Heat Recovery Systems & ClIP Vol. 10, No. 4, pp. 399..405, 1990 Printed in Great Britain
08904332/90 $3.00 + .00 Pergamon Press pk
A NEW APPROACH TO HEAT EXCHANGER NETWORK SYNTHESIS Y. P. WANG) Z. H. CHL~2 and M. G R O L L 3 ~Shanghai Institute of Mechanical Engineering, 516 JunGong Road, Shanghai, P.R.C., now as a visiting scholar at IKE, University of Stuttgart, F.R.G., 2Shanghai Institute of Mechanical Engineering, 516 JunGong Road, Shanghai, P.R.C., and 3Institut ffir Kemenergetik & Energiesysteme (IKE), University of Stuttgart, F.R.G. (Received 10 January 1989and/n revisedform 17 July 1989)
AlmtraetA novel annual cost model is proposedfor the optimal synthesisof heat exchanger networks. With this model a tradeoffbetweenthe capital costs and the energycosts can be wellhandledand the minimum temperaturedifferencein heat exchangers,heaters and coolerscan be optimized.Two numerical examples are presented which demonstrate the efl~cieneyof the proposed model.
INTRODUCTION With heat exchanger networks hot streams and cold streams in the production processes of industrial plants can be matched in such a way that the expense for heating cold streams and cooling hot streams is reduced. Heat exchanger network synthesis has become an active research field for energy saving in industrial plants. The objective of the synthesis problem is to develop a network of countercurrent heat exchangers that minimizes investment and operational costs. In the past 10 years much progress has been made in tackling the problem. There are two major design targets: minimum utility usage (energy consumption) and minimum number of units. These two targets were first identified by Hohman [1] and later significantly developed by Linnhoff and Flower [2]. Based on these two targets the design method for heat exchanger networks [3] is composed of three steps: (1) the minimum utility target allows to obtain the schemes of heat exchanger networks, which save as much energy as possible; (2) the minimum unit number target allows to select from these schemes the heat exchanger networks, which are characterized by minimum investment costs; (3) from these "optimized" heat exchanger networks a final network is determined on the basis of the design engineer's experience and/or ingenuity. The same result as per steps (1) and (2) can be obtained through the application of a suitable mathematical programming method (transshipment model) for this problem. Several formulations of the transshipment model were proposed in the literature [4]. The unit number of heat exchangers is minimized on the condition that the utility usage (energy cost) has been minimized. To some extent the utility usage will increase as the unit number decreases. Similarly the energy costs will increase and the capital costs will decrease as the minimum temperature difference in heat exchangers, heaters and coolers A T B increases. The tradeoff between the capital costs and the energy costs involves the tradeoff between the unit number and the energy costs as well as the optimization of ATm. Recently a supertargeting method is proposed by Linnhoff and Ahmad [5] for the optimization of ATtain. With this method the minimum captial costs can be determined for a given & T u before heat exchanger networks are designed. Then AT,~ can be optimized in terms of the annual costs (including the capital costs and the energy costs). Floudas et al. [6] proposed the superstructure concept and a nonlinear programming formulation to determine automatically the optimal configuration of heat exchanger networks. Together with the linear programming and mixedinteger linear programming transshipment models, the nonlinear programming formulation is applied to optimize ATm. Though these efforts make the synthesis method of heat exchanger networks more successful, the tradeoff between the energy costs and the unit number as well as the optimization of ATtain simultaneously with synthesizing heat exchanger networks remains to be further studied. 399
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400
In this paper a novel annual cost model is proposed for the synthesis of heat exchanger networks. In this model the annual costs of heat exchanger networks, which consist of the energy costs and the capital costs, are taken as the objective, and ATtar, is taken as a variable to be optimized. Thus the tradeoff between the capital costs and the energy costs as well as the optimization of ATmm can be better handled than before. The validity of this approach will be demonstrated with the help of two examples, one of which is the standard reference case TC3 [3], the other is a practical example related to a plant in Shanghai. THE A N N U A L COST M O D E L The heat exchanger network synthesis problem can be stated as follows. In a processing system there is a set H = {ili = l, N H } of hot streams that have to be cooled, and a set C = {JlJ = l, N C } of cold streams that have to be heated. Each hot stream i has a heat capacity flow rate (FC)i and has to be cooled from supply temperature T,'. to target temperature T~. Each cold stream j has a heat capacity flow rate (FC)j and has to be heated from supply temperature T] to target temperature T~. Auxiliary heating and cooling are assumed to be available .from a set S = {m I m = 1, NS} of hot "utilities" (e.g. fuel, steam), and a set W = {n In = l, N W } of cold "utilities" (e.g. cooling water, refrigerant). The objective of the synthesis problem is to develop a network of countercurrent heat exchangers with minimum annual costs. This paper tries to treat the problem from the viewpoint of mathematical programming. The temperature range of each hot stream i can be partitioned into a set U = {k Ik = l, N K  l } of temperature intervals. The end temperatures t,k and ti.k+~ for each temperature interval k are t~k=T~
when
k=l,
ti.k+l=T~
when
k=NK1.
Similarly, the temperature range of each cold stream j can be partitioned into a set V = { I I I = 1, N L  1} of temperature intervals. The end temperatures t~ and t~.t+, for each temperature interval I are tit=T]
when
1=1
tj.l+l=T~
when
I=NL1.
Let Q~,/~be the heat quantity transferred from temperature interval k of hot stream i to temperature interval I of cold stream j, Q,,jt from hot utility m to temperature interval I of cold stream j and Q~k, from temperature interval k of hot stream i to cold utility n. Then the total capital costs of hea t exchangers are
Z=d E E E v ,. ( ,..
;.¢
e,,J, Y,
(1)
where K, is the overall heat transfer coefficient of the heat exchangers, At~/~is the mean temperature difference in the heat exchangers, At~kjl=
(t/k
tj, l+ I ) 
(tiok + I  lfl)
,
(2)
In t~,  t/.l+ i tj.k + 1 tj/ and d, g the cost coefficients in terms of heat transfer area. The total capital costs of heaters are
where K~ is the overall heat transfer coeffuzient of the heaters, At.,/~ is the mean temperature
difference in the heaters, At,~ji = tjl t2.t+l , tmtj.t+l In
and tm is the temperature of hot utility m.
tm m tj I
(4)
Heat exchanger network synthesis
401
The total capital costs of coolers are
,..~"k.uE~.w\K~At~]
Ic=d
where Kc is the overall heat transfer coefficientof the coolers, At~ is the mean temperature difference in the coolers, Ate, =
(t~


ti~n)   ( f L k + I  
T.w)
,
(6)
In t ~  t~, li, k + 1   Tnw
and T,. is the inlet temperature of cold utility n, t~, is the outlet temperature of cold utility n to cool temperature interval k of hot stream i. The annual energy costs are
e , = E ~ E P.Qm, t+ ~'. E ~'. P,(CP)~,,, m~zS j i C l e F
(7)
n~W iilf kGU
where P . is the annual cost of hot utility m for unit heat power Q . j (assumption: heating is accomplished by condensation of steam), P. is the annual cost of cold utility n for unit heat capacity flow rate (CP)~ of cold utility n to cool the temperature interval k of hot stream L The annual costs of the heat exchanger network are taken as the objective function to be minimized. An annual cost model is then developed as follows: Minimize
z =~(L+Ih+L)+E.
(8)
subject to
(t~  tj.t+ t  ATtar.) Q~kjt>I 0,
(9)
(tj.~+ i tjt  ATm) Qikjti> 0,
(10)
(t~  t~,  ATm) Q ~ t> 0,
(11)
(tl.~+ I  T,,  ATm) Q ~ >t 0,
(12)
(t'~  r , ~ ) ( c e ) ~  Q~ = o,
(13)
(ti.k+lt~)(FC)i+ ~ Q~ + ~. ~. Qt~jt=O,
(14)
nGW
(tj,,÷,

tj,)tec)j E
jeCleV
E E
raGS
=o
(15)
t~,  T,u, ~<0
(16)
ieHkeU
Q,~j~,Q~.,Q,.j~>~O,
(17)
for all
iEH
keU
j~C
l~V
m~S
neW,
where ~ is the annual rate of return, AT,~ is the minimum temperature difference in heat exchangers, heaters and coolers. The constraints are explained as follows: inequalities (9), (10) represent that heat can be exchanged only when temperature differences between hot streams and cold streams are not less than ATm; inequalities (! 1), (12) represent that heat can be exchanged only when temperature differences between hot streams and cold utilities are not less than ATm; equations (13), (14) and (15) represent energy balances around intervals of hot streams, cold streams and cold utilities, respectively; inequality (16) represents that the outlet
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temperature t~. of cold utilities cannot be larger than the maximum value permitted, Tou,; inequality (17) represents that variables Q~,jt, Q~,. and Q.,jt cannot be negative. The model is a nonlinear programming model. An optimal solution can be obtained by applying appropriate algorithms. NUMERICAL EXAMPLES The application of the annual cost model presented in this paper is demonstrated with the help of two example problems.
Example No. I The TCl, TC2, TC3 problems [2, 3] are generally accepted as reference problems for heat exchanger network synthesis. The design data [2] for them are shown in Table 1. Here the TC3 problem, which has two hot and two cold process streams, is taken as the first example. The stream data for it are shown in Table 2. In the literature [3] this problem has been solved with the classical pinch technology; when ATR is chosen to be 20 K the respective heat exchanger network with annual costs of $16,433 is depicted in Fig. 1. When the supertargeting method is applied to the TC3 problem the optimal minimum temperature difference ATm can be obtained to be 12.5 K. The respective network with annual costs of $10,638 is shown in Fig. 2. The network has fewer heaters and coolers than that in Fig. 1. The energy costs are reduced by 44% and although the capital costs are increased by 2%, the annual costs are reduced by 35% while A T ~ is reduced from 20 K to 12.5 K (see Table 3). Table I. D ~ i ~ data for the reference problems Steam pressure
45 bar 258°C 1676 kJ/kg T~. = 30°C 30°C ~ T~. ~ 80°C 1000 Wirez K (heaters) 750 W/re' K (excJumgers and m o k n ) 0.006 $/kg 0.00015 S/ks 8500 h/yr 300~a) o's $ (a in m 2) 0.1
Temperature Latent heat Cooling water temperature Overall heat transfer coefficient Cost of steam Cost of enolins water Equipment availability Cost of heat transfer area a Annual rate of return
Table 2. Stream data for the TC3 problem Stream number and type (I) Hot (2) Hot (3) Cold (4) Cold
Heat capacity flow rate (FC) kW/K 2.0 8.0 2.5 3.0
7"
T'
°C 150 90 20 25
°C 60 60 125 100
()
ITI
,e0"c 40 kW , e0*~'
17.5 k W
120 k W
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105 k W
20 k W
H'Q[]
185 kW
Fig. 1. Heat exchanger network from the literature [3] for the TC3 problem according to the classic.al pinch technology (AT,,~, chosen to he 20 K).
Heat exchanger network synthesis
403
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Heat exchanger network for the TC3 problem according to the supertargeting method (ATm •ffi 12.5 K, calculated). Table 3. Cost comparbon for the TC3 problem Annual costs Capital costs Energy costs $ S $ Network in Fig. I 16,433 31,945 13,238 Network in Fig. 2 10,638 32,433 7395 Network in Fig. 3 10,562 31,666 7395
ATm~. K 20 (given) 12.5 (calculated) 11.9 (calculated)
By the annual cost model the TC3 problem can be solved with the help of the MINOS mathematical programming software. The optimal ATm is 11.9 K and the respective network with annual costs of $10,562 is shown in Fig. 3. Comparing the network with that in Fig. 2, it can be seen that there is the same number of heat exchangers. Although ATm is reduced from 12.5 K to 11.9 K, the energy costs do not change. In fact when ATm ~< 12.5 K, the energy costs remain constant. The capital costs for the network in Fig. 3 are insignificantly lower than those for the network in Fig. 2 (see Table 3). This is due to the fact that despite one additional small heater in the Fig. 3 network, the two networks are very similar. It is illustrated that the results obtained by the novel approach lead to reduced annual costs compared to the pinch technology and that the annual costs are about the same as those obtained by the supertargeting method. Example No. 2 The second example is a practical problem from a plant in Shanghai, which has three hot and three cold streams. The stream data and design data for this problem are shown respectively in Table 4, Table 5. The optimal heat exchanger network (as shown in Fig. 4) can be obtained with the annual cost model. Comparison with the results obtained by the pinch technology (as shown in Fig. 5) indicates hardly any difference (see Table 6). A small A T ~ is expected to be optimum, because of the very low heat transfer area unit cost. The equality of the ATm's obtained by the annual cost model and given for the pinch technology analysis is accidental. However, it should be mentioned that 150°C
• 00oc
[ wc
,
2o'c~
41.7 k W
125°O ~ 1.8 k W
138.: kW
60"0
80.7 kW
lOO.C
(
=5"ClTI
~ . ? kW 159.: kW Fig. 3. Heat exchanger network for the TC3 problem according to the annual cost model (ATm ffi 11.9 K,
calculated).
404
Y . P . WAuo et al.
Stream number and type (I) (2) (3) (4) (5) (6)
Table 4. Stream data for example No. 2. Heat capacity flow rate (FC) kW/K
Hot Hot Hot Cold Cold Cold
8.1 1.7 2.3 1.7 1.7 0.58
T' °C
T' °C
60 40 20 20  l0 15
l0 15 10 100 10 40
Table 5. D a i s n data for example No. 2 Steam pressure 10 bar Temperature 180°C Latent heat 2Ol5kJ/kS ~. = 2o~c; 20oc ~ r . <40oc Refrigeram temperature Overall heat transfer eoet~ent 814 W/m2K 151.9 RMB Yuan/kW.yr Cost of steam 387.4 ILMB Yuan/kW. yr Cost of cooling water 685.8(a) °'~ Yuan (a in m 2) Cost of heat transfer area a Annual rate of return 0.I
Network in Fig. 4
Network in Fig. 5 Network in Fis. 6
Table 6. Cmt comparison for example No. 2 Almttal costs Capital costs Eemlgy eeat* RMB Yuan RMB Yuan RMB Yuan 176,890 14,505 175,440 176,986 15,460 175,440 231,625 13,079 230,317
ATm K 10 (calculated) 10 (siren) 10 (siren)
some problems were encountered with the employed algorithm software. ATm ffi 10 K is probably not the absolute minimum but only a local minimum. Comparing the actual heating and cooling system in the plant (as shown in Fig. 6) with the networks in Fig. 4 and Fig. 5, it can be seen that by adding three heat exchangers and removing two heaters the annual costs and the energy costs are reduced significantly althoush the capital costs increase somewhat (see Table 6). This practical example shows that heat exchanger network synthesis is a good means to save energy in industrial plants whether the pinch technology and the supertargeting method or the annual cost model are used. CONCLUSIONS From the above, it can be concluded that the developed novel annual cost model allows the
simultaneous optimization of the scheme and the major parameter AT=m of heat exchanser networks. The validity of the model has been demonstrated by comparing with the clussiml pinch technology and the supertarjeting method. The new approach is useful to study further the synthesis of heat exchanser networks with mathematical programming methods. However, the annual cost model is a little complicated; simplification is recommended as a future task. ~,60°C
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Heat exchanger network synthesis
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Fig. 6. Actual heating and cooling system in the plant. REFERENCES 1. E. C. Hohmann, Optimum networks for heat exchange. Ph.D. Thesis, University of California, Los Ani~lcs (1971). 2. B. Linnhoff and J. R. Flower, Synthesis of heat exchanger networks. AIChE Journal 24, 633654, (1978). 3. B. Linnhoff and E. Hindmarsh, The pinch design method for heat exchanger networks. Chem. Eng. Sci. 38, 745763, (1983). 4. S. A. Papoulias and I. E. Grossmann, A stntctura] optimization approach in process synthesisll. Heat recovery networks. Computers ,4 Chemical Eng. 7, 707721, (1983). 5. B. Linnhoff and S. Abroad, Supertargntins: Optimum synthesis of energy management systems. ASME Winter Meeting, Anaheim, December 1986. 6. C. A. Floudas, A. R. Ciric and I. E. Grossmann, Automatic Synthesis of Optimum Heat Exchanger Network Configurations. AIChE 3. 32, 276290, February 1986.