Mathematical Modelling of Transient Response of Plate Fin and Tube Heat Exchanger

Transcription

1 Proceedgs of e Iteratoal Coferece o Heat rasfer ad Flud Flo Prague Czech Republc August - 4 Paper o. 9 Maematcal Modellg of raset Respose of Plate F ad ube Heat Exchager Aa Korzeń Craco Uversty of echology Isttute of hermal Poer Egeerg al. Jaa Pała II Craco Polad korze@mech.pk.edu.pl Dad aler Craco Uversty of echology Isttute of hermal Egeerg ad Ar Protecto ul. Warszaska Craco Polad dtaler@pk.edu.pl Abstract - Cross-flo tubular heat exchagers are appled as codesers ad evaporators ar codtoers ad heat pumps or as ar heaters heatg systems. hey are also appled as ater coolers so called dry ater coolg systems of poer plats as ell as car radators. here are aalytcal ad umercal maematcal models of heat exchagers of at type to determe e steady state temperature dstrbuto of fluds ad e rate of heat trasferred betee fluds. I ve of e de rage of applcatos practce ese heat exchagers ere expermetally examed steady-state codtos mostly to determe e overall heat trasfer coeffcet or e correlato for e heat trasfer coeffcets o e ar sde ad o e teral surface of e tubes. here exst may refereces o e traset respose of heat exchagers. Most of em hoever focus o e o-steady-state heat trasfer processes parallel ad couter flo heat exchagers. I s paper e e equato set descrbg traset heat trasfer process tube ad f cross-flo tube exchager ll be gve ad subsequetly solved usg e fte volume meod. Keyords: ube ad plate-f heat exchager raset respose Plate-f maematcal model Expermetal valdato.. Itroducto I cotrast to e exstg meods for modellg traset respose of heat exchagers exteded surfaces hch e eghted steady-state heat trasfer coeffcet o e fed tube sde s used e traset temperature dstrbuto ll be calculated each fs )Sm 997; Roetzel Xua 998; aler 9). hs allos for computg more exactly e heat flo rate from e fs to e flog gas. he axal heat coducto e tube all ll be also accouted for. raset temperature dstrbutos cotuous fs attached to oval tubes ll be calculated usg e fte volume fte elemet meod (aler Korzeń a; aler Korzeń b). A system of dfferetal equatos of e frst order for traset temperature at e odes ll be solved usg e explct fte dfferece meod. he developed meod for e determg traset temperature dstrbutos fs s used e traset aalyss of compact heat exchagers to calculate correctly e heat flo rate trasferred from e fed surface to e flud. A traset respose of e cross-flo tube ad f heat exchagers s aalyzed. raset equatos for bo hot ad cold flud are solved usg e fte volume meod. A maematcal model of a car radator ll be developed ad examed expermetally. he automotve radator for e spark-gto combusto ege a cubc capacty of 58 cm3 s a double-ro topass plate-fed heat exchager. he radator cossts of alumum tubes of oval cross-secto. he coolg lqud flos parallel rough bo tube ros. A traset respose of e tube ad plate f heat exchager due to step chage of ar temperature ll be modeled usg e developed meod. he 9-

2 umercal model as valdated by comparso of outlet ater ad ar temperatures obtaed from e umercal smulato e expermetal data. Good agreemet betee e umercal predctos ad expermetal results has bee foud.. Maematcal Model of Oe-ro Heat Exchager A umercal model of a cross flo tubular heat exchager hch ar flos trasversally rough a ro of tubes (Fg. ) ll be preseted. he system of partal dfferetal equatos descrbg e space ad tme chages of: lqud tube all ad ar temperatures are respectvely x τ t () mc k U U c h U h U t s t L x f f f m m ch o z y t () τ (3) here deotes e mea ar temperature over e ro ckess defed as x t x y t dy. (4) he symbols x x Lch ad y y p he umbers of heat trasfer uts ad are gve by equatos (-3) stad for dmesoless coordates. h Arg m c p h Azrg (5) m c p here: s Lch / f Arg r U Lch Azrg r Uz Lch. he tme costats τ ad τ are τ m c p p τ h Arg h Azrg m c. (6) he symbols equatos (-6) deote: m r A Lch m r p p Aoval s f f m r Um Lch m p p A A π ab A π a δ b δ U U U /. f oval f he subscrpt refers to e all f to e f ad m to e mea value. he eghted heat trasfer coeffcet h o s defed by oval m z 9-

3 A A h o t h t f h t Azrg Azrg mf f. (7) he tal temperatures of bo fluds are equal ad amout to. he tal codtos are x t x (8) x t t x (9) x y t t x y t he boudary codtos have e follog form x t f t x y t f t x () () y () x x x x Lch (3) (4) here ad are fuctos of tme descrbg e varato of e let lqud temperature ad let ar temperature. he tal boudary problem formulated above ( 4) apples to heat exchagers made of bare tubes ad also from fed oes. For bare tubes mf s equal to zero. a) b) Fg.. Scheme of e aalyzed heat exchager (a) ad cotrol volume (b). he traset fluds ad tube all temperature dstrbutos e oe-ro heat exchager (Fg. ) ll be determed by e explct fte dfferece meod. o calculate tme depedet effcecy ηf of e rectagular f attached to a oval tube e fte volume fte elemet meod ll be used... Fte Dfferece Model of e Heat Exchager Whe actual heat exchagers are calculated e ermo-physcal propertes of e fluds ad e heat trasfer coeffcets deped o e temperature of e flud ad e tal boudary problem (-4) s a 9-3

4 o-lear oe. I such cases e Laplace trasform caot be appled. he temperature dstrbuto (x + t) (x + t) ad (x + y + t) ca e be foud by e explct fte dfferece meod. I at meod e tme dervatve s approxmated by a forard dfferece hle e spatal dervatves are approxmated by backard or cetral fte dffereces. he equatos ( 3) are approxmated usg e explct fte dfferece meod as follos (Fg. b): x t (5) mc f f f f f f Umc t s t t (6) ku m h U ho U z L ch x (7) t here (8) he otato used Equatos (5-8) s llustrated Fg.. Sce e secod dervatve Eq. as approxmated by e cetral dfferece quotet e magary odes at e eds of e tube are ecessary. he boudary codtos (3) ad (4) are approxmated as follos x x (9) () Solvg Eqs (9) ad () for ad gves ad. () Equato () s accouted for Eqs (6) for = ad =. he umbers of odes are sho Fg.. I P I I I. W he otato Fg. s as follos: P R 9-4

5 Fg.. Fte dfferece grd used e calculato of temperature dstrbuto; P(I) let ar temperature R(I) tube all temperature P(I) outlet ar temperature. he uko temperature s foud from Eq. (5) from Eq. (6) ad from Eq. (7): t t x t ku m h U f mc f f L ch Umc x s m f c f f ho U z s t t t (4) () (3) here x /. Equato (3) as derved assumg at sce e tme step t s small ad e dfferece f f betee f ad f s eglgble. he tal codtos (8 ) ad e boudary codtos (-) assume e form: - tal codtos x x x x... (5) (6) (7)... (8) here x x - boudary codtos 9-5

6 f (9) f... (3) here f f t f f t.... I order to esure stablty of e calculatos by e explct fte dfferece meod e codtos of Courat Fredrchs Ley must be satsfed (Press et al. 7) t x (3) t (3) o avod umercal dffuso e tme step should ot be too small (Press et al. 7). Because of e hgh ar flo velocty e tme step t resultg from e codto (3) should be very small e rage of tes of ousads of a secod. he lqud ar ad tube all temperature dstrbutos are calculated usg e formulas () ad (4) takg to cosderato e tal (5-8) e boudary codtos () (9-3) startg at =. o compare smulato ad measuremet results a smple model of e ermocouple as used to calculate ermocouple respose (t) he flud temperature a (t) s ko from e umercal smulato. he traset respose of e ermocouple ca be descrbed by a smple dfferetal equato (Jaremkecz et al. 9) d a (33) dt here m c / h A s e tme costat of e ermocouple. he symbols equato (33) are: m - ermocouple mass c - specfc heat capacty of e ermocouple h - heat trasfer coeffcet o e surface of e ermocouple ad A - area of e outsde surface of e ermocouple. Approxmatg e tme dervatve Eq. (33) by e cetral dfferece quotet ad trasformg Eq. (33) gves a t t (34) t here t t t t ad t s e samplg tme terval durg temperature measuremet by meas of e data acqusto system. he tme costat of e ermocouple s a fucto of ar velocty a at e cross secto here e ar temperature as measured (Jaremkecz et al. 9). hs tme costat of e ermocouple depeds o ar velocty ad as approxmated by e follog fucto a (35) here e tme costat s expressed s ad ar velocty a m/s. 9-6

7 .. Modellg of raset Heat rasfer rough Rectagular Fs he meod preseted (aler Korzeń a; aler Korzeń b) as used to calculate traset temperature dstrbuto e fs ad subsequetly tme depedet f effceces. raset temperature dstrbutos cotuous fs attached to oval tubes ere calculated usg e fte volume fte elemet meod. he cotuous plate f s broke to e magary fs. he magary f model s dvded to tragular elemets ad e fte volumes ere formed aroud e odes by coectg tragle gravty cetres sde cetres of tragles (Fg. 3a). After calculatg e temperature dstrbuto e heat trasferred from e f to e evromet ad f effcecy ll be computed ad compared e results obtaed by usg e commercal softare ASYS.. he f as dvded to 9 fte volumes (Fg. 3a). he lateral surfaces: are ermally sulated hle o e surfaces:4-6-3 ad -5- covecto heat trasfer occurs. he computatos ere carred out for e follog data: c = 896 J/(kg K) = 77 kg/m 3 k = 7 W/(m K) δ f =.8 mm f = o C b = o C = o C h = 5 W/(m K). a) b) Fg. 3. F attached to oval tube (a) ad tme chages of f temperature at odes ad. he f effcecy as also determed based o e temperature dstrbuto obtaed by e developed meod f j Af hb f A A 3 o f l j l j f l (36) here e symbols deote: A 3 surface area of e tragle o f temperature at e gravty ceter of e tragle umber of tragles A lj area of e j lateral surface e ckess δ f mea temperature of e j lateral surface e ckess δ f l umber of lateral surfaces l j e ckess δ f.he comparso of e f effcecy calculated from e expressos (36) for e FVM-FEM mesh sho Fg. 3a ad ASYS results obtaed for very fe mesh s preseted able. able.. Comparso of e f effcecy obtaed usg e FVM-FEM (Eq.(36)) ad ASYS softare. h W/(m K) Eq.(36) ASYS he accuracy of e preset meod s very satsfactory. I spte of e coarse fte volume mesh used preset meod (Fg. 3a.) e cocdece of e calculated effceces s very good. he traset 9-7

8 respose of e f llustrates Fg. 3b. he agreemet of e results obtaed by e developed meod ad ASYS s good. Usg e preset meod for calculatg e traset temperature dstrbuto ad f effcecy e traset heat flo rate trasferred from e hot lqud to cold ar ca be calculated more accurately. 3. he umercal Model of e Heat Exchager he automotve radator for e spark-gto combusto ege a cubc capacty of 58 cm 3 s a double-ro to-pass plate-fed heat exchager. he radator cossts of alumum tubes of oval cross-secto. he ater flos parallel rough bo tube ros. Fgure 4a shos a dagram of e topass cross-flo radator to ros of tubes. he heat exchager cossts of e alumum tubes of oval cross-secto. here are u = tubes e upper pass u e frst ad secod ro. Smlarly ere are l = 8 tubes e frst ad secod ros e loer pass l each of em. Mass flo rate of e lqud at passes rough e frst ro of tubes e upper pass s equal to half of e total m flo rate (Fg. 4a). Fg. 4b shos e dvso of e frst pass (upper pass) to cotrol volumes. I order to crease e accuracy of e calculatos a staggered mesh as appled. Water temperatures at e cotrol volume odes are deoted by W(I) ad W(I) for e frst ad secod ros of tubes respectvely. P(I) deotes ar temperature t frot of e radator P(I) deotes e ar temperature um um after e frst ro of tubes ad P3(I) deotes e ar temperature am um after e secod ro of tubes e - cotrol volume. Usg e otato sho Fg. 4a e boudary codtos ca be rtte e follog form: W W t t t P I t I.... (37) am he let ar velocty ad mass flo rate m are also fuctos of tme. I e smulato program e tme varatos of t am t t ad m t ere terpolated usg atural sples of e rd degree. he temperature m t s a temperature of e ater at e outlet of e upper pass here e ater of temperature W from e frst ro of tubes has bee mxed e ater of temperature W flog out of e secod ro of tubes. I e case of e automotve radator temperature t deotes e ater temperature at e let to e radator hereas t deotes ar temperature frot of e radator. a) b) am Fg. 4. Flo arragemet of to ro cross-flo heat exchager to passes (a) ad dvso of e frst pass of e car radator to cotrol volumes (b); frst tube ro upper pass - secod tube ro upper pass 3 - frst tube ro loer pass 4 - secod tube ro loer pass - ar temperature - ater temperature. 9-8

9 Havg determed e mea temperatures of e ar um ad lm hle leavg e secod ro of tubes e upper ad loer pass respectvely a mea temperature of e ar behd e hole radator am u um l lm / t as calculated. If e ater ad ar temperatures are ko e heat trasfer rate e frst ad secod ros of tubes e upper ad loer passes ca be determed. he total heat trasfer rate for e radator as calculated usg e formula Qchl m mac pa am am. he umercal model of e heat exchager descrbed brefly above s used to smulate ts traset operato. Before startg traset smulato e steadystate temperature dstrbuto of ater tube all ad ar as calculated usg e steady-state maematcal model of e heat exchager. 4. Expermetal Verfcato I order to valdate e developed model of e heat exchager a expermetal test stad as bult. he measuremets ere carred out a ope aerodyamc tuel. he expermetal setup as desged to obta heat trasfer ad pressure drop data from commercally avalable automotve radators. Ar s forced rough e ope-loop d tuel by a varable speed axal fa. he ar flo passes rough e hole frot cross-secto of e radator. A persoal computer-based data-acqusto system as used to measure store ad terpret e data. he relatve dfferece betee e ar-sde ad lqud-sde heat trasfer rate as less a 3%. Extesve heat trasfer measuremets uder steady-state codtos ere coducted to fd e correlatos for e ar- ad ater-sde usselt umbers hch eable calculato of heat trasfer coeffcets. Based o 47 measuremet seres e follog correlatos ere detfed: / d r u.re Pr 35 Re (38) L ch.7 /3 u.73re Pr 6 Re 35 (39) a a a a here e symbol d r stads for e hydraulc dameter of e oval tube. a) b) c) Fg. 5. me varatos of measured data (a) computed outlet ater ad ar temperatures (b) ad f temperature at selectve odes (c); t - ater let temperature ta - ar let temperature t - ater outlet temperature ta - ar outlet temperature V - ater mass flo rate - ar velocty before e heat exchager b f base temperature curve umbers Fg. 5c correspod to ode umbers Fg. 3a. 9-9

10 he ater-sde Reyolds umber Re d r / s based o e hydraulc dameter dr 4 A / U here A deotes sde cross secto area of e oval tube. he hydraulc dameter for e vestgated radator s: d r = m. he ar-sde Reyolds umber s defed as Re a max dh / a here max s mea axal velocty of ar e mmum free flo area ad d h =.4-3 m s e ar-sde hydraulc dameter. he physcal propertes of ar ad ater ere approxmated usg smple fuctos. he effect of temperature-depedet propertes s accouted for by evaluatg all e propertes at e mea temperature of ar ad ater respectvely. he e traset respose of e heat exchager as aalyzed. Usg e measured values of e let ater temperature e let ar temperature am ar velocty frot of e radator ad e ater volumetrc flo rate V e ater tube all ad ar temperatures are determed usg e preset explct fte dfferece meod. Durg e expermet e ar let velocty as suddely decreased (Fg. 5a). he calculato results ad er comparso expermetal data are sho Fg. 5b. he agreemet betee e calculated ad measured ater ad ar temperatures at e outlet of e heat exchager s very good. I e case of ater temperature measuremet e tme costat of e ermocouple s very small sce e heat trasfer coeffcet e ermocouple surface s very hgh ad e temperature dcated by e ermocouple ad e real ater temperature are very close. 5. Coclusos he umercal model of a cross-flo tube heat exchager hch eables heat trasfer smulato uder traset codtos as developed. Frst e traset temperature dstrbutos of fluds tube alls ad fs e oe ro tubular cross flo heat exchager ere determed usg e fte dfferece meod. raset heat trasfer rough rectagular fs attached to e oval tubes as modeled usg e Fte Volume Meod Fte Elemet Meod. he e umercal model of e to ro heat exchager to passes as preseted. he umercal model as valdated by comparso of outlet ater ad ar temperatures obtaed from e umercal smulato e expermetal data. he dscrepacy betee umercal ad expermetal results s very small. Refereces Jaremkecz M. aler D. Sobota. (9). Measurg traset temperature of e medum poer egeerg maches ad stallatos Appled hermal Egeerg vol. 9 pp Press W.H. eukolsky S.A. Vetterg W.. Flaery B.P. (7). umercal Recpes he Art of Scetfc Computg. hrd Edto Cambrdge Uversty Press Cambrdge. Roetzel W. Xua Y. (998). Dyamc behavour of heat exchagers Computatoal Mechacs Publcatos Vol.3 WI Press Souampto. Sm E. M. (997). hermal Desg of Heat Exchager Joh Wley & Sos Chchester. aler D. (9). Dyamcs of ube Heat Exchagers Publshg House of AGH Uversty of Scece ad echolygy (UWD) Craco ( Polsh). aler D. Korzeń A. (a). umercal modelg of heat trasfer plate fs hermodyamcs Scece ad echology ed. Bogusłask L.Procedgs of e -st Iteratoal Cogress o hermodyamcs Pozań Polad Sep. 4-7 pp aler D. Korzeń A. (b). Modelg of heat trasfer plate fs of complex shape Ryek Eerg o. 6 pp ( Polsh) 9-