Entrance and Wall Conduction Effects in Parallel Flow Heat Exchangers

Transcription

1 Proceedngs of Ffth Interntonl Conference on Enhnced, Compct nd Ultr-Compct Het Echngers: Scence, Engneerng nd Technology, Eds. R.K. Shh, M. Ishzuk, T.M. Rudy, nd V.V. Wdekr, Engneerng Conferences Interntonl, Hoboken, NJ, USA, September 5. CHE5 Entrnce nd Wll Conducton Effects n Prllel Flow Het Echngers Ahmd Fkher Deprtment of Mechncl Engneerng, Brdley Unversty, Peor, IL, USA hmd@brdley.edu Hussen Al-Bkht, Mossvlle Engne Center, Cterpllr Inc., Mossvlle, IL ABSTRACT Het echnger nlyss tools such s F correcton fctor chrts nd ε-ntu reltons ssume tht the overll het trnsfer coeffcent s constnt cross the het echnger. The short lengths nd the comprtvely thck wlls n mcrochnnel het echngers preclude the estence of thermlly fully developed flow over lrge porton of the het echnger nd ncrese the sgnfcnce of het conducton n the wll. In ths study, prllel flow het echnger s smulted numerclly to determne the mpct of entrnce effects nd conducton het trnsfer through the wlls on the performnce of the het echnger, ncludng the number of trnsfer unts nd the effectveness. The mpct of wll conducton on the het trnsfer s studed for dfferent wll thckness nd therml conductvty. It s shown tht entrnce nd wll effects cn be ncorported n the longtudnlly verged number of trnsfer unts nd ccountng for them sgnfcntly chnges the het echnger sze for to gven effectveness. KEYWORDS Mcrochnnel, Het Echngers, Conjugte, Developng flow, Effectveness, NTU, Prllel Flow NOMENCLATURE ducts wdth, m A Het Trnsfer Are, m A c Duct cross sectonl Are, m α lower duct spect rto (=b /) α upper duct spect rto (=b /) b lower duct heght, m b upper duct heght, m C C = m Ý c p, Flud cpcty, W/K D h hydrulc dmeter D h = 4A c p h k K r convectve het trnsfer coeffcent, W/m k therml conductvty, w/m k K r = k s k L duct length, m N N = k nond. hot flud/sold conductvty k s t N N = k nond. cold flud/sold conductvty k s t Nu Nusselt number P P P = ρ u,n nondm. pressure p heted permeter (), m Pr Pe Re Prndtl number (= c p µ/k) nondm Peclet number, Pe = ρ u µ c p = u µ k α Reynolds number, Re = ρ u T flud temperture, K T flud temperture t nlet of the lower duct, K T flud temperture t nlet of the upper duct, K t conductng wll thckness, m T T = T T,n, nondm. T,n T,n = nondm. coordntes + Gretz Number, + = D h Pe, nondm. y y = y Nondm. y coordntes z z = z Nondm. z coordntes µ 93

2 U overll het trnsfer coeffcent, W/m k u u = u u,n nondm. -component of velocty v v = v. nondm. y-component of velocty u,n w w = w. nondm. z-component of velocty u,n Greek Symbols ρ densty, kg/m 3 α therml dffusvty, m /s µ dynmc vscosty, kg/m s ν Knemtcs vscosty, m /s = T m,h T m,c, Nondm. T,n T,n Subscrpts nd Superscrpts nondmensonl quntty lower duct (hot flud) upper duct (cold flud) m men H Hot C Cold s refers to sold n refers to nlet. INTRODUCTION Full numercl soluton of het echngers re computtonlly prohbtve becuse the flow nd temperture felds must be determned smultneously n t lest two fluds nd the sold seprtng the two. The het echnger nlyss nd desgn re, therefore, bsed on solutons obtned for flows n ndvdul ducts subject to sotherml or unform het flu boundry condtons nd epermentl mesurements. The F correcton fctor chrts, or the ε-ntu reltons, wdely used for het echnger nlyss, re bsed on the ssumpton tht the overll het trnsfer coeffcent s constnt. In mcrochnnel het echngers the re of the sold s cross secton perpendculr to the drecton of the flow my be s lrge s the cross sectonl re vlble for the flow. For these het echngers, thermlly fully developed flow my not est over lrge porton of the het echnger due to the comprtvely thck wlls through whch het s conducted, nd the short length of the het echnger. These necesstte emnton of the detled flow felds to determne the vldty of the ssumptons typclly mde, ncludng constnt overll het trnsfer coeffcent. Yn nd Bu (99) studed flow between nfnte prllel pltes nd crculr ppes to study the effect of l conducton on the performnce of mcro-chnnel het echngers. They used fully developed velocty feld nd nlytclly solved for the temperture felds n the chnnel nd sold wll. They found tht l conducton plys n mportnt role t the entrnce regon, were locl vlues Nusselt Number re hgher thn when l conducton s gnored for unform wll temperture. Wng nd Shyu (99) epermentlly studed the effect of chnnel sze nd wll therml conductvty n mcro het echngers for hot/cold wter test loop. They found tht the chnnel sze nd wll mterl hve strong nfluence on the het trnsfer cpblty of mcro het echnger. The effect of sold conducton on the het echnger performnce hs been studed etensvely. Stef et. l (999) numerclly nvestgted the effect of sold therml conductvty n mcro het echngers. They showed tht the reducton of conductvty of the wll mterl cn mprove the het trnsfer effcency of the echnger due to nfluence of l het conducton n the chnnel wlls. Vektrthnm nd Nrynn (999) lso found tht the performnce of the het echnger s lrgely dependent on the het conducton tht tkes plce through the wlls of the het echngers used n mnture refrgertors. They used two dmensonl energy blnce to ccount for the conducton through wll nd convecton through the flud. They ssumed tht the temperture of the flud to be unform t ny flow cross secton. Rvgururjn et. l. (996) bult n epermentl set up to study the therml performnce chrcterstcs of snglephse flow n prllel mcro het echnger, Refrgernt- 34 ws used s n epermentl flud. He ttrbuted the ncrese n het trnsfer coeffcent to the thnnng of the boundry lyer n the nrrow chnnels, whch lowers therml resstnce. Dvs nd Gll (969) ncluded het conducton n the wll for Poseulle-Couette flow, where het s trnsferred through hetng zone n sttonry wll of fnte thckness, nd the het flu s unformly specfed t the outer surfce of the hetng zone. They concluded tht the nterfce temperture dstrbutons nd locl Nusselt number dstrbutons were ffected by the flow condtons, the wll thckness, the rto of therml conductvty of the wll to the flud, nd the wdth of the duct (spect rto). Ther epermentl observtons lso ponted out tht wll conducton ws sgnfcnt n the het trnsfer phenomen. Net nd Echhorn (983) used the fnte dfference method to solve for hydrodynmclly nd thermlly developng flow n squre duct. They neglected the l momentum nd energy dffuson nd ssumed tht the pressure grdent vred lnerly n the l drecton. They lso neglected the wll conducton, presentng results for the centerlne veloctes, pressure drop nd Nusslet number for the cse of Pr = 6. for n sotherml duct. Aprecdo nd Cott (99) studed the thermlly developng flow nsde rectngulr ducts nlytclly by etendng the generlzed ntegrl trnsform technque to solve the energy equton for wde rnge of spect rtos. Ther results for locl nd verge Nusselt number n the entrnce regon re used for vldton of numercl solutons. 94

3 Fkher et. l. (994) numerclly nvestgted the hydrodynmclly nd thermlly developng flows n rectngulr ducts wth conductng wlls. Bsed on ther numercl soluton, they cme up wth the followng correlton T m = e whch ppromtes the men flud temperture n the entrnce regon of n sotherml rectngulr duct for + >. to wthn %. They showed the sgnfcnce of conducton on men flud temperture nd Nusselt numbers. Chndurpth, et. l. (977) numerclly solved for lmnr het trnsfer of Newtonn nd non-newtonn flud n the therml entrnce regon of squre duct for dfferent boundry condtons. Shh nd London (978) compled comprehensve soluton for lmnr forced convecton flow het trnsfer n wde rnge of duct szes nd shpes. Al-Bkht nd Fkher (4) consdered the problem of smultneously developng flows n prllel rectngulr ducts ncludng conducton through the sold boundres. In tht study, the uthors emned the mpct of the ncluson of the wll conducton nd entrnce effects on the performnce of the het echnger. The wll ws ssumed to be thn, so tht the wll temperture vrton n the z drecton cn be neglected, but the het conducton n the z drecton ws ncluded (thn fn ssumpton). It ws shown tht there s sgnfcnt chnge n the overll het trnsfer coeffcent n the developng regon nd tht the threedmensonl het trnsfer n the het echnger wll must be ncluded n the nlyss. The present study etends ths work to thck wll, encountered n mcrochnnels, where the z vrton of the temperture n the wll seprtng the two chnnels s ncluded.. ANALYSIS In ths study, the flow feld n prllel flow het echnger s numerclly smulted to determne the mpct of dfferent flow prmeters on het trnsfer nd determne the ccurcy of the ssumpton of constnt overll het trnsfer coeffcent. The nlyss s for developng flow n ducts of prllel flow het echnger. u,n, T,n u,n T,n b b t y z L 95 Fgure Three-dmensonl sketch of the rectngulr duct prllel flow het echnger In the confgurton shown n Fgure, the hot flud enters the lower ducts wth unform velocty u,n nd unform temperture T,n whle the cold flud enters the ducts n the upper row t u,n, nd T,n. Het s trnsferred from the hot flud to the cold flud through the chnnel wlls tht hve thckness t. Although the hot nd cold flud ducts my hve dfferent spect rtos, the results presented here re for the cses where both ducts hve the sme spect rto. The gol of the nvestgton s to numerclly solve the three-dmensonl het trnsfer for thermlly developng lmnr flows n the two prllel rectngulr chnnels, nd to look nto prmeters tht ffect the het trnsfer between the two fluds. For lmnr, 3-D, stedy, ncompressble flow, the nondmensonl governng equtons become u + v y + w z = () u u + v u y + w u z = P + u Re () v u + v v y + w v z = P y + v Re (3) w u + v w y + w w z = P y + w Re (4) T u + v T y + w T z = T Pe (5) where = nd refer to the lower nd upper (hot nd cold) chnnels respectvely. The three-dmensonl stedy stte conducton equton through the sold s T s =. (6) Nondmensonl boundry condtons re For z < b = u =, v = w =,T = (7) = L = L u = v = w = T = (8) y = u = v = w = T y = (9) y = u y = v y = w y = T y = () u z =, z = v z = w z = T z = () For b +t < z < b +b +t = u = u,n,v = w =,T = () u,n = L = L u = v = w = T = (3) y = u = v = w = T y = (4) y = u y = v y = w y = T y = (5)

4 z = b + b + t, u z = v z = w z = T z = (6) At the sold lqud boundres z = b T z = k s T s (7) k z z = b + t T z = k s T s (8) k z Once the temperture dstrbuton s determned, the overll het trnsfer coeffcent for prllel flow het echnger cn be obtned by dong n elementl energy blnce t gven cross secton to get dq = U(T H T C )d = C C dt C = C H dt H (9) then dt ( H T C )= dq + () C H C C whch s then smplfed to d ln T ( H T C )= + C r Upd () C mn Where p s the crcumference of the hot duct through whch het s trnsferred ( n ths cse) nd C r = C mn s the C m rto of the therml cpctes of the two fluds. The bove equton cn be rerrnged nto ( ) d ( ) d + d ln T H T C d ln T H T C ( ) NTU = + C r = + C r ( ) NTU + () + (3) Solvng for the locl vlue of the Number of Trnsfer Unts, NTU + ( ) NTU + = + d ln T H T C (4) + C r d + The longtudnlly verged vlue of the NTU s defned n terms of the verge vlue of the overll het trnsfer coeffcent from the entrnce of the het echnger to ny locton from the nlet NTU vg, = U A C mn Where the verge overll het trnsfer coeffcent s defned s U = Ud (5) Integrtng Eq.(9) ln T ( H T C )= + C r p Ud = C mn + C r C mn p Ud Whch cn be solved for = + C r C mn AU (6) ( ) (7) NTU vg, = ln T H T C + C r Therefore, f the vrton of the men temperture of the cold nd hot fluds re known, the locl vlue of the number of Trnsfer Unts cn be determned from Eq. () nd the verge vlue from the nlet of the het echnger to ny locton from Eq. (5). The performnce of het echngers s chrcterzed by the concept of the het echnger effectveness whch s the rto of the ctul het trnsfer through the het echnger to the mmum possble het tht cn be trnsferred ε = q q m (8) ε = C H (T H,n T H ) C mn (T H,n T H ) = C C (T C T C,n ) C mn (T H,n T H ) (9) snce C H (T H,n T H ) = C C (T C T C,n ) (3) Then Eqs. (8) nd (5) cn be rerrnged C c T C +T H = (3) C H T H T C = e ( +C r )NTU vg, (3) To fnd the relton between the Number of Trnsfer Unts nd the het echnger effectveness, when the het trnsfer coeffcent chnges long the het echnger, we consder two cses,. C mn =C c Then from Eq. (7) (33) ε = T C,out T C,n = T C (34) T H,n T C,n nd from Eq. (9) C r T C + T H = (35) Substtutng these bck nto Eq. () nd smplfyng results n ε = ( e +C r )NTU vg, (36) + C r b. C mn =C h Then from Eq. (7) (37) 96

5 ε = T H,n T H = T H (38) T H,n T C,n nd from Eq. (9) T C + T H = (39) C r Substtutng these bck nto Eq. () nd smplfyng results n ε = e ( +C r )NTU vg, (4) + C r As s shown, Eq. (8) s the sme s Eq. (4) nd nterestngly they both re the sme s the epresson gven for the relton between ε nd NTU for prllel flow het echnger when the overll het trnsfer coeffcent s ssumed constnt. Ths shows tht the sme relton between effectveness nd Number of Trnsfer Unts holds when the NTU used n the reltonshp s the longtudnlly verged vlue of the Number of Trnsfer Unts for the length of the het echnger. Ths number, of NTU vg, gven by Eq. (5), ccounts for the developng effects nd het conducton through the wlls. 3. RESULTS The governng equtons re solved numerclly usng hybrd pproch, where the momentum equton s solved usng Fluent 6. nd the velocty dstrbuton s then nputted nto code developed for solvng the energy equton for the conjugte het trnsfer usng the fnte dfference method. The detls of the numercl soluton re gven n (Al-Bkht, H. nd Fkher, A., 4; Al-Bkht, H., 4). The numercl grd vred wth the flow condtons, but ws typclly composed of grd bout nodes n the drecton, bout 5 nodes n the y nd z drectons n ech chnnel, nd dependng on the wll conductvty to 5 nodes n the z drecton n the wll seprtng the two ducts. Etensve verfctons wth the vlble results were mde (Al-Bkht, H. nd Fkher, A., 4) to ensure the ccurcy of the solutons. The nondmensonl velocty feld ws clculted for vrous spect rtos nd Reynolds numbers. For dfferent spect rtos, the clculted vlues of the centerlne velocty n the fully developed regon were found to be wthn.%. of the publshed results of by Shh nd London (978). The vrton of the locl vlue of the men tempertures of hot nd cold flud for dfferent Reynolds number s shown n Fgure. The results re for blnced flow het echnger wth squre chnnels, mde of low thermlly conductng wlls (K r =). As cn be seen, ecept for the very low Peclet number of 7 (blue lne), the men temperture vrton s ndependent of the Reynolds number, when the results re epressed n terms of Gretz number. Also, s shown before, the men temperture of 97 the cold flud s the sme s the het echnger effectveness, nd therefore, for gven wll conductvty nd Gretz number, the effectveness s the sme regrdless of the vlue of the Reynolds number. Fgure 3 shows the temperture vrton of the cold nd hot fluds long the het echnger for blnced flow ( C r = ). The results re for dfferent spect rto ducts when the wll therml conductvty to flud conductvty s hgh (K r =) Kr= Re = Re=5 Re= Re= X+ Fgure Longtudnl vrton of the hot nd cold flud men temperture For these cses, the temperture vrton n the duct wll cn be neglected nd the wll wll be t unform temperture, nd n the lmt of n nfntely long het echnger from Eq. () T C = T H = T w then from Eq. (9) T w = + C c C H (4) (4) whch for blnced flow het echnger, hvng thn wlls nd/or wlls mde of hgh therml conductvty mterl T w =.5. Tm == ==.5 == X+ Fgure 3 Longtudnl vrtons of the men tempertures of hot nd cold fluds for K r =

6 Fgure 3 shows the longtudnl vrton of the men temperture for the dfferent spect rtos for het echnger hvng thn wlls nd/or very hgh therml conductvty, where the temperture vrton n the wll s smll. As cn be seen the tempertures chnge monotonclly nd symptote to the vlue of.5 for very long het echnger. It cn lso be seen tht tht for gven Reynolds number, the het echnger effectveness ncreses sgnfcntly wth decresng the spect rto. For the cse of lquds flowng n low therml conductvty ducts such s cermc het echngers nd or when the wll thckness to the duct wdth rto s of the order of one, s n the cses of mcrochnnel het echnger then conducton n the sold must be tken nto consderton. Tm Tm Kr= Kr= Kr=4 Kr= Kr= Kr= Kr=4 Kr= α= α=.5 Tm Kr= Kr= Kr=4 Kr= α=.. Fgure 4 Longtudnl vrtons of the men tempertures for rectngulr ducts Fgures 4 show the longtudnl vrton of the men temperture for the dfferent spect rtos nd dfferent wll therml conductvtes. In ll the cses shown, the het echnger s blnced flow one. The mpct of therml conductvty of the sold wll on the het echnger performnce hs been nvestgted by chngng K r whch s the rto of the sold to flud conductvty. For gven spect rto, more het s trnsferred s the wll therml conductvty s ncresed up to certn lmt (K r = 4), fter whch ncresng the therml conductvty wll not enhnce the het trnsfer. Beyond ths vlue, the wll wll behve s n nfntely conductng wll wth neglgble temperture grdent, nd the het trnsfer between the two fluds wll be ndependent of the wll propertes, dependng only on the fluds condtons nd the het echnger geometry As shown before, the vrton of the het trnsfer coeffcent t the entrnce regon of the het echnger or the mpct of conducton het trnsfer through the wlls cn be studed by how these effects mpct the Number of Trnsfer Unts. Fgure 5 shows the vrton of the verge NTU long the het echnger for wde rnge of Peclet numbers nd therml conductvtes. As cn be seen from Fgure 5, the soluton s only functon of conductvty rto nd ndependent of Reynolds number. For gven het echnger length, (re) ncresng the wll conductvty, ncreses NTU nd therefore the het echnger effectveness, up to conductvty rto of bout 4, beyond whch the wll mterl propertes do not pper to mpct the performnce of the het echnger. Also, for the sme mterl, nd het echnger sze, ncresng the Reynolds number, decreses, + nd thus NTU nd the het echnger effectveness. 98

7 Kr= Kr= Kr=.5. + Fgure 5 Vrton of the Averge NTU long blnced prllel flow het echnger (α=) 4. CONCLUSION In ths study, prllel flow het echnger s smulted numerclly to determne the mpct of entrnce effects nd conducton het trnsfer through the wlls on the performnce of the het echnger, ncludng the number of trnsfer unts nd the effectveness. The mpct of wll conducton on the het trnsfer s studed for dfferent wll thckness nd therml conductvty. It s shown tht ll such mpcts re reflected n the verge number of trnsfer unts. It s shown tht there s sgnfcnt chnge n the het echnger effectveness n the developng regon nd tht the three-dmensonl het trnsfer n the het echnger wlls must be ncluded n the nlyss. Accountng for the vrton n the overll het trnsfer coeffcent wll sgnfcntly reduce the het echnger length requred to rech the mmum possble effectveness. For conventonl het echngers, the thn fn ssumpton s resonble ppromton, nd therefore the performnce of the het echnger s prmrly dependent on the flow n the ducts,.e. the flud propertes, mss flow rte, nd spect rtos. For mcrochnnel het echngers, the conducton n the wll s mportnt, nd the thn fn ppromton wll not be stsfed for prctcl mcrochnnel het echngers. Ths requres the soluton of the three dmensonl equtons n both sold nd flud. Usng hgh conductve mterl wll not hve n effect on ncresng the het echnger effectveness snce the het echnger effectveness wll be ndependent of the wll therml conductvty for K r > REFERENCES Al-Bkht, H. nd Fkher, A., 4, Numercl Smulton of Het Trnsfer n Smultneously Developng Flows n Prllel Rectngulr Ducts, Proceedng of 4 ASME Het Trnsfer/Fluds Engneerng Summer Conference, Chrlotte, NC, USA Al-Bkht, H., 4, Full Numercl Smulton of Prllel Flow Het Echngers, Mster Thess, Brdley Unversty, Peor IL Aprecdo J. B nd. Cott R. M, 99, Thermlly Developng Lmnr Flow Insde Rectngulr Ducts, Int. J. Het Mss Trnsfer, Vol. 33, pp Chndurpth, A. R. nd V. M. K. Sstr., 977, Lmnr Forced Convecton Het Trnsfer of NON- Newtonn Flud n Squre Duct, Int. J. Het Mss Trnsfer, Vol., pp ,. Dvs E. nd Gll N., 969, The Effect of Al Conducton n the Het Trnsfer wth Lmnr Flow, Int. J. Het Mss Trnsfer, Vol. 3, pp Fkher, A. Zhu, J., nd Azeem, M., 994, Lmnr Developng Flow Insde Het Conductng Rectngulr Ducts ASME Interntonl Mechncl Engneerng Congress & Ehbton, Chcgo Illnos, USA Nov 6- Net, S., nd Echhorn, R., 983, Combned Hydrodynmc nd Therml Development n Squre Duct, Numercl Het Trnsfer, Vol. 6, pp Rvgururjn, T. S., Cut, J., McDonld, C. E,, Drost, M. K, 996, Sngle-phse flow therml performnce chrcterstcs of prllel mcro-chnnel het echnger. ASME, HTD, Vol. 39, pp Shh, R. K., nd London, A. L., 978, Lmnr Flow Forced Convecton n Ducts, Adv. Het Trnsfer Suppl. Stef, T., Lnger, O. U., nd Schubert, K., 999, Numercl Investgton of optml het conductvty n mcro het echngers Chem Eng Tech, Vol., pp Vektrthnm, G. Nrynn, S., 999, Performnce of counter flow het echnger wth het loss through the wll t the cold end, Cryogencs, Vol. 39, pp 43-5 Wng, J. H., Shyu, R. J., 99, Therml-hydrulc chrcterstcs of mcro het echngers ASME Wnter Annul Meetng Atlnt, GA, USA Dec -6 Yn, X., nd Bu, H. H., 99, Al Conducton Effect Performnce of Mcro-het Echngers ASME Wnter Annul Meetng, New Orlens, Lousn USA Nov. 8- Dec 3. 99