Nusselt number correlations for simultaneously developing laminar duct flows of liquids with temperature dependent properties

Transcription

1 Journal of Physics: Confrnc Sris OPEN ACCESS Nusslt numbr corrlations for simultanously dvloping laminar duct flows of liquids with tmpratur dpndnt proprtis To cit this articl: Stfano Dl Giudic t al 2014 J. Phys.: Conf. Sr Viw th articl onlin for updats and nhancmnts. Rlatd contnt - Entranc and tmpratur dpndnt proprty ffcts in th laminar forcd convction in straight ducts with uniform wall tmpratur Stfano Dl Giudic, Stfano Savino and Carlo Nonino - Numrical Invstigation of Viscous Dissipation in Elliptic Microducts P Vocal, G Pucctti, G L Morini t al. - Passiv tchniqus for th nhancmnt of convctiv hat transfr in singl phas duct flow S Rainiri, F Bozzoli and L Cattani This contnt was downloadd from IP addrss on 28/01/2018 at 10:20

2 Nusslt numbr corrlations for simultanously dvloping laminar duct flows of liquids with tmpratur dpndnt proprtis Stfano Dl Giudic, Stfano Savino and Carlo Nonino Dipartimnto di Inggnria Elttrica, Gstional Mccanica, Univrsità dgli Studi di Udin,Via dll Scinz 206, UDINE, Italy Abstract. Nw corrlations, suitabl for nginring applications, for th man Nusslt numbr in th ntranc rgion of circular tubs and squar ducts with uniform hat flux boundary conditions spcifid at th walls ar proposd. Ths corrlations ar obtaind on th basis of th rsults of a prvious paramtric invstigation on th ffcts of tmpratur dpndnt viscosity and thrmal conductivity in simultanously dvloping laminar flows of liquids in straight ducts of constant cross-sctions. In ths studis, a finit lmnt procdur has bn mployd for th numrical solution of th parabolizd momntum and nrgy quations. Viscosity and thrmal conductivity ar assumd to vary with tmpratur according to an xponntial and to a linar rlation, rspctivly, whil th othr fluid proprtis ar hld constant. Axial distributions of th man Nusslt numbr, obtaind by numrical intgration from thos of th local Nusslt numbr, ar usd as input data in th drivation of th proposd corrlations. A suprposition mthod is provd to b applicabl in ordr to stimat th Nusslt numbr by considring sparatly th ffcts of tmpratur dpndnt viscosity and thrmal conductivity. Thrfor, for ach of th considrd cross-sctional gomtris, two distinct corrlations ar proposd for flows of liquids with tmpratur dpndnt viscosity and with tmpratur dpndnt thrmal conductivity, in addition to that obtaind for constant proprty flows. 1. Introduction In many laminar duct flows, ntranc ffcts on fluid flow and hat transfr must b tan into account, sinc th total lngth of th duct is comparabl with that of th ntranc rgion. Morovr, th dvlopmnt of th vlocity and tmpratur filds can b strongly affctd by tmpratur dpndnc of fluid proprtis. If th fluid is a liquid, th rlativ variations of viscosity with tmpratur ar th most rlvant, whil thos of thrmal conductivity ar, in gnral, much smallr, and thos of dnsity and spcific hat capacity ar almost ngligibl [1, 2]. As far as vlocity distribution and prssur drop ar concrnd, th main ffcts of tmpratur dpndnt fluid proprtis can b rtaind vn if only viscosity is considrd to vary with tmpratur, whil th othr proprtis ar assumd constant [1, 3]. Instad, if hat transfr charactristics, as th Nusslt numbr, hav to b valuatd, also th tmpratur dpndnc of thrmal conductivity must b tan into account [2]. Contnt from this wor may b usd undr th trms of th Crativ Commons Attribution 3.0 licnc. Any furthr distribution of this wor must maintain attribution to th authors) and th titl of th wor, journal citation and DOI. Publishd undr licnc by Ltd 1

3 In prvious articls, w carrid out systmatic analyss of th ffcts of tmpratur dpndnt viscosity and thrmal conductivity on forcd convction in simultanously dvloping laminar flows of liquids in straight ducts with uniform wall hat flux boundary conditions [4, 5]. In particular, w obtaind axial distributions of th local valu of th priphrally avragd Nusslt numbr for ducts of diffrnt cross-sctions. W assumd that viscosity dcrass with incrasing tmpratur according to an xponntial rlation, whil thrmal conductivity varis linarly with tmpratur. Suitabl dimnsionlss paramtrs, namly, th viscosity and thrmal conductivity Parson numbrs, wr usd to quantitativly xprss tmpratur dpndnc of th corrsponding proprtis. A finit lmnt procdur was mployd for th stp-by-stp solution of th parabolisd momntum and nrgy quations in a domain corrsponding to th cross-sction of th duct [6]. Finally, a suprposition mthod was provd to b applicabl in ordr to obtain accurat stimats of th local Nusslt numbr by considring sparatly th ffcts of tmpratur dpndnt viscosity and thrmal conductivity [4, 5]. In this papr, first w us prvious numrical rsults for th local Nusslt numbr for circular and squar cross-sctional gomtris [5] to obtain by numrical intgration th corrsponding axial distributions of th man, i.., longitudinally avragd, Nusslt numbr. Thn, on th basis of such distributions, w obtain vry accurat corrlations, suitabl for nginring applications. Actually, taing advantag of th abov mntiond suprposition mthod, w propos for ach of th considrd cross-sctional gomtris a corrlation valid for constant proprty flows and two distinct corrlations for flows of liquids with tmpratur dpndnt viscosity and with tmpratur dpndnt thrmal conductivity. Finally, w dmonstrat that, by proprly assmbling ths corrlations, accurat stimats of th man Nusslt numbr can b obtaind in wid rangs of oprativ conditions. No corrlations ar prsntd hr for prssur drop paramtrs sinc, as dmonstratd in prvious wors [4, 5], th influnc of thrmal conductivity variation on prssur drop is almost ngligibl if compard with that of viscosity variations. Thrfor, th corrlations for prssur drop paramtrs alrady proposd by th authors for liquids with tmpratur dpndnt viscosity and constant thrmal conductivity [7] can b usd for all liquids with tmpratur dpndnt proprtis. 2. Statmnt of th problm Th laminar forcd convction in th ntranc rgion of straight ducts of constant cross-sctions with uniform wall hat flux q w > 0 fluid hating) is studid. Sinc th liquid hating is assumd to bgin at th duct inlt, uniform valus of th axial vlocity componnt u and of th tmpratur t i.., t = t, u = u = u and v = w = 0) ar spcifid as th appropriat inlt conditions, bing u th avrag axial vlocity and v and w th transvrs vlocity componnts. Th hat flux Numann) boundary condition q = t/ n = q w is applid at th solid walls, whr is th thrmal conductivity and n is th outward normal to th boundary. Sinc th fluids considrd hr ar liquids, th dynamic viscosity is assumd to dcras with incrasing tmpratur according to th widly usd xponntial formula [3] = xp[ β t t )] 1) whr is th valu of at t and β = d/dt)/ = const is positiv. Th thrmal conductivity can b assumd both to incras or dcras with incrasing tmpratur, dpnding on th fluid considrd, according to th linar rlation = [1 + α t t )] 2) whr is th valu of at t and α = d/dt)/ = const can b both positiv or ngativ. By mans of simpl manipulations, quations 1) and 2) can b cast in th following dimnsionlss forms = xp Pn T ) 3) 2

4 = 1 + Pn T 4) whr T = t t ) /q w D h ) is th dimnsionlss tmpratur, D h is th hydraulic diamtr, Pn = β q w D h / is th viscosity Parson numbr rprsnting th ratio of th charactristic procss tmpratur diffrnc q w D h / to th charactristic tmpratur diffrnc 1/β that can produc apprciabl viscosity variations) and Pn = α q w D h / is th thrmal conductivity Parson numbr rprsnting th ratio of th charactristic procss tmpratur diffrnc q w D h / to th charactristic tmpratur diffrnc 1/α that can produc apprciabl thrmal conductivity variations). It is worth noting that, sinc th dnsity ρ and th spcific hat c ar constant, th local Rynolds numbr R = ρ u D h /, th local Prandtl numbr Pr = c/ and th local Péclt numbr P = R Pr = ρ c u D h / all vary with tmpratur bcaus of th variations of, of th ratio / and of, rspctivly. Thrfor w hav / = R /R and / = P /P, whr R and P ar th Rynolds and Péclt numbrs valuatd at t. Morovr, sinc th viscosity of liquids dcrass with incrasing tmpratur β > 0), in th cas of fluid hating w hav Pn > 0 and R /R < 1, whil Pn = 0 and R /R = 1 rfr to constant viscosity fluids β = 0). Instad, sinc th thrmal conductivity can ithr incras α > 0) or dcras α < 0) with incrasing tmpratur, w hav Pn > 0 and P /P > 1 in th first cas and Pn < 0 and P /P < 1 in th scond on, whil Pn = 0 and P /P = 1 rfr to constant thrmal conductivity fluids α = 0). As alrady pointd out, as th fluid tmpratur riss along th duct, th viscosity dcrass whil R incrass. Thrfor, to nsur laminar flow conditions, th local Rynolds numbr R b valuatd at th bul tmpratur is only allowd to rach th maximum valu R b ) max = 2, 000, corrsponding to th maximum valu x max of th axial coordinat x, whrupon computations ar stoppd. Thrfor, taing into account quation 3) and th appropriat hat balanc for th duct, th following xprssion for th maximum valu Xmax of th dimnsionlss axial coordinat X = x/d h P ) can b obtaind [4, 5, 7] Xmax = 1 [ ] Rb ) max ln 4 Pn R Numrical rsults of intrst for th prsnt study concrn th axial distributions of th priphrally avragd local Nusslt numbr Nu = hd h /, whr h is th priphrally avragd local convctiv hat transfr cofficint dfind as 5) h = q w t w t b = D h T w T b ) 6) In quation 6), t w and t b ar th priphrally avragd wall tmpratur and th man bul tmpratur, rspctivly, and T w and T b ar thir dimnsionlss forms. According to quation 6), th priphrally avragd local Nusslt numbr can b xprssd as Nu = 1 T w T b 7) Axial distributions of th longitudinally avragd Nusslt numbr, dfind as [1] Nu = 1 x x 0 Nu dx = 1 X X Nu dx 8) 0 can b obtaind by mans of an appropriat numrical intgration rul. 3

5 3. Numrical procdur If th ffcts of axial diffusion can b nglctd R > 50 and P > 50) and thr is no rcirculation in th longitudinal dirction, stady-stat flow and hat transfr in straight ducts of constant cross-sctions ar govrnd by th continuity and th parabolizd Navir-Stos and nrgy quations [8, 9]. Ths quations ar not rportd hr for lac of spac, but thy ar rportd lswr togthr with th boundary conditions spcifid on th boundaris of th computational domain [5, 7]. A finit lmnt procdur for th analysis of th forcd convction of fluids with tmpratur dpndnt proprtis in th ntranc rgion of straight ducts [6, 7, 10] is usd to solv th modl quations. Th adoptd procdur is basd on a sgrgatd approach which implis th squntial solution of th momntum and nrgy quations on a two-dimnsional domain in th cas of thr-dimnsional gomtris and on a on-dimnsional domain in axisymmtric problms, corrsponding to th cross-sction of th duct. A marching mthod is thn usd to mov forward in th axial dirction. Th prssur-vlocity coupling is dalt with using an improvd projction algorithm alrady mployd by on of th authors C.N.) for th solution of th Navir-Stos quations in thir lliptic form [11]. Th procdur has alrady bn validatd, with rfrnc to both constant and tmpratur dpndnt proprty fluids, by comparing hat transfr and prssur drop rsults with xisting litratur data for simultanously dvloping laminar flows in straight ducts [6, 10, 12, 13]. 4. Numrical rsults In this study, two cross-sctional gomtris ar considrd, namly circular and squar, chosn as rprsntativ of axisymmtric and thr-dimnsional duct gomtris, rspctivly. Th corrsponding computational domains, dfind taing into account xisting symmtris, ar on-dimnsional and two-dimnsional for th circular and th squar cross-sctions, rspctivly. Dtails of finit lmnt domain and tim discrtisations ar rportd lswr [4, 5]. Th sam valu of th Rynolds numbr R = 100 is assumd in all th computations. Instad, th valus Pr = 5, 20 and 100 of th Prandtl numbr at t ar slctd to ta into account th bhaviours of diffrnt liquids. Th corrsponding valus of th Péclt numbr at t ar P = 500, 2000 and Th valus of th dynamic viscosity Parson numbr Pn = 0, 1, 2 and 4 ar considrd to account for rasonabl viscosity tmpratur dpndncs. Thus, for th assumd R, th maximum valus of th dimnsionlss axial coordinat givn by quation 5) ar Xmax = , and for Pn = 1, 2 and 4, rspctivly. For ach nonzro valu of Pn ight valus of Pn ar slctd four positiv and four ngativ), giving th corrsponding valus of th ratio Pn /Pn = β/α = ±10, ±20, ±40 and ±80. Thus, on th whol, th valus Pn = 0, ±0.0125, ±0.025, ±0.05, ±0.1, ±0.2 and ±0.4 ar considrd. Th ffcts of tmpratur dpndnt proprtis viscosity and thrmal conductivity) on hat transfr can b illustratd by comparing th local Nusslt numbr Nu, obtaind for givn nonzro valus of Pn and Pn, with th corrsponding local Nusslt numbr, computd for simultanously dvloping constant proprty flows Pn = Pn = 0). Thrfor, w can assum th valu of th ratio Nu )/ = Nu / ) 1 as a masur of th ffcts of both tmpratur dpndnt viscosity and thrmal conductivity. Axial distributions of th ratio Nu / with rfrnc to th dimnsionlss coordinat X, obtaind by mans of th numrical procdur dscribd in Sction 3 for th sam cross-sctional gomtris and valus of dimnsionlss input paramtrs considrd in this papr, ar rportd lswr [4, 5]. Th conclusion rachd thr is that both tmpratur dpndnt viscosity and thrmal conductivity can hav comparabl ffcts on th Nusslt numbr, according to th valus of Pn and Pn. Morovr, it has bn dmonstratd that if th ffcts of tmpratur dpndnt viscosity and thrmal conductivity ar considrd sparatly to comput Nu undr th assumptions of tmpratur dpndnt viscosity and constant thrmal conductivity, i.., Pn > 0 and Pn = 0) 4

6 and Nu undr th assumptions of tmpratur dpndnt thrmal conductivity and constant viscosity, i.., Pn 0 and Pn = 0), a suprposition mthod is applicabl in ordr to obtain approximat valus of th Nusslt numbr Nu, according to th rlation Nu = Nu ) + Nu ) 9) with an accuracy which can b considrd satisfactory in most situations. With rfrnc to th ratios Nu /, Nu / and Nu /, quation 9) can b rcast in th form [4, 5] Nu ) Nu = = Nu + Nu 1 10) whr Nu / ) is th approximat valu of Nu / givn by th suprposition mthod. Thrfor, th valus of th diffrncs Nu / ) 1 and Nu / ) 1 approximatly masur th sparat ffcts of tmpratur dpndnt viscosity and thrmal conductivity, rspctivly. In this wor, prvious numrical rsults concrning axial distributions of th ratios Nu /, Nu / and Nu / [5] hav bn usd to obtain, by mans of a suitabl numrical intgration rul according to quation 8), th corrsponding distributions of th ratios Nu /, Nu / and Nu /. Thn, it has bn vrifid that th following rlation, obtaind by combining quations 8) and 9), still holds tru with an accuracy which can b considrd satisfactory in most situations Nu = Nu ) = Nu + Nu 1 11) In quation 11), Nu / ) rprsnts th approximat valu of Nu / givn by th suprposition mthod. Axial distributions of th ratio Nu / ) for th highst valus of Pn, i.., Pn = 2, and Pn = 4, diffrnt valus of Pn and Pr = 20 ar rportd in figurs 1 and 2 for circular tubs and squar ducts, rspctivly, to allow th comparison with th corrsponding distributions of Nu /. As can b sn, vn for th highst valus of Pn and th highst/lowst valus of Pn th dashd curvs, rprsnting axial distributions of th ratio Nu / ), ar vry clos to th solid ons giving th numrical solutions for Nu /, Nu Pn = 2 Nu Pn = 4 Pn = 0.4 Pn Nu = 0.2 c Nu / Nu 0.1 c Nu / 0.1 Nu ) Nu ) / 0.05 / X * X * 1 a) b) ) Figur 1. Comparison of axial distributions of th ratios Nu, / and Nu, / for simultanously dvloping laminar flows in circular tubs with Pr = 20, diffrnt valus of Pn and: a) Pn = 2 and b) Pn = 4. 5

7 1.3 Nu Pn = 2 Nu / Nu / ) Pn = Nu Pn = 4 Nu / Nu / ) Pn = X * X * 1 a) b) ) Figur 2. Comparison of axial distributions of th ratios Nu, / and Nu, / for simultanously dvloping laminar flows in squar ducts with Pr = 20, diffrnt valus of Pn and: a) Pn = 2 and b) Pn = 4. thus confirming th validity of quation 11). Morovr, th maximum positiv and ngativ rlativ rrors in th approximation of Nu / by mans of Nu / ) ar ε + max = 0.15% and ε max = 0.28% for circular ducts and ε + max = 0.42% and ε max = 1.39% for squar ducts in th rang 10 4 X X max with Pr = 5, 20 and 100 and diffrnt valus of Pn and Pn. Thrfor, w can conclud that th agrmnt btwn computd and approximat rsults is vry good. 5. Corrlations for th man Nusslt numbr Taing advantag of quation 11), a corrlation for Nu has bn obtaind by assmbling th diffrnt corrlations for, Nu / and Nu / obtaind from computd rsults for flows of liquids with constant proprty Pn = Pn = 0), tmpratur dpndnt viscosity Pn = 0 and Pn > 0) and tmpratur dpndnt thrmal conductivity Pn = 0 and Pn 0), rspctivly Corrlations for th man Nusslt numbr for flows of liquids with constant proprtis Th rsults obtaind for circular and squar cross-sctional gomtris undr th assumption of constant proprty flow Pn = Pn = 0) hav bn usd to obtain corrlations for th man Nusslt numbr as a function of X in th form = ) fd + a X ) m 1 + b X ) n 12) whr ) fd, i.., th asymptotic valu of for fully dvlopd conditions, th cofficints a and b and th xponnt m only dpnd on th cross-sctional gomtry, whil th xponnt n also dpnds on th Prandtl numbr Pr according to th rlation n = n 1 Pr s n 2 Pr 13) whos cofficints n 1 and n 2 and th xponnt s only dpnd on th cross-sctional gomtry. Th appropriat forms of quations 12) and 13) for circular and squar cross-sctions, valid 6

8 Tabl 1. Corrlations for corrsponding to quations 12) and 13) for circular and squar cross-sctions. Cross-sction circular squar Corrlations for = X ) X ) n n = Pr Pr = X ) X ) n n = Pr Pr for 5 Pr 100 and 10 4 X X max, ar rportd in tabl 1. Sinc, for Pr = 5, 20 and 100, th maximum positiv and ngativ rlativ rrors in th approximation of by mans of quation 12) ar ε + max = 0.80% and ε max = 1.29% for circular ducts and ε + max = 2.20% and ε max = 1.31% for squar ducts, w can conclud that th agrmnt btwn computd and approximat rsults is vry good Corrlations for th man Nusslt numbr for flows of liquids with tmpratur dpndnt viscosity Th rsults obtaind for circular and squar cross-sctional gomtris for Pn = 0 and Pn > 0 hav bn usd to obtain corrlations for th ratio Nu / as a function of X in th form Nu = 1 + [ Nu ) fd 1 ] {1 xp [ a X ) m b X ) n ]} 14) In quation 14), Nu / ) fd is th asymptotic valu for fully dvlopd conditions of th ratio Nu /, which can b valuatd by mans of th vry accurat formula ) Nu = 1 + d 1 Pn + d 2 Pn 2 15) fd whos cofficints d 1 and d 2 dpnd on th cross-sctional gomtry. Bsids, still with rfrnc to quation 14), th cofficint b and th xponnt n only dpnd on th cross-sctional gomtry, whil th cofficint a and th xponnt m also dpnd on th viscosity Parson numbr Pn and on th Prandtl numbr Pr, rspctivly, and can b xprssd as a = a 1 + a 2 Pn 16) m = m 1 Pr s 17) whr th cofficints a 1, a 2 and m 1 and th xponnt s only dpnd on th cross-sctional gomtry. For nginring application, th appropriat forms of quations 14) to 17) for circular and squar cross-sctions, valid for 5 Pr 100, 1 Pn 4 and 10 4 X Xmax, ar rportd in tabl 2. For Pr = 5, 20 and 100, th maximum positiv and ngativ rlativ rrors in th approximation of Nu / by mans of quation 14) ar ε + max = 0.14% and ε max = 0.21% for circular ducts and ε + max = 0.31% and ε max = 0.28% for squar ducts. Thrfor, w can conclud that th agrmnt btwn computd and approximat rsults is vry good. 7

9 Tabl 2. Corrlations for Nu / corrsponding to quations 14) to 17) for circular and squar cross-sctions. Cross-sction circular Corrlations for Nu / [ Nu ) Nu { = 1 + 1] 1 xp [ a X ) m 2.0 X ) 0.6]} fd ) Nu = Pn Pn 2 fd a = Pn m = 0.48 Pr squar [ Nu ) Nu { = 1 + 1] 1 xp [ a X ) m X ) 0.8]} fd ) Nu = Pn Pn 2 fd a = Pn m = 0.57 Pr Corrlations for th man Nusslt numbr for flows of liquids with tmpratur dpndnt thrmal conductivity Th rsults obtaind for th considrd cross-sctional gomtris for Pn = 0 and Pn 0 hav bn usd to obtain corrlations for th ratio Nu / as a function of X in th form whr th cofficint a can b xprssd as Nu ) = 1 + a Pn + b 1 Pn + b 2 Pn 2 X 18) a = a 1 Pr d {1 xp [ a 2 X ) n ]} 19) In quations 18) and 19) th cofficints a 1, a 2, b 1 and b 2 and th xponnt d only dpnd on th cross-sctional gomtry and on th sign of Pn, whil th xponnt n also dpnds on th Prandtl numbr Pr according to th rlation n = n 1 Pr s 20) whr th cofficint n 1 and th xponnt s only dpnd on th cross-sctional gomtry. For nginring application, th appropriat forms of quations 18) to 20) for circular and squar cross-sctions, valid for 5 Pr 100, Pn 0.4 and 10 4 X Xmax, ar rportd in tabl 3. For Pr = 5, 20 and 100, th maximum positiv and ngativ rlativ rrors in th approximation of Nu / by mans of quation 18) ar ε + max = 0.53% and ε max = 0.21% for circular ducts and ε + max = 0.30% and ε max = 0.17% for squar ducts. Thrfor, w can conclud that th agrmnt btwn computd and approximat rsults is vry good. 8

10 Tabl 3. Corrlations for Nu / corrsponding to quations 18) and 19) for diffrnt cross-sctions and signs of Pn. Cross-sction Pn Corrlations for Nu / circular squar > 0 < 0 > 0 < 0 Nu ) = 1 + a Pn Pn Pn 2 X a = Pr {1 xp [ 28 X ) n ]} n = 0.60 Pr Nu ) = 1 + a Pn Pn Pn 2 X a = Pr {1 xp [ 28 X ) n ]} n = 0.60 Pr Nu ) = 1 + a Pn Pn Pn 2 X a = Pr {1 xp [ 18 X ) n ]} n = 0.60 Pr Nu ) = 1 + a Pn Pn Pn 2 X a = Pr {1 xp [ 18 X ) n ]} n = 0.60 Pr Estimation of th man Nusslt numbr for flows of liquids with tmpratur dpndnt proprtis by mans of th proposd corrlations Th corrlations proposd in th prvious sctions can b usd to prdict th valu of Nu. In fact, according to quation 11), w can writ Nu = Nu = Nuc Nu ) Nu = + Nu ) 1 whr, Nu / and Nu / ar givn by quations 12), 14) and 18), rspctivly. Th maximum absolut valus ε max of th rlativ rror in th approximation of Nu by mans of quation 21) for Pr = 5, 20 and 100 and diffrnt valus of Pn and Pn /Pn ar rportd in tabl 4 for circular and squar cross-sctional gomtris. As can b sn, th agrmnt btwn computd and approximat rsults is vry good. 6. Conclusions Nw corrlations, suitabl for nginring applications, for th man Nusslt numbr in th ntranc rgion of circular tubs and squar ducts with uniform hat flux boundary conditions spcifid at th walls hav bn proposd. Ths corrlations hav bn obtaind on th basis of th rsults of a prvious paramtric invstigation on th ffcts of tmpratur dpndnt viscosity and thrmal conductivity in simultanously dvloping laminar flows of liquids in straight ducts of constant cross-sctions. In ths studis, a finit lmnt procdur has 21) 9

11 Tabl 4. Maximum absolut valus ε max %) of th rlativ rror in th approximation of Nu by mans of quation 21) for simultanously dvloping flows in circular and squar ducts with diffrnt viscosity and thrmal conductivity Parson numbrs. Cross-sction Pn Pn /Pn circular squar bn mployd for th numrical solution of th parabolizd momntum and nrgy quations. Viscosity and thrmal conductivity hav bn assumd to vary with tmpratur according to an xponntial and to a linar rlation, rspctivly, whil th othr fluid proprtis ar hld constant. Th tmpratur dpndncs of viscosity and thrmal conductivity hav bn quantitativly xprssd by th corrsponding Parson numbrs. Axial distributions of th man Nusslt numbr, obtaind by numrical intgration from thos of th local Nusslt numbr, hav bn usd as input data in th drivation of th proposd corrlations. A suprposition mthod has bn provd to b applicabl in ordr to stimat th Nusslt numbr by considring sparatly th ffcts of tmpratur dpndnt viscosity and thrmal conductivity. Thrfor, for ach of th considrd cross-sctional gomtris, two distinct corrlations hav bn proposd for flows of liquids with tmpratur dpndnt viscosity and with tmpratur dpndnt thrmal conductivity, in addition to that obtaind for constant proprty flows. Acnowldgmnts This wor was fundd by MIUR PRIN/COFIN 2009 projct). Rfrncs [1] Shah R K and London A L 1978 Nw Yor: Acadmic Prss) [2] Hrwig H 1985 Int. J. Hat Mass Transfr [3] Kaaç S 1987 Handboo of Singl-Phas Convctiv Hat Transfr d S Kaaç, R K Shah and W Aung Nw Yor: Wily) chaptr 18 [4] Dl Giudic S, Savino S and Nonino C 2010 Proc. 28 th UIT Hat Transfr Congrss Brscia, Italy) p 231 [5] Dl Giudic S, Savino S and Nonino C 2012 Entranc and tmpratur dpndnt proprty ffcts in laminar duct flows of liquids to appar) [6] Nonino C, Dl Giudic S and Comini G 1988 Numr. Hat Transfr [7] Dl Giudic S, Savino S and Nonino C 2011 J. Hat Transfr, Trans. ASME [8] Patanar S V and Spalding D B 1972 Int. J. Hat Mass Transfr [9] Hirsh C 1988 vol 1 Nw Yor: Wily) p 70 [10] Nonino C, Dl Giudic S and Savino S 2006 Int. J. Hat Mass Transfr [11] Nonino C 2003 Numr. Hat Transfr, Part B [12] Dl Giudic S, Nonino C and Savino S 2007 Int. J. Hat and Fluid Flow [13] Nonino C, Dl Giudic S and Savino S 2007 J. Hat Transfr, Trans. ASME