MODELING OF PLATE HEAT EXCHANGERS WITH GENERALIZED CONFIGURATIONS

Transcription

1 XV OGRESSO BRASLERO DE EGEHARA MEÂA 6t BRAZLA OGRESS OF MEHAAL EGEERG MODELG OF LATE HEAT EXHAGERS WTH GEERALZED OFGURATOS Jorge Andrey Wlelms Gut Department of emal Engneerng - Unversty of São aulo Av. rof. Luano Gualberto, trav. 3, 380 / São aulo - S / / BRAZL andrey@lsp.pq.ep.usp.br José Maurío nto Department of emal Engneerng - Unversty of São aulo Av. rof. Luano Gualberto, trav. 3, 380 / São aulo - S / / BRAZL jompnto@usp.br Abstrat. A matematal model s developed for te smulaton of gasketed plate eat exangers, operatng n steady state, wt a general onfguraton. Te man purposes of ts model are to study te nfluene of te onfguraton on te equpment performane and to furter develop a metod for optmzng te exanger onfguraton. Te onfguraton s defned by sx parameters, w are as follows: number of annels, number of passes at ea sde, flud loatons, feed onneton loatons and type of annel-flow. Te matematal model s developed troug an assemblng algortm, sne t s not possble to represent te model expltly as a funton of te sx parameters. Te resultng system s omposed of ordnary dfferental equatons of te boundary value type, w s solved by te fnte dfferene metod, usng te software groms (roess Systems Enterprse, 00). Te man smulaton results are te followng: temperature profles n all annels, termal effetveness, dstrbuton of te overall eat transfer oeffent along te exanger and flud pressure drops. Examples of eat exangers wt approxmately 50 plates yeld algebra models wt tousands of equatons, w are solved wtn mnutes. Moreover, te assumpton of nvarant overall eat transfer oeffent s verfed. Keywords: plate eat exanger, matematal modelng, fnte dfferene metod, eat exanger onfguraton. ntroduton For beng ompat, easy to lean, effent and very flexble, te gasketed plate eat exanger (HE) s wdely employed n te emal, food and parmaeutal proess ndustres. Te HE onssts of a pak of gasketed orrugated metal plates, pressed togeter n a frame (see Fg. ). Te gaskets on te orners of te plates form a seres of parallel flow annels, were te fluds flow alternately and exange eat troug te tn metal plates. Te gasket desgn and te losed ports of te plates determne te flud flow dstrbuton, w an be parallel, seres or any of ter varous possble ombnatons. Te number of plates, flow dstrbuton, type of gaskets and te flud feed loatons araterze te exanger onfguraton. Fgure. Te plate eat exanger assemblage, examples of flow dstrbutons and plate man dmensons. Te smplfed termal modelng of a HE n steady state yelds a lnear system of frst order ordnary dfferental equatons, omprsng te energy balane for ea annel and te requred boundary ondtons. Te man assumptons are as follows: plug-flow nsde te annels, onstant overall eat transfer oeffent along te exanger, unform dstrbuton of flow n te annels, no eat loss and no eat exange n te flow dreton. Ts bas model was presented by MKllop & Dunkley (960) for 3 dfferent onfguratons and also by Masubu & to (977), were te

2 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 377 dynam responses of some usual onfguratons were studed. n bot works, te Runge-Kutta-Gll ntegraton metod was used to solve te system of equatons. Te ntegraton s non-trval beause te boundary ondtons are defned at dfferent extremes of te annel. Approxmate solutons were developed by Settar & Venart (97) n polynomal form, and by Zalesk & Klepaka (99) n exponental form. Bot metods lead to good approxmatons of te exat soluton, but tey may not be relable wen tere s a large dfferene between flud eat apates. Kandlkar & Sa (989b) developed a metod to alulate an approxmated termal effetveness for large exangers, were te effets of te end plates and of te anges of passes an be negleted. n ts ase, te exanger s dvded nto a group of smpler exangers tat are nteronneted, wt known effetveness. Te analytal soluton of te system of equatons n matrx form was studed by Zalesk & Jarzebsk (973) and Zalesk (984) for exangers wt seres and parallel arrangements. Ts soluton metod may lead to numeral problems on te alulaton of egenvalues and egenvetors, and t s not reommended for large szed exangers. Kandlkar & Sa (989a) and Georgads et al. (998) used te fnte dfferene metod for te smulaton of HEs. Kandlkar & Sa (989a) smulated and ompared several onfguratons. t was verfed tat ger effetveness s aeved wen te exanger s symmetral, wt te same numbers of passes for bot streams, beause te annels tat are next to te anges of passes as well as te end annels ave a lower effetveness. However, wen te fluds ave very dfferent flow rates or eat apates, a non-symmetral onfguraton must be used. n su ases, tere s no rgorous desgn metod to selet te best onfguraton, w s made by omparson among te usual onfguratons from termal effetveness and pressure drop vewponts. Georgads et al. (998) presented a detaled modelng of a HE used for mlk pasteurzaton tat ouples te dynam termal model wt te proten-foulng model. Tree dfferent onfguratons were ompared and te reduton of te overall eat transfer oeffent, aused by te proten adeson on te plates, was studed. Te model was solved wt te fnte dfferene metod, mplemented n te software groms (roess System Enterprse, 00). To te autors knowledge tere s no rgorous desgn metod for HEs n te open lterature, as tere are for te sell-and-tube exangers (Taborek, 983). Sa & Foke (988) ave presented a detaled step-by-step desgn proedure for ratng and szng a HE, w s owever restrted to parallel flow arrangements. n all of tose works, te overall eat transfer oeffent was onsdered nvarable along te exanger beause ts assumpton brngs a great smplfaton for te soluton of te system of equatons, lnearzng te dfferental equatons of te annel flud temperatures. However, tere may be a onsderable varaton of te overall oeffent for some ases, su as te seres flow arrangement wt equal flow rates of te fluds (Buonopane et al, 963). Te am of ts work s to present a HE modelng framework tat s sutable for any onfguraton. Te purpose of developng su model s to study te nfluene of te onfguraton on te exanger performane and to furter develop an optmzaton metod for onfguraton seleton. Te frst step for te modelng onerns te parameterzaton of te dfferent onfguratons, followng te work of gnott & Tamborenea (988). Te varaton of te overall eat transfer oeffent along te exanger s also studed n ts work, wt respet to te assumpton of a onstant value. omenlature a gener model parameter A effetve eat transfer area (m ) b annel average tkness (m) p flud spef eat at onstant pressure (J/kg.K) D e equvalent dameter of annel (m) D port dameter (m) E exanger termal effetveness (%) f Fannng frton fator g gravtatonal aeleraton (g 9,8 m/s ) G annel mass veloty (kg/m.s) G port mass veloty (kg/m.s) onvetve eat transfer oeffent (W/m.K) k flud termal ondutvty (W/m.K) k plate termal ondutvty (W/m.K) L effetve plate lengt, measured between ports (m) M dagonal matrx defned wt Eq. (0) number of annels per pass number of annels u usselt number number of passes r randtl number R flud foulng fator (m.k/w) Re Reynolds number s annel flow dreton parameter (s + or ) T temperature (K) U overall eat transfer oeffent (W/m.K) w effetve plate wdt, measured between gaskets (m) W flud mass flow rate (kg/s) x oordnate, tangental to annel flud flow (m) Y f bnary parameter for type of annel-flow Y bnary parameter for ot flud loaton α dmensonless varable defned wt Eq. (9) β evron orrugaton nlnaton angle (degrees) flud pressure drop (a) ρ flud densty (kg/m 3 ) ε maxmum devaton for alulated termal effetveness ε tkness of metal plate (m) φ parameter for feed onneton relatve loaton Φ enlargement fator of plate η dmensonless oordnate, tangental to annel flud flow µ flud vsosty (a.s) θ dmensonless flud temperature Subsrpts n max mn out w old flud ot flud gener element flud nlet maxmum value mnmum value flud outlet at te plate wall Supersrpts sde of exanger sde of exanger

3 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 378. onfguraton araterzaton Te onfguraton of a HE s defned by te nformaton tat allows te detalng of te equpment assemblage, nludng te onnetons on te fxed and moveable overs, te losed and open ports n ea plate and te type and poston of ea gasket. To araterze su onfguratons, sx dstnt parameters are used:,,, φ, Y and Y f. Tese parameters are defned wt detals n Tab. (). Table. araterzaton of te sx onfguraton parameters. : umber of annels arameter desrpton Te spae omprsed between two plates s a annel, and te HE an be represented by a row of annels, numbered from to. Te odd-numbered annels belong to sde, and te even-numbered ones belong to sde (as an analogy to te tube and sell sdes n a sell-and-tube exanger). and are te numbers of annels n ea sde. f s even, bot sdes ave te same number of annels, oterwse sde as one more annel. Allowable values:, 3, 4, 5 and : umber of passes at sdes and A pass s a set of annels were te stream s splt and dstrbuted. For a regular onfguraton, ea sde of te HE s splt nto passes wt te same number of annels per pass ( and ). asses wt dfferent numbers of annels are unusual (Kakaç & Lu, 998). Te relatonsp between, and s gven n Tab. (). Allowable values: from te fatorzaton of and respetvely φ : Feed onneton relatve loaton Te feed onneton of sde s arbtrarly set n annel at η0. Te relatve poston of te feed of sde s gven by te parameter φ, as sown n te dagram (gnott & Tamborenea, 988). Te dmensonless lengt η s not assoated wt te top and bottom of te HE, neter annel s assoated wt te fxed over. Te onfguraton an be freely rotated or mrrored. Allowable values:,, 3 and 4 Y : Hot flud loaton Ts bnary parameter assgns te fluds to te exanger sdes: llustratve fgure - f Y : te ot flud s at sde, and te old flud at sde - f Y 0 : te old flud s at sde, and te ot flud at sde Y f : Type of flow n annels Y f s a bnary parameter tat defnes te type of flow nsde te annels. As sown n te dagram, te flow an be stragt or rossed, dependng on te gasket type. Te rossed flow avods te formaton of stagnaton areas, but te stragt flow type s easer to assemble. t s not possble to use bot types togeter. - f Y f : te flow s rossed n all annels - f Y f 0 : te flow s stragt n all annels Te sx parameters an represent any regular onfguraton. For a fxed number of annels, te fve remanng parameters ave a known set of allowable values. Te ombnaton gves a fnte number of possble regular onfguratons for ea value of, as presented n Fg. (). Te dsperse pattern s due to te varaton of te number of nteger fators of and for ea value of. A onfguraton example s sown n Fg. (3) for llustraton. t represents an egt-plate HE, were te ot flud n sde (Y ) makes passes ( ) and te old flud on sde makes 3 passes ( 3). n ts example, te nlet of

4 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 379 sde s loated next to te nlet of sde (φ) and te type of annel-flow s rossed flow (Y f ). Te parameter Y f for type of annel-flow s mostly useful for te exanger pysal onstruton and t may not be neessary for te smulaton sne ts nfluene over te onvetve oeffents and frton fators s usually unknown. Table. Relatonsp between numbers of annels and passes. Man equatons +.. f s even:.. + f s odd: +.. 7,000 umber of possble regular onfguratons 6,000 5,000 4,000 3,000,000, umber of annels - Fgure. umber of possble regular onfguratons as a funton of te number of annels. Fgure 3. Example of onfguraton for a HE wt egt plates. 3. Equvalent onfguratons For a gven value of number of annels and a fxed type of flow, te exstene of equvalent onfguratons (tat ave te same termal effetveness and pressure drops) s possble. Te dentfaton of te equvalent onfguratons s mportant to avod unneessary smulatons. Te equvalene ours due to 3 remarks, as follows: A) Aordng to te property of flow reversblty (gnott & Tamborenea, 988), te nverson of te flud flow dreton n bot sdes does not alter te effetveness of te HE. A) Wen tere s a sngle pass n a sde, te flow dreton s te same n all annels, regardless f te feed s loated at te frst annel or at te last one. A3) Smply nvertng te dreton of η, or numberng te annels n reverse order, may yeld to a new set of onfguraton parameters. A metodology to detet equvalent onfguratons s presented n Tab. (3). For ea set,, and Y f tere are groups of te parameter φ tat result n equvalent onfguratons. n te ase of an even-numbered, tere may be equvaleny between Y 0 and Y, beause sdes and ave te same number of annels and terefore an support te same passes. An example of equvaleny between 4 dfferent onfguratons s sown n Fg. (4).

5 Table 3. dentfaton of equvalent onfguratons for gven values of and Y f. roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 380 (, ) Groups of equvalent values of φ Reduton n te number of smulatons (, ) ; (, odd) ; (odd, ) {, 3} ; {, 4} 50 % odd (, even) ; (even, ) {,, 3, 4} 75 % (odd, odd) ; (even, even) {} ; {} ; {3} ; {4} 0 % (odd, even) ; (even, odd) {, } ; {3, 4} 50 % (, ) ; (, odd) ; (odd, ) {, 3,, 3} ; {, 4,, 4} 75 % (, even) {, 4,, 4} ; {, 3,, 3} 75 % even (even, ) {, 3,, 3} ; {, 4,, 4} 75 % (odd, odd) ; (even, even) {, } ; {, } ; {3, 3} ; {4, 4} 50 % (odd, even) ; (even, odd) {, } ; {, } ; {3, 3} ; {4, 4} 50 % ote: wen s even, denotes Y and denotes Y 0. Wen s odd, all equvalent onfguratons ave te same value for Y. Fgure 4. Example of four equvalent onfguratons wt 6 and Y f. 4. Modelng of te HE Te followng assumptons were made n order to derve te matematal model: B) Steady state operaton. B) o eat loss to surroundngs. B3) o eat exange on te dreton of flow. B4) lug-flow nsde te annels. B5) Unform dstrbuton of flow troug te annels of a pass. B6) erfet mxture of flud n te end of a pass. B7) Fluds wt ewtonan beavor. B8) o pase anges. Te flud nsde a annel exanges eat wt te negbor annels troug te tn metal plates, as sown n Fg. (5). Te effetve eat exange area s A Φ.w. L, were Φ s te area enlargement fator, w aounts for te orrugaton wrnkles. Te lengt of te pat x and te flud temperature T(x) an be onverted nto dmensonless form as sown n Eqs () and (). x η( x ) 0 η () L T T,n θ (T ) 0 θ () T T,n,n

6 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 38 Applyng te energy balane to te ontrol volume sown n Fg. (5) (MKllop & Dunkley, 960) t s possble to derve te dfferental equatons for te annel temperature (Eqs 3, 4 and 5). Te overall eat transfer oeffent U between annels and + s gven by Eq. (6) as a funton of te flud onvetve eat transfer oeffent, te plate termal ondutvty k and te foulng fator R for ot and old streams. Te mass flow rate nsde annel s alulated by Eq. (7) aordng to assumpton B5, were sde() refers to te sde tat ontans annel. t s known tat ts assumpton may not old for passes wt a large number of annels (Bassouny & Martn, 984), but tere s no rgorous modelng to ts beavor. Te flow rates n ea sde, W and W, are assoated to te ot and old flud flow rates troug te parameter Y. Te dreton of te flow n annel s gven by te varable s. f te flow follows te dreton of η, ten s +, oterwse s. For te frst annel: dθ s.a U. θ dη W.p [ ( θ )] (3) For annel : ( < < ) dθ s.a [ U.( θ θ ) + U.( θ+ θ )] (4) dη W.p For te last annel: dθ s.a dη W.p [ U.( θ θ )] (5) U + + ε + k + R + R, ( ) (6) Fgure 5. ontrol volume for dervaton of energy balane nsde an upward flow annel. sde( ) W W, sde( ) sde() {, } (7) Te overall eat transfer oeffent s a funton of te flud temperature n te annels (Eq. 6), w depends on η. Sne ts makes U also a funton of η, te soluton of te system of dfferental Eqs (3), (4) and (5) s not smple. However, f te flud pysal propertes are assumed onstant, Eq. (4) an be smplfed to Eq. (8), were te oeffents α and α are gven by Eq. (9) as a funton of te onstant overall eat transfer oeffent U. onsequently, te system of equatons s redued to a lnear system of ordnary dfferental equatons, w s represented n matrx form n Eq. (0), were M s a trdagonal matrx and θ s te vetor of annel dmensonless temperatures θ (η). Te analytal soluton of Eq. (0) was studed by Zalesk & Jarzebsk (973). dθ dη s. α sde( ).( θ. θ + θ ) +, sde() {, } (8) sde( ) sde( ) A.U. α, sde( ) sde( ) sde() {, } (9) W.p dθ M. θ dη (0) Te neessary boundary ondtons for te smplfed model, defned by Eq. (8), and for te rgorous model, defned by Eqs (3) troug (6), are presented n Tab. (4). Every annel requres a boundary ondton equaton for ts nlet temperature. Te nlet poston of annel s gven by te value of s. f s + te nlet s loated at poston η 0, oterwse te nlet s loated at η. Te number and struture of te requred boundary equatons are funtons of te onfguraton parameters. For te performane evaluaton of te eat exanger, te termal effetveness and te pressure drop alulatons are needed. One te system of equatons s solved, te termal effetveness E an be alulated by Eq. () usng te average value of te flud spef eats, were W and W are related to W sde() by te parameter Y. W.p E mn W.( θ ) (.p,w.p ) mn( W.p,W.p ),out W.p. θ,out ()

7 Table 4. Types of termal boundary ondtons for te HE annels. roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 38 Boundary ondton Flud entrane: te temperature at te entrane of te frst pass s te same as te stream nlet temperature. anges of pass: tere s a perfet mxture of te flud leavng te annels of a pass, before enterng te next one. Flud ext: te stream outlet temperature results from a perfet mxture of te flud leavng te last pass. θ η) Equaton Form θ ( θ, frst pass flud,n ( η ). θ j ( η ) j prevous pass θ flud,out. θ j ( η ) j last pass, urrent pass Te flud pressure drops at sdes and, and, an be alulated by Eq. () tat reles on assumptons B7 and B8 (Sa & Foke, 988; Kakaç & Lu, 998). Te frst term n te rgt-and sde evaluates te frton loss nsde te annels, were G denotes te annel mass veloty (Eq. 3a). Te seond term represents te pressure drop for port flow, were G s te port mass veloty (Eq. 3b). Te last term s te pressure varaton due to an elevaton ange. Sne te gravtatonal aeleraton dreton s not assoated to te vertal dmenson η and tere s no nformaton on te pump loaton, ts term s always onsdered. Terefore te pressure drop may be overestmated..( L + D ). f..g G +, ρ.g. +.D e. ρ ρ ( L D ) for sdes and () W G,.b.w G 4W. for sdes and (3) π.d Te neessary onsttutve equatons for te alulaton of onvetve oeffents and frton fator, aordng to assumpton B7, are presented n Tab. (5), as well as te neessary dmensonless numbers and te annel equvalent dameter D e. Usual values for te empral parameters a to a 6 of te onsttutve equatons an be found on Saunders (988) and Sa & Foke (988). Table 5. Equatons for onvetve oeffents and frton fator alulaton, for sdes and. onsttutve Equatons Dmensonless umbers Equvalent Dameter u a.re a.r a3 a f a + Re. 5 4 a6 µ µ w 0, 7.De u, k. r k µ D e Re G. µ 4.b.w.b D e.( b + w. Φ) Φ 5. Smulaton of te HE Te matematal modelng of a HE, for te alulaton of ts termal effetveness and flud pressure drops, was presented n te prevous seton. However, t s not possble to derve a model tat s expltly a funton of te onfguraton parameters, espeally beause of te bnary varable s and of te requred boundary ondtons (te equatons n Tab. 4). To overome ts lmtaton, te modelng was developed n te form of an assemblng algortm. Gven te onfguraton parameters, ts algortm gudes te onstruton of te omplete matematal model of te HE and ts smulaton at steady state operaton. n ts work, te soluton of te system of equatons s arred out by te software groms (roess Systems Enterprse, 00), usng te seond order entered fnte dfferene metod. n ts metod, all varables dependng on te plate lengt η are dsretzed wt respet to ts dmenson. Several semes were tested and, wt 0 dsretzaton ntervals wtn η [0,], an exellent approxmaton of te analytal result was aeved. Te smplfed model tat reles on te assumpton of onstant overall eat transfer oeffent and te rgorous model developed by relaxng ts assumpton were tested and ompared. Te man struture of te assemblng algortm for te smplfed model s presented n Fg. (6). Te algortm as 4 steps as follows:

8 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 383 Fgure 6. Man struture of te model assemblng algortm. ) Te requred data for te HE (L, w, b, D, ε, Φ, k ), for te ot and old fluds (T n, W, ρ, µ,, k, R, a, a 6 ) and te onfguraton parameters (,,, φ, Y, Y f ) are read. ) Te parameter Y assgns all flud data to sdes and. 3) umbers of annels per pass and are alulated dependng on te value of (see Tab. ). 4) ressure drops are alulated for bot sdes of te HE wt Eqs (), (3) and te equatons n Tab. (5). 5) oeffents α and α are obtaned by Eq. (9), usng te onstant overall eat transfer oeffent U, w s obtaned from Eq. (6) usng onstant eat transfer oeffents for sdes and, and. 6) Te values of s (, ) are determned. Tese depends on te onfguratons parameters,, and φ. 7) Te assemblng of te system of equaton starts wt te dfferental equatons for te HE nner annels (Eq. 8). 8) Dfferental equatons for te temperature n te frst and last annels are nluded n te system. 9) Boundary ondtons for te annels n sde are generated n te format sown n Tab. (4). Te flud pat nsde te exanger, from nlet to outlet, needs to be followed n order to determne te onnetons between annels. 0) Boundary ondtons for sde are also generated, but ea value of te parameter φ requres a spef treatment beause ts parameter determnes te flow dreton nsde annels. ) Equatons of termal effetveness for bot sdes (E and E ) are nluded n te dmensonless form of Eqs (4a) and (4b), w are obtaned usng Eqs (9) and (). E.max α α,. θ n θ out, α E.max α α,. θ n θ out (4) α ) Te resultng system of dfferental equatons, defned by te dfferental equatons on te temperature n ea annel, te boundary ondtons equatons and te effetveness equatons, s solved by numeral or analytal metods. 3) f a numeral soluton metod s used, te overall energy onservaton an be verfed by te effetveness values E and E. f E E > ε, were ε s te maxmum allowable devaton, te parameters of te numeral metod and/or te nput data sould be revsed. 4) Te man smulaton results, su as te pressure drops and outlet temperatures for sdes and (w are assgned to te ot and old streams usng Y ), as well as te HE termal effetveness, are obtaned. A omputer program was developed to run steps troug of te assemblng algortm n Fg. (6). After readng te data, te program makes all neessary alulatons, generates a report and reates te formatted nput fle for groms. Ts proedure made te smulaton of dfferent onfguratons mu smpler and effetve. Te dervaton of te assemblng algortm for te rgorous model s stragtforward. Step 5 sould be removed and all te neessary equatons for te alulaton of te annel overall eat transfer oeffent U (Eq. 6), su as dmensonless numbers, onsttutve equatons and te flud pysal propertes dependene upon annel temperature, sould be nserted n steps 7 and 8. Equaton (4) sould also be wrtten n te form of Eq. () for te effetveness alulaton n step. Tese modfatons nrease te sze and omplexty of te system of equatons and make te soluton more dffult beause of te larger number of dsretzed varables. 6. Smulaton Example An example of a HE applaton s presented to sow te smulaton results. A ot stream of toluene exanges eat wt a old stream of benzene n a medum-szed HE wt 5 plates (see Fg. 7). Bot rgorous and smplfed models were smulated and te man results are presented n Tab. (6) and Fgs (8), (9) and (0). For te smplfed model, te flud average temperatures were estmated targetng a termal effetveness of 90%.

9 roeedngs of OBEM 00, Energy and Termal Systems, Vol.4, 384 Table 6. Man smulaton results for te example Varable Rgorous Model Smplfed Model Devaton (%) HE termal effetveness (%) Benzene outlet temp. ( ) Toluene outlet temp. ( ) Benzene pressure drop (ka) Toluene pressure drop (ka) umber of varables after dsretzaton 4, U tme on workstaton (s) ,700,650 U 3,600 U7 U U6 U (W/m.K),550 U, U 9,500 U8 U Fgure 7. aratersts of te HE used for smulaton example,400,450 Rgorous model Smplfed model η Fgure 8. Dstrbuton of te overall eat transfer oeffent along te exanger 75.0 Sde - Benzene 75.0 Sde - Toluene nd ass st ass T ( o ) T ( o ) st ass nd ass rd ass η Fgure 9. Temperature profles for sde annels (rgorous model) η Fgure 0. Temperature profles for sde annels (rgorous model) Te temperature dstrbuton along te exanger, obtaned by te rgorous model, s sown n Fgs (8) and (9), were te stream passes and annel numbers are ndated. Te temperature varaton n te outer annels and 4 are lower tan n te oter annels n te same pass beause eat s exanged wt only one negbor annel. Te dstrbuton of te overall eat transfer oeffent U along te exanger, obtaned by bot models, s presented n Fg. (8). Te results from te rgorous model smulaton sowed tat te oeffent U vares from,464 to,665 W/m.K, wle te smplfed model was solved wt an average value of,56 W/m.K. Despte ts sgnfant

10 dfferene, te man smulaton results obtaned by bot models are very lose (Tab. 6), wt a devaton of only 0.35% for te exanger effetveness. Several examples ave sown tat, even wt a remarkable dfferene n te dstrbuton of te overall eat transfer oeffent, tere s lttle dfferene between te man smulaton results obtaned from bot rgorous and smplfed models. Te obtaned devatons were under.5%, wt respet to E. Te number of varables after te dsretzaton by te fnte dfferene metod s approxmately 9 tmes larger for te rgorous model. To make te soluton of large exangers relable, te blok deomposton opton of groms s used, were te system of equatons s deomposed nto smaller systems. Even wt ts opton, te soluton of large exangers (e.g. 50 plates) usng te rgorous model requres 3 to 5 mn n a DE-UX workstaton. For te ase of smaller exangers, as te one presented n ts paper, te soluton s aeved n a matter of seonds (Tab. 6). 7. onlusons Te onfguraton of a plate eat exanger (HE) was araterzed by a set of sx parameters and a metodology to detet equvalent onfguratons was presented. Based on ts parameterzaton, a detaled model for te smulaton of a HE n steady state wt a general onfguraton was developed n algortm form. Te developed assemblng algortm made te smulaton and omparson of dfferent onfguratons more flexble. An mportant feature of te proposed algortm s tat t may be oupled to any proedure to solve te system of dfferental and algebra equatons. Te assumpton of onstant overall eat transfer oeffent along te exanger, often used for te matematal modelng, was tested and sowed lttle nfluene over te man smulaton results for eat exange (termal effetveness and outlet temperatures). Te presented algortm s an mportant tool for te study of te nfluene of te onfguraton over te exanger performane, and an be furter used to develop an optmzaton metod for seletng te pleat eat exanger onfguraton. A resear n ts subjet s urrently n development. 8. Aknowledgments Te autors would lke to tank te fnanal support from FAES (grants 98/5808- and 00/3635-4). 9. Referenes Bassouny, M.K.; Martn, H., Flow Dstrbuton and ressure Drop n late Heat Exangers -, U-Type Arrangement, emal Engng Sene, v.39, n.4, pp , 984. Buonopane, R.A.; Troupe, R.A.; Morgan, J.., Heat Transfer Desgn Metod for late Heat Exangers, emal Engng rogress, v.57, n.7, pp.57-6, July 963. Georgads, M..; Rotsten, G.E.; Maetto, S., Modelng and Smulaton of omplex late Heat Exanger Arrangements under Mlk Foulng omputers & em. Engng, v. sups, Mar/Aprl, pp.s33-s338, 998. Kakaç, S.; Lu, H., Heat Exangers: Seleton, Ratng and Termal Desgn, R ress, ew York, 998. Kandlkar, S.G.; Sa, R.K., Multpass late Heat Exangers - Effetveness-TU Results and Gudelnes for Seletng ass Arrangements, ASME Journal of Heat Transfer, v., pp , 989a. Kandlkar, S.G.; Sa, R.K., Asymptot Effetveness-TU Formulas for Multpass late Heat Exangers, ASME Journal of Heat Transfer, v., pp.34-3, 989b. Masubu, M.; to, A., Dynam Analyss of a late Heat Exanger System, Bulletn of te JSME, v.0, n.4, pp , Aprl 977. MKllop, A.A.; Dunkley, W.L., late Heat Exangers: Heat Transfer, ndustral and Engng emstry, v.5, n.9, pp , Sept gnott, A.; Tamborenea,.., Termal Effetveness of Multpass late Exangers, nt. Journal of Heat and Mass Transfer, v.3, n.0, pp , Ot roess Systems Enterprse Ltd, groms ntrodutory User Gude, Release.0, London, 00. Saunders, E.A.D., Heat Exangers: Seleton, Desgn & onstruton, Longman S.&T., ew York, 988. Settar, A.;Venart, J.E.S., Approxmate Metod for te Soluton to te Equatons for arallel and Mxed-Flow Mult- annel Heat Exangers, nt. Journal of Heat Mass Transfer, v.5, pp.89-89, 97. Sa, R.K.; Foke, W.W., late Heat Exangers and ter Desgn Teory, n: Sa, R.K.; Subbarao, E..; Maselkar, R.A., Edtors, Heat Transfer Equpment Desgn, pp.7-54, Hemspere.., ew York, 988. Taborek, J., Sell-and-tube Heat Exangers: Sngle-pase Flow, n: Hewtt, G.F., Edtor, Handbook of Heat Exanger Desgn, s.3.3, Begell House, ew York, 99. Zalesk, T., A General Matematal-Model of arallel-flow, Multannel Heat-Exangers and Analyss of ts ropertes, emal Engng Sene, v.39, n.7/8, pp.5-60, 984. Zalesk, T.; Jarzebsk, A.B., Remarks on Some ropertes of Equaton of Heat-Transfer n Multannel Exangers, nt. Journal of Heat Mass Transfer, v.6, n.8, pp , 973. Zalesk, T.; Klepaka, K., Approxmate Metods of Solvng Equatons for late Heat-Exangers, nt. Journal of Heat Mass Transfer, v.35, pp.5-30, 99.