Published in International Journal of Heat Exchangers, 2004, Vol 5, Iss 2, P OPTIMUM DESIGN OF CROSSFLOW PLATE-FIN HEAT EXCHANGERS THROUGH

Transcription

1 Pulished in Interntionl Journl of Het Exchngers, 2004, Vol 5, Iss 2, P Archived with Dspce@NITR, OPTIMUM DESIGN OF CROSSFLOW PLATE-FIN HEAT EXCHANGERS THROUGH GENETIC ALGORITHM Mnish Mishr Mechnicl Engineering Deprtment, Indin Institute of Technology, Khrgpur, Indi mishr_md@yhoo.com Phone: , Fx: Prsnt Kumr Ds* Mechnicl Engineering Deprtment, Indin Institute of Technology, Khrgpur, Indi Phone: , Fx: pkd@mech.iitkgp.ernet.in Sunil Srngi Cryogenic Engineering Deprtment, Indin Institute of Technology, Khrgpur, Indi Phone: , Fx: ssrngi@hijli.iitkgp.ernet.in

2 OPTIMUM DESIGN OF CROSSFLOW PLATE-FIN HEAT EXCHANGERS THROUGH GENETIC ALGORITHM ABSTRACT A genetic lgorithm sed optimistion technique hs een developed for crossflow plte-fin het exchngers. The optimistion progrm ims t minimising the totl nnul cost for specified het duty under given spce nd flow restrictions. A multilyer plte-fin het exchnger hs een considered nd the optimum vlues of the design vriles consisting of the core nd the fin geometricl prmeters re otined for the minimum totl cost. For the vlidtion, optimistion of reduced model of two-lyer het exchnger hs een compred with the solution otined y the conventionl optimistion technique. Comprison of the solutions hs een mde etween the cses when no restriction is there on the upper limit of Reynolds numer nd with the lminr flow restriction. The effect of fixing the het exchnger dimensions on the optimum solution hs lso een studied. Key Words: crossflow, genetic lgorithm, het exchnger, optimistion, plte-fin, totl cost. 2

3 1. INTODUCTION Crossflow plte-fin het exchngers re widely used in erospce, utomoile nd chemicl process plnts. They offer vrious dvntges like low weight, high efficiency nd ility to hndle mny strems. Often the design of such het exchngers hs to meet the stringent requirements of low initil nd operting cost ssocited with superior therml performnce. Thus there is strong motivtion for optimising the design of plte-fin het exchngers to give desired performnce with minimum cost, volume or weight or with comintion of these properties. Optimistion of het exchngers is very ctive field of design reserch in therml engineering. A multitude of techniques rnging from clssicl techniques like Lgrnge multiplier, geometric progrmming, dynmic progrmming nd different non-liner progrmming methods to vrious non-clssicl methods s discrete mximum principle, rndom serch s well s the method of cse study hve een dopted for such purpose. A comprehensive review of different methods dopted for optimum design of het exchngers till erly 90 s hs een given y Ro (1991). In this review the merit of ll the ville techniques hve een criticlly judged nd their limittions for the optimistion of het exchngers hve een highlighted. Ro et l. (1996) otined the optiml design of shell nd tue het exchnger y twostge technique, where optimistion of the geometric design ws first done decoupling the geometricl nd the het trnsfer spects. Next, the geometric optimistion prolem ws linked 3

4 to therml rting module to otin therml design for ny given het duty. Bsed on Lgrngin multiplier technique Venktrthnm (1991) optimised the design of mtrix het exchnger. Armzon nd Ostersetzer (1993) put forwrd n itertive solution method termed s Omeg method for optimistion. They hve pplied this technique for optimum design of plte-fin het exchnger s well s cold pltes for electronic component cooling nd compred the results with those otined from rndom serch lgorithms. Dzyuenko et l. (1993) discussed method sed on pressure drop nd het trnsfer experimentl dt to select optimum het trnsfer surfce for spce ppliction. Hesselgreves (1993) suggested n nlyticl method for clculting optimum size nd weight of plte-fin het exchnger for given het duty using dimensionless design prmeters. Murlikrishn nd Shenoy (2001) pointed out the difficulties for determining the optimum design of shell nd tue het exchngers. They hve suggested methodology sed on grphicl technique where region of fesile design is identified on pressure drop digrm. On the digrm, curves corresponding to constnt het exchnger re or totl cost cn e plotted nd the optimum solution cn e picked up. Gonzles et l. (2001) determined the optimum vlues of ten operting nd geometric vriles for determining the minimum cost of n ir-cooled het exchnger using successive qudrtic progrmming. Though the technique produces non-integer vlues of the integer vriles, the uthors suggest its use s good strting point. Out of the different techniques of optimistion clculus sed techniques re well known for their mthemticl rigor nd elegnce. Though they re time tested nd suitle for multiple vriles, the complexity of ny lgorithm sed on this method increses with the incresing numer of vriles. In the sence of proper initil guess the solution my converge onto some locl extrem or my even diverge finlly. To gurd the convergence of the clculus sed 4

5 lgorithm onto locl extrem, one my need to try different strting points for the itertion setting initil vlues of the vriles. Moreover, the clculus-sed methods re not very convenient for hndling discrete vriles. Different versions of rnch nd ound techniques re suitle for non-liner optimistion prolems contining discrete-continuous vriles (Gupt nd Rvindrn, 1983; Sljegheh nd Vnderplts, 1993). These methods in generl tret ll the vriles s continuous nd susequently select fesile discrete solution to identify the optimum. While doing so the originl optimistion prolem is expnded to lrge numer of su-optimistion prolem. On the other hnd in the methods sed on penlty function pproch (Gisvold nd Moe, 1972), the diversity of locl optim my not gurntee convergence to fesile discrete optiml point. In most of the cses the penlty prmeters need further djustment to continue serch itertions. Different serch techniques could e good lterntives for optimistion prolem contining discrete or discrete-continuous vriles. However, the conventionl technique (Stoecker, 1999) ecomes very cumersome nd lorious when the extremum is sought for multivrile prolem hving numer of constrints. In recent times, some proilistic serch lgorithms nmely genetic lgorithm (GA) nd simulted nneling (SA) re eing pplied to the optimistion of vrious engineering systems in generl nd to thermo-processes nd fluid pplictions in prticulr. These techniques cn overcome the ove-mentioned difficulties to lrge extent. Genetic lgorithm hs een pplied successfully for the nlysis nd optimum design of diverse therml systems nd components nmely convectively cooled electronic components (Queipo et l., 1994) nd cooling chnnels (Wolfersdorf et l., 1997), flow oiling 5

6 (Cstrogiovnni nd Sforz, 1997), fin profiles (Fri, 1997; Younes nd Potiron, 2001), finned surfce nd finned nnulr ducts (Fri, 1998), compct high performnce coolers (Schmit et l., 1996) nd shell nd tue het exchngers (Tyl et l. 1999). In n effort of predicting het exchnger performnce Pcheco-Veg et l. (2001) recently demonstrted the superiority of GA over the conventionl lest squre technique. The uthors commented tht s GA works on glol serch, it out performs the conventionl locl grdient-sed methods. In cse of grdientsed methods there is lwys risk to converge t locl extrem unless one tries multi-initil vlues. On the contrry, GA strts with popultion of possile solutions, which minimizes the risk of premture convergence. However, it needs to e mentioned tht for convergence GA needs lrge numer of itertions. It posses gret demnd on computtionl time nd renders the ppliction of GA unsuitle for simpler prolems. Using the sic frmework of GA, technique for multiconstrint minimistion hs een developed in the present work. The technique hs een pplied to otin the design of crossflow plte-fin het exchngers for the lowest totl nnul cost (TAC). To check the ccurcy of the developed method initilly simplified design hving only two geometricl vriles hs een considered. The optimum solutions of this prolem s otined y the present technique greed closely with n ccurte solution otined y grdient serch method. Different cses of optimum design hve een studied next. In ll these exercises minimistion of TAC hs een trgeted for specified het duty constrint under different comintions of spce nd flow restrictions. Finlly comprison etween the optimum designs ttined under different design constrints hs een mde. The effect of GA prmeters on the optiml solution hs een seen. Further, the effect of different constrints on the solution hs lso een discussed. The methodology used is not new, 6

7 ut the system like plte-fin het exchnger where it hs een pplied nd the wy it hs een used is new to the reserchers working in this re. 2. OUTLINE OF THE SOLUTION METHODOLOGY Genetic lgorithm is serch procedure sed on the principles of genetics nd nturl selection. An elorte description of this technique is ville in numer of references, for exmple, Hollnd (1975), Mitchell (1998) nd Golderg (2000). 2.1 Bsic Algorithm In the simplest form GA cn e used to mximise the ojective function f(x), which in turn depends on numer of vriles. Following is the sttement of the prolem. Mximise f(x), where, X = x i, i=1,2,..,k nd x i, min x i x i, mx (1) The voculry used in GA elongs originlly to genetics. A fesile solution is represented y inry coded string known s chromosome. The vriles x i s re first coded in some string structure. In simple GA (Golderg, 2000) inry coded strings consisting of 0 s nd 1 s re mostly used. The length of the string depends on the desired solution ccurcy. The vrile x i is coded in sustring s i of length l i. The decoded vlue of inry sustring s i is l 1 clculted s 2 i s i, where s i (0,1) nd the string is represented s (s l-1 s l-2 s 2 s 1 s 0 ). For i= 0 exmple, four it sustring (0111) hs decoded vlue equl to [(1)2 0 + (1)2 1 + (1)2 2 + (0)2 3 ] or 7. If there re two vriles then it needs totl 8 its ( ). A set of fesile solutions is 7

8 known s popultion. The vlue of the ojective function for prticulr memer decides its merit (competitiveness) in comprison with its counterprts. In GA lnguge this is termed s fitness function. After creting n initil popultion simple GA works with three opertors: reproduction, crossover nd muttion. Reproduction, which constitutes selection procedure wherey individul strings re selected for mting sed on their fitness vlues reltive to the fitness of the other memers. Individuls with higher fitness vlues hve higher proility of eing selected for mting nd for susequent genetic production of offsprings. This opertor, which wekly mimics the Drwinin principl of the survivl of the fittest, is n rtificil version of nturl selection. The reproduction opertor used here cretes roulette wheel where ech string in the popultion is ssigned slot in the wheel sized in proportion to its fitness. Since the popultion size is usully kept fixed, the sum of the proility of ech string eing selected must e one. Therefore the proility for selecting the i th string is f i p =, where N p is the popultion size. (2) i N p j= 1 f i The evolution is chieved y mens of crossover nd muttion. After reproduction, the crossover opertor lters the composition of the offspring y exchnging prt of strings from the prents nd hence cretes new strings. Though different types of crossover techniques re common in prctice, in the present nlysis single point crossover is used (Figure 1). Crossover opertion tkes plce in two steps. In the first step, selection of two rndom strems (chromosomes) tkes plce from the mting pool generted y the reproduction opertor. Next crossover site is selected t rndom long the string length, nd lleles (gene vlues in chromosomes) re swpped etween the two strings etween the crossover site nd the end of the strings. 8

9 FIGURE 1 HERE Muttion is secondry opertor, which increses the vriility of the popultion. For GA using inry lphet to represent chromosome, muttion provides vrition to the popultion y chnging it of the string from 0 to 1 or vice vers with smll muttion proility p m (Figure 2). The need for muttion is to crete point in the vicinity of the current point to prevent the solution from flling into locl optimum, therey chieving locl serch round the current solution, which sometimes is not possile y reproduction nd crossover. FIGURE 2 HERE A genertion or n itertion from the computtionl point of view is completed when the offspring replces the prents from the preceding genertion. A simple flow chrt for GA sed optimistion procedure is given in figure 3. FIGURE 3 HERE GA s do not gurntee convergence to glol optimum solution nd so require suitle stopping criteri. The GA cn e terminted when there is no improvement in the ojective function (fitness) for defined numer of consecutive genertions within prescried tolernce rnge, or when it covers prespecified mximum numer of genertions. 2.2 Modifiction for Constrined Minimistion 9

10 If there re numer of constrint conditions nd the ojective function needs to e minimised, the prolem is modified s follows: Minimise f(x), X=[x 1, x k ] (3) Where, g j (X) 0, j=1,,m (4) nd x i, min x i x i,mx, i=1,,k. (5) The prolem cn e recst into unconstrined mximistion prolem nd the solution my e otined s outlined erlier. The first step is to convert the constrined optimistion prolem into n unconstrined one y dding penlty function term. m Minimise f(x) + = Φ (gi(x)), (6) i 1 suject to x i, min x i x i,mx, i =1,,k. (7) Where Φ is penlty function defined s, Φ(g(X)) = R. g(x) 2. (8) R is the penlty prmeter hving n ritrry lrge vlue. The second step is to convert the minimistion prolem to mximistion one. This is done redefining the ojective function such tht the optimum point remins unchnged. The conversion used in the present work is s follows Mximise F(X), (9) m where, F(X) = 1 / { f(x) + = Φ (gi(x)) }. (10) i 1 10

11 The ove lgorithm cn e used for minimising the totl nnul cost of crossflow plte-fin het exchngers. 3. GEOMETRICAL, THERMOHYDRAULIC AND COST PARAMETERS OF PLATE- FIN HEAT EXCHANGERS FIGURE 4 HERE Figure 4 depicts schemtic view of crossflow plte-fin het exchnger with offsetstrip fins. The initil nd running costs of such equipments depend on the geometricl specifictions nd thermohydrulic performnce prmeters. These detils re estimted sed on the following ssumptions. 1. The stedy stte condition is ssumed to e previling. 2. Offset-strip fins hving the sme specifictions re used for oth the fluids. 3. Het trnsfer coefficients nd the re distriution re ssumed to e uniform nd constnt. 4. Property vrition of the fluids with temperture is neglected. 5. When the design consists more thn two lyers of finned pssges, numer of fin lyers for fluid (which hs men temperture closer to tmospheric temperture) is ssumed to e one more thn tht of fluid (N =N +1). 6. In generl, the fin effectiveness for compct plte-fin het exchngers re quite high (more thn 90%). However, in the present exercise, clcultions hve een done tking 11

12 100% fin efficiency. If required one my redily introduce fin efficiency in the present formultion. 3.1 Geometricl Prmeters For the geometricl detils shown in figure 4, one my get the free flow res s Aff = (H t ).(1 n t ).L. N, (11) Aff = (H t ).(1 n t ).L. N. (12) Similrly het trnsfer res for the two sides cn e otined s given elow. A = L.L. N [1+ 2.n.(H -t )] (13) A = L.L.N [1+ 2.n.(H -t )] (14) Totl het trnsfer re, A HT = A + A = L.L.[ N {1+ 2.n (H -t )}+N {1+2.n (H -t )}] (15) Hydrulic dimeter (Joshi nd We, 1987) of the finned pssges is given y 2(s t)(h t) Dh =, (16) (H t)t {s + (H t)} + l f where s = (1/n t) (17) 3.2 Thermo-hydrulic Prmeters The rte of het trnsfer my e clculted s follows Q = m.cp.(t,in T,out ) = m.cp.(t,in T,out ) (18) Q = UA(F.LMTD) (19) The LMTD (log men temperture difference) cn e given y T1 T2 LMTD =, (20) T1 log e T2 where 12

13 T 1 = T,in - T,out nd T 2 = T,out - T,in Neglecting the therml resistnce due to the metl wll, overll het trnsfer etween the two fluids cn e expressed s, 1 UA 1 1 = + (21) (ha) (ha ) The het trnsfer coefficient cn e otined in terms of Colurn j fctor s j = St.Pr 2/3 = h. Pr 2/3 (22) G.Cp Sustituting h, A nd UA in eq. (21) the equlity constrint for the het duty my e expressed s j m Cp 1 Pr 2 / 3 (H t )(1 n t. (1 + 2.n h )L ) + j m Cp 1 Pr 2 / 3 (H t )(1 n t. (1 + 2.n h ) L ) F(LMTD) = = Zq (23) Q Pressure drop for the two fluid strems cn e clculted redily s 2 4.f.L.G 2.f.m L P = =, (24) ρ.d ρ D.L.N.(H t ) (1 n t ) h, 2 h, 2 4.f.L.G 2.f.m L P = =. (25) ρ.d ρ D.L.N.(H t ) (1 n t ) h, 2 h, j nd f fctors my e evluted from ville correltions (Joshi nd We, 1987). For lminr flow (Re 1500) j = 0.53(Re) (l f / D h ) {s /(H t)}. (26) f = 8.12(Re) (l f / D h ) {s /(H t)}. (27) For turulent flow (Re>1500) j = 0.21(Re) (l f / D h ) (t / D h ). (28) 13

14 f = 1.12(Re) (l f / D h ) (t / D h ). (29) Where, GD Re = µ h m.d h =. (30) Aff. µ 3.3 Cost Estimtion The method of defining the totl nnul cost my vry depending upon the ppliction. However, it should comprise of the initil cost of the equipments nmely the het exchnger nd the prime movers for the fluid strems nd the running cost. Cost of oth the het exchnger nd the prime movers will hve fixed nd vrile component s Z=kA+k 0 (Zuir et l., 1987). The vrile component (ka) for the het exchnger my e ssumed to depend on the totl het trnsfer re s the type of the het trnsfer surfce hs een specified. In cse of prime movers the vrile component of the cost will depend on the product of cpcity nd pressure drop. The running cost on the other hnd will depend on the power consumption. Such sis for cost estimtion hs lso een tken y Murlikrishn nd Shenoy (2000). Totl nnul cost, TAC = Initil cost of (het exchnger core + pump + pump ) + Operting cost of (pump + pump ) m TAC = Af.[{C + C.A c HT } + {Ce + Cf. (. P ρ d ) m d }+ {Ce + Cf. (. P ) }] ρ Cpow.(Time/ yer) m m + [ P + P ] η ρ ρ pump (31) Where, P nd P re in kp. As specific exmple following vlues re selected for the cost fctors (Murlikrishn nd Shenoy, 2000) nd other operting prmeters. c=0.8, d=0.68, Af=0.322, C=30000, C=750, Ce=2000, Cf=5, C pow = $/W-hr, η pump = 0.7, totl opertion time/yer = 8000 hours, specified het duty, Q = 160 kw 14

15 Operting conditions re sed on design prolem y Shh (1980). m = kg/s, ρ = kg/m 3, Cp = kj/kg-c, Pr = 0.687, T,in = 240 C m = kg/s, ρ = kg/m 3, Cp = kj/kg-c, Pr = 0.694, T,in = 4 C 4. OPTIMUM DESIGN THROUGH GA DIFFERENT CASES Optimum design of plte-fin het exchngers hve een chieved sed on the methodology descried in the preceding sections. Three different cses, s elorted elow, hve een considered. 4.1 Cse I - Het Exchnger With Two Fin Lyers FIGURE 5 HERE At the outset n effort hs een mde to compre the results otined through GA with those computed using the grdient serch technique ville with MATLAB. For this the originl prolem hs een simplified sustntilly. Only two fin lyers hve een considered with fixed fin geometry nd specified coefficients of het trnsfer s well s frictionl pressure drop. The length (L ) nd redth (L ) re the only vriles, which re to e optimised. The sttement of the prolem is s follows. N = N = 1 Totl nnul cost cn e expressed s TAC = Z + Z.L c c.l + Zc. L d.l -2d + Zd. L -2d.L d + Ze. L.L -2 + Zf. L -2.L, (32) where, Z = Af.(C + 2.Ce), Z = Af.C, m d Kp d Zc = Af.Cf.( ) ( ), m d Kp d Zd = Af.Cf.( ) ( ), ρ 1000 ρ

16 C.Time / yer m Kp Ze = pow, η ρ 1000 pump C.Time / yer m Kp Zf = pow, η ρ 1000 pump Kp 2 2.f m =, 2 ρ.dh K Aff Kp 2 2.f m =, 2 ρ.dh K Aff K Aff = Aff / L, nd K Aff = Aff / L. The optimistion prolem then ecomes minimistion of the ojective function f(x) = Z + Z.L c c.l + Zc. L d.l -2d + Zd. L -2d.L d + Ze. L.L -2 + Zf. L -2.L, (33) sujected to constrints: g1(x) 0.13 L 2; (34) g2(x) 0.12 L 2. (35) When required het duty is specified, n dditionl equlity constrint comes s, g3(x) ξ(x) Zq = 0. (36) Where ξ(x) is the LHS of the eqution (23). Bsed on the ove formultion optimum solution is sought through GA s well s through grdient serch technique using the following prmetric vlues. H=6.35 mm, t=0.152 mm, l f =3.18 mm, n=615 fins/m (15.62 fins per inch). j =j =0.015, f =f =0.062 Penlty prmeter for GA hs een selected s, R=10 6. Optimum dimensions of the het exchnger nd corresponding totl cost re otined using the grdient serch technique nd GA oth without nd with the het duty constrint. In figure 6() nd 6() the optimum solutions otined from grdient serch technique nd GA re depicted for cses without nd with het duty constrints. The GA solutions of the ove prolem hve een otined y chnging the GA prmeters (like popultion size N p, crossover proility p c nd muttion proility p m ). 16

17 FIGURE 6 HERE As GA is technique sed on stochstic methods the resulting solutions will not e unique one s shown in figure 6. With the vrition of GA prmeters results re not exctly identicl ut re very close to one nother. To ring out this feture clerly the GA result long with tht otined from MATLAB re once gin plotted in figure 7 on spce ound y the physicl limits of L nd L used in the prolem. Additionlly constnt cost contours re lso plotted on the figures. The close greement etween the solutions otined from grdient serch technique nd tht from GA is ovious in the figures. All the GA solutions stisfy the constrint conditions while there is slight vrition in the corresponding cost function. Though GA does not produce unique solution it gives numer of ner optiml solutions nd ultimtely offers more flexiility to the designer. Finlly n verge vlue of ll the GA solutions hs een tulted in tle 1. It compres very well with the solution otined through grdient serch technique. However, it needs to e mentioned tht time tken for the solution through GA is much more compred to tht through grdient serch technique. FIGURE 7 HERE TABLE 1 HERE 4.2 Cse II - Multilyer Het Exchnger With Inequlity Constrints For Het Duty nd Flow Rtes 17

18 After gining confidence through simplified design in the previous exmple, GA hs een pplied for the optimum design of the plte-fin het exchnger hving multiple lyers. The sttement of the optimistion prolem is s follows. Minimise f(x)=tac, (37) Sujected to the constrints: g1(x) 0.1 L 1; g2(x) 0.1 L 1; g3(x) H 0.01; g4(x) 100 n 1000; g5(x) t ; g6(x) l f 0.010; g7(x) 1 N 10. (38) The minimum het duty generted is given y g8(x) ξ(x) Zq 0, (39) where ξ(x) is the LHS of the eqution (23). In most of the pplictions of plte-fin het exchngers, the flow remins either in lminr or in the lower turulent rnge. Therefore, dditionl constrints hve een introduced, to limit the Reynolds numer elow 1500 for oth the fluids. g9(x) Re 1500, nd g10(x) Re (40) Though the designer hs some independence in selecting the GA prmeters, it hs een shown tht selection of proper GA prmeters (Grefenstette, 1986; Wolfersdorf et l., 1997) renders quick convergence of the lgorithm. The proper GA prmeters re prolem specific. 18

19 Therefore initilly n exercise hs een mde following the methodology of the Wolfersdorf et l. (1997) to select the optimum GA prmeters for the present prolem. Figure 8 shows the vrition of mximum fitness function nd the totl cost with the popultion size, crossover nd muttion proilities nd penlty prmeter. Except for penlty prmeter R1, in ll the cses the minimum cost corresponds to the mximum vlue of the fitness function. Tking minimum cost s the selection criteri following prmetric vlues re selected for GA, popultion size 90, crossover proility 0.8, muttion proility 0.01, nd penlty prmeter R1=4000, R2=500 nd R3=1000. FIGURE 8 HERE The optimum solution sed on the optimum GA prmeters re listed in tle 2. TABLE 2 HERE 4.3 Cse III Effect of Higher Flow Rtes nd Equlity Constrint on Het Duty In the present exercise the constrints on Reynolds numer hve een relxed while the het duty constrint is mde more restrictive s follows. Q=160 kw. The GA prmeters for this modified prolem hs een selected following the methodology descried efore. The vlues re s follows: popultion size = 30, crossover proility = 0.8, muttion proility = 0.01 nd penlty prmeter = The optimum solution sed on these GA prmeters re listed in tle 3. 19

20 TABLE 3 HERE 5. COMMENTS ON THE RESULTS On the sis of the optimum solution for different cses given in the erlier section, few importnt oservtions cn e summrised s follows. 5.1 Effect of Constrint Conditions on Optimum Design A comprison of tle 2 nd 3 revels numer of interesting points. Restriction on Reynolds numer gives het exchnger with lrge length, width nd higher numer of fin lyers nd t the sme time provides lrger rte of het trnsfer. On the other hnd if the restriction on Reynolds numer is relxed, the het exchnger cn e designed for required lower therml performnce nd t the sme time its cost cn e reduced. This fct cn e explined etter with help of constnt cost nd constnt het duty contours s depicted in figure 9. In this figure iso-cost nd iso-het duty curves re constructed s functions of L nd L while tking ll the geometricl prmeters of the het exchnger from tle 2. Due to the restriction put in the Reynolds numer on the two sides, the solution spce is limited to re OABC nd the solution is otined t point O, which gives the limiting vlues of the Reynolds numers. Though the solution gives much higher het duty it lso corresponds to much higher nnul cost for the equipment. FIGURE 9 HERE 20

21 In similr fshion the cost nd the het duty contours for the second cse is shown in figure 10. Corresponding to equlity constrint Q=160 kw one gets much lower cost of the het exchnger. FIGURE 10 HERE 5.2 Imposition of Additionl Constrints Due to different prcticl resons sometimes there my e spce restrictions. As result one of the dimensions of the het exchnger my hve to e fixed priori. This cts s n dditionl constrint. The effect of fixing the het exchnger lengths on the minimum totl cost for cse II is shown in figure 11. In generl the totl cost increses with the increse of L s depicted in figure 11(). Minimum cost is otined t L = m, which corresponds to the optimum design when no restriction ws put on the het exchnger length. The sme figure gives n dditionl informtion of the pressure drops occurring on the two sides of the het exchnger due to chnge of L. In figure 11() the vrition of totl minimum cost s function of L long with the corresponding pressure drops for the two fluids is depicted. The curves exhiit similr nture to those shown in the previous figure. The totl cost increses with L, the minimum eing t point where no restriction on length is imposed. FIGURE 11 HERE Next, the effect of dditionl constrints on optimum design for cse III hs een studied. The totl minimum cost is determined vrying L, L nd numer of lyers, N individully. The 21

22 results re shown in figure 12 (), () nd (c) respectively. In ll these three figures the minimum vlue of the respective vrile corresponds to the optimum vlue given in tle 3. The vritions of pressure drop for oth the fluids with the vrition of L, L nd N hve lso een depicted in the respective curves. It my e noted the pressure vritions shown oth in figure 11 nd 12 do not follow ny prticulr trend. This is ecuse the pressure drop vlues correspond to the optimum design condition. The optimum design condition gives comintion of prmetric vlues, which my chnge sustntilly if prticulr prmeter is vried. Therefore the oserved ehviour of P curves is not unexpected. FIGURE 12 HERE Figure 13 gives comprison of the vlues of TAC nd the corresponding het duty produced for cse II nd III for vrition of one of the lengths of the het exchnger. It shows clerly tht the optimum solution is very sensitive to the vrition of het exchnger lengths for cse II, where the upper limit of Reynolds numer is restricted. FIGURE 13 HERE 6. CONCLUSION A methodology sed on GA hs een developed for the optimistion of multilyer pltefin het exchngers with lrge numer of design vriles of oth discrete nd continuous nture. Initilly two-lyer het exchnger with given fin specifictions hs een considered. The scheme determines optimum vlues of length nd width of the het exchnger, which minimise 22

23 the totl nnul cost. Solution otined for different comintions of GA prmeters gve different set of optimum vlues for the length nd the width. However, the optimum vlues otined from ll the GA exercises re close enough. The sme prolem hs lso een solved grphiclly s well s through grdient serch technique. The solutions generted y GA gree very closely to the grphicl solution s well s tht otined from grdient serch technique. Optimistion of multilyer plte-fin het exchngers hs een considered next. Two different cses hve een tken up. In the first cse the lower limit of the het duty ws specified nd the het exchnger ws designed for lminr flow conditions. In the second cse the constrint on fluid Reynolds numer ws relxed while the design ws mde to meet the specified het duty exctly. By imposing the lminr flow constrints, the effective domin in the fesile design spce reduces nd the size of the het exchnger increses, which leds to increse in totl cost nd lso the corresponding het duty produced. Further, the effect of fixing ny of the min geometricl prmeters of the het exchnger on its optimum design hs een investigted. In generl this dditionl constrint increses totl nnul cost of the het exchnger. However, the effect of this dditionl constrint is more significnt when the design is mde for lminr flow conditions. ACKNOWLEDGEMENT We would like to cknowledge the reviewers for their vlid suggestions which hve helped us to modify the communiction in the present form. NOMENCLATURE A, A HT het trnsfer re, m 2 Af cost fctor 23

24 Aff free flow re, m 2 C het cpcity rte (mcp), J/K C, C, Ce, Cf cost fctors Cp specific het of fluid C pow - cost of power, $/W-hr D h hydrulic dimeter, m f Fnning friction fctor f mx mximum fitness prmeter f(x) - ojective function F crossflow correction fctor g(x) - constrint G mss flux velocity (= m/aff), kg/ m 2 -s h het trnsfer coefficient H - height of the fin, m j - Colurn fctor l f lnce length of the fin, m L - het exchnger length, m l i - length of sustring N p popultion size. NTU numer of trnsfer units p proility Pr - Prndtl numer P pressure drop, N/ m 2 Q rte of het trnsfer, W R, R1, R2, R3 penlty prmeters Re Reynolds numer s fin spcing (1/n-t), m s i - inry su-string St Stnton numer [=h/(gcp)] t fin thickness, m T-Temperture, K TAC totl nnul cost, $ Time/yer nnul opertionl time, hours U overll het trnsfer coefficient, W/ m 2 K x i - vrile X (x 1, x 2, x k ) LMTD - log men temperture difference m mss flow rte of fluid, kg/s n - fin frequency, fins per meter N, N numer of lyers of finned pssges N G numer of genertions Greek symols ρ density, kg/ m 3 µ - viscosity, N/ m 2 -s φ(.) penlty function 24

25 η pump efficiency of pump in inlet m muttion Suscripts, fluid nd c - crossover mx -mximum min minimum out exit i - vrile numer REFERENCES Armzon, B. Ostersetzer, S Optiml Therml nd Hydrulic Design of Compct Het Exchngers nd Cold Pltes for Cooling of Electronic Components, in Aerospce Het Exchnger Technology - Proc First Interntionl Conference on Aerospce Het Exchnger Technology, eds. R.K.Shh nd A. Hshemi, pp Plo Alto, CA, USA. Elsevier. Cstrogiovnni, A., Sforz, P.M A Genetic Algorithm Model for High Het Flux Flow Boiling. Experimentl Therml nd Fluid Science. 15(3): Dzyuenko, B.V., Dreitser, G.A., Ykimenko, R.I Method of Optimum Het Trnsfer Surfce Choosing for Spce Het Exchngers, in Aerospce Het Exchnger Technology - Proc the First Interntionl Conference on Aerospce Het Exchnger Technology, eds. R.K.Shh nd A. Hshemi, pp Plo Alto, CA, USA. Elsevier. Fri, G A Genetic Algorithm for Fin Profile Optimistion. Interntionl Journl of Het nd Mss Trnsfer. 40(9): Fri, G Het Trnsfer Optimistion in finned Annulr Ducts Under Lminr Flow Conditions. Het Trnsfer Engineering. 19(4):

26 Gisvold, K.M., Moe, J A Method for Nonliner Mixer Integer Progrmming nd its ppliction to Design Prolems. Journl of Engineering for Industry. 94(2): Golderg, D.E Genetic Algorithms in Serch, Optimiztion, nd Mchine Lerning. Addison-Wesley Longmn, Inc. Gonzlez, M.T., Petrcci N.C., Uricn M Air-Cooled Het Exchnger Design Using Successive Qudrtic Progrmming (SQP). Het Trnsfer Engineering. 22: Grefenstette, J.J Optimiztion of Control Prmeters for Genetic Algorithms. IEEE Trns. Of Systems, Mn nd Cyerntics. 16(1): Gupt, O.K., Rvindrn, A Nonliner Integer Progrmming nd Discrete Optimistion. Journl of Mechnisms, Trnsmission nd Automtion in Design. 105: Hesselgreves, J.E Optimum Size nd Weight of Plte-Fin Het Exchngers, in Aerospce Het Exchnger Technology - Proc First Interntionl Conference on Aerospce Het Exchnger Technology, eds. R.K.Shh nd A. Hshemi, pp Plo Alto, CA, USA. Elsevier. Hollnd, J Adpttion in Nturl nd Artificil System. Ann Aror: University of Michign Press. Joshi, H.M., We, R.L Het Trnsfer nd Friction in the Offset Strip-fin Het Exchnger. Interntionl Journl of Het & Mss Trnsfer. 30(1): Mitchell, M An Introduction to Genetic Algorithm. Prentice Hll of Indi Pvt. Ltd. Murlikrishn, K., Shenoy, U.V Het Exchnger Design Trgets for Minimum Are nd Cost. Trnsctions of IchemE. 78(A): Pcheco-Veg, A., Sen, M., Yng, K.T., McClin, R.L. 2001, Correltions of Fin-Tue Het exchnger Performnce dt Using Genetic lgorithms, Simulted nneling nd intervl 26

27 Methods, Proc. of ASME Interntionl Mechnicl Engineering congress nd Exposition Queipo, N., Devrkond, R., Humphery, J.A.C Genetic Algorithms For Thermosciences Reserch: Appliction to the Optimized Cooling of Electronic Components. Interntionl. Journl of Het nd Mss Trnsfer. 37(6): Ro, K. R Optiml Synthesis of Shell nd Tue Het Exchngers. Ph.D. Thesis, Indin Institute of Science, Bnglore. Ro, K. R. Shrinivs, U. nd Srinivsn, J Constrined optimistion of het exchngers. Interntionl Journl of Chemicl Engineering. 103: Sljegheh, E., Vnderplts, G.N Efficient Optimum Design of Structures with Discrete Design Vriles. Interntionl Journl for Spce Structures. 8(3): Schmit, T.S., Dhingr, A.K., Lndis, F., Kojsoy, G A Genetic Algorithm Optimiztion Technique for Compct high Intensity Cooler Design. Journl of Enhnced Het Trnsfer. 3(4): Shh, R.K Compct Het Exchnger Design Procedure, in Het exchngers, Therml Hydrulic Fundmentls nd Design, eds. S. Kkc, A.E. Bergles nd F. Myinger, Wshington: Hemisphere Pulishing Corportion. Stoecker, W.F Design of Therml Systems. McGrw-Hill Book Compny. Tyl, M.C., Fu, Y., Diwekr, U.M Optimum Design of Het Exchngers: A Genetic Algorithm Frmework. Industril Engineering Chemicl Reserch. 38: Venktrthnm, G Mtrix Het Exchngers. Ph.D. Thesis, Indin Institute of Technology, Khrgpur. 27

28 von Spkovsky, M.R., Evns, R.B The Design nd Performnce Optimiztion of Therml System Components. Journl of Energy Resources Technology. 111: Wolfersdorf, J.V., Achermnn, E., Weignd, B Shpe Optimiztion of Cooling Chnnels Using Genetic Algorithms. Trnsctions of ASME Journl of Het Trnsfer. 119: Younes, M., Potiron, A. A Genetic Algorithm for the Shpe Optimistion of Prts Sujected to Therml Loding. Numericl Het Trnsfer, Prt A. 39: Zuir, S.M., Kd, P.V., Evns, R.B Second-Lw-Bsed Thermoeconomic Optimistion of Two-Phse Het Exchngers. Trnsctions of ASME Journl of Het Trnsfer. 109:

29 Memer 1 SINGLE POINT CROSSOVER Memer Before Crossover After Crossover FIGURE 1. Schemtic representtion of crossover technique MUTATION Before Muttion After Muttion FIGURE 2. Schemtic representtion of muttion technique Genertion of initil popultion Evlution of fitness vlue Reproduction, Crossover nd Muttion Genertion of new popultion Select individul hving mximum fitness yes convergence stisfied? no N G = N G +1 numer of genertions N G N G,mx? no END yes FIGURE 3. Flowchrt for Genetic Algorithm computtion 29

30 () Fluid () Fluid 1/n s l H h L t L FIGURE 4. () crossflow plte-fin het exchnger () offset-strip fin. Fluid L L Fluid FIGURE 5. Crossflow Plte-Fin Het Exchnger with two lyers () GA solutions grdient serch technique () 0.35 Q=160 kw 0.34 GA solutions grdient serch technique L, m L, m L, m L, m FIGURE 6. Different GA runs () without het duty () with het duty constrint. 30

31 L, m () 2.05E4 2E4 2.1E4 1.7E4 1.65E4 GA solutions grdient serch technique 2E4 1.8E4 1.75E4 2.1E4 1.9E4 1.85E4 2.05E4 1.95E L, m L, m () 2.2E4 3.4E4 3.8E4 2.8E4 2.4E4 4.6E4 GA solutions grdient serch technique 4.2E4 5E4 Q=160 kw L, m FIGURE 7. Totl cost contour () without het duty () with het duty constrint long with conventionl nd GA solutions f mx () Mximum fitness, f mx Totl Cost, TAC f mx TAC TAC - $ f mx () Mximum fitness, f mx Totl Cost, TAC f mx TAC TAC - $ Popultion Size Crossover Proility, P c f mx (c) f mx Mximum fitness, f mx Totl Cost, TAC TAC TAC - $ f mx (d) f mx TAC TAC - $ E Muttion Proility, P m Mximum fitness, f mx Totl Cost, TAC Penlty Prmeter, R

32 f mx (e) TAC f mx Mximum fitness, f mx Totl Cost, TAC Penlty Prmeter, R TAC - $ f mx (f) f mx TAC Mximum fitness, f mx Totl Cost, TAC Penlty Prmeter, R TAC - $ FIGURE 8. Effect of different GA prmeters, () popultion size () crossover proility (c) muttion proility, nd penlty prmeters (d) R1, (e) R2, nd (f) R3 on mximum fitness nd totl nnul cost Re A E4 Re 2.5E TAC 3E4 8E5 B L - m E4 4.8E5 O 1.6E5 Q Re C Re L - m FIGURE 9. Totl nnul cost nd het duty contours in the design spce with flow restrictions. 32

33 E5 2.4E5 3.2E5 4E5 2E4 1.9E4 1.8E4 0.7 L - m E4 Q 1.6E4 1.7E TAC 1.4E4 1.5E E4 1.7E4 1.8E L - m FIGURE 10. Totl nnul cost nd het duty contours without ny flow restriction. Optimum Totl Cost, TAC - $ () TAC L - m P P Pressure Drops, P nd P - kp Optimum Totl Cost, TAC - $ () TAC P P L - m Pressure Drops, P nd P - kp FIGURE 11. Effect of vrition of length on totl cost nd pressure drops. () Vrition of L, nd () vrition of L. 33

34 Optimum Totl Cost, TAC - $ () TAC P P L - m Pressure Drops, P nd P - kp Optimum Totl Cost, TAC - $ () TAC P P L - m Pressure Drops, P nd P - kp Optimum Totl Cost, TAC - $ TAC P P Pressure Drops, P nd P - kp Numer of Lyers, N FIGURE 12. Effect of vrition of L (), L (), nd N (c) on optimum totl cost nd corresponding pressure drops on the two sides. Optimum Cost, TAC - $ () with lminr flow constrint TAC Q TAC without lminr flow constrint Q L - m Het Duty, Q - Wtts Optimum Cost, TAC - $ () with lminr flow constrint TAC Q TAC without lminr flow constrint Q L - m Het Duty, Q - Wtts FIGURE 13. Vrition of optimum totl cost nd het duty with length () L, nd () L. 34

35 TABLE 1. Solution for two lyer het exchnger. L, m L, m Totl cost (TAC), $ Without het grdient serch technique duty constrint GA With het duty grdient serch technique constrint GA TABLE 2. Solution for multilyer het exchnger with flow constrint. L, m L, m H, mm n, fins/m t, mm l f, mm N TAC, $ Q, kw TABLE 3. Solution for multilyer het exchnger without flow constrint. L, m L, m H, mm n, fins/m t, mm l f, mm N TAC, $ Q, kw