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Wall conduction effects in laminar counterflow parallel-plate heatexchangersAlberto E. Quintero, Marcos Vera, Bernardo Rivero-de-AguilarDept. de Ingeniera Trmica y de Fluidos, Universidad Carlos III de Madrid, 28911 Legans, Spaina r t i c l ei n f oArticle history:Received 23 July 2013Received in revised form 15 November 2013Accepted 20 November 2013Available online 20 December 2013Keywords:Heat exchangersCounterflowParallel-plateLaminarEigenfunction expansionsWall thermal resistancea b s t r a c tWall conduction effects in multilayered, counterflow, parallel-plate heat exchangers are analyzedtheoretically and numerically. The analysis, carried out for constant property fluids, considers a hydrody-namically developed laminar flow and neglects axial conduction both in the fluids and in the plates. Thetemperature field is expanded as an infinite series in terms of a complete set of eigenfunctions associatedwith sets of both positive and negative eigenvalues. In addition to the exact solution, an approximatesolution that retains only the first two terms in the eigenfunction expansion is considered. The approx-imate two-term solution, which still incorporates the effect of higher order modes through apparent tem-perature offsets introduced at the inlet/outlet sections, provides an accurate representation for thetemperature field away from the thermal entrance regions, thereby enabling simplified expressions forthe wall and bulk temperatures, local Nusselt numbers, and overall heat-transfer coefficient. As main out-come of the analysis, it is seen that increasing the wall thermal resistance lowers the absolute value ofboth positive and negative eigenvaluesthus reducing heat-exchanger effectivenessand increases theNusselt number of the fluid with lower heat-capacity flow rate bringing it closer to its theoretical value140=17 8:2353 corresponding to a constant heat flux boundary condition. Moreover, the proposed two-term solution is seen to reproduce with great accuracy the dependence of the outlet bulk temperatureswith the wall thermal resistance. The asymptotic solution for nearly-balanced heat exchangers is alsoobtained, providing closed-form analytical expressions for this limiting case of practical interest.? 2013 Elsevier Ltd. All rights reserved.1. IntroductionParallel-plate heat exchangers are widely used in chemical,pharmaceutical,foodprocessing,andmanyotherindustrialapplications. More recently, they have also found application in avariety of emerging branches of thermal engineering. Thus, theyare currently used in miniaturized reaction systems involving het-erogeneously catalyzed gas-phase reactions 1, in thermoelectricgenerators that convert low-grade thermal energy into electricalpower 2,3, and in thermoacoustic engines and refrigerators 4.In addition, they are a key component of many cryogenic systems57.In recent years, there has also been a growing interest in thedevelopment of polymer heat exchangers, due particularly to theirhigh resistance to fouling and corrosion. Specifically, the use ofpolymers offers substantial weight, volume, space, and mainte-nance cost savings in many applications over metallic heatexchangers 8. Nevertheless, the low thermal conductivity ofpolymer materials typically results in a dominant wall heat trans-fer resistance, which imposes serious restrictions on the thermaldesign and operation of these devices 911.Progress in the analysis of parallel-plate heat exchangers hasbeen significant in the last decades due to their simple geometryand well established flow conditions 12. In particular, the analy-sis of the steady-state laminar heat transfer between differentstreamscoupledthroughcompatibilityconditionsattheboundaries constitutes the so-called conjugated Graetz problem1315. Under certain simplifying assumptionsconstant prop-erty fluids and fully developed laminar flowthe problem becomeslinear and is amenable to an elegant solution based on eigenfunc-tion expansions, which in counterflow systems involves sets ofpositive and negative eigenfunctions associated with sets of posi-tive and negative eigenvalues 1517.The aim of this paper is to generalize the recent work on lami-nar counterflow parallel-plate heat exchangers carried out by Veraand Lian 18,19 so as to include the effect of a finite wall thermalresistance. The analysis, based on the seminal contributions byNunge and Gill 16,17, uses symbolic algebra to write closed formanalytical expressions for the eigenfunctions, leading to an exactanalyticaleigenconditionthatprovidestheeigenvalues0017-9310/$ - see front matter ? 2013 Elsevier Ltd. All rights reserved./10.1016/j.ijheatmasstransfer.2013.11.063Corresponding author. Tel.: +34 916249987; fax: +34 916249430.E-mail address: marcos.verauc3m.es (M. Vera).URL: http:/fluidos.uc3m.es/people/mvcoello/ (M. Vera).International Journal of Heat and Mass Transfer 70 (2014) 939953Contents lists available at ScienceDirectInternational Journal of Heat and Mass Transferjournal homepage: /locate/ijhmtnumerically. In addition to the exact solutionexpanded as an infi-nite series using the complete set of eigenfunctionsan approxi-mate solution that retains only the first two modes in theeigenfunction expansion is derived. This approach, which stillincorporates the effect of the higher order eigenfunctions throughapparent temperature offsets induced at the inlet/outlet sections,provides accurate representations for the temperature field awayfrom the thermal entrance regions. The accompanying expressionsfor the wall and bulk temperatures, local heat-transfer rate, overallheat-transfercoefficient,Nusseltnumbers,andoutletbulktemperatures may be useful for engineering applications even formoderately short heat exchangers.The paper is organized as follows. In Section 2, we present thedimensionless formulation of the problem, introducing the govern-ing dimensionless parameters. In Section 3, we expand the solutionas an infinite series of eigenfunctions and derive the linear systemfor the expansion coefficients. In Section 4, we propose anapproximate two-term solution that involves only the lowest ordereigenvalue and eigenfunction. In Section 5, we comment on the rel-evance of our results in the context of classical heat exchangeranalysis. In Section 6, we validate the exact and approximate solu-tions against numerical solutions, and discuss the main effects ofthe wall thermal resistance on heat-exchanger performance. Final-ly, in Section 7, the main conclusions of the analysis are presented.As additional material, to be used in the discussion of results, inAppendix A we give the asymptotic solution in the limiting case ofnearly-balanced heat exchangers, while in Appendix B we addressthe degenerate case of highly-unbalanced heat exchangers.2. Problem formulationIn this paper we analyze the heat transfer between two con-stant-property newtonian fluids flowing through a multilayeredcounterflow parallel-plate heat exchanger composed by a rela-tively large number of channels separated by plates of finite thick-ness, dw, and thermal conductivity, kw. The conducting plates allowthe exchange of heat through a section of length L, presenting insu-lated regions at both ends of the heat exchange region, where noheat transfer is allowed 20. In the configuration considered here,the two fluids, denoted by 1 and 2 (hereafter, the subscript i will beused indistinctly for both fluids, i 1, 2), flow in oppositedirections in adjacent channels. Then, if the characteristic cross-sectional dimension of the heat exchanger is large compared withthe channel width, 2ai, the temperature field, as seen with thisscale, appears as periodic in the transverse direction, with period2a1 a2. As a result, when describing the temperature field inthe unitary cell of the heat exchanger we can use symmetryboundary conditions at the channel symmetry planes, where nothermal energy can be transferred.Fig. 1 presents a sketch of the theoretical model under study,including the coordinate system, the velocity profiles, the inletNomenclatureAexpansion coefficient?AO1 expansion coefficient in the limit j?j ? 1?BO1 expansion coefficient in the limit j?j ? 1aichannel half-width of fluid iCnexpansion coefficient corresponding to the n-th eigen-functioncispecific heat of fluid ifnn-th eigenfunction in the limitjw! 1gnn-th eigenfunction for finitejwGn;icontribution of order?nto the 0-th eigenfunction in thelimit j?j ? 1hiheat-transfer coefficient of fluid ikdimensionless parameter, a1k2=a2k1kithermal conductivity of fluid ikwthermal conductivity of the walllncontribution of order?nto the eigenvalue k0in the limitj?j ? 1Llength of the heat exchangerMWhittakers function, Mj;lzmdimensionless parameter, a2Pe2=a1Pe1NuiNusselt number of fluid i;hi4ai=kiPeiPeclet number of fluid i;2aiVi=aiPriPrandtl number of fluid i;mi=aiReiReynolds number of fluid i;2aiVi=miTtemperatureeUdimensionless overall heat-transfer coefficientuilongitudinal velocity of fluid iViaverage velocity of fluid iWWhittakers function, Wj;lzwyiweight function, 3=41 ? y2iXlongitudinal distance from the inlet of fluid 1Yitransverse distance from channel i symmetry planeyidimensionless transverse coordinate, Yi=aiGreek lettersaithermal diffusivity of fluid i;ki=qiciDibulk temperature offset of fluid i at the inletdwthickness of the wall?small parameter, 1 ? mk?1eheat exchanger effectivenessCGamma function,Czjfirst argument of Whittaker functionsjwdimensionless parameter, a1kw=dwk1jlumped variable, m1=3kknn-th eigenvaluelsecond argument of Whittaker functionsmdimensionless local heat-transfer rate, h1=y1jy11mikinematic viscosity of fluid iqidensity of fluid ihidimensionless temperature of fluid indimensionless longitudinal coordinateX?ncoefficients defined by Eq. (41)Subscriptsisubscript used indistinctly for fluids 1 and 2ininletLlength of the heat exchangermbulk, or mixing-cup, temperaturencorresponding to the n-th eigenvalue/eigenfunctionoutoutletwheat-exchanging wallSuperscripts(0)zeroth-order two-term solution(1)first-order two-term solutionN?2N 1-term truncated exact solution940A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953conditions, and the physical properties of fluids i 1;2. Hereafter,X represents the longitudinal coordinate measured from the inletof fluid 1, and Yithe transverse coordinate measured from thesymmetry plane of channel i. The Reynolds numbers of the flowin the channels, Rei 2aiVi=mi, based on the average flow velocity,Vi, and kinematic viscosity,mi, of fluid i, are assumed to be suffi-ciently small for the flow to remain laminar and steady 21. Onthe other hand, the Prandtl numbers, Primi=ai, defined in termsof the thermal diffusivity,ai ki=qici, of fluid i, are assumed tobe large compared to unity, a good approximation for most non-metallic liquids, so that the thermal entry length is large comparedto the hydrodynamic entry length. Accordingly, the flow can beconsidered to be a fully developed Poiseuille flow which, due tothe constant property assumption, is independent of the tempera-ture field, uiYi ?3=2Vi1 ? Yi=ai2?, where the (?) signholds for fluid 1 (2).In addition, we shall assume here that the Peclet numbers of theflow in the channels, Pei ReiPri 2aiVi=ai, are both large com-pared to unity. In this case, axial heat conduction in the fluids isconfined to small regions located at both ends of the heat exchangeregion, close to the heat conducting plates, whose characteristicsize, of order aiai=Vi1=2 Pe?1=2iai, is small compared to aiwhenPei? 1. In this limiting case, axial conduction can be neglectedin the fluids in first approximation, hence we shall assume thatthe inlet temperature of fluid i is uniform and equal to Ti;in. Previ-ous analyses of counterflow heat exchangers concluded that this isa good approximation for Peclet numbers of order 100 or more22,23. For these Peclet numbers, axial conduction in the platesis also anticipated to be negligible, introducing only noticeable ef-fects at very small distances from the ends of the heat exchange re-gion. This is the case when the thickness of the heat conductingplates, dw, is sufficiently small compared to the channel width,2ai, a condition that often holds in macro-heat transfer devices24.To write the problem in non-dimensional form, we introducethedimensionlessvariablesn X=Pe1a1;yi Yi=ai,andhi Ti? T1;in=T2;in? T1;in. Then, under the assumptions statedabove, the energy equation reduces, in first approximation, to abalance between axial convection and transverse conduction341 ? y21h1n2h1y211?m341 ? y22h2n2h2y222in 0 n nL;0 yi 1, where the dimensionless parametersm a2a1Pe2Pe1andnLLa11Pe13are both assumed to be of order unity. Note that due to the existingcounterflow pattern, the mathematical problem is elliptical ratherthan parabolic even though longitudinal heat conduction effectsare not taken into account.The above equations have to be integrated with symmetryboundary conditions at the channel symmetry planesh1y1h2y2 0at 0 n nL;yi 04the thermal coupling conditions at the heat conducting wallh2? h11jwh1y1mnjwh1y1 ?kh2y2?mnat 0 n nL;yi 15and the specified dimensionless temperatures at the inlet sectionsh1 h1;in 0at n 0;0 6 y1 1;6h2 h2;in 1at n nL;0 6 y2 1:7Note that the continuity of heat fluxes at the heat conductingwall stated in (5) introduces two additional parametersk a1a2k2k1andjwa1dwkwk18which represent the normalized fluid-to-fluid and wall-to-fluidconductivity ratios, respectively.The solution to the problem (1)(8) provides the temperaturefield, hin;yi, for given values of m;k;jwand nL. It also providesthe local heat-transfer rate from fluid 2 to fluid 1;mn, the interfa-cial wall-fluid temperature distributionshwin ? hin;19the local wall temperature jumphw2n ? hw1n mnjw10and the temperature profiles at the outletsh1;outy1 ? h1nL;y1;11h2;outy1 ? h20;y2:12Note that the absence of longitudinal heat conduction effectsfor Pei? 1 implies that at both ends of the heat exchanging regionFig. 1. Unitary cell of the idealized counterflow parallel-plate heat exchanger showing the coordinate system, the velocity profiles, the inlet conditions, and the physicalproperties of fluids i 1;2. A heat conducting wall, of finite thickness, dw, and thermal conductivity, kw, separates both fluids. The domain under study is shaded gray.A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953941the wall temperature is imposed by the fluid entering the exchan-ger in that section, so that we may anticipate that hw10 0 andhw2nL 1.Another interesting result is the distribution of the bulkormixing-cuptemperatures of both streamshmin Z10321 ? y2ihin;yidyi;13defined as the uniform temperature that would eventually be at-tained if the fluid at a particular section n was allowed to evolve adi-abatically. Specifically, the main outcome of the analysis are thevaluesoftheoutletbulktemperaturesofbothstreams,hm1nL ? hm1;outand hm20 ? hm2;out 1 ? mk?1hm1;out, closely re-lated to the heat exchanger effectiveness 18.To conclude this section, we shall further comment on thehypothesis of negligible axial heat conduction in the wall. Thefollowing discussion is based on the so-called axial conductionnumber, M dwkw=L=2a1V1q1c1, defined as the ratio of the ax-ial heat transfer by conduction in the wall, dwkwT2;in? T1;in=L, tothe convective heat transfer in the flow channels, 2a1V1q1c1T2;in? T1;in. Numerical computations by Maranzana et al. 24have shown that axial conduction has a negligible effect in heat-exchanger performance as long as M 10?2, a condition thatmay be rewritten asdwa1kwk1 10?2Pe21nL;ordwa1 10?1Pe1ffiffiffiffiffiffiffinLjws14in terms of the dimensionless parameters Pe1;nL, andjw. Thisinequality provides an upper limit for the thickness of the wall be-low which the effects of axial heat conduction can be safely ne-glected.Forinstance,forPe1 100;nL 1=8 0:125,andjw f2;10;100;1000g, we have dw=a1 0) and negative (n 1 and negative for mk 0. In this case, use ofexpansion (16) in Eqs. (1) and (2) leads to the following equationsfor the n-th eigenfunction? kn341 ? y21gn;1d2gn;1dy21;17knm341 ? y22gn;2d2gn;2dy22;18in 0 yi 1, to be integrated with the boundary conditionsdgn;1dy1dgn;2dy2 0at yi 019andgn;2? gn;11jwdgn;1dy1?g0n;11jwdgn;1dy1 ?kdgn;2dy2? g0n;11at yi 1;20following respectively from Eqs. (4) and (5), where the value, g0n;11,of dgn;1=dy1at y1 1 is to be determined as part of the solution.Note that due to the linearity of the problem we must also specifya normalization condition. Then, without loss of generality, we setgn;2 gn;12 1at yi 1:213.2. Determination of the eigenfunctions3.2.1. Casejw! 1In the thermally thin wall regime,jw! 1, studied in detail byVera and Lian 18, the boundary condition at the wall (20) re-duces to the continuity of temperatures and heat fluxes, and thenormalization condition (21) becomes fn;1 fn;2 1 at yi 1. Theresulting problem admits an analytical solution that can be writtenusing standard symbolic mathematical software packages, such asMathematica or Maple, in the form 18fn;iyi 1ffiffiffiffiyip2ffiffiffiffippMjn;i;144jn;iy2i An;iWjn;i;144jn;iy2i2ffiffiffiffippMjn;i;144jn;i An;iWjn;i;144jn;i();22where, maintaining the notation introduced in 18, we denote byfn;ithesolutionoftheproblem(17)(21)correspondingto942A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953jw! 1. In the above expression Mj;lz and Wj;lz are the Whit-taker functions 25,26, and we have defined the complex dimen-sionless parametersAn;iC14?jn;i?;jn;1ffiffiffiffiffiffiffiffi3knp8;jn;2 iffiffiffiffiffiffiffiffiffiffiffiffi3mknp8;23whereCz is the Gamma function and i is the imaginary unit. Sim-ilarly, we shall use the notation 18f0n;i1 ?dfn;idy1?yi1 ?122jn;i2ffiffiffiffipp3=4jn;iMjn;i1;144jn;i?An;iWjn;i1;144jn;i2ffiffiffiffippMjn;i;144jn;iAn;iWjn;i;144jn;i()24for the derivatives of the eigenfunctions fn;iat the wall, with An;iandjn;igiven in terms of knand m (only for i 2) by Eq. (23).3.2.2. Case of finitejwWhen the heat exchanging wall has a finite thermal resistance,0 jw 0. Interestingly, asymptotic expressionsfor the negative eigenvalues, k?n, can be readily obtained from(32)(35) using the symmetry relationk?nm;k;jw ?m?1knm?1;k?1;jwk?1:36The results presented in Fig. 2 and in Tables 13 show that theeffect of increasing the wall thermal resistance 1=jw(i.e., ofdecreasingjw) is always to reduce the absolute value of both thepositive and negative eigenvalues. This trend is particularly appar-ent for the positive eigenvalues in the region mk 1, i.e., whenfluid 2 has a higher heat-capacity flow rate than fluid 1. Thus,Fig. 3. Eigenfunctions gn;iyi for n 0;?1;.;?4 and selected values ofjwcorresponding to m 1;k 1 (left) and m 1;k 2 (right). The eigenfunction g0;i? 1 is notshown on the left plots (see Appendix A for details). Dash-dotted lines (right plots): approximate form of g0;igiven by (A.2) together with (A.6)(A.13) and (A.14) forj1 ? mk?1j j?j ? 1.Fig. 4. Variation of the outlet bulk temperature of fluid 1 with the heat exchangerlength corresponding to m 1 and k 2 given by truncated exact solution, Eq. (42),for N 5 (12-term solution) and different values ofjw.A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953945the lowest order eigenvalue k0is roughly reduced by 50% whenjwdrops from 1 (thermally thin wall regime) to 2 (when the thermalresistance of the wall becomes comparable to that of the fluid; seeEq. (69) for details), while it is further reduced to 0 asjw! 0. Byway of contrast, higher order eigenvalues show comparativelysmaller changes; e.g., k1is reduced by 60% whenjwdrops from1 to 0 in the region mk ? 1 (see Appendix B for details), whereasfor k ? 1 it remains almost constant regardless ofjw.The eigenvalues reported in Tables 13 were then used to calcu-late the eigenfunctions gn;iyi shown in Fig. 3. As clearly seen in thefigure, the main effect of the wall thermal resistance is to introducea finite jump at the wall in the n-th eigenfunction. The magnitudeof this jump, g0n;11=jw, increases with n due to the increasinglylarger gradients of gn;1y1 at yi 1. Note also that the n-th eigen-function has exactly jnj zeros, located in 0 y1 0, and in0 y2 1 for n 0 28.3.4. Orthogonality condition for the eigenfunctionsAs discussed by Nunge and Gill 16,17, the eigenfunctions gnand gqsatisfy the orthogonality conditionhgn;gqi Z10wy1gn;1y1gq;1y1dy1? mkZ10wy2gn;2y2gq;2y2dy2 0for n q;37where wyi 3=41 ? y2i is the weight function and gnis theeigenfunction defined by gn;1y1 for 0 y1 1 and gn;2y2 for0 y2 1 coincides with the heat ex-changereffectivenessincreasesmonotonicallywithnL,approaching unity as nL! 1. By contrast, the effect of the wallthermal resistance 1=jwis to reduce the heat exchanger efficiencyfor a fixed heat exchanger length. Other interesting feature of thetruncated exact solution, seen in the inset of Fig. 4, is that the pre-dicted outlet bulk temperature of fluid 1 does not vanish as nL! 018. Instead, we haveDN?1? limnL!0hN?m1;outnL 0;with limN!1DN?1 0:43As discussed by Vera and Lin 18, the value ofDN?1constitutes ameasure of the additional heat that would have been exchanged ifwe had retained higher-order eigenfunctions in the exact solution.This extra heat transfer, which takes place at the near-inlet regions,decreases for larger wall thermal resistances 1=jwdue to theincreasingly tighter boundmn 6jwimposed on the heat transferrate by the correspondingly smaller values ofjw.Interestingly enough, for the two-term truncated exact solution,N 0, the linear system (39) and (40) can be solved analytically togiveD0?121 mkg020;11 mkX?0? g00;11X0=2jw?k208g020;11 1 mkX?0? g00;11X0=2jw?k2044which providesD0?1from the corresponding values of g00;11;X?0,and k0calculated earlier. This result will be used in Section 4.2 toincorporate the effect of higher order modes in the approximatedtwo-term solution by introducing apparent temperature offsets atthe inlet/outlet sections, which, as we shall see, leads to a signifi-cant improvement in numerical accuracy.As a final remark, it should be noted that as the balanced heatexchanger regime is approached, mk ! 1, the linear system forthe expansion coefficients (39) and (40) becomes more and moreill-conditioned. The rigorous mathematical analysis of this singularregime, in which k0 0 becomes an algebraically double eigen-value, is out of the scope of this work. In that case, the liner system(39) and (40) should be derived again using as starting point thealternative series expansion (A.5) given in Appendix A.4. Approximate two-term solutionThe gray shaded areas in the center plots of Fig. 2 indicate re-gions in the mk plane where the second lowest eigenvalue inabsolute value (k?1for mk 1;k1for mk 1) is larger than 25%(light gray), 50% (gray), and 100% (dark gray) of the absolute valueof k0. Outside these regions, the approximate solution obtained bytruncating the eigenfunction expansion to the first two modeshin;yi A C0e?k0ng0;iyi45is expected to give an accurate description of the temperature fieldaway from the inlets, i.e., for k?11 n 0(mk 1),andhm1;out mkandhm2;out 0 for k0 0 (mk 1), regardless of the value ofjw.Remember also that (57) and (58) are not valid for k0 0(mk 1), a singular case treated separately in Appendix A.4.2. First-order solutionThe accuracy of the two-term solution can be improved byreplacing the lumped inlet conditions (49) byh1m10 D1;h1m2nL 1 ?D259where we introduce the inlet bulk temperature offsetsD1 h0m2;outD0?12;D2 mk?11 ? h0m1;out?D0?1260so as to include the effect of higher-order eigenfunctions at bothends of the heat exchanger 18.Due to the linearity of the problem, the temperature field thatsatisfies the inlet conditions (59) can be written ash1in;yi D1 1 ?D1?D2?h0in;yi;61in terms of the zeroth-order two-term solution h0in;yi, given in(51). This leads to the following expressionsh1w1n D1 1 ?D1?D2?h0w1n;62h1w2n D1 1 ?D1?D2?h0w2n;63h1m1n D1 1 ?D1?D2?h0m1n;64h1m2n D1 1 ?D1?D2?h0m2n;65m1n 1 ?D1?D2?m0n;66which give significantly better approximations in 0 n nLthan(52)(56), specially for moderately short heat exchangers, nLK1.It should be noted, however, that (64) and (65) can not be usedto evaluate the outlet bulk temperatures, since the temperatureoffsets that take place at the outlets are not included. Instead, wemust useh1m1;outD1 1 ?D1?D2?h0m1;out mkD2;67h1m2;outD1 1 ?D1?D2?h0m2;out? mk?1D1;68where the last term in both expressions represents the outlet tem-perature offset written in terms of the corresponding inlet temper-ature offset (60) 18.5. Relevance to classical heat-exchanger analysis5.1. Nusselt numbers and overall heat-transfer coefficientA prerequisite for the application of thee-NTU method is theknowledge of the overall heat-transfer coefficient, which in thecase of finitejwconsidered here is given byA.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 9399539471eU4Nu11jw4kNu2;69according to the concept of additivity of thermal resistances, whereNui hi4ai=kirepresents the Nusselt number based on the heat-transfer coefficient, hi, and hydraulic diameter, 4ai, of fluid i. Intro-ducing dimensionless variables, from (45)(47) we getNu14mhw1? hm1 41 ? g00;11=2jwg00;112k0#?1;70eU mhm2? hm1k0=21 ? mk?1:71whereas Nu2can be conveniently calculated from (69)(71) asNu24mkhm2? hw24k1eU?4Nu1?1jw?;72Note that the last expressions in (70) and (71), obtained from thetruncated two-term solution, are indeed independent of the expan-sion coefficients. Thus, they are anticipated to give excellentapproximations for Nu1andeU far away from the thermal entry re-gions, i.e., for k?11 n nL? jk?1j?1. In fact, they can be viewed asexact asymptotic results valid for sufficiently long heat exchangers17.Fig. 5 shows contour plots of Nu1;eU, and Nu2in the mk planeobtained from (70)(72). It is seen that regardless of the value ofjw, for mk 1, when the heat-capacity flow rates of fluids 1 and2 are perfectly balanced, Nu1takes on the value 140=17 8:235corresponding to a constant heat flux boundary condition (seeAppendix A for details). By way of contrast, for mk ? 1, when fluid2 has a much higher heat-capacity flow rate than fluid 1, Nu1tendsto a limiting value,gNu1, that varies withjwmonotonically be-tween the constant wall temperature limit, 7.541, reached in thethermally thin wall regime,jw! 1 18, and the constant heatFig. 5. Contour plots of NuiandeU in the mk plane given by the two-term solution, Eqs. (70) and (71), for different values ofjw. Dashed lines: approximate solution forj1 ? mk?1j j?j ? 1 given by (A.17) and (A.18). The two-term solution is expected to fail in the gray shaded areas (see caption of Fig. 2). (For interpretation of the referencesto color in this figure legend, the reader is referred to the web version of this article.)948A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953flux limit, 8.235, obtained asjw! 0 for all mk P 1 (see AppendixB for details).6. Numerical validation and discussion of resultsTo asses the numerical accuracy of the analytical solutions re-ported in the previous sections, finite-difference solutions of theproblem stated by Eqs. (1)(8) were computed for selected valuesof m;k;jw, and nL. The details of the numerical method can befound elsewhere and will not be repeated here for brevity 18.Truncated exact solutions corresponding to N 5 (12-term solu-tion) and 10 (22-term solution) were also obtained for referencepurposes.In this paper we shall compare the different solution methodsin terms of the outlet bulk temperature of fluid 1, which formk P 1, a condition that holds for all the cases presented below,coincides with the heat exchanger effectiveness, hm1;out?e18. Inorder to make it more understandable and easier to follow in thetext, the different methodologies used to determine hm1;outaresummarized in Fig. 6 in the form of flow chart. Further compari-sons in terms of spatial distributions of bulk temperatures, interfa-cial wall-fluid temperatures, local heat-transfer rates, and overallheat-transfer coefficients, show similar agreements to those re-ported in 18 in the limit of thermally thin walls and, therefore,will be omitted from the present discussion.Fig. 7 shows the variation of hm1;outwith the wall thermal resis-tance 1=jwfor selected values of m;k, and nLdetermined numeri-cally and using three different analytical solutions, namely thezeroth-order two-term solution (57), the first-order two-termsolution (67), and the truncated exact solution (42) for N 5.The results shown in the figure indicate that increasing the wallthermal resistance significantly penalizes the heat exchanger effec-tiveness, with a larger effect (in relative terms) for small heat ex-changer lengths.Table 4 compares the values of hm1;outcorresponding tom 1;k 2 obtained numerically with those predicted by thezeroth- and first-order two-term solutions, and by the truncatedexact solutions for N 5 and 10. It also shows the relative errorcommitted by the different solution approaches as compared withthe truncated exact solution for N 10, taken here as benchmarkcase. As can be seen, the relative error associated with the trun-cated solution for N 5 is simply too small to be appreciable ex-cept for very short heat exchangers with negligible wall thermalresistance, where it reaches values of order 0.01%. The numericalsolution also shows very good agreement with the reference solu-tion, exhibiting relative errors below 0.1% in all cases. However,what is particularly noteworthy is that the first-order two-termsolution is also able to describe, with errors below 1%, the variationof the heat exchanger performance with both nLand 1=jweven forFig. 6. Summary of the procedure for determining hm1;outusing the three methodsdiscussed in the text: the zeroth- and first-order two-term solutions, and the2N 2-term truncated exact solution.Fig. 7. Variation with 1=jwof the outlet bulk temperature of fluid 1 correspondingto m 1;k 2 (left) and m 2;k 1 (right) for selected heat exchanger lengths.Results obtained from the numerical integration of the problem (?), the truncatedexact solution (?), Eq. (42), with N 5 (12-term solution), the first-order two-termsolution (), Eq. (67), and the zeroth-order two-term solution (?), Eq. (57).A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953949moderately short heat exchangers (nLJ1=8). By way of contrast,the zeroth-order solution leads to comparatively much largererrors.Specifically, the relative errors committed by the zeroth- andfirst-order two-term solutions are plotted in Fig. 8 as a functionof the wall thermal resistance, 1=jw, for selected values of nL. Ascan be seen, not only the first-order solution exhibits much smallererrors, but these are reduced at a much faster rate for increasingvalues of the wall thermal resistance. Thus, while the error of thezeroth-order solution is about 4.5 to 5 times larger to that of thefirst-order solution for thermally thin walls, 1=jw 0, this ratiogrows to about 9 to 12 for the largest wall thermal resistance,1=jw 0:5.It should be remembered that the different performance of thezeroth- and first-order two-term solutions arises only from the dif-ferent inlet boundary conditions 18. While the zeroth-order solu-tion replaces the exact inlet conditions (6) and (7) by the lumpedconditions (49), the first-order solution still makes use of the exactinlet conditions through the corrected inlet temperatures given by(44), (59) and (60). Accordingly, the results shown in Fig. 8 showthat the error introduced by using the lumped conditions (49) in-stead of (59) is largely more important to that due to neglectinghigher order modes, whose effect can to a great extent be replacedby the apparent inlet/oulet temperature offsetsD1andD2.7. ConclusionsAn analytical and numerical study of laminar, counterflow, par-allel-plate heat exchangers is presented. The analysis, valid for con-stant property fluids, assumes that the Peclet numbers of bothstreams are large compared to unity. As a result, upstream heatconduction is neglected both in the fluids and in the plates, whichare assumed to have a finite thermal resistance. Under these con-ditions, four dimensionless parameters emerge in the mathemati-cal formulation of the problem: m;k;nL, and the wall-to-fluidconductivity ratio,jw a1kw=dwk1, whose inverse representsthe dimensionless wall thermal resistance.The analytical model was solved in terms of eigenfunctionexpansions involving infinite sets of both positive and negativeTable 4Outlet bulk temperature of fluid 1 as obtained numerically (NUM), from the truncated exact solution for N 10 (22-term solution, 22TS) and N 5 (12-term solution, 12TS), andfrom the approximate two-term solutions (2TS), corresponding to different heat exchanger lengths, nL, and wall thermal resistances, 1=jw. The results correspond to thebenchmark case m 1;k 2 represented in the left plot of Fig. 7. The table also includes the relative error of the different solution approaches compared to the truncated exactsolution for N 10, taken here as reference solution. Numerically computed values are shown in boldface type.1jw00.40.5MethodnLhm1;outErrorhm1;outErrorhm1;outErrorhm1;outErrorhm1;outErrorhm1;outError180.31260.064%0.27770.072%0.25000.080%0.22730.088%0.20840.096%0.19230.052%NUM0.3124a0.2775a0.2498a0.2271a0.2082a0.1922a22TS exact0.3124a0.009%0.2775a0.002%0.2498a0.001%0.2271a0.001%0.2082a0.000%0.1922a0.000%12TS exact0.3041b2.674%0.2722b1.933%0.2461b1.458%0.2245b1.128%0.2063b0.891%0.1908b0.714%2TS 1st order0.2700c13.57%0.2440c12.09%0.2225c10.92%0.2044c9.975%0.1891c9.183%0.1759c8.510%2TS 0th order140.47650.063%0.43520.046%0.40070.100%0.37110.081%0.34560.087%0.32320.031%NUM0.4762a0.4350a0.4003a0.3708a0.3453a0.3231a22TS exact0.4762a0.003%0.4350a0.001%0.4003a0.000%0.3708a0.000%0.3453a0.000%0.3231a0.000%12TS exact0.4701b1.279%0.4308b0.954%0.3973b0.744%0.3686b0.599%0.3436b0.492%0.3217b0.412%2TS 1st order0.4470c6.142%0.4109c5.522%0.3802c5.035%0.3536c4.636%0.3304c4.299%0.3101c4.009%2TS 0th order120.67610.044%0.63470.032%0.59800.050%0.56510.071%0.53540.056%0.50860.039%NUM0.6758a0.6345a0.5977a0.5647a0.5351a0.5084a22TS exact0.6758a0.001%0.6345a0.000%0.5977a0.000%0.5647a0.000%0.5351a0.000%0.5084a0.000%12TS exact0.6725b0.495%0.6321b0.377%0.5959b0.300%0.5634b0.245%0.5340b0.204%0.5075b0.173%2TS 1st order0.6602c2.311%0.6210c2.125%0.5859c1.976%0.5543c1.850%0.5258c1.740%0.5000c1.643%2TS 0th order10.85850.035%0.82770.024%0.79830.038%0.77030.039%0.74380.040%0.71870.042%NUM0.8582a0.8275a0.7980a0.7700a0.7435a0.7184a22TS exact0.8582a0.000%0.8275a0.000%0.7980a0.000%0.7700a0.000%0.7435a0.000%0.7184a0.000%12TS exact0.8570b0.150%0.8265b0.119%0.7972b0.098%0.7694b0.083%0.7429b0.071%0.7180b0.061%2TS 1st order0.8524c0.682%0.8221c0.657%0.7930c0.634%0.7653c0.612%0.7391c0.591%0.7143c0.571%2TS 0th order20.96700.000%0.95360.011%0.93890.011%0.92360.011%0.90780.011%0.89200.022%NUM0.9670a0.9535a0.9388a0.9235a0.9077a0.8918a22TS exact0.9670a0.000%0.9535a0.000%0.9388a0.000%0.9235a0.000%0.9077a0.000%0.8918a0.000%12TS exact0.9667b0.028%0.9532b0.025%0.9386b0.023%0.9233b0.020%0.9075b0.018%0.8916b0.017%2TS 1st order0.9658c0.127%0.9522c0.137%0.9375c0.144%0.9221c0.148%0.9063c0.151%0.8904c0.153%2TS 0th ordera(42) with A and Cn;n 0;?1;?2;.;?N obtained from the solution of the 2N 1-term truncated system (39)(41).b(67) and (60) withD0?1given by (44) and h0m1;outevaluated with k0and g00;11 obtained numerically (see, e.g., Table 1).c(57) evaluated with k0obtained numerically (see, e.g., Table 1).Fig. 8. Variation with 1=jwof the relative error committed by the approximatezeroth-order (left) and first-order (right) two-term solutions corresponding tom 1;k 2 for selected heat exchanger lengths.950A.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953eigenvalues 16. Analytical expressions for the eigenfunctionswere obtained in terms of Whittaker functions, which were thenused to evaluate the eigenvalues numerically. To simplify the cal-culations, asymptotic expressions for the eigenvalues were alsoprovided. Contour plots of the lowest order eigenvalues and plotsof the corresponding eigenfunctions were presented, along withhighly precise numerical representations of the eigenvalues for se-lected values of m;k, andjw.Since higher-order eigenfunctions decay exponentially fastaway from the inlets, an approximate two-term solution whichretains only the effect of the first non-vanishing eigenvalue hasbeen studied. The expansion coefficients were obtained using inletconditions of increasing levels of complexity, leading to the zeroth-and first-order two-term solutions discussed in the text. The two-term solution leads to analytical expressions for the outlet bulktemperaturesor, equivalently, heat exchanger effectivenessaswell as for the spatial distributions of bulk (or mixing-cup) temper-atures, hmi, interfacial wall-fluid temperatures, hwi, local heat-trans-fer rate,m, Nusselt numbers, Nui, and overall heat-transfercoefficient,eU. These expressions, which involve the parametersm;k;jw, and nL, and the corresponding values of k0;g00;11, andX?0, were accompanied by closed form asymptotic expressionsinvolving only k;jwand 1 ? mk?1?in the limit of nearly bal-anced heat exchangers, j?j ? 1.The analytical expressions resulting from the first-order two-term solution were compared with finite-difference numericalsimulations, showing excellent agreement even for moderatelyshort heat exchangers in terms of outlet bulk temperatures, bulkand wall temperature distributions, local heat-transfer rate, overallheat-transfer coefficient, and Nusselt numbers, which suggests thepossibility of using this approximate solution as a useful engineer-ing design tool.This investigation could serve as starting point to build anapproximate model that includes the effect of axial conduction inthe heat conducting plates. Extensions of the present model tocylindrical geometries and/or non-Newtonian flows are also worthfuture consideration.AcknowledgementsThis work was supported by Project ENE2011-24574 of theSpanish Ministerio de Economa y Competitividad.Appendix A. Solution for nearly-balanced heat exchangersA.1. Asymptotic expansions for k0and g0;iyiWhen the ratio of the heat-capacity flow rates of bothstreams is close to unity, a regular perturbation analysis similarto that presented in 18 leads to the following asymptoticexpansionsk0?l1?2l2 O?3;A:1g0;iyi 1 ?G1;iyi ?2G2;iyi O?3;i 1;2A:2for the lowest eigenvalue and the corresponding eigenfunction interms of the small parameter 1 ? mk?1?. Introducing (A.1)and (A.2) in (16) we may writehin;yi A C0e?l1?2l2O?3?n1 ?G1;iyi ?2G2;iyi O?3?X1n?1n0Cne?knngn;iyi;which, expanding the exponential in Taylor series, reduces tohin;yi A C0?C0G1;iyi ? l1n?2C0G2;iyi ? l1nG1;iyi ? l2n l21n22# O?3C0 X1n?1n0Cne?knngn;iyi:A:4According to this, in the limit?! 0 the solution would reduce to aconstant unless we assume the scaling?C0 O1. Thus, in order toobtain a physically meaningful solution we must rewrite (A.4) ashin;yi ?A ?BG1;iyi ? l1n?G2;iyi ? l1nG1;iyi ? l2n l21n22#(O?2?X1n?1n0Cne?knngn;iyiA:5where both?A A C0and?B ?C0are O1 coefficients in the limit?! 0. In brief, the alternative series expansion (A.5) represents theasymptotic form of the solution in the case of nearly-balanced heatexchangers, obtained by taking the formal limit? 1 ? mk?1? 1in the series solution for mk 1 (16) obtained in Section 3.Rewriting m 1 ?1k?1and substituting expansion (A.5) inEqs. (1) and (2), in the boundary conditions (4) and (5), and inthe normalization condition (21), rewritten here as Gn;i1 0,we obtain a hierarchy of problems to be solved successively at dif-ferent orders for the unknowns lnand Gn;i. Thus, to first-order, wehavel17017kk 11 12jw7017kk 1?1;A:6G1;1y1 l1U1y1 14jw?;A:7G1;2y2 ?l1U1y2k14jw?;A:8U1yi y4i16?38y2i516A:9and, to second-order,l2l1 ?8239537k ? 1k 11 12jw7017kk 1?2;A:10G2;1y1 l2l1l1U1y1 l21U2y1 U1yi4jw? K;A:11G2;2y2 ?l1k1 l2l1?U1y2 l21k2U2y2 kU1yi4jw? K; A:12U2yi 3y8i3584?7y6i64011y4i256?15y2i128151117920;A:13whereK 1560j2w34 35jw?l21 140l2l1l1?A:14and the coefficient lnis obtained by imposing the continuity of heatfluxes at the heat conducting wall at order n 1.A.2. Asymptotic expansion forX?0To get an asymptotic expansion forX?0we apply formal deriva-tion with respect to knto Eqs. (18) and (17) as well as the boundarycondition (19) and normalization condition (21). Substituting theabove expansions for k0and g0;iyi, rewriting m 1 ?1k?1,and expanding the result in series of?, we obtain a new hierarchyof problems that can be solved to giveA.E. Quintero et al./International Journal of Heat and Mass Transfer 70 (2014) 939953951X?0 ?1 ?3k ? 12k 114jw7017kk 1?1 12jw7017kk 1?1 O?2A:15andX0 ?12?239794j2wk2 137445jwk2 121275k213217jwk 1 35k238148j2wk 58905jwk ? 1646j2w13217jwk 1 35k2# O?3:A:16A.3. Heat exchanger performanceA.3.1. Zeroth-order solutionEven though Eqs. (50)(58) apply for any non-zero value of?,they exhibit a singular behavior for? 1 that can be solved usingthe asymptotic expansions for k0and g0;iyi given above. This leadsto asymptotic expansions for the coefficients, wall and bulk tem-perature distributions, local heat-transfer rate and outlet bulktemperatures.A.3.2. First-order correctionsApproximate analytical expressions for the first-order two-termsolution may also be obtained, which requires the use of an ana-lytic expression forD0?1in (60). This expression can be obtainedby substituting expansions (A.1), (A.2), (A.15) and (A.16) fork0;g00;11, andX?0in (44).A.3.3. Nusselt numbers and overall heat-transfer coefficientUsing the asymptotic expansions for k0and g0;iyi given above,the following closed-form asymptotic expressions may be writtenfor the Nusselt number of fluid 1Nu1140171 ?82339270l1 O?2?;A:17and the overall heat-transfer coefficienteU l121 ?l2l1 O?2?;A:18which readily yield the corresponding value of Nu2through Eq. (72).It is interesting to note that in (A.17) and (A.18) the dependence onjwappears only implicitly through the dependence of l1and l2onjwgiven by (A.6) and (A.10). Note also that although the bulk tem-perature differences and local heat-transfer rate are functions of n,both Nu1;eU, and Nu2are all constant, up to order?, throughoutthe heat exchanger.It should be noted that in the derivation of (A.17) and (A.18) itwas necessary to retain, respectively, terms up to order?3and?2in(A.1) and (A.2). Retaining terms up to order?2in the first case leadsto the incorrect expression for Nu1given in 18, Eq. (91). However,it should be noted that the comparison between numerical andasymptotic values of Nu1presented in 18, Fig. 5, as those shownhere in the left plots of Fig. 5, was indeed generated using the cor-rect expression (A.17).Appendix B. Case of highly-unbalanced heat exchangersIn the limit of highly-unbalanced heat exchangers, mk ! 1, theinlet temperature of fluid 2 is imposed throughout the heat ex-changer, so that h2n;y2 hw2n h2;in 1. The problem statedin (1)(8) is then reduced to the conjugate heat transfer betweena hydrodynamically developed laminar flow and two flat plateswith finite thermal resistance, subject to a constant temperaturecondition at the outer surfaces of the plates 29. Rewriting the ori-ginal problem in terms ofh1 1 ? h1, and substituting h2? 1where needed, we get341 ? y21h1n2h1y21in 0 n nL;0 y1 1;B:1to be integrated with the symmetry condition h1=y1 0 at y1 0,the coupling condition h1=y1 ?jwh1at the fluid-wall interfacey1 1, and the inlet conditionh1 1 at n 0.The solution to this problem can be written ash1n;y1 X1n0eCne?knnfn;1y1;B:2where, in the absence of fluid 2, only the positive eigenvalues mustbe included. Introducing (B.2) in (B.1) and making use of the accom-panying boundary conditions leads to the following equation for theeigenfunctions?kn341 ? y21fn;1d2fn;1dy21in 0 y1 1;B:3subject to the symmetry condition dfn;1=dy1 0 at y1 0. If we fur-ther impose the normalization condition fn;1 1 at y1 1, it is easyto check that the solution to this problem is that given by Eqs. (22)and (23) with i 1.Once the eigenfunctions are known, the coupling condition atthe fluid-wall interface,f0n;11;kn ?jw;B:4with f0n;11;kn given analytically by (23) and (24), can be used todetermine the eigenvaluesknnumerically.Table 5 presents the values ofk0andk1thus obtained for se-lected values ofjwranging from 1 (thermally thin wall) to 0(adiabatic wall). As can be seen, the main effect of the wall ther-mal resistance is to damp the downstream evolution of the tem-perature field in the channel, the first eigenvalue dropping from3:77 to 1:99, and the second from 42:86 to 31:76, whenjwisreduced, e.g., from 1 to 2. Note that these values are preciselythose reached asymptotically by k0and k1for mk ? 1 as clearlyseen in Fig. 2.Another interesting result of the analysis is the asymptotic Nus-selt number of fluid 1,gNu1, in the thermally developed flow down-stream the thermal entry length, n ? jk1j?1. According to thedefinition of the Nusselt number, we havegNu14mhw1? hm1 42k0?1jw?1;B:5which can be used to evaluate the asymptotic value ofgNu1from theknown values ofjwandk0. The numerical results are shown in Ta-ble 5. Also included in the table is the relative error of the approx-imate Nusselt number given by the correlationTable 5Asymptotic values ofk0;k1, andgNu1obtained in the limit mk ! 1 for different valuesofjw. The last column shows the relative error of the approximate Nusselt numbergiven by correlation (B.6) as compared to the exact valuegNu1.jwk0k1gNu1gNucor1?gNu1?=gNu113:770350436442:86304309717:54070087420103:210828133438:84767767327:6497594018?1:5 ? 10?421:985020549831:76426633147:8810552617?4:2 ? 10?411:333333333328:90615402878:0000000000?6:8 ? 10?40:50:801883066426:95035663588:0950482309?6:6 ? 10?40:10:190700137725:03997326958:2022779223?2:4 ? 10?40024:50706364598:23529411760952A.E. Quintero et al./International Journal of Heat and Mas
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