Termal interation between free onvetion and fored onvetion along a vertial onduting wall J.-J. Su, I. Pop Heat and Mass Transfer 35 (1999) 33±38 Ó Springer-Verlag 1999 Abstrat A teoretial study is presented in tis paper to investigate te onjugate eat transfer aross a vertial nite wall separating two fored and free onvetion ows at different temperatures. It is assumed tat te eat ondution in te wall is only in te transversal diretion. We also assume tat ounterurrent boundary layers are formed on te bot sides of te wall. Te governing equations of tis problem and teir orresponding boundary onditions are all ast into a dimensionless form by using a non-similarity transformation. Tese resulting equations, wi are singular at te points n ˆ 0 and 1, are solved numerially using a very ef ient singular perturbation metod. Te effets of te resistane parameters and of te Prandtl numbers on eat transfer arateristis are investigated and presented in a table and ten gures. List of symbols A onstant b tikness of te plate C onstant f redued stream funtion g aeleration due to gravity x loal eat transfer oef ient k termal ondutivity L lengt of te plate Nu x loal Nusselt number Nu average Nusselt number Reeived on 8 April 1998 J.-J. Su Department of Meanial Engineering University of Hong Kong Te Pokfulam Road Hong Kong I. Pop Faulty of Matematis University of Cluj R-3400 Cluj, CP 53 Romania Correspondene to: J.-J. Su Te autors greatly appreiate te valuable omments made by te referee wi led to te improvement of tis paper. Pr Prandtl number q x loal eat ux Q overall eat ux Re Reynolds number R t fored onvetion termal resistane parameter R t free to fored onvetion parameter t ambient temperature Dt arateristi ambient temperature T uid temperature u; v veloity omponents U 1 free stream veloity x; y Cartesian oordinates Greek letters a termal diffusivity b oef ient of termal expansion dimensionless temperature k dummy variable m kinemati visosity n; g redued oordinates w stream funtion Subsripts old uid system ot uid system s solid wall w ondition at te wall 1 Introdution Te study of termal interation between two semi-in nite uid reservoirs at different temperatures troug a vertial ondutive wall is a very important topi in eat transfer beause of its numerous engineering appliations. Tis eat transfer proess applies to reator ooling, eat exangers, termal insulation, nulear reator safety, et. Additionally, su interation meanisms is, for te most part, inerent in te design of eat transfer apparatus. On te oter and, it is wort mentioning tat reent demands in eat transfer engineering ave requested researers to develop new types of equipments wit superior performanes, espeially ompat and ligtweigt ones. Te need for small-size units, requires detailed studies on te effets of interation between te termal eld in bot uids and of te wall ondution, wi usually degradates te eat exanger performane. 33
34 Te problem of eat exange between two free onvetion systems separated by a nite vertial ondutive wall as not been studied extensively beause of dif ulties in solving te developments of ow and termal boundary layers simultaneously. Lok and Ko [1] applied te loal similarity metod to investigate teoretially te effet of termal oupling produed by ondution troug a nite vertial wall separating two free onvetion systems. Te study was done under te assumption tat te wall ondution is only in te transversal diretion. Te governing boundary layer equations were transformed by introduing semi-similar variables and ten solved numerially using a nite-differenes metod. Tis problem was also treated by Viskanta and Lankford [] following a more simple analysis based on a super-position metod. Te autors ave also onduted interferometri experiments wi on rm te validity of teir approximated teoretial results. Next, Anderson and Bejan [3] ave employed te modi ed Oseen linearized metod to solve su a onjugate problem wen te Prandtl number is very large. It was sown tat te overall eat transfer rate is relatively independent by te Prandtl number. Sakakibara et al. [4] ave reently extended te problem of oupled eat transfer between two free onvetion systems separated by a nite vertial onduting wall assuming te twodimensional ondution equation in te wall. In oter words, te wall ondution takes plae in bot axial and transversal diretions. Numerial solutions for bot free onvetion systems and te analytial solution for te wall ondution were ombined to obtain nal solutions for te ow and eat transfer arateristis wi t te onjugate boundary onditions at bot sides of te wall. Experiments were also onduted for air-air systems wit te onduting wall made of aluminum or glass. It was found tat teoretial results desribe well te experimental temperature distributions. More reently, in a very interesting paper, TrevinÄo et al. [5] reported numerial and asymptoti solutions of te free onvetion boundary layers on bot sides of a vertial onduting wall for all possible values of two main parameters. We notie to tis end tat tere were also publised several papers by Poulikakos [6], MeÂndez and TrevinÄo [7], and Cen and Cang [8, 9] on te problem of termal interation between laminar lm ondensation of a saturated vapor and a fored or free onvetion system separated by a vertial onduting wall. In tis paper we intend to propose a new teoretial (matematial) metod to predit te eat transfer between free onvetion on one side of a nite vertial onduting wall and fored onvetion ow on te oter side of te wall wit te onsideration of te wall termal resistane. It is assumed tat te ondution in te wall is in te transversal diretion only. Sine bot te plate temperature and te eat ux troug te plate are unknown a priori in tis problem, te boundary layer equations on bot sides of te wall and te onedimensional eat ondution equation for te wall are solved simultaneously. Te numerial metod used is a new ompreensive and non-iterative seme based on te singular perturbation metod desribed in a reent paper by Su and Pop [10]. We notie tat tis metod differs by te iterative guessing tenique proposed by Cen and Cang [8, 9]. Heat transfer arateristis ave been derived for some main parameters entering tis problem. Basi equations Te pysial model under onsideration along wit te oordinate systems is sown in Fig. 1, were te vertial plate wit lengt L and tikness b separates two semiin nite uid reservoirs at different temperatures. Te warmer reservoir ontains a stagnant uid wit temperature t, wile te ambient temperature on te old side of te plate is t. Obviously, t is iger tan t. Te upper left orner of te plate oinides wit te origin of a Cartesian oordinate system wose y axis points in te diretion normal to te plate, wile te x axis points downward in te plate's longitudinal diretion. Due to gravity, a free onvetion laminar boundary layer appears on te ot side of te plate and ows downward along te plate. A fored onvetion ow of te ooling uid wit veloity U 1 is imposed on te rigt lateral surfae of te plate tus generating a fored onvetion boundary layer on tis surfae, wi develops wit inreasing tikness downstream. Aordingly, two uid streams move in opposite diretions. Due to tis assumption, te present problem an be formulated in terms of te boundary layer equations for two different eat transfer systems. Tese governing differential equations need to be onsidered separately and tey are: Hot uid ou ox ov oy ˆ 0 1 ou u ox v ou oy ˆ m o u oy gb T t ot u ox v ot oy ˆ a o T oy 3 were u and v denote te veloity omponents of te ot uid in te x and y diretions, respetively, T is te temperature of te ot uid, g is te gravitational aeleration, b is te termal expansion oef ient of te ot uid, and m and a are te kinemati visosity and termal diffusivity of te eat uid, respetively. Fig. 1. Semati diagram of te studied pysial model
Te boundary onditions for te free onvetion system are u ˆ v ˆ 0; T ˆ T w x on y ˆ 0 u! 0; T! t as y!1 4 were T w x denotes te wall temperature faing te ot side of te plate. Cold uid ou ox ov oy ˆ 0 u ou ox v ou oy ˆ m o u oy u ot ox v ot oy ˆ a o T oy 5 6 7 were x and y are Cartesian oordinates on te fored onvetion side, u and v denote te veloity omponents of te old uid in te x and y diretions, and T ; m and a are te temperature, kinemati visosity and termal diffusivity of te old uid, respetively. Te boundary onditions for te fored onvetion system are u ˆ v ˆ 0; T ˆ T w x on y ˆ 0 u! U 1 ; T! t as y!1 8 were x ˆ L x and T w x denotes te wall temperature faing te fored onvetion side. It is assumed tat eat ondution along te plate is negleted in omparison wit transverse eat ondution. Under tis ondition, te eat ux entering te left fae of te plate will be equal to tat leaving te rigt fae at any given vertial position x, i.e. k s T w T w b ˆ k ot oy yˆ0 ˆ k ot oy ˆ q x yˆ0 9 were k s ; k and k denote te termal ondutivities of te solid plate, ot uid and old uid, respetively, and q x is te loal eat ux troug te plate. A orrelation between T w and T w an be obtained from Eq. (9) as T w ˆ T w bk ot : 10 k s oy yˆ0 3 Solution To solve Eqs. (1)±(3) and (5)±(7), we introdue te following dimensionless variables n ˆ x=l; g ˆ C y = Ln 1=4 w ˆ 4m C n 3=4 f n; g ; n; g ˆ T t t =Š=Dt n ˆ x =L ˆ 1 n; g ˆ y Re 1= = Ln 1= w ˆ U 1 m Ln 1= f n ; g ; n ; g ˆ T t t =Š=Dt 11 were C ˆ gbdtl 3 = 4m Š1=4, Dt ˆ t t ; Re ˆ U 1 L=m is te Reynolds number for te fored onvetion ow, and w and w are te stream funtions of te ot and old uids, respetively, wi are de ned as u ˆ ow oy ; u ˆ ow oy ; v ˆ ow ox v ˆ ow ox 1 Due to te de nition of (11) and (1), Eqs. (1)±(3) and (5)±(7) an be transformed into te following form o 3 f og 3 3f ˆ 4n of og 1 o f ogon o f of og on og ˆ 4n of o og on o of og on of o f o f of og og on og on of o o of og on og on o f og of og 1 o Pr og 3f o o 3 f og 3 1 f o f og ˆ n 1 Pr o og 1 f o ˆ n og 13 14 15 16 were Pr ˆ m =a and Pr ˆ m =a are te Prandtl numbers of ot and old uids, respetively. Te boundary onditions (4) and (8) beome f ˆ of og ˆ 0; ˆ w n at g ˆ 0 of og! 0;! 1 17 as g!1 f ˆ of ˆ 0; ˆ w n R t R o t og n 1=4 at g og ˆ 0 gˆ0 of og! 1;! 1 as g!1 18 were R t ˆ bk Re 1= = k s L denotes te termal resistane ratio of te fored onvetion ow to te wall, R t ˆ k C = k Re 1= an be regarded as te termal resistane of te ot uid to te old uid, w ˆ T w t t =Š=Dt and w ˆ T w t t =Š=Dt. Substituting variables (11) into Eq. (9), we get R t n 1= =n 1=4 o og o og gˆ0 gˆ0 ˆ 0 at any given position x : 19 Based on Eq. (9), te loal eat transfer oef ient x for te fored onvetion system an be expressed as 35
36 ot x ˆ4 k oy yˆ0 ˆ q x = T x ; 0 t Š : 3, 5 T x ; 0 t Š 0 Te loal Nusselt number for te fored onvetion system an be expressed as Nu x ˆ x x ˆ Re 1= n 1= o k og w n 1 : gˆ0 1 Te total eat ux Q troug te surfae faing te old uid is obtained by integrating te loal eat ux over te entire eigt of te plate and an be expressed as Z 3 L Q ˆ k ot 4 5 dx : oy 0 yˆ0 Substituting (11) into () yields te average Nusselt number for te fored onvetion system as Nu ˆ Q Z 1 k Dt ˆ Re1= o n 1= dn 0 og were A ˆ Z 1 0 ˆ ARe 1= o og gˆ0 gˆ0 3 n 1= dn : 4 To solve Eq. (13) to (16), a ompreensive and non-iterative numerial seme is proposed, wi is in ontrast wit te iterative proess purposed by Cen and Cang [8, 9] using a guessing strategy. Based on Eq. (19), te boundary onditions an be rewritten as f ˆ of og ˆ 0; o og ˆ n1=4 k n at g ˆ 0 of og! 0;! 1 5 as g!1 f ˆ of o ˆ 0; ˆ R t og og n1= k n at g ˆ 0 of! 1;! 1 as g og!1 6 and k ˆ 0 at g ˆ 0; g ˆ 0 7 were te dummy variable k is de ned as k n; g ˆR t R t k n : 8 Te systems (13)±(16) and (8), togeter wit te boundary onditions (5), (6) and (7), are ten solved using te singular perturbation metod and te dif ulties assoiated wit te guessed interfaial onditions ave been obviated. Sine tis proedure was desribed in a reent paper by Su and Pop [10], we will not repeat it ere. Note tat te points n ˆ 0 and 1 are singular wi make te problem more dif ult. 4 Results and disussion In tis setion we disuss te effets of te Prandtl numbers Pr and Pr, and resistane parameters R t and R t on te interfae temperatures, eat transfer rates and Nusselt numbers. Figures and 3 sow variation of w n wit n for some values of R t and R t wen te two working uids ave Pr ˆ Pr ˆ 1. Te results of tese gures sow tat for inreasing values of R t te temperature of te old side of te wall dereases. It appens beause wen R t is inreased te wall beomes more effetive insulation between te two fored and free onvetion ows. In ontrast, w n inreases as te free Fig.. Effet of R t on w n for R t ˆ Pr ˆ Pr ˆ 1 Fig. 3. Effet of R t on w n for R t ˆ Pr ˆ Pr ˆ 1
onvetion beomes dominant, i.e. wen te parameter R t inreases. It turns out tat an inrease of R t leads to a redution of te eat transfer rates at te old side of te wall and to an inrease of tese rates wit inreasing R t, as an be seen from Fig. 4 and 5. It is evident from Fig. 6 and 7 tat te effets of R t and R t on te loal Nusselt number are insigni ant. Furter, Fig. 8 to 11 illustrate te variation of w n and Nu x n wit te distane n along te wall. As an be seen from Fig. 9 and 11 te temperature and loal Nusselt number of te old side of te wall are less affeted by Pr. To tis end it sould be mentioned tat te same trends persist for te eat transfer arateristis of te free onvetion system. But, tey are not presented ere for te sake of spae onservation. Finally, values of te average Nusselt number are given in Table 1 for R t ˆ R t ˆ 1 and some values of Pr and Pr. We notie from tis table tat Nu =Re 1= inreases wit dereasing Pr or inreasing Pr. Fig. 6. Effet of R t on Nu x n =Re 1= for R t ˆ Pr ˆ Pr ˆ 1 37 Fig. 4. Effet of R t on o og n ; 0 for R t ˆ Pr ˆ Pr ˆ 1 Fig. 7. Effet of R t on Nu x n =Re 1= for R t ˆ Pr ˆ Pr ˆ 1 Fig. 5. Effet of R t on o og n ; 0 for R t ˆ Pr ˆ Pr ˆ 1 Fig. 8. Effet of Pr on w n for R t ˆ R t ˆ Pr ˆ 1
Table 1. Values of Nu =Re 1= for R t ˆ R t ˆ 1 Pr ˆ 0:1 Pr ˆ 1 Pr ˆ 10 Pr ˆ 0:1 0.4945 0.4941 0.4899 Pr ˆ 1 0.4974 0.4970 0.497 Pr ˆ 10 0.5168 0.5164 0.5116 38 Fig. 9. Effet of Pr on w n for R t ˆ R t ˆ Pr ˆ 1 Fig. 10. Effet of Pr on Nu x n =Re 1= for R t ˆ R t ˆ Pr ˆ 1 Fig. 11. Effet of Pr on Nu x n =Re 1= for R t ˆ R t ˆ Pr ˆ 1 5 Conlusions A onjugate problem of eat transfer between a laminar fored onvetion ow and a laminar free onvetion separated by a vertial nite wall was studied teoretially. Te axial termal ondution in te wall was negleted. Te governing boundary layer equations subjet to onjugate boundary onditions were solved numerially using a very ef ient metod wi differs from te one used by oter autors. As a result te temperature distributions and eat transfer rates at bot sides of te wall ave been determined. Te results sow tat te resistane parameters in uene substantially te interative eat transfer arateristis. We ave given partiular attention to te ase wen te resistane parameters R t ˆ R t ˆ 1 wi inlude te situation in wi te uids at bot sides of te wall are te same. It is wort mentioning tat te results reported in tis paper are in general in agreement wit tose from te open literature. However, te auray of tese results an be furter evidened troug experiments. Referenes 1. Lok GSH; Ko RS (1973) Coupling troug a wall between two free onvetive systems. Int J Heat and Mass Transfer 16: 087±096. Viskanta R; Lankford DW (1981) Coupling of eat transfer between two natural onvetion systems separated by a vertial wall. Int J Heat and Mass Transfer 4: 1171±1177 3. Anderson R; Bejan A (1980) Natural onvetion on bot sides of a vertial wall separating uids at different temperatures. Transations of te ASME, J Heat Transfer 10: 630± 635 4. Sakakibara M; Amaya H; Mori S; Tanimoto A (199) Conjugate eat transfer between two natural onvetions separated by a vertial plate. Int J Heat and Mass Transfer 35: 89±97 5. TrevinÄo C; MeÂndez F; Higuera FJ (1996) Heat transfer aross a vertial wall separating two uids at different temperatures. Int J Heat and Mass Transfer 39: 31±41 6. Poulikakos D (1986) Interation between lm ondensation on one side of a vertial wall and natural onvetion on te oter side. Transations of te ASME, J Heat Transfer 108: 560±566 7. MeÂndez F; TrevinÄo C(1996) Film ondensation generated by a fored ooling uid. European J Meanis, B/Fluids 15: 17±40 8. Cen H-T; Cang S-M (1996) Termal interation between laminar lm ondensation and fored onvetion along a onduting wall. Ata Meania 118: 13±6 9. Cen H-T; Cang S-M (1997) Coupling between laminar lm ondensation and natural onvetion on opposite sides of a vertial plate. Int J Num Met Fluids 4: 319±336 10. Su J-J; Pop I (1998) Termal interation between free onvetion and fored onvetion along a onduting plate embedded in a porous medium. Hybrid Metods in Engineering: ± Modeling, Programming, Analysis, Animation 1: 55±66