Effectiveness - NTU Data and Analysis for Air Conditioning and Refrigeration Air Coils

Transcription

1 Hélio Aparcido Navarro t al. Hélio Aparcido Navarro han@sc.usp.br Univrsidad d São Paulo USP Escola d Engnharia d São Carlos Dpartamnto d Engnharia Mcânica São Carlos, SP, Brazil Lubn Cabzas-Gómz lubn@pucminas.br Pontifícia Univrsidad Católica d Minas Grais Dpartamnto d Engnharia Mcânica Blo Horizont, MG, Brazil João R. Bastos Zoghbi Filho jrzoghbi@sc.usp.br Effctivnss - Data and Analysis for Air Conditioning and Rfrigration Air Coils A simulation program basd on a control volum analysis has bn usd in th valuation of th (ε, ) rlationship for coils of complx gomtry and flow arrangmnt. Th simulation program has bn valuatd through simpl gomtry and flow arrangmnt coils. Th program rsults compar vry wll with corrlations for simpl cross flow coils, and a numbr of rows up to four. It has also bn dtrmind that closd form corrlations dvlopd for coils of an infinit numbr of tub rows ar inadquat for thos with numbr of rows in th rang btwn 5 and. In addition, it has bn found that closd form (ε, ) corrlations for cross flow coils with th sam tub arrangmnt and numbr of rows might lad to inaccuracis highr than % in th valuation of th ffctivnss of coils of complx flow arrangmnt. Kywords: ffctivnss,, air coil Ghrhardt Ribatski ribatski@sc.usp.br Univrsidad d São Paulo USP Escola d Engnharia d São Carlos Dpartamnto d Engnharia Mcânica São Carlos, SP, Brazil José Maria Saiz-Jabardo mjabardo@cdf.udc.s Univrsidad d la Coruña Escula Politécnica Suprior 543 Frrol, Coruña, Spain Introduction Rfrigration and air conditioning air coils prsnt a wid rang of gomtric configurations and flow arrangmnts. Thy ar part of th family of th so-calld compact hat xchangrs, Kays and London (998). Ovrall analysis of this kind of hat xchangr has bn gnrally mad through th so-calld (ε, ) procdur, as suggstd by Kays and London (998). On of th problms facd by th dsignr in using this procdur is to find adquat corrlations btwn th coil ffctivnss, ε, and th Numbr of Transfr Units,. Ths corrlations involv othr coil paramtrs such as th ratio btwn fluids hat capacity rats, th gomtry, and th rlativ flow arrangmnt of th fluids. During th past fifty yars, svral studis hav bn ddicatd to th dvlopmnt of (ε, ) corrlations for compact hat xchangrs. On of th bst-known publications is th book by Kays and London (998), rfrrd abov, which contains graphical and tabl information rlating to th ffctivnss and th Numbr of Transfr Units for numrous compact hat xchangrs for diffrnt gomtris and flow arrangmnts. Dspit its compltnss, th Kays and London txt dos not covr som of th coil arrangmnts usd in th rfrigration and air conditioning industry. Bowman t al. (94) addrssd th problm by calculating th factor F, for th man logarithmic tmpratur diffrnc, for cross flow, svral pass hat xchangrs. Latr on, Stvns t al. (957) proposd closd form sris corrlations and plots for cross flow, multipl Papr accptd January,. Tchnical Editor: José A. dos Ris Paris row hat xchangrs, including th thr possibl arrangmnts of th fluids: unmixd/unmixd, mixd/unmixd, and both mixd. Pignotti and Cordro (983) proposd closd form xprssions for th factor F of th logarithmic man tmpratur for svral arrangmnts of cross flow compact hat xchangrs. Latr on Pignotti (988) suggstd a matrix formalism for th valuation of th thrmal ffctivnss of complx hat xchangrs configurations that can b brokn into simpl constitutiv parts, connctd to ach othr by unmixd strams. Baclic (99) providd a list of closd form (ε, ) corrlations for a numbr of flow arrangmnts usd in compact hat xchangrs. Pignotti and Shah (99) discussd som mthods for th dtrmination of th (ε, ) rlationship for complx hat xchangr flow arrangmnts, among thm th Domingos ruls (969), th chain rul and th ruls for hat xchangrs with a mixd fluid. Eightn (ε, ) closd form corrlations wr dvlopd from ths ruls. Pignotti and Shah (993) considrd complx hat xchangr flow arrangmnts and rlatd thm to simpl ons for which a closd form ithr is availabl or an approximat solution can b obtaind. Wang t al. () rfrrd to svral (ε, ) corrlations dvlopd by th 998 ESDU (Enginring Scinc Data Unit) vrsion for coils of complx gomtry, including thos with paralll circuits of th tub fluid and svral rows, typical of thos usd in th rfrigration and air conditioning industry. Th computr program prsntd hr is basd on a control volum formulation. Similar approach has bn usd in th past in th simulation of tub-plat fin coils. Domanski (99) suggstd a modl basd on dividing coil in succssiv finit volums following th tub fluid path. Bnsafi t al. (997) proposd a similar modl, 8 / Vol. XXXII, No. 3, July-Sptmbr ABCM

2 Effctivnss - Data and Analysis for Air Conditioning and Rfrigration Air Coils but usd local hat transfr cofficints instad of avrag ons xtnsiv to th ovrall hat xchangr ara. Vardham and Dhar (997) proposd a modl that, similarly to th prvious ons, divids th coil into finit volums along th tub fluid path and carris out itrativ marchs btwn th tub fluid ntranc and xit, whil simultanously updating th air-sid proprtis. In ach volum lmnt, th ffctivnss is computd as if it wr a mixd/unmixd cross flow hat xchangr. Corbrán and Mlón (998) usd similar approach as th othrs to simulat vaporation and condnsation of rfrigrant R-34a in ordr to chck th prformanc of diffrnt chang of phas corrlations. In th prsnt study, a computr simulation program has bn usd to rais ffctivnss data for coils of complx gomtry and flow arrangmnts. Th prsnt papr uss th sam approach of works publishd by Navarro and Cabzas-Gómz (5) and Cabzas-Gómz t al. (7). To follow a dscription of th working modl and computr simulation program is providd for radr information. Thn, th program prformanc is valuatd for coils of simplr gomtry for which closd form (ε, ) rlations ar availabl. Finally, this program is usd in dvloping (ε, ) plots for air conditioning and rfrigration air coils with complx flow arrangmnt. Nomnclatur A = ovrall hat transfr ara of th coil, m C = hat capacity rat, W/K C* = hat capacity ratio, Cmin/Cmax, dimnsionlss L = lngth of th tub, m m& = mass flow rat, Kg/s Nc = numbr of tub fluid circuits N = numbr of lmnts pr tub Nr = numbr of rows Nt = numbr of tubs pr row = numbr of transfr units, (UA/Cmín) q = hat transfr rat, kw Pl = longitudinal pitch of tubs, m Pt = transvrsal pitch of tubs, m T = tmpratur, C U = ovrall hat transfr cofficint, W/(m K) Grk Symbols Δ = dnots diffrnc ε = ffctivnss of hat xchangr Γ = ffctivnss of th tub lmnt δ = dviation Subscripts a air sid av avrag i inlt max highr valu min lowr valu o outlt t tub sid Suprscripts lmnt Govrning Equations for Tub Elmnts Th gomtry of th air coil considrd in th prsnt study is schmatically shown in Fig.. Th fluid flowing insid th tubs is dsignatd as tub fluid. Air flows xtrnally to th tubs, in th spac btwn fins, in a gnral cross flow configuration. Th modl consists in dividing th air coil in small (finit) volums dsignatd as tub lmnts, ach ncompassing a sgmnt of tub and th corrsponding fins, as illustratd in Figs. and. Th cntr of a tub lmnt is a nodal point, its spatial location bing charactrizd by th triplt (i, j, k), corrsponding to th spac dirctions, as shown in Fig.. Th numbr of tub lmnts in ach dirction is rspctivly qual to N, Nt and Nr, with th first bing th numbr of lmnts in a tub, th scond th numbr of tubs in a row (tubs in th normal dirction to th air flow), and th third th numbr of rows in th air coil. Th tub fluid is distributd into Nc paralll circuits. Finally, th following additional assumptions ar considrd in modl dvlopmnt: constant ovrall hat transfr cofficint and thrmodynamic and transport proprtis; adiabatic rturn bnds; th inlt air is at a uniform tmpratur and its vlocity is vnly distributd through th fac ara. Figur. Schmatic rprsntation of th air coil considrd in th prsnt study; a tub lmnt. Enrgy balancs for th tub fluid and air in a tub lmnt can b writtn as q q Ct ( Tt,o Tt,i ) ( T T ) = Ca = a,o a,i Th suprscript rfrs to th tub lmnt (i, j, k). Th product of th lmnt mass flow rat of ach fluid by its spcific hat hav bn dsignatd by C t and C a, rspctivly for th tub fluid and for th air, and will b rfrrd to as hat capacity rats for simplification purposs. Th mass flow rat of ach fluid in th lmnt can b dtrmind from th corrsponding ovrall mass flow rat according to th following quations: ma m& a NNt & = (a) mt m& t Nc & = Sinc th tub lmnts ar rlativly short (th shortr th bttr), th tub fluid tmpratur variation is small. Thus, as a first approximation, th tub fluid tmpratur could b assumd constant in th lmnt. This tmpratur is assignd to th corrsponding nodal point, and can b dtrmind as ( T T ) T t =. 5 t,i + (3) t,o Thus, for all practical purposs, tub lmnts can b considrd as hat xchangrs with th tmpratur of on of th fluids (tub) J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright by ABCM July-Sptmbr, Vol. XXXII, No. 3 / 9

3 Hélio Aparcido Navarro t al. rmaining constant. In such a cas, th individual thrmal ffctivnss of ach tub lmnt can b writtn as (UA ) C ( ) a Γ = = (4) Actually, th air hat capacity rat, C a, is clarly smallr than th tub fluid on, sinc th air flow rat is of th sam ordr as th lngth of th tub lmnt. Th product of th ovrall hat transfr cofficint by th hat transfr ara of th lmnt, (UA), is givn by UA ( UA ) = (5) N N t N r It must b notd that th ovrall hat transfr cofficint, U, is assumd constant ovr th air coil. Thus, th rmaining trms of th right hand sid of Eq. (5) corrspond to th hat transfr ara of ach tub lmnt. Finally, introducing th dfinition of hat xchangr ffctivnss, a rlationship btwn th tmpraturs of th fluids and th lmnt ffctivnss, Γ, can b stablishd according to th following quation: ( T T ) q a,o a,i Γ = = (6) q máx (T t, i T a, i ) Th xit tmpraturs of both fluids in ach tub lmnt can b obtaind through th solution of th abov st of quations, providing that th following ovrall paramtrs ar known: th gomtry and th ovrall hat transfr cofficint, mass flow rat of both fluids, th numbr of tubs in a row, th numbr of rows and th numbr of paralll circuits of th tub fluid along with th numbr of tub lmnts for ach tub (arbitrarily chosn) and th inlt tmpratur of both fluids in th lmnt. Computational Procdur Th air coil simulation program has bn dvlopd from th st of govrning quations for ach tub lmnt dscribd in th prcding sction. Whn th objctiv is th air coil simulation, th input paramtrs ar th following: gomtry, mass flow rats and inlt tmpraturs of both fluids, and th ovrall hat transfr cofficint. Th xit tmpraturs of both fluids along with th coil ffctivnss would rsult from this mod of application. In th prsnt papr, th objctiv is to dtrmin th (ε, ) rlationship. In this cas, using th mass flow rat as an input paramtr would not b adquat. As a rsult, th ovrall air-coil and th hat capacity rats substitut for th mass flow rats of both fluids as input paramtrs. Individual tub lmnt hat capacitis can thus b dtrmind from th following quations: If C min = C a UA/ NNt C a = and UA/ C t = (7a) * C N If C min = C t UA / * C N Nt c C a = and UA / C t = (7b) N c A block diagram of th algorithm for th ffctivnss dtrmination (valuation) is prsntd in Figs. and 3. A stp-bystp discussion will b prsntd nxt in ordr to mak it asir to th radr to follow th adoptd procdur. Stp Rad th input data. Th air coil gomtry is rad from a fil containing information such as th numbr of rows, tubs, and th numbr of circuits of th tub fluid and thir arrangmnt. Th givn valu of is introducd along with th capacity ratio, C*, and th valu of C min. Th tub lmnt siz (lngth) is also valuatd at this stag. Its valu is dtrmind from a trial and rror procdur consisting in running th program for an incrasing numbr of tub lmnts and chcking th obtaind rsults. Th adquat numbr of tub lmnts would b th on that causs no furthr variation in th dtrmind paramtrs. An adquat numbr of lmnts has bn found to b as low as. Th numbr of lmnts actually usd throughout th prsnt invstigation xcdd th amount considrd adquat by a factor of thr or mor, sinc in most of th runs, th numbr of lmnts was of th ordr of. Stp Arbitrary valus ar assignd to th ovrall air coil ntranc tmpraturs of both fluids, T a,i and T t,i, and to th product of th ovrall hat transfr cofficint by th hat transfr ara, UA. Stp 3 Valus of (UA),, C a C t and Γ for th tub lmnt ar valuatd. Stp 4 At this point, th tmpratur distribution along th air coil is valuatd through th subroutin TEMPERATURE, dscribd in th block diagram of Fig. 3. Starting from th ntranc of th tub fluid in on of th paralll circuits, th procdur in this subroutin consists in dtrmining th outlt tmpraturs of both fluids in succssiv tub lmnts in th path of th tub fluid. Th starting tmpraturs ar always th air coil inlt tmpraturs. It must b notd that, dpnding on th flow arrangmnt, th actual starting tub lmnt could corrspond to on in which th inlt air tmpratur might not b th air coil inlt tmpratur. Thus, as a gnral procdur, th inlt air tmpratur of th first tub lmnt is assumd to b th on at th air coil ntranc, and a trial procdur must b carrid out in ordr to dtrmin th xit air and tub fluid tmpraturs, as dscribd in th block diagram of Fig.. Th air and tub fluid tmpraturs in th tub lmnts along th particular circuit ar dtrmind in th subroutin TEMPERATURE along with th rat of hat transfr in ach tub lmnt. In th prsnt stp, th first trial is prformd. Stp 5 Th first trial xit avrag air tmpratur is dtrmind, according to th following quation: Ta,o T a, o ( i, j, N r ) i, j N t N = (8) / Vol. XXXII, No. 3, July-Sptmbr ABCM

4 Effctivnss - Data and Analysis for Air Conditioning and Rfrigration Air Coils Stp 6 Figur. Block diagram of th main program. Th scond trial is prformd calling th subroutin TEMPERATURE, using as input data in th tub lmnts th inlt air tmpraturs obtaind in th first trial. Th nw xit avrag air tmpratur is dtrmind and compard with th old on (prvious stp). Nw trials will b prformd until th rlativ diffrnc btwn th avrag xit air tmpratur in succssiv trials is lowr than a givn valu, δp. Th program procds to dtrmin ovrall air coil paramtrs such as avrag xit tub fluid tmpratur, ffctivnss, and ovrall rat of hat transfr, according to th following quations: Tt,o Nc Tt,o,m m Nc = = (9) q = q i, j,k = C t(tt,o Tt,i ) = Ca(Ta,o Ta, i ) () i, j,k q ΔTmax = = q T max t,i Ta,i ε () Th valus of th diffrnt trms in Eq. (), dtrmind indpndntly, must b clos nough to ach othr to warrant th nrgy consrvation. Figur 3. Block diagram of th subroutin TEMPERATURE. Program Prformanc Evaluation A sris of multi-row air coils with in-lin tub distribution has bn considrd for valuation of th prformanc of th proposd modl. Th arrangmnt is th typical cross-flow, as shown in Fig. 4, whr on can also not that th tub fluid is distributd in as many circuits as th numbr of tub rows. Both fluids flow in an unmixd/unmixd arrangmnt, xcpt in th coil of Fig. 4, for a singl tub-fluid circuit. Closd form (ε, NUT) quations ar availabl in th litratur for this kind of coils. Som of th ons listd in Tabl hav bn proposd by ESDU 985 (998) and ar valid for diffrnt numbr of tub rows. Ths corrlations also dpnd on th thrmal capacity ratio, C*, and on which of th fluids is th on with lowr thrmal capacity (ithr th air or th tub fluid). Two corrlations hav bn considrd whn th numbr of rows is highr than 4. Both of thm hav bn dvlopd for unmixd/unmixd flow arrangmnt and an infinit numbr of tub rows. Stvns t al. (957), basd on a prvious work by Mason (955), suggstd an infinit sris form, Eq. (6). According to Wang t al. (), this corrlation could b rducd to a simpl and straightforward form suggstd by ESDU 985 (998), Eq. (7). Effctivnss has bn dtrmind according to Eqs. (6) and (7) for hat capacitis ratio and valus totaling 6 calculation points. Th maximum and avrag absolut rlativ dviations (s blow for dfinitions) btwn rsults from ths corrlations with rspct to Eq. (6) ar of th ordr of 3.78% and.683% rspctivly. Th maximum dviation was obtaind for C* = and =.3. Ths dviations ar rathr small and confirm th J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright by ABCM July-Sptmbr, Vol. XXXII, No. 3 /

5 Hélio Aparcido Navarro t al. adquacy of th simplifid form, Eq. (7), for coils with a larg numbr of rows. Figur 4. Cross-flow coils gomtry for simulation program valuation. Two diffrnt paramtrs hav bn usd in th simulation program valuation: th absolut rlativ dviation, δ, and th avrag absolut rlativ dviation, δav, both of thm rlatd to th dviation of simulation program with rspct to corrlation rsults. Thir xprssions ar as follows: δ ε ε s t = (8) ε t N ε s ε t δ av = (9) N ε t Th program has bn run for valus of th hat capacity ratio, C*, varying in th rang btwn. and, and from. up to a maximum of 6, with incrmnts of. for ach paramtr, prforming a total of 6 computr program runs. Tabls and 3 prsnt a summary of th prformd comparisons. Rsults for coils up to 4 rows ar prsntd in Tabl. Columns and 3, for th two possibl conditions of minimum hat capacity rat, prsnt information rgarding th maximum absolut rlativ dviation and th hat capacity ratio and th for which this dviation has bn obtaind. Th rsulting dviations can b considrd as ngligibly small for all practical purposs. Tabl 3 prsnts comparisons of rsults from th simulation program with thos from Eq. (6), usd as rfrnc for that purpos. Th avrag absolut rlativ dviation, xtnsiv to all th data points, has bn includd in this tabl in addition to th paramtrs of Tabl. It can b notd that dviations diminish with th numbr of rows, as should b xpctd. Though rlativly small, dviations for coils with a numbr of rows lowr than ar much highr than thos obtaind for th othr coils. In fact, for th 5 rows coil, th maximum obtaind dviation is of th ordr of.45%, thrfold that for th 9 rows coil. Rsults from th simulation program and thos from th Stvns t al. (957) corrlation compar vry wll for coils with mor than rows, sinc, in such cass, th avrag dviations ar lowr than.3%. Tabl. Closd form (ε, ) corrlations for coils with diffrnt numbr of rows according to ESDU (998). N r C min Corrlation Equation Air C*( ) ε = [ ] C* (a) Tub fluid ( C* ) / C* ε = Air KC* ε = [ ( + C * K )] C * (3a) / K = Tub fluid = K / C* K ε + C * (3b) C * K = C* / 3 Air KC* ε = + C * K ( 3 K ) Tub fluid C * 3K ε = ( C *) K + / 3 K = / C* K = ( 3 K ) K + C * C* / 3 4 3K + ( C *) 4 Air KC* 4 ε = + C * K ( 6 4K + K ) + 4( C *) K ( K ) ( C *) K + 3 / 4 K = 4 4K / C* ( ) ( ) Tub fluid K 6 4K + K 4K K ε = + + ε *From Stvns t al. (957). C * K = C* / K 3 ( C *) 3( C *) (4a) (4b) (5a) (5b) n m ( ) n m ( ) ε C* C * = (6)* C * n= m= m! m=.. C* 78 / C* = m! (7) / Vol. XXXII, No. 3, July-Sptmbr ABCM

6 Effctivnss - Data and Analysis for Air Conditioning and Rfrigration Air Coils Tabl. Comparison in trms of th maximum absolut rlativ dviation of th ffctivnss dtrmind according to th prsnt modl, with rspct to Tabl corrlations for coils with numbr of rows from to 4. Coil Equation Maximum absolut dviation, % (C *, )* C min = C air, kw/k C min = C t, kw/k Fig. 4 and.7 x -5, (, 4.6).77 x -5, (.35, 6) Fig. 4 3 and.7 x -5, (.99, 5.8).9 x -5, (.38, 6) Fig. 4(c) 4 and.8 x -5, (, 6).95 x -5, (.4, 6) Fig. 4(d) 5 and.8 x -5, (, 5.6).97 x -5, (.48, 6) * C* and corrsponding to th maximum dviation. Tabl 3. Comparison in trms of th avrag and maximum absolut rlativ dviations of th ffctivnss dtrmind according to th simulation program, with rspct to that from th Stvns t al. (957) corrlation for coils with diffrnt numbr of rows. N r Equation Avrag and maximum absolut rlativ dviations, % (C *, )* C min = C air, kw/k C min = C t, kw/k ,.45, (,6).5,.45, (,6) 6 6.5,., (,6).7,., (,6) 7 6.9,.76, (,6).3,.76, (,6) 8 6.4,.58, (,6).97,.58, (,6) 9 6.,.46, (,6).78,.46, (,6) 6.9,.38, (,6).63,.38, (,6) 6.3,.95, (,6).5,.95, (,6) ,.5, (,6).5,.5, (,6) * C* and corrsponding to th maximum dviation. Effctivnss - Rlations for Coils of Complx Flow Arrangmnt Closd form accurat (ε, ) xprssions for coils with simpl gomtry and flow arrangmnt wr considrd in th prcding sction. Contrary to gomtry and flow arrangmnt of thos coils, rfrigration and air conditioning coils can b rathr complx. In such cass, accurat (ε, ) xprssions ar not radily availabl, and simulation programs lik th on considrd in this papr could provid usful rsults in that rspct. In ordr to illustrat such a capability, th simulation program has bn applid to th four coils shown schmatically in Fig. 5. Thr of ths coils,, (c) and (d), ar similar to th ons considrd by Rich (975). Th numbr of volum lmnts considrd in ach run was st qual to on hundrd, a vry high valu considring commnts mad undr th dvlopmnt of th simulation program. For ach coil, th program has bn run for hat capacitis ratio and Numbr of Transfr Units in th following rangs: C* and. 6, with corrsponding stps of.5 and. rspctivly. Whn th tub fluid undrgos a phas chang, th hat capacity ratio, C*, is qual to zro, and th (ε, ) corrlation is th sam as that of a countr flow hat xchangr, rgardlss of th flow arrangmnt () For th sam valu of, th coil ffctivnss incrass with th numbr of rows. This is an xpctd rsult givn that th hat transfr ara incrass with th numbr of rows and, as a rsult, so dos th xit tmpratur of both fluids. () Th (ε, ) rlationship dpnds on which of th fluids is th on with th lowst hat capacity rat. This trnd is clarly rproducd by th corrlations of Tabl, and by th simulation program rsults as plots and diffr from ach othr. ) ε = ( () which corrsponds to th maximum hat xchangr ffctivnss. All th corrlations of Tabl satisfy this asymptotic limit and so do th simulation program rsults. Figurs 6 to 9 prsnt th plots of ffctivnss vrsus th Numbr of Transfr Units for th coils of Fig. 5. Two plots ar includd in ach figur corrsponding to ithr th tub fluid or th air with th lowr hat capacity rat. Th uppr continuous lin in ach plot corrsponds to C* =, Eq. (). For comparison purposs, rsults from Tabl corrlations hav bn suprposd with thos from th program in ach plot (brokn lins). Th corrlation from Tabl usd for comparison in ach plot is th on with th sam numbr of rows as th corrsponding coil in Fig. 5. A clos xamination of ths plots allows on to draw svral conclusions, which can b summarizd as follows: Figur 5. Tub fluid flow arrangmnt of th coils. Th air flows from lft to right. Z-shap cross-flow Nt = ; staggrd two-row and twocircuit arrangmnt, Nt = ; (c) staggrd thr-row two-circuit arrangmnt, Nt = ; (d) staggrd six-row fiv-circuit arrangmnt, Nt =. (3) Figurs 6 and 7 for two-row staggrd-tub coils clarly display diffrncs in ffctivnss, though ths coils basically diffr from ach othr just in th flow arrangmnt of th tub fluid. This trnd is not capturd by th availabl corrlations, but it is apparnt whn th computr program rsults ar compard. In th prsnt cas, th arrangmnt of Fig. 5 is clarly mor fficint than th on of Fig. 5. J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright by ABCM July-Sptmbr, Vol. XXXII, No. 3 / 3

7 Hélio Aparcido Navarro t al. (4) As a gnral rul, in th rang of highr (roughly >.5), th corrlations tnd to undrstimat th coil ffctivnss. An xcption to this rul is th gomtry of Fig. 5, sinc, in this cas, Eqs. 3 and, for two row coils, ovr-prdict th ffctivnss. (5) In th rang of lowr, corrlations rsults with rspct to thos from th simulation program do not follow a clar trnd, and th rlativ bhavior dpnds on th particular gomtry and flow arrangmnt. For xampl, in th cas of th gomtry of Fig. 5(d), for both conditions of lowr hat capacity rat, Figs. 9 and clarly display a trnd shift in th rlativ bhavior of th (ε, ) rlationship. In fact, in th lowr rang, th corrlation ffctivnss slightly ovrstimats th simulation program. This trnd changs for highr valus, with th shifting point dpnding on th hat capacity ratio. Th lowr th lattr, th highr th shifting. Notic that th sam conclusion applis to Figs. 7 and 8 and th shift point bhavior is also th sam. (6) Dviations of corrlation with rspct to program rsults ar prsntd in Tabl 4, raching valus as high as.%, obtaind for th coil of Fig. 5. Dviations for th othr coils ar limitd to a maximum of 8.39% for th Fig. 5(d) coil. Th last dviations occur for th coil of Fig. 5, with two paralll tub fluid circuits. (7) Th dviations for cass and of th plots of Figs. 6 to 9 tnd to b vry clos to ach othr. (8) It is intrsting to not, as clarly shown in Tabl 4, that th maximum dviations occur prfrably for a hat capacity ratio of th ordr of on, and rlativly high valus (of th ordr of 5). An xcption to this rul is th gomtry of Fig. 5, which coincidntly is th on with lowr dviations. In this cas, th maximum dviation is obtaind at a hat capacity ratio of. and a qual to.. (9) A comparison of th Stvns t al. (957) corrlation rsults, Eq. (6), with thos from th simulation program, for th coil of Fig. 5(d), has also bn includd in th last row of Tabl 4. Dviations ar similar to thos obtaind for th ESDU (998) corrlation, Eq. (7). Th maximum dviation of rsults from ths corrlations with rspct to thos from th simulation program ar of th ordr of 8%, a valu that maks thir us qustionabl for coils with this flow arrangmnt and numbr of tub rows. Tabl 4. Dviations of corrlations from Tabl with rspct to simulation program rsults for th coils of Fig. 5. Coil Equation Avrag rlativ rror, %, maximum rlativ rror, % (C *, )* C min = C air, kw/k C min = C t, kw/k Fig. 5 3 and 6.4,., (.9, 6) 4.49,.93, (, 6) Fig. 5 3 and.8, 4.6, (.,.).4, 4.5, (.,.) Fig. 5(c) 4 and 3., 6.9, (, 5.) 3.8, 6.9, (, 5.) Fig. 5(d) 7 3.9, 8.3, (, 6) 3.45, 8.39, (.9, 6) Fig. 5(d) , 7.8, (, 4.6) 3.86, 7.8, (, 4.6) * C* and corrsponding to th maximum dviation. Effctivnss C * = C o i l F i g. 5 ( a )/ C m i n = C a S i m u l a t i o n p r o g r a m E q. ( 3. a ) C*=.5 C*=.5 C*=.75 C*= Effctivnss C*= Coil F i g. 5 ( a )/ C m i n = C t Simu l a t i o n p r o g r a m Eq.( 3. b ) C*=.5 C*=.5 C*=.75 C*= Figur 6. Effctivnss variation with for th coil of Fig. 5. Corrlation rsults ar plottd as brokn lins (two rows). Ca < Ct; Ct < Ca. 4 / Vol. XXXII, No. 3, July-Sptmbr ABCM

8 Effctivnss - Data and Analysis for Air Conditioning and Rfrigration Air Coils Effctivnss C*= Coil Fig.5/ C min = C a Simulation program Eq. (3.a) C*=.5 C*=.5 C*=.75 C*= Effctivnss C*= Coil Fig. 5(d)/ C min = C a Simulation program Eq. (6) C*=.5 C*=.5 C*=.75 C*= Effctivnss C*= Coil Fig. 5/ C min = C t Simulation program Eq. (3.b) C*=.5 C*=.5 C*=.75 C*= Effctivnss C*= Coil Fig 5(d)/ C min = C t Simulation program Eq. (6) C*=.5 C*=.5 C*=.75 C*= Figur 7. Effctivnss variation with for th coil of Fig. 5. Corrlation rsults ar plottd as brokn lins (two rows). Ca < Ct; Ct < Ca Figur 9. Effctivnss variation with for th coil of Fig. 5(d). Corrlation rsults ar plottd as brokn lins (six rows). Ca < Ct; Ct < Ca. Effctivnss Effctivnss C * = C o i l F i g. 5 ( c )/ C m i n = C a S i m u l a t i on p r o g r a m E q. ( 4. a ) C*=.5 C*=.5 C*=.75 C*= C * = C o i l F i g. 5 ( c )/ C m i n = C t S i m u l a ti o n p r o g r a m E q. ( 4. b ) C*=.5 C*=.5 C*=.75 C*= Figur 8. Effctivnss variation with for th coil of Fig. 5(c). Corrlation rsults ar plottd as brokn lins (thr rows). Ca < Ct; Ct < Ca. Conclusions Th computr simulation program dscribd hrin has bn applid in th valuation of (ε, ) rlationships for air conditioning and rfrigration coils. It has bn dtrmind that, for th cas of strictly cross-flow gomtris, th availabl (ε, ) corrlations ar adquat up to four tub rows. Corrlations for an infinit numbr of tub rows, such as th Stvns t al. (957), ar rlativly inaccurat whn applid to coils with rows varying in th rang btwn 5 and. Caution must b xrcisd whn applying (ε, ) closd form corrlations to complx flow arrangmnt and gomtry coils, sinc it has bn shown that modrat inaccuracis might rsult. In such cass, th us of simulation programs lik th on of th prsnt papr is rcommndd. As a concluding rmark, it must b strssd that th rsults discussd hrin rgarding coils of complx gomtry allow on to conclud that th indiscriminat us of closd form corrlations could lad to unaccptabl inaccuracis in th dtrmination of ithr th ffctivnss or th. An xampl of th lattr would b th cas whn air-sid hat transfr data ar bing dtrmind from xprimnts involving flow of watr insid th tubs. Sinc th tub-sid hat transfr charactristics ar radily known, th air-sid ons ar obtaind from th ovrall coil conductanc (UA), which in turn rsults from th (ε, ) rlationship. Thus, inaccuracis could caus sam ordr inaccuracis in th air-sid hat transfr charactristics. Acknowldgmnts Th first author acknowldgs th financial support by Brazilian rsarch funding agncy CNPq (Conslho Nacional d J. of th Braz. Soc. of Mch. Sci. & Eng. Copyright by ABCM July-Sptmbr, Vol. XXXII, No. 3 / 5

9 Hélio Aparcido Navarro t al. Dsnvolvimnto Cintífico). Th scond author thanks CNPq and FAPEMIG (Fundação d Amparo à Psquisa do Estado d Minas Grais). G. Ribatski acknowldgs th support givn by Th Stat of São Paulo Rsarch Foundation FAPESP, Brazil. Rfrncs Baclic, B.S., 99, ε- analysis of complicatd flow arrangmnts. In: R.K. Shah, A.D. Kraus, and D. Mtzgr (ditors) Compact Hat Exchangrs. Hmisphr Publishing, Nw York, pp.3-9. Bansal, P.K. and Purkayastha, B., 998, An -ε modl for altrnativ rfrigrants, Intrnational Journal of Rfrigration, Vol., No. 5, pp Bnsafi, A., Borgand, S. and Parnt, D., 997, CYRANO: a computational modl for th dtaild dsign of plat-fin-and-tub hat xchangrs using pur and mixd rfrigrants, Intrnational Journal of Rfrigration, Vol., No. 3, pp Bowman, R.A., Mullr, A.C. and Nagl, W.M., 94, Man tmpratur diffrnc in dsign, ASME Transactions, Vol. 6, pp Cabzas-Gómz, L., Navarro, H.A. and Saiz-Jabardo, J.M., 7, Thrmal prformanc of multi-pass paralll and countr cross-flow hat xchangrs, ASME Hat and Mass Transfr (in prss). Corbrán, J.M. and Mlón, M.G., 998, Modlling of plat finnd tub vaporators and condnsrs working with R34a, Intrnational Journal of Rfrigration, Vol., No. 4, pp Domanski, P.A., 99, Simulation of an vaporator with non-uniform on-dimnsional air distribution, ASHRAE Transactions, Vol. 97, No., pp Domingos, J.D., 969, Analysis of complx assmblis of hat xchangrs, Intrnational Journal of Hat and Mass Transfr, Vol., pp ESDU 985, 998, Dsign and Prformanc Evaluation of Hat Exchangrs: Th Effctivnss Mthod. Enginring Scinc Data Unit 985, London: ESDU Intrnational Publishing, July, pp Kays, W.M. and London, A.L., 998, Compact Hat Exchangrs. Nw York: McGraw-Hill. Mason, J.L., 955, Hat transfr in crossflow, Procdings nd US National Congrss of Applid Mchanics, ASME, Nw York, pp Navarro, H.A. and Cabzas-Gómz, L., 5, A nw approach for thrmal prformanc calculation of cross-flow hat xchangrs, Intrnational Journal of Hat Mass Transfr, Vol. 48, pp Pignotti, A., 988, Linar matrix oprator formalism for basic hat xchangr thrmal dsign, ASME Journal of Hat Transfr, Vol., pp Pignotti, A and Shah, R.K., 99, Effctivnss-numbr of transfr units rlationships for hat xchangr complx flow arrangmnts, Intrnational Journal of Hat and Mass Transfr, Vol. 35, pp Pignotti, A. and Cordro, G.O., 983, Man tmpratur diffrnc in multipass crossflow, ASME Journal of Hat Transfr, Vol. 5, pp Rich, D.G., 975, Th ffct of th numbr of tub rows on hat transfr prformanc of smooth plat fin-and-tub hat xchangrs, ASHRAE Transactions, Vol. 8, pp Shah, R.K. and Pignotti, A., 993, Thrmal analysis of complx crossflow xchangrs in trms of standard configurations, ASME Journal of Hat Transfr, Vol. 5, pp Stvns, R.A., Frnandz, J. and Woolf, J.R., 957, Man tmpratur diffrnc in on, two and thr-pass crossflow hat xchangrs, Transactions ASME, Vol. 79, pp Vardhan, A. and Dhar, P.L., 998, A nw procdur for prformanc prdiction of air conditioning coils, Intrnational Journal of Rfrigration, Vol., No., pp Wang, C.C., Wbb, R.L. and Chi, K.Y.,, Data rduction for airsid prformanc of fin-and-tub hat xchangrs, Exprimntal Thrmal and Fluid Scinc, Vol., pp / Vol. XXXII, No. 3, July-Sptmbr ABCM