimum. One cycle of the last three operations is called a generation. To start with, we define our objective function f(y) to be optimized, where y = (y1 , . . ., yn ) is a vector of n decision variables. The aim is to find a vector y in the given search space, for which the value of the objective function is an optimum. The search space is first defined by providing the lower and upper bounds for each of the n decision variables of y, i.e., ymin ≤ y ≤ ymax. In the initialization step, NP vectors, each of n dimensions, are randomly initialized. The parameters are encoded as floating point numbers: ypm

= ymin + rand(0, 1) × (ymax − ymin ) m

m

m

(11)

where p = 1, . . ., NP, m = 1, . . ., n, rand(0,1) denotes a random number generated between 0 and 1. Mutation is basically a search mechanism, which, together with recombination and selection, directs the search towards potential areas of optimal solution. In this step, three distinct target vectors ya , yb and yc are randomly chosen from the NP parent population on the basis of three random numbers a, b and c. One of the vectors yc is the base of the mutated vector. To this is added the weighted difference of the remaining two vectors, i.e. (ya − yb ) to generate a noisy random vector, np : np = yc + F × (ya − yb )

(12)

where p = 1, . . ., NP and a, b, c are chosen between 1 and NP. F is termed the scaling factor and it is user-supplied. It is usually in the range 0–1.2. This mutation process is repeated to create a mate for each member of the parent population. Mutation ensures an efficient search of the solution space in each dimension. In the recombination (crossover) operation, each target vector of the parent population is allowed to undergo recombination by mating with a mutated vector. Thus, vector yp is recombined with the noisy random vector, np to generate a trial vector, tp . Each element of the trial vector (tpm , where p = 1, . . ., NP and m = 1, . . ., n), is determined by a binomial experiment, whose success or failure is determined by the user-supplied crossover factor, CR. The parameter CR is used to control the rate at which the crossover takes place: tpm = nm p tpm

=

ypm

if rand(0, 1) < CR or m = rand(1, n) otherwise

(13)

with p = 1, . . . NP and m = 1, . . ., n. Therefore, trial vector, tp , is the child of two parent vectors: noisy random vector, np and the target vector, yp . DE performs a non-uniform crossover, that determines which trial vector parameters are inherited from which parent.

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It is sometimes possible that some particular combinations of the three target vectors from the parent population and the scaling factor F would result in noisy vector values, which are outside the bounds, set for the decision variables. It is necessary, therefore, to bring such values within the bounds. For this reason, the value of each element of the trial vector is checked at the end of the recombination step. If it violates the bounds, it is heuristically brought back to lie within the bounded region: u × (ymaxm − yminm ) tpm = yminm + 2.0 × v if tpm > ymaxm with u = (tpm − ymaxm ) and v = (tpm − yminm ) tpm = yminm + 2.0 × if tpm < yminm

(14) u

× (ymaxm − yminm ) v with u = (yminm − tpm ) and

v = (ymaxm − tpm )

(15)

It is in the last stage of the ‘selection’ that the constrained optimization version of the DE differs from the DE version for the unconstrained optimization case. Here, the trial vector is pitted against the target vector and the fitness is tested and constraint violations, if any, are noted. In the normal case, where both tp and yp do not violate any constraints or when both tp and yp violate same number of constraints, fitter of the two vectors survives and proceeds to the next generation. If both tp and yp violate the constraints, the one with fewer constraint violations survives. If either tp or yp violates the constraints, the vector with no constraint violations survives. These are called “tournament rules’ and this procedure is similar to the ‘tournament selection’ employed for handling constraints using the Genetic Algorithm (Deb, 2000): yp = tp

if Viol(yp ) > Viol(tp ) or if Viol(yp ) = Viol(tp ) and f (yp ) > f (tp )

(16)

where p = 1, . . ., NP. In the above equation, Viol(. . .) stands for the number of constraint violations. After NP competitions of this kind in each generation, one will have a new population, which is fitter than the population one has started with. This evolution procedure consisting of the above three steps, namely, mutation, crossover and selection, is repeated over several generations until the termination condition is reached, i.e. when the objective function attains a prescribed optimum or a specified number of generations are completed, whichever happens earlier. 4. Development of the HENS algorithm The algorithm is divided into four parts, one for each operation of DE. In the initialization part, the general procedure for calculating the intermittent temperatures of the streams from the heat loads of the exchangers is discussed. This procedure is followed, wherever temperatures at the ends of the exchangers are needed. Next, the three DE operations, mutation, recombination

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and selection are dealt with for the string variables and other variables. Here, a specific notation is followed for the optimized variables: ‘y’ precedes a target vector, ‘m’, a mutated vector and ‘t’, a trial vector. Fig. 1. Stage wise representation of a string vector.

4.1. Initialization During the initialization step, a population of NP individual heat exchanger networks is generated. This population is called the target population. Each target individual p has the following attributes: (1) The string vector ySp . (2) The heat load vector yQp . (3) The heat capacity flowrate vectors of split streams, yfhp and yfcp. These are the variables that need to be optimized for the resulting cost-optimal network. 4.1.1. String The concept of the string Sp , which denotes the arrangement of heat exchangers in a particular network, is similar to the one used by Lewin et al. (1998). A sequence of integers is used to represent a sequence of heat exchangers placed on the cold streams. The length of this string, denoted by LEN, is equal to the maximum number of exchangers that can be placed on the cold streams, assuming that each cold stream split can have a maximum of one exchanger in a particular stage. For example, if we are considering a network of three hot streams and two cold streams, and the number of stream splits allowed is two, then the maximum number of heat exchangers possible for each cold stream in a stage is two (one on each of the splits). For the two cold streams, a maximum of four exchangers is possible in each stage. And, if we were to have three stages in our network, the total length of the string would be 4 × 3 = 12. Therefore, the string length and hence the maximum number of heat exchangers is a product of the number of cold streams NC, the number of splits SP allowed per cold stream and the number of stages ST allowed per network: LEN = NC × SP × ST

(17)

Note that at the end of each stage, the split streams are mixed to form a single stream. If necessary, they are split again in the beginning of the next stage. For any individual p in the target population, a string value ySkp denotes the hot stream Hotstrk exchanging heat with the cold stream split associated with exchanger k in the string: Hotstrk = ySkp

(18)

The cold stream associated with the exchanger k is computed from the value of k:

(k − 1) (k − 1) k Colstr = int − int × NC + 1 SP NC × SP (19)

where ‘int’ is an operator for calculating the integer part of a real value. If ySkp at any k is 0, then the cold stream split corresponding to exchanger k does not exist, i.e. the cold stream is split into one less number of splits. An example of a string is given in Fig. 1, where the broken lines separate the stages and the continuous lines separate the cold streams. Here yS1p is 0. Hence, there is no exchanger on the first split of the first cold stream. This implies that cold stream 1 is un-split in stage1 and it exchanges heat with hot stream 1 (since yS2p is equal to 1). Similarly, the values of the last two positions of the string are 0, which indicate that there are no exchangers on both splits of the cold stream 2 in stage 3. From here onwards, a string vector would be denoted by a series of integers, separated by commas and enclosed within parentheses. Accordingly the above string can be written as ySp = (0, 1, 3, 1, 0, 2, 1, 0, 3, 1, 0, 0) In the initialization step of the algorithm, the target string values ySkp are generated randomly for each individual p in the population. The initial target string vector is denoted by ySp , and the string variables ySkp can have any integral value between 0 and NH with equal probability: ySpk = rand{0, NH}

(20)

It is ensured, however, that a hot stream in a stage does not exchange heat more times than the maximum number of splits it is allowed to have. This is a result of our assumption of allowing no more than one exchanger per stream split. This condition restricts the number of occurrences of any integer (except 0) in a stage to a maximum of SP. If the number of occurrences of a non-zero integer in a stage exceeds SP, then the subsequent occurrences of the integer in that stage are made equal to 0. For example, if a particular initial string for “three hot and two cold” stream problem is ySp = (2, 2, 0, 2, 1, 1, 0, 3, 0, 0, 0, 2) And if SP is fixed as two, then clearly, the network given by the above string is not acceptable to the algorithm. This is because, in the first stage, the hot stream 2 exchanges heat three times and hence needs to be split three times, which is greater than the value of SP. The algorithm corrects this by removing the last occurrence of hot stream 2 in stage 1: ySp = (2, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 2) Now this string represents a feasible HEN structure. User-defined constraints like, forbidden matches and compulsory matches can likewise be handled using the present notation for strings.

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4.1.2. Heat loads The string generates a skeletal structure of the network indicating the placement of exchangers between different streams. Each ySpk across a string represents an exchanger and hence there are heat loads yQkp associated with each such exchanger. Every positive string value represents a real heat exchanger, with non-negative heat load. ySpk = 0 indicates the absence of an exchanger and hence zero heat load at that position in the string. During the initialization process, the values of the heat load vector yQp are generated randomly. However, in order to reduce the occurrences of unusually large heat load values and consequently to minimize the computational time for the solution to converge, a limit for the maximum heat load that a particular heat exchanger can have is fixed. This maximum value Max Q is set equal to the total heat available for exchange in the particular hot stream of the exchanger or the total heat required by the corresponding cold stream of the exchanger, whichever is minimum:

And for cold stream j in stage s, the split heat capacity flowrates yfckp are

yQkp = rand{0, Max Q}

4.1.4. Design of the network After initializing the values of the string variables, heat loads and the split stream heat flows, the total cost of a network needs to be calculated. The total cost of the network for the target population is given by Eq. (7). Generally, the placement of exchangers starts from the outlet end of hot streams and inlet ends of cold streams. In the present method however, placement of heat exchangers starts from the inlet ends of the hot streams and the outlet ends of the cold streams (because we are assuming countercurrent operation). Since traditionally heaters are placed at the outlet end of the cold streams, we put HU1 for each cold stream at the outlet end. This is done at the beginning of the program before the cold streams exchange heat with any of the hot streams. After all the ‘NC × SP × ST’ exchangers have been assigned the hot stream numbers, the heat loads, the split fractions, there is usually some heat that is left to be satisfied on the cold streams. This is given by Eq. (26). In order to provide this heat to the cold streams, so that they reach their inlet temperatures (we may remember that we are assigning heat exchangers and heat loads while moving from the outlet end of the cold streams to their inlet ends), we put another heater HU2 at the inlet end of these ‘unsatisfied’ cold streams. However, it is to be noted that HU2 is only used to satisfy the inlet temperature constraints on the cold streams for suboptimial HENs. For an optimized structure of heat exchangers, the heat load on HU2 for all cold streams would become zero. This is because of the fact that any positive heat load on HU2 would lead to uneconomical use of energy and hence increases the cost of the whole network. Thus, adding HU2 would not make any difference, as the optimized structures would not have them.

where Max Q = min(Fhi × (T ini − T outi ), Fcj × (toutj − tinj ))

(21)

and i = Hotstrk ; j = Colstrk A hot utility exchanger HU1 is placed at the outlet end of each j,HU1 cold stream j. The heat load on this exchanger yQp is also a variable that needs to be optimized (there will be a total of LEN + NC heat loads which have to be optimized) and is a randomly generated value between 0 and the maximum heat available in that particular cold stream: j,HU1

= rand{0, Max Q} where Max Q = Fcj × (toutj − tinj ) yQp

(22)

4.1.3. Heat ﬂows of split streams Also associated with each exchanger k is the information about the splitting of the streams exchanging heat in that exchanger. If the process streams have to be split, then the heat capacity flowrates of the split streams need to be optimized for obtaining a cost-optimal network. The heat capacity flowrate of the hot stream Hotstrk (whether it is a split stream or not) in exchanger k is denoted by yfhkp , and the heat capacity flowrate of the cold stream Colstrk is denoted by yfckp . Heat capacity flowrates yfhkp are generated randomly during the initialization procedure. It is ensured, however, that the heat flows of all the splits of a stream in a stage add up to the total heat capacity flowrate of the particular stream. Thus, for hot stream i in stage s, the split heat capacity flowrates yfhkp are yfhkp = rand{0, Fhi }

and

NC×SP×s k=[NC×SP×(s−1)]+1

NC×SP×(s−1)+j×SP

yfckp

= rand{0, Fcj }

and

yfckp = Fcj

(24)

k=NC×SP×(s−1)+(j−1)×SP+1

It may be noted that the random numbers generated are not directly used; instead they are modified to satisfy the constraints 23 and 24. The modification is done using an intuitive approach: (a) The split heat flows corresponding to the random numbers are sequentially added one after another until the sum becomes greater than the heat capacity flowrate of the particular stream under consideration. At this point, the last split fraction is adjusted by assigning to it a value equal to the difference between the stream heat capacity flowrate and the sum of the split fractions hitherto generated. (b) The random numbers for the later split fractions are made 0.

yfhkp = Fhi subject to ySkp = i

(23)

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This is also evident from Figs. 5–10, which present various cost-optimal networks obtained for the five case studies, that none has a hot utility at the inlet of the cold streams. j,HU2 The end utility heat loads Qi,CU and Qp can be written in p k terms of the exchanger heat loads yQp and the HU1 heat loads as yQj,HU1 p = Fhi × (T ini − T outi ) Qi,CU p LEN

−

yQkp

Fig. 3. Schematic representation of stream temperatures.

subject to ySpk = i

(25)

k=1

< 0, then Qi,CU =0 if Qi,CU p p j,HU2

Qp

= Fcj × (toutj − tinj ) −

LEN

yQkp − yQj,HU1 p

k=1

subject to Colstrk = j j,HU2

if Qp

< 0,

j,HU2

then Qp

(26)

=0

Eqs. (25) and (26) facilitate the computation of terms 4 and 5 in Eq. (7). The calculation of the capital cost needs the knowledge of dtk for each exchanger k, which is the Chen’s approximation for LMTD. To calculate dtk , the values of the temperatures of the hot and cold streams at either end of the exchanger k are needed. A procedure for calculating these temperatures from the heat loads of the exchangers is as follows. All the exchangers are operated in a counter-current manner and the algorithm traverses from the inlet side of the hot streams (or the outlet side of the cold streams) to the outlet side of the hot streams (or the inlet side of the cold streams) for calculating the temperatures. This implies that the first exchanger encountered on a cold stream is always a heater. The outlet temperature, j,HU1 top for this heater is the final outlet temperature, toutj that needs to be realized for the cold stream. Therefore, for any cold stream j of pth individual in the target population (see Fig. 2): toj,HU1 p

= toutj

(27) j,HU1

And the inlet temperature, tip using the heat balance:

for this heater is calculated

j,HU1

tij,HU1 p

=

toj,HU1 p

yQp − Fcj

(28)

This becomes the outlet temperature of the cold stream j in the first stage: j,HU1 toj,1 p = tip

(29)

After the heaters are placed, the cold streams exchange heat with the hot streams. The sequence in which these exchangers

are placed is dictated by the string variable values ySpk . In some cases, the streams need to be split in order to realize the structure given by the string. The assumption that a stream branch can have at most only one exchanger implies that all the exchangers placed on a hot stream i in a stage s have the same hot inlet temperature. Similarly the exchangers placed on a cold stream j in a stage s have the same cold inlet temperature. If T ii,s p denotes the inlet temperature of hot stream i in stage s, then the hot stream inlet temperatures of each exchanger k placed on hot stream i in that stage is equal to T ii,s p (See Fig. 3): T ikp = T ii,s p

(30)

Similarly, if tij,s p denotes the inlet temperature of cold stream j in stage s, then the cold stream inlet temperature of each exchanger k placed on cold stream j in that stage is equal to tij,s p : tikp = tij,s p

(31)

tij,s p is calculated from the knowledge of the heat loads of the exchangers placed on cold stream j in stage s and the outlet j,s temperature of the mixed stream j in that stage, denoted by top : NC×SP×(s−1)+j×SP

yQkp

k=NC×SP×(s−1)+(j−1)×SP+1

Fcj

j,s tij,s p = top −

The outlet temperature of the hot stream from exchanger k is denoted by T okp , and the outlet temperature of the cold stream is denoted by tokp . These are calculated using the following heat balances: T okp = Tikp − tokp = tikp +

yQkp

(33)

yfhkp

yQkp

(34)

yfckp

At the end of a stage, the split branches of a stream if any, are mixed, and if necessary split again in the next stage. Therefore, the outlet temperature of a mixed hot stream i in a stage s is its inlet temperature in the next stage s + 1: NC×SP×s

T oi,s p =

Fig. 2. Representation of a heater on a cold stream j.

(32)

T okp × yfhkp

k=[NC×SP×(s−1)]+1

= T oi,s T ii,s+1 p p

Fhi

subject to ySpk = i (35) (36)

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

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Similarly, for any cold stream j, the inlet temperature in a stage s is the outlet temperature of the mixed stream in the next stage s + 1:

The string variables thus generated are adjusted heuristically to lie within the limits {0, NH}.

= tij,s toj,s+1 p p

4.2.2. Heat loads and split stream heat ﬂows These are non-integer variables. The heat loads (including those of the heaters HU1) and the split stream heat capacity flowrates of the three individuals a, b and c are combined using the general rule for mutation to get the heat loads mQkp and the split stream heat capacity flowrates mfhkp and mfckp of the mutated individual p:

(37)

Proceeding in this way for all the stages, the inlet and outlet temperatures of the streams in each exchanger including the heater are calculated. If the target temperatures of the streams are not reached, then coolers are placed at the outlet end in case of hot streams and additional heaters denoted by HU2 are placed at the inlet end in case of cold streams. The loads of these exchangers are calculated from the total load on all the exchangers placed on that stream (refer to Eqs. (25) and (26)). Since, all the temperatures are known at both ends of each exchanger, we can calculate the temperature difference t as t1 = T ikp − tokp t2 = T okp − tikp

(38)

The temperature differences are calculated likewise for the utility exchangers. Using dtpk and heat loads, the total cost associated with the specific network is calculated. At the end of the initialization step, we thus have a target population of NP individuals and their total costs. 4.2. Mutation Mutation operation leads to the generation of a mutated population from the individuals of the target population. Three individuals a, b and c from the target population are selected and the various variables associated with these three individuals are combined to give the corresponding variables of a mutated individual p (see Fig. 4). NP such individuals are generated and these individuals make up the mutated population. Formulation of mutated individuals in the case of string and other variables is discussed as follows. 4.2.1. String The values of the string variables are integers. Using a modified mutation operation suitable for integer variables, the values of the string variables mSp of a mutated individual p, are calculated as mSpk = int[ySck + F × (ySak − ySbk ) ] The value of the scaling factor F is fixed at 0.5.

Fig. 4. Mutation operation performed on string variables.

(39)

mXp = yXc + F × (yXa − yXb )

(40)

where X = Q (or) fh (or) fc, as the case may be. If the set of three random numbers chosen for the generation of the noisy vector at any time result in negative values, the sign before the value is simply changed from negative to positive. This is tantamount to selecting a different set of random numbers, which will result in the positive number just created. 4.3. Recombination Recombination is an operation in which the individuals of the mutated population are mated with the target population individuals to give rise to a new population called the trial population. The crossover of the different variables is performed based on the value of the crossover factor (fixed at 0.75 in the present work), CR. This operation is discussed for string and other variables as follows: 4.3.1. String For every string value tSpk , a random number (rand) between 0 and 1 is generated and is compared with the crossover factor CR. If ‘rand’ is less than CR, then the value of the string variable tSpk of the trial individual p, is set equal to the string variable mSpk of the corresponding mutated individual. Else, the value of tSpk is made equal to the string variable ySpk in the target population:

tSpk

=

mSpk

if rand(0, 1) < CR or k = rand(1, LEN)

ySpk

otherwise

(41)

with p = 1, . . ., NP and k = 1, . . ., LEN. The new string so obtained is checked for any inconsistencies with respect to the number of hot stream splits. If the number of occurrences of any hot stream is greater than the maximum number of splits allowed, then the subsequent occurrences are made equal to 0. 4.3.2. Heat loads and split stream heat ﬂows Recombination operation for the heat loads and the heat capacity flowrates of split streams is similar to the operation for string variables. A random number ‘rand’ is generated for each heat load and each split stream heat capacity flowrate of the trial individual p, and a decision on cross-over is made after

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comparison with CR:

mXpk if rand(0, 1) < CR or k = rand(1, LEN) tXpk = (42) yXp otherwise X = Qk

Qj,HU1

where p = 1,. . ., NP and k = 1,. . ., LEN and (or) (or) fhk (or) fck , as the case may be. The values of the heat loads of the trial individuals are adjusted to lie within (0,Max Q), where Max Q, as before, is the total heat available for exchange in the particular hot stream of the exchanger or the total heat required by the corresponding cold stream of the exchanger, whichever is minimum. The values of the split heat capacity flowrates of each stream in a stage are adjusted to add up to the total heat capacity flowrate of the stream. After the string variables tSp , heat loads tQp , and the split heat capacity flowrates tfhp and tfcp are calculated for each individual p in the trial population, the temperatures at the ends of each exchanger are calculated. The procedure for this is similar to the one employed for the target population in the initialization step. From these temperatures, and the heat loads, the total cost of each network is found for the trial population.

In the selection step of the algorithm, the corresponding individuals in the trial population and the target population are compared according to the tournament rules for constrained optimization. The fitter of the two individuals is retained in the target population. The tournament rules employed in the selection process are if Viol(ySp , yQp , yfhp , yfcp )>Viol(tSp , tQp , tfhp , tfcp ),

= Viol(tSp , tQp , tfhp , tfcp ) and

Stream

Tin (K)

Tout (K)

CP (kW K−1 )

H1 H2 C1 C2 Steam Water

443 423 293 353 450 293

333 303 408 413 450 313

30 15 20 40 – –

U = 0.8 kW m−2 K−1 for all matches except those involving steam; U = 1.2 kW m−2 K−1 for matches involving steam; Annual cost of heat exchangers and coolers (US$) = 6250 + 83.26 × S [S = area in m2 ]; annual cost of heaters (US$) = 6250 + 99.91 × S [S = area in m2 ]; cost of steam = 80 US$ kW−1 year−1 ; cost of water = 20 US$ kW−1 year−1 .

the target vector yp violates and Viol(tp ) refers to the number of thermodynamic constraints that the trial vector tp violates. The three steps described in Sections 4.2, 4.3 and 4.4 constitute one generation. The updated target population goes through the mutation, recombination and selection steps yet again to spawn the next generation. This process goes on and on until the termination condition is reached. 5. Application of the DE model

4.4. Selection

then : ⎧ ⎫ ySp = tSp ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ yQ = tQ ⎪ ⎬ p p ⎪ yfhp = tfhp ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ yfcp = tfcp if Viol(ySp , yQp , yfhp , yfcp )

Table 1 Input data for the case study 1

The differential evolution-based model (DEM) developed in the present study for HENS has been applied to some of the HENS problems reported in the published literature to test its performance vis-`a-vis other methodologies. The results of this exercise are presented below for different case studies in terms of the Minimum approach temperature (Tmin ), Heat Exchanger Network structure, cost, load on the utilities, exchanger areas etc., as the case may be. 5.1. Case study 1

(43)

if TC(ySp , yQp , yfhp , yfcp ) > TC(tSp , tQp , tfhp , tfcp ) then : ⎧ ⎫ ySp = tSp ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ yQ = tQ ⎪ ⎬ p p ⎪ yfhp = tfhp ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ yfcp = tfcp where p = 1, . . ., NP; TC(. . .) is the function for the total annualized cost of the network. The constraint violations mentioned above refer to the violations of the Eqs. (8)–(10). These equations are the thermodynamic constraints, which the algorithm explicitly checks for. All other constraints are logical constraints and are taken care of through the logic of the algorithm. In Eq. (16), Viol(yp ) refers to the number of thermodynamic constraints (Eqs. (8)–(10)) that

This is a 4SP1 problem with two hot and two cold streams (Zamora & Grossmann, 1998). The data for the problem is given in Table 1. For the no stream-split case, DEM provided a solution identical to the one obtained by Randomized Algorithm of Pariyani, Gupta, and Ghosh (2006). However, allowing for a maximum of two splits on the cold and hot streams, DEM gives the best solution reported so far. The results of the DEM are compared with other available values in Table 2. The cost of the network for DEM (shown in Fig. 5) is $84,222, which is about $1,100 lower than the value obtained by Pariyani Table 2 Results of the case study 1 HENS algorithm

Splits: hot (cold)

No. of units

QC (kW) Annual cost ($ year−1 )

Zamora and Grossmann (1998) Pariyani et al. (2006) Pariyani et al. (2006) This work (DEM) This work (Fig. 5) (DEM)

0 (0)

5

400

85,968

0 (0) 0 (2) 0 (0) 0 (2)

5 6 6 6

400 400 400 400

85,972 85,307 85,972 84,222

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

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Fig. 5. DEM solution for cost-optimal network for case study 1. The heat loads are given in parentheses. The network costs US$ 84,222. Table 3 Input data for the case study 2

There are two variants for this problem:

Stream

Tin (◦ F)

Tout (◦ F)

CP (Btu/(h ◦ F))

H1 H2 C1 C2 S1 W1

320 480 140 240 540 100

200 280 320 500 540 180

16,666.8 20,000 14,450.1 11,530 – –

Cost of hot utility S1 = $12.76 kBTU−1 yr−1 . Cost of cold utility W1 = $5.24 kBTU−1 yr−1 . Annual cost of heat exchangers ($) = 35 × S0.6 (S = area in ft2 ). U = 150 Btu/ft2 ◦ F for all matches, except ones involving steam. U = 200 Btu/ft2 ◦ F for matches involving steam.

et al. (2006). The solutions obtained by both DE and Pariyani et al. (2006) have the same basic network structure. Nevertheless, DE efficiently fine-tunes the heat loads of the exchangers in the network to give a better result. The lower cost for DEM is attributed to the lower exchanger area requirement (465 m2 ) than that for Pariyani et al. (478 m2 ). The computational time for the network synthesis on a Pentium-4 PC was 75 s. It has to be noted that the Chen’s approximation for LMTD slightly underestimates the real value of LMTD. The DE results shown in Table 2 are based on this approximation. If actual LMTD were used in place of Chen’s approximation, as has been done by Pariyani et al., DEM results would be still better. This has indeed proved to be the case. Based on actual LMTD, DE model gives a network cost of $83,942 and exchanger area of 462 m2 . 5.2. Case study 2 This case study is also a 4SP1 problem and it is taken from Lee, Masso, and Rudd (1970). It involves two hot and two cold streams along with steam and cooling water as utilities. The input data for this problem is given in Table 3.

(a) This deals with a general situation in which there are no restrictions on the matches. The solution obtained by Lee et al. (1970) for this case gives a network, which costs $13,481 for a minimum approach temperature fixed at 18 ◦ F. The cost of the network obtained from the DEM solution is $10,782, which is approximately $2,500 less than the one reported by Lee et al. The network by Lee et al. was obtained in a sequential manner using Block Synthesis, which apparently yields a network that is not cost-optimal. Fig. 6 shows the network synthesized by the DEM method. The optimal Tmin value for this network is 2.1 ◦ F. The calculations took about 5 s on a Pentium-4 PC. Another network involving two splits on the hot stream 2 in the first stage has also been synthesized by the DEM but its cost is $10,788.5. (b) In the second variation of the 4SP1 problem, which is solved by many workers, e.g. Dolan et al. (1987), Yee and Grossmann (1990), and Papoulias and Grossmann (1983), the match between hot stream H1 and cold stream C1 is forbidden. Dolan et al. (1987) and Yee and Grossmann (1990) have used additionally cold stream-cold stream matching, whereas Papoulias and Grossmann (1983) have not considered this aspect. Since the present algorithm also does not handle cold-cold matching, the DEM results are compared with those given by Papoulias and Grossmann (1983). Fig. 7 shows the network obtained from DEM solution, which needed a computational time of 4 s. The cost of this network is $18,705, whereas the cost of the network reported by Papoulias and Grossmann (1983) is $21,100. In a different study, a Heat Exchanger Network has been synthesized using the DEM methodology for a fixed Tmin of 18 ◦ F and this has been found to be identical to the one obtained by Papoulias and Grossmann.

Fig. 6. DEM solution for cost-optimal network for case study 2(a). The heat loads are given in parentheses. The network costs US$ 10,782.

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Fig. 7. Cost-optimal network for case study 2(b) with match H1-C1 forbidden. The heat loads are given in parentheses. The network costs US$ 18,705.

Table 4 Input data for the case study 3 Stream

Tin (K)

Tout (K)

H1 H2 H3 H4 H5 C1 C2 C3 C4 C5 Steam Water

433 522 544 500 472 355 366 311 333 389 509 311

366 411 422 339 339 450 478 494 433 495 509 355

CP (kW K−1 ) 8.79 10.55 12.56 14.77 17.73 17.28 13.90 8.44 7.62 6.08 – –

U = 0.852 kW m−2 K−1 for all matches except for those involvfor all matches involving ing steam; U = 1.136 kW m−2 K−1 steam; heat exchanger annual cost (US$) = 145.63 × S0.6 [S = area in m2 ]; cost of steam = 37.64 US$ kW−1 year−1 ; cost of cooling water = 18.12 US$ kW−1 year−1 .

5.3. Case study 3 This is the popular 10 SP1 problem, studied by many research workers. The input data for the problem is given in Table 4. The results of the DEM are presented in Table 5 along with other published values. The network structure obtained for this case is shown in Fig. 8. The cost of the network obtained using DEM is $43,538, which compares very well with other reported values. The computational time for this case study was 4980 s. DEM also synthesized another network with two splits on the hot stream 2 in stage 1 only, out of four stages considered but its cost is much higher at $79,449. Table 5 Results of the case study 3 Method

Splits: hot (cold)

No. of units

QC (kW)

Annual cost ($ year−1 )

Linnhoff and Flower (1978) Papoulias and Grossmann (1983) Lewin et al. (1998) Lin and Miller (2004) Pariyani et al. (2006) This work (Fig. 8)

0

10

1975

43,934

0 0 3 (2) 0 0

43,934 10 10 10 10

1879 1879 1879 1879

43,452 43,329 43,439 43,538

A point worth noting here is that the annual costs for the last four studies in Table 5 differ from one another, even when the utility loads and the number of exchangers are the same. This is due to the fact that there is another level of optimization involved with the distribution of the heat loads among the different exchangers. This optimization is needed to decrease the capital cost of the network by minimizing the areas needed for the individual exchangers. This can be done by finding the optimum approach temperatures for each exchanger. So, at a lower level, the algorithm has to optimize for the lowest approach temperatures for the exchangers in order to have the network with the lowest utility heat loads (minimizing operating costs) and the lowest area costs (minimizing capital costs). The DEM does well in this aspect as is evident from the near global solutions obtained for this case study. In this context, it is prudent to ponder whether HEN designs obtained without splitting of streams are not a better choice than those with split-streams, when the difference between the two alternatives is not significant. More so, when the costing model adopted for the split-stream case does not take into account various costs associated with splitting of streams, such as costs of valves, piping, control system, etc. 5.4. Case study 4 This is another 10-stream problem, studied first by Ahmad (1985). Ravagnani et al. (2005) reported much better results for this problem using a combination of the Pinch method and the Table 6 Input data for the case study 4 Stream

Tin (◦ C)

Tout (◦ C)

CP (kW K−1 )

H1 H2 H3 H4 H5 H6 C1 C2 C3 C4 Hot utility Cold utility

85 120 125 56 90 225 40 55 65 10 200 15

45 40 35 46 86 75 55 65 165 170 198 25

156.3 50.0 23.9 1250.0 1500.0 50.0 466.7 600 180 81.3 – –

U = 0.025 kW m−2 K−1 ; capital cost of heat exchanger (US$) = 60 × S [S = area in m2 ]; cost of hot utility = 100 US$ kW−1 year−1 ; cost of cold utility = 15 US$ kW−1 year−1 .

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

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Fig. 8. Cost-optimal network for case study 3. The heat loads are given in parentheses. The network costs US$ 43,538. Table 7 Results of the case study 4

Hot utility (kW) Cold utility (kW) Total area (m2 ) Energy cost ($ year−1 ) Capital cost ($ year−1 ) Global cost ($ year−1 )

Table 8 Input data for the case study 5

Ahmad

Ravagnani et al.

This work (Fig. 9)

Stream

Tin (◦ C)

Tout (◦ C)

CP (kW K−1 )

h (kW m−2 K−1 )

15,400 9,796 – 1,686,940 5,387,060 7,074,000

20,529.3 14,923.8 56,600.56 2,276,787 3,396,034 5,672,821

20745.4 15139.9 56085.25 2,301,641 3,365,115 5,666,756

H1 H2 H3 H4 C1 C2 C3 C4 C5 Hot oil Water

327 220 220 160 100 35 85 60 140 330 15

40 160 60 45 300 164 138 170 300 250 30

100 160 60 400 100 70 350 60 200 – –

0.50 0.40 0.14 0.30 0.35 0.70 0.50 0.14 0.60 0.50 0.50

Genetic Algorithm. The input data for the problem is given in Table 6. The results of the DEM along with those of Ahmad (1985) and Ravagnani et al. (2005) are presented in Table 7. The network structure obtained using DEM for this case is shown in Fig. 9. Ahmad (1985) employed a fixed heat recovery approach temperature (HRAT) of 10 ◦ C for synthesizing his network and reported a network cost of $7,074,000. Ravagnani et al. (2005) have obtained a network costing $5,672,821, for an ‘optimized’ HRAT of 24 ◦ C. Their work involved find-

Lifetime of plant = 5 years; rate of interest = 0%. Exchanger cost (US$) = 10,000 + 350 × S [S = area in m2 ]. Cost of hot oil = 60 US$ kW−1 year−1 ; cost of water = 6 US$ kW−1 year−1 .

ing the optimal value for the minimum approach temperature using a combination of the pinch method and the Genetic Algorithm and subsequently synthesizing a cost-optimal HEN.

Fig. 9. Cost-optimal network for case study 4. The heat loads are given in parentheses. The network costs US$ 5,666,756.

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Fig. 10. Cost-optimal network for case study 5. The heat loads are given in parentheses. The network costs US$ 2.942 × 106 . Table 9 Results of the case study 5 Method

Stream splits

Linnhoff and Ahmad (1990) Zhu, O’Neill, Roach, & Wood (1995) Zhu et al. (1995) Lewin (1998) Lewin (1998) Pettersson (2005) This work (Fig. 10)

0 2 0 2 0 7 0

Exchanger area (m2 ) [no. of units] 17,400 [13] 16,630 [14] 16,380 [10] 17,050 [12] 16,880 [11] 17,473 [17] 16,536 [15]

As mentioned earlier, the DEM obtains an optimal value for the minimum approach temperature (EMAT), during the simultaneous optimization of the network structure and other parameters. In this case study, it was found that the costoptimal network (Fig. 9) has a minimum approach temperature of 19.46 ◦ C between hot stream 3 and cold stream 1. The computations took 607 s in this case study. As may be seen from Table 7, DEM finds a new network, which costs less than that reported by Ravagnani et al. (2005). The energy cost here is slightly higher than for Ravagnani et al. but the reduction in the capital cost more than offsets the increase in the energy cost, resulting in a network, which is roughly $6000 cheaper. A network involving split-streams has also been synthesized by the DE methodology and it has two splits on the cold stream 1 in each of the stages 1, 2, 3 and 4 out of 5 stages considered but the cost is much higher ($6,497,145) than the previous solution without any split streams.

QH (MW)

QC (MW)

Cost (M$ year−1 )

25.31 26.22 26.83 25.09 25.69 24.27 25.88

33.03 33.94 34.55 32.81 33.41 31.99 33.60

2.960 2.970 2.980 2.936 2.946 2.905 2.942

sponding network structure is shown in Fig. 10. The calculations needed a total computation time of 23289 s in this case study. The network obtained in the present work has the maximum number (15) of heat exchangers but still has the lowest cost among all the unsplit cases. This is largely due to the linear nature of the objective function with respect to the exchanger surface areas. However, the network obtained by Pettersson (2005) using stream splitting has the lowest cost among all the reported solutions. Another network obtained using the DEM method has two splits on the hot streams 1 and 4 in stages 2 and 3, respectively, out of four stages considered but its cost is higher ($3,146,540). This case study also helps in validating the usefulness of the algorithm in handling streams with different heat transfer coefficients. From the results of the above case studies, it is evident that the performance of the DEM is quite encouraging. In some instances, the HENS algorithm has been able to discover networks not found by other methods.

5.5. Case study 5 The popular Aromatics problem is the subject for this case study. The problem involves finding a cost-optimal network of exchangers for four hot streams and five cold streams having different heat transfer coefficients. The input data for the aromatics problem is given in Table 8. The DEM solution is compared with the results available in the literature in Table 9. The corre-

6. Conclusions A new generation optimization method called ‘differential evolution’ (DE) has been used in the present work to develop a computational algorithm for heat exchange network synthesis (HENS). The salient features of the HENS model include stream

K.M. Yerramsetty, C.V.S. Murty / Computers and Chemical Engineering 32 (2008) 1861–1876

splitting, non-isothermal mixing of split streams, handling of compulsory and forbidden matches. The solution algorithm employs a simultaneous method of approach to optimize the heat exchanger network. The variables optimized during the synthesis are the string variables associated with the network structure, the heat duties and heat capacity flowrates of the split streams for the heat exchangers in the network. The minimum approach temperature, Tmin , is a crucial parameter, which affects the cost of the heat exchanger network. It has been recognized that there is an optimum value for Tmin for any cost-optimal network. In the present work, the optimal Tmin is evaluated implicitly in the solution process. Exchangers are allowed to have any minimum approach temperature greater than a specified value close to zero, e.g. 0.1. Since the algorithm considers the global cost of the network as the fitness criterion in the selection process, networks with exchangers having very low minimum approach temperature would be costlier and would be weeded out in the selection process, unless these exchangers led to a proportionate decrease in the operating costs. On the other hand are the networks, with high minimum approach temperatures. Such networks have lower capital costs, but the amount of heat that is transferred from the hot streams to the cold streams is limited. Hence, they would again lead to higher total costs. Thus, networks with very low or very high minimum approach temperatures do not pass the fitness test and are eliminated automatically. As a result, the search will be directed towards a network with the optimal trade-off between the capital and operating costs. The HENS model has been applied to some well-known examples available in the published literature. The results of these case studies show that the present DE-based model is both robust and accurate in finding out global optima. In some instances the model has come up with even better HENs than what has been reported in the literature. Cost models presently used for HENS do not take into account the costs of the paraphernalia associated with the splitting of streams, such as valves, piping, control system etc. Until such comprehensive cost models become available, it is rather difficult to make an objective evaluation of cost-optimality of HENs involving stream splitting. However, it is safe to state that HEN solutions obtained with unsplit streams are preferable, if the costs are not significantly higher than those with stream splitting. References Ahmad, S. (1985). Heat exchanger networks: Cost tradeoffs in energy and capital. Ph.D. Thesis. UK: UMIST Manchester. Androulakis, I. P., & Venkatasubramanian, V. (1991). A Genetic Algorithm framework for process design and optimization. Computers & Chemical Engineering, 15(4), 217–228. Athier, G., Floquet, P., Pibouleau, L., & Domenech, S. (1996). Optimization of heat exchanger networks by coupled simulated annealing and NLP Procedures. Computers & Chemical Engineering, 20(Suppl.), S13–S18. Athier, G., Floquet, P., Pibouleau, L., & Domenech, S. (1997a). Process optimization by simulated annealing and NLP Procedures. Application to heat exchanger network synthesis. Computers & Chemical Engineering, 21(Suppl.), S475–S480.

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