Analysis of Thermally Developing Flow in Porous Media Under Local

International Journal of Heat and Mass Transfer 67 (2013) 768–775

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International Journal of Heat and Mass Transfer

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Analysis of thermally developing ﬂow in porous media under local thermal non-equilibrium conditions ⇑ Xiao-Long Ouyang a,b, Kambiz Vafai b, , Pei-Xue Jiang a a Key Laboratory of CO2 Utilization and Reduction Technology, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China b Department of Mechanical Engineering, University of California, Riverside, CA 92521-0425, USA article info abstract

Article history: In the present study, analytical solutions for thermally developing ﬂows in porous media under local Received 18 July 2013 thermal non-equilibrium (LTNE) condition are derived for different fundamental models. The analytical Received in revised form 19 August 2013 solutions are validated by comparison with the numerical results. Based on the analytical solutions, Accepted 21 August 2013 the local Nusselt number is obtained, and the dimensionless thermal entry length nc is predicted. The results are presented in terms of the pertinent physical parameters such as the Biot number and the effective thermal conductivity ratio of the ﬂuid to solid phases. Keywords: Ó 2013 Elsevier Ltd. All rights reserved. Local thermal non-equilibrium Thermally developing Porous media Analytical solution

1. Introduction confined channel flow in porous media, Haji-Sheikh and Vafai [3] and Haji-Sheikh et al. [4] derived the analytical solutions of ther- Thermally developing section usually has a larger heat transfer mally developing flows in porous media. capacity than the corresponding thermally fully developed section Amiri and Vafai [5] analyzed the dispersion effects, non-Darcian in a confined channel flow since it is dominated by the thermal effects, variable porosity and the LTNE effects for flow through por- boundary effect. The thermal development problem for a laminar ous media. They presented error maps to establish conditions un- flow in a tube was investigated by Graetz [1]. In a channel fully der which LTE model can be utilized. Jiang et al. [6,7] investigated filled with a porous medium, the thermally developing section the convection heat transfer in channels filled with packed parti- can also play a significant role. Two characteristics appear due to cles or sintered porous media using the LTNE and LTE models, the existence of a porous medium. First, in contrast to the case and compared their results with experiments [8,9]. They found without a porous medium, the momentum boundarypffiffiffiffiffiffiffiffi layer thick- good agreement between the LTNE model and the experimental re- ness is confined to a region of the order of K=e and the flow sults while the LTE model did not match with their data. Nield and boundary effect is less important in most cases in the presence of Kuznetsov [10] presented a semi-analytical solution of thermally a porous medium, as pointed out by Vafai and Tien [2]. Second, developing flow in a porous medium using the LTNE model, but the local temperature difference between the solid and fluid no explicit expression was presented. The first analytical solution phases can substantially influence the heat transfer process. for the thermal developing region in porous media while incorpo- There are two primary ways to model the heat transfer process rating LTNE model condition was derived by Yang and Vafai [11] in porous media: the local thermal equilibrium (LTE) model and for an isothermal wall boundary condition. the local thermal non-equilibrium (LTNE) model. In the LTE model In this work, we will visit the work analyzed by Yang and Vafai the interphase temperature difference is neglected, while the LTNE [11] under constant heat flux boundary condition. The constant model accounts for the interphase temperature differences. Early wall heat flux boundary condition is quite involved when incorpo- investigations have focused more on the LTE model coupled with rating the LTNE aspects, as introduced by Alazmi and Vafai [12] non-Darcian effects. Vafai and Tien [2] analyzed the forced convec- and Ouyang et al. [13]. tion in porous media with the LTE model and discussed the The analysis for thermally fully developed flow while incorpo- non-Darcian effects on temperature and Nusselt number. For the rating the LTNE conditions with an imposed constant wall heat flux has been established. Lee and Vafai [14] studied analytically the

⇑ Corresponding author. Tel.: +1 951 827 2135; fax: +1 951 827 2899. thermally fully developed region of a channel with an imposed E-mail address: [email protected] (K. Vafai). constant heat ﬂux condition, based on the LTNE and LTE models.

0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.08.056 X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775 769

Nomenclature

ah H2 Bi Bi¼ sf , Biot number g g ¼ y, dimensionless transverse coordinate ks;eff H k cp heat capacity [J/kg K] j j ¼ f ;eff , effective thermal conductivity ratio of the fluid 2 ks;eff hfs interfacial heat transfer coefficient [W/m K] H half of the height of the channel [m] to solid phases j dimensionless parameter defined by Eq. (18) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m,n k k ¼ Bið1 þ jÞ=j, dimensionless parameter k thermal conductivity [W/m K] q density [kg/m3] K permeability [m2] 1þj 2 Nu Nussult number rn rn ¼ j ðÞnp , dimensionless parameter q c uH Pe Pe¼ f pf , Pectlet number h ðTTf 0Þks;eff , dimensionless temperature kf ;eff qwH 2 qw wall heat flux [W/m ] ha dimensionless averaged temperature defined by Eq. (12) T temperature [K] x dimensionless decay factor for thermally developing Tf0 inlet fluid temperature [K] component x u Darcy velocity [m/s] n n ¼ PeH, dimensionless axial coordinate x axial coordinate [m] y transverse coordinate [m] Subscripts A,B,C Models A, B and C Greek symbols c the transition point where the thermally developing a phase interface area per unit volume [m1] flow transits to the thermally fully developed flow 2 2 2 eff effective cm cm =(1+j)[(0.5 + m) p + k ] f fluid phase D dimensionless temperature difference defined by Eq. LTE the LTE model (12) s solid phase e volume fraction

They found that the Biot number, Bi and the effective thermal con- where Tf and Ts denote the fluid and solid temperatures, kf,eff and ductivity ratio, j are the key parameters for the validity of LTE ks,eff the effective fluid and solid conductivities, and u the Darcy model and had presented an error map establishing the validity re- velocity. gion of the LTE model in terms of Bi and j. The phenomenon of For the boundary conditions at the heated wall with constant heat flux bifurcation was introduced by Yang and Vafai [11,15– heat flux, the following three models are considered, which are 17]. In the present work, the corresponding theory for the ther- introduced by Amiri and Vafai [18]. These models are based on dif- mally developing flow is analyzed, especially the transition point ferent assumptions. where the thermally developing flow transits to the thermally fully Model A: The wall temperature of each phase at the wall is con- developed flow in a channel filled with a porous medium. In the sidered to be equal to each other. present work, analytical solutions are established and the transi- @T @T tion point is predicted. T ¼ T ; k f k s ¼ q at y ¼ 0 ð2Þ f s f ;eff @y s;eff @y w 2. Modeling and formulation Model B: Each phase is considered to receive the same imposed heat flux at the wall. Convective transport through a channel filled with a porous medium is considered, as shown in Fig. 1. Only the upper half of @Tf @Ts kf ;eff ¼ks;eff ¼ qw at y ¼ 0 ð3Þ the channel is considered due to the symmetry. A fluid at Tf0 flows @y @y through the channel and each wall of the channel receives a con- Model C: The intrinsic averaged wall heat fluxes of the two stant heat flux qw. Thus, the thermal boundary layers develop along phases are considered to be equal. the axial coordinate until x reaches the transition point x = xc. The portion of channel with x < xc, i.e., the thermally developing section kf ;eff @Tf ks;eff @Ts ¼ ¼ qw at y ¼ 0 ð4Þ is the focus of the current study. The value of xc represents the e @y 1 e @y thermal entry length. The following assumptions are adopted in the present study: The other boundary conditions at the central line and inlet can be presented as: ( (1) The flow is steady and incompressible and can be repre- @T f ¼ @Ts ¼ 0aty ¼ H sented by the Darcy flow model. @y @y ð5Þ (2) Axial conduction is neglected in the governing equations. Tf ¼ Tf 0 at x ¼ 0 (3) Natural convection and radiative heat transfer are negligible. (3) Properties such as porosity, specific heat, density and ther- 2.1. Normalization mal conductivity are assumed to be constant. The following non-dimensional variables have been introduced: Based on these assumptions, the following energy equations are obtained from Amiri and Vafai [5] employing the LTNE model: ðT T Þk ah H2 k 8 h ¼ f 0 s;eff ; Bi ¼ sf ; j ¼ f ;eff ; @T @2T < f f qwH ks;eff ks;eff qf cpf u @x ¼ kf ;eff @ 2 þ ahsf ðTs Tf Þ ð6Þ y ð1Þ : 2 y x qf cpf uH @ Ts g ¼ ; n ¼ ; Pe ¼ 0 ¼ ks;eff @ 2 þ ahsf ðTf TsÞ y H PeH kf ;eff 770 X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775

Fig. 1. Schematic diagram for a thermally developing ﬂow through a channel ﬁlled with a porous medium and the corresponding coordinate system.

8 2 The governing equations and boundary conditions, Eqs. (1)–(5), can < @ha 1þj @ ha 1 @D @n ¼ j @g2 þ 1þj @n then be written as: 2 ð13Þ : 1 @D @ D 1þj @ha 8 1þj @n ¼ @g2 Bi j D þ @n 2 < @hf @ hf j ¼ j 2 þ Biðhs hf Þ @n @g ð7Þ while the boundary conditions given by Eqs. (8) and (11) become 2 : @ hs 0 ¼ 2 þ Biðhf hsÞ 8 @g @ha <> ðÞ1 þ j @g ¼ 1; D ¼ 0atg ¼ 0 Model A: @h a ¼ @D ¼ 0atg ¼ 1 ð14Þ :> @g @g @h @h ha ¼ D ¼ 0atn ¼ 0 h ¼ h ; j f s ¼ 1atg ¼ 0 ð8Þ f s @ @ g g A series distribution utilized for ha and D as given below Model B: 8 "# > X1 > a ðnÞ > 1 hp;0ffiffi < ha ¼ 1þj þ ah;nðnÞ cos ðÞnpg @h @h 2 j f ¼ s ¼ 1atg ¼ 0 ð9Þ n¼1 ð15Þ @ @ > X1 g g > : D ¼ aD;mðnÞ sinðð0:5 þ mÞpgÞ Model C: m¼0 Integrating Eq. (13) with respect to g after multiplying it by cos @h @h j f ¼ e; s ¼ 1 e at g ¼ 0 ð10Þ (npg) or sin ((0.5 + m)pg) results @ @ g g 8 > pffiffiffi X1 > dah;0 daD;m Other boundary conditions: > ¼ 2 1þj þ j ; n ¼ 0 > dn j m;0 dn ( > m¼0 <> @hf @hs X1 ¼ ¼ 0atg ¼ 1 da da @g @g h;n ¼ 2 1þj r a þ j D;m ; n ¼ 1; 2; 3; ... ð11Þ > dn j n h;n m;n dn ð16Þ h ¼ 0atn ¼ 0 > m¼0 f > > X1 > daD;m dah;n : dn ¼cmaD;m þ jm;n dn ; m ¼ 0; 1; 2; ... 3. Analysis n¼0 where The following linear transformation is utilized to recast the hi dimensionless temperatures 1 þ j 2 2 2 2 rn ¼ ðnpÞ ; cm ¼ð1 þ jÞð0:5 þ mÞ p þ k ; k ( ( qjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hsþjhf D h ¼ hf ¼ ha a 1þj ; 1þj 12 Bi 1 = 17 jD ð Þ ¼ ð þ jÞ j ð Þ D ¼ hs hf hs ¼ ha þ 1þj 8 pffiffiffiR pffiffi < 1 2 1 where ha denotes the dimensionless local averaged temperature 2 0 sinðð0:5 þmÞpgÞdg ¼ p 0:5þm ; n ¼ 0 weighed by effective conductivities, and D the dimensionless local jm;n ¼ R :2 1 sinðð0:5 þmÞpgÞcosðnpgÞdg ¼ 2 0:5þm ; n ¼ 1;2;3;... temperature difference. These two variables are also important for 0 p ð0:5þmÞ2n2 the subsequent analysis. ð18Þ

and j is a special set with the following orthogonality 3.1. Analytical solution for Model A m,n characteristic The analytical solution for Model A has to incorporate the fact X1 1; m ¼ k X1 1; n ¼ k j j ¼ ; j j ¼ ð19Þ that the temperature variables are coupled with the boundary con- m;n k;n 0; m – k m;n m;k 0; n – k dition given by Eq. (8). Applying the transformation given by Eq. n¼0 m¼0 (12), the governing equations given in Eq. (7) can be represented as Deﬁning the inﬁnite vectors: X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775 771

2 3 2 3 2 pffiffiffi 3 ah;0 aD;0 2 s þ rn s T s T 6 7 6 7 6 7 6 7 6 7 6 7 Uh ¼ D ½j0;n½j0;n ½j1;n½j1;n 6 ah;1 7 6 aD;1 7 6 7 s s þ c s þ c 6 7 6 7 6 2 7 0 2 1 3 6 7 6 7 1 þ j 6 7 2 6 ah;2 7 6 aD;2 7 6 7 ahðnÞ¼6 7; aDðnÞ¼6 7; b ¼ 6 2 7; j j j 0 0 6 7 6 7 j 6 7 0;0 0;0 0;1 6 7 6 7 6 7 6 7 6 . 7 6 . 7 6 . 7 6 7 4 . 5 4 . 5 4 . 5 6 2 7 6 j0;0j0;1 j0;1 0 0 7 2 s þ r s 6 7 ah;n aD;m ... D n 6 7 2 pffiffiffi 3 2 3 ð20Þ 6 000 0 7 2=2 sinð0:5pgÞ s s þ c0 6 7 6 7 6 7 . . . . . 6 7 6 7 6 . . . . . 7 6 cosðpgÞ 7 6 sinð1:5pgÞ 7 4 . . . . . 5 6 7 6 7 6 7 6 7 ð28Þ 6 cos 2 7 6 sinð2:5pgÞ 7 fhðgÞ¼6 ð pgÞ 7; fDðgÞ¼6 7; m ¼ n ¼ 0; 1; 2; ... 000 0 6 7 6 7 2 3 6 . 7 6 . 7 2 6 . 7 6 . 7 j j j 0 0 4 . 5 4 5 6 1;0 1;0 1;1 7 6 7 cosðnpgÞ sinðð0:5 þ mÞpgÞ 6 2 7 6 j1;0j1;1 j1;1 0 0 7 s 6 7 6 7 as well as defining the infinite matrices: 6 000 0 7 s þ c1 6 7 2 3 2 3 6 ...... 7 j0;0 j0;1 j0;n w 0 0 6 7 0 4 . . . . . 5 6 7 6 7 6 j j j 7 6 0 w 0 7 6 1;0 1;1 1;n 7 6 1 7 000 0 J 6 7; D w 6 7; m n 0; 1; 2; ... 21 ¼ 6 . . . . 7 ð nÞ¼6 . . . . 7 ¼ ¼ ð Þ 6 . . . . 7 6 . . . . 7 4 . . . . 5 4 . . . . 5 s þ cm s T s T j j j 00 wn UD ¼ D ½j ½j ½j ½j m;0 m;1 m;n s s þ r m;0 m;0 s þ r m;1 m;1 0 2 1 3 j2 j j 0 0 where J is an orthogonal matrix due to Eq. (19) 6 0;0 0;0 1;0 7 6 7 T T 6 2 7 JJ ¼ J J ¼ I ð22Þ 6 j0;0j1;0 j1;0 0 0 7 s s 6 7 þ cm 6 7 ... D 6 000 0 7 while D(w ) denotes the diagonal matrix of an infinite series w . s s þ r0 6 7 n n 6 ...... 7 As such Eq. (16) can be simplified to 4 . . . . . 5 ( ð29Þ dah T daD 000 0 dn ¼ b DðrnÞah þ J dn 2 3 ð23Þ 2 da da D h j ; j0;1j1;1 0 0 dn ¼DðcmÞaD þ J dn 6 0 1 7 6 7 6 2 7 6 j0;1j1;1 j1;1 0 0 7 Eq. (23) cannot be solved directly because D(r ) is a singular matrix. s 6 7 n 6 000 0 7 This problem can be solved by using the Laplace transformation 6 7 s þ r1 6 7 6 . . . . . 7 with respect to n: 4 ...... 5 ( b T 000 0 Dðs þ rnÞLfahg¼ þ sJ LfaDg s ð24Þ 8 no Dðs þ c ÞLfa g¼sJLfa g m D h <> L1 1 b ahðnÞ¼ Uh s2 noÂÃ ÂÃ ð30Þ > U1 Thus, after solving Eq. (24), the exact solution can be expressed as : 1 D b0 b1 aDðnÞL j þ j s sþr0 m;0 sþr1 m;1 ( a ðnÞf ðgÞ h ¼ h h where [wn] denotes the column vector of an infinite series wn. This a 1þj ð25Þ D ¼ a ðnÞf ðgÞ leads to D D 8 2 2 > n ðÞ1 g 2 1=3 j j > h ¼ þ þ 0;0 þ 1;0 where > a > j 21ðÞþ j jc0 jc1 > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > 8 no > Fully developed component > 1 > < a ðnÞ¼L1 U b > h h s2 > D1 expðxA0nÞD2 expðxA1nÞ no > > ð26Þ > : L1 1 s b > ½D3 expðxA0nÞþD4 expðxA1nÞ cosðpgÞ aDðnÞ¼ U JD 2 > D sþrn s > P > 1 2 > expðrnnÞ cosðnpgÞ > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n¼2 jrn <> s þ rn T s Developing component Uh ¼ D J D J > s s þ c > 1 coshðkð1 gÞÞ m ð27Þ > D ¼ 1 > Bið1 þ Þ coshðkÞ s þ cm s T > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}j UD ¼ D JD J > > Fully developed component s s þ rn > > > D exp n D exp n D exp n sin 0:5 > ½ 5 ðxA0 Þþ 6 ðxA1 Þþ 7 ðr1 Þ ð pgÞ 1 > However, it is very difficult to obtain the exact expressions for Uh > > ½D8 expðxA0nÞþD9 expðxA1nÞ sinð1:5pgÞ 1 > and UD . Therefore, no explicit exact solution for this problem can > P > 1þj 1 jm;0 be found here. > j m¼2 c expðcmnÞ sinðð0:5 þ mÞpgÞ :> |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflm fflfflfflfflfflffl} Developing component 3.2. Approximate solution for Model A ð31Þ where We obtain an approximate solution by dropping a few higher pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ordered components in Eqs. (26) and (27). That is xA0 ¼ B B 4AC =2A; xA1 ¼ B þ B 4AC =2A ð32Þ 772 X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775 2 2 2 2 2 2 where bf = bs = 1 for Model B, and bf = e and bs =1 e for Model C. As A ¼ 1 j1;0 j1;1 c0 þ 1 j0;1 j0;0 c1 þ 1 j1;0 j0;0 r1 such the solutions for Models B and C can be obtained simulta- 2 2 B ¼ 1 j1;0 r1c0 þ 1 j0;0 r1c1 þ c1c0 neously based on Eq. (37). Again, with the transformation given by Eq. (12), the governing equations and the boundary conditions C ¼ r c c 1 0 1 given by Eqs. (11) and (37) become ð33Þ 8 2 < @ha 1þj @ ha 1 @D @n ¼ j @g2 þ 1þj @n and ð38Þ : 2 2 0 ¼ @ D þ 1þj @ ha Bi 1þj D 1 r c c 1 @g2 j @g2 j D ¼ 1 0 1 þ ðr c þ r c þ c c ÞE 1 jAðx x Þ x2 x 1 0 1 1 0 1 1 A1 A0 A0 A0 8 > ð1 þ jÞ @ha ¼ b þ b ; j @D ¼ jb b at g ¼ 0 1 r1c0c1 1 > @g s f @g s f D2 ¼ ðr1c þ r1c þ c c ÞþE1 < 2 0 1 0 1 @ha @D jAðxA1 xA0Þ xA1 xA1 ¼ ¼ 0atg ¼ 1 > @g @g ffiffiffi ð39Þ 1 2c c > p 0 1 : bs coshffiffiffi ðÞBið1ffiffiffigÞ D3 ¼ E2 ð1 þ jÞha ¼ D ¼ p p at n ¼ 0 jAðxA1 xA0Þ xA0 Bi sinh ðÞBi 1 2c c D ¼ 0 1 þ E Utilizing a series representation for h and D as 4 jAðx x Þ x 2 a A1 A0 ffiffiffi A1 !8 p X1 2j 2j E > 1 þ j 0;0r1c1 0;1r1c1 xA0 4 > h n n D5 ¼ E3 <> a ¼ ch;0ð Þþ ch;nð Þ cosðnpgÞ jAðxA1 xA0Þ xA0 r1 xA0 n¼1 ffiffiffi ! ð40Þ p > X1 2j 2j E > 1 þ j 0;0r1c1 0;1r1c1 xA1 4 :> D ¼ cD;0ðnÞþ cD;nðnÞ cosðnpgÞ D6 ¼ þ E3 jAðxA1 xA0Þ xA1 r1 xA0 n¼1 ÀÁ r1ð1 þ jÞ2j c þ E4 The governing equations given by Eq. (38) can be integrated with D ¼ 0;1 1 7 respect to g after multiplying it by cos (npg) jAðr1 xA0Þðr 1 xA1Þ ! pffiffiffi 8 b b 1 þ j 2j1;0r1c0 > dch;0 sþ f 1 dcD;0 D ¼ E > dn ¼ j þ 1þj dn ; n ¼ 0 8 jAðx x Þ x 5 > A1 A0 A0 <> pffiffiffi ! 0 ¼ bs BicD;0; n ¼ 0 1 þ j 2j r1c dc 2 bsþb dc 1;0 0 > h;n ðÞf 1þj 2 1 D;n D9 ¼ þ E5 > dn ¼ j j ðnpÞ ch;n þ 1þj dn ; n ¼ 1; 2; 3; ... jAðxA1 xA0Þ xA1 > hi :> 2 2 0 ¼ 2b 1þj 1þj ðnpÞ c ðnpÞ þ Bi 1þj c ; n ¼ 1; 2; 3; ... ð34Þ s j j h;n j D;n ð41Þ hipffiffiffi hipffiffiffi E ¼ r þ c 1 j2 þ 2j j þ c 1 j2 þ 2j j 1 1 0 1;1 1;0 1;1 1 0;1 0;0 0;1 Thus, after solving Eq. (41), the coefficients for Eq. (40) are obtained hipffiffiffi hipffiffiffi 2 2 as E2 ¼ c0 21 j1;0 þ 2j1;0j1;1 þ c1 21 j0;0 þ 2j0;0j0;1 hi 8 pffiffiffi pffiffiffi b b 2 2 f þ s bs > c ; ¼ n þ ; n ¼ 0 E3 ¼ 2j0;0 r1 1 j þ c þ 2j0;0j r1 > h 0 j Bið1þjÞ hi1;0 1 1;0 > < bs 2 cD;0 ¼ Bi ; n ¼ 0 E4 ¼ 2j r1 1 j þ c þ 2j j j r1 0;1 1;0 1 0;0 1;1 1;0 2 ð42Þ hi > ch;n ¼dh;n expðxBorC; nnÞþ ; n ¼ 1; 2; 3; ... pffiffiffi pffiffiffi > jrn 2 2 > E5 ¼ 2j1;0 r1 1 j1;0 þ c0 þ 2j0;0j1;0r1 : jbsbf cD;n ¼dD;n expðxBorC; nnÞþ ; n ¼ 1; 2; 3; ... j½ðnpÞ2þk2 ð35Þ where When removing more high ordered components as shown in Eqs. hi (28)–(30), some detailed information near the entrance will get lost. ðnpÞ2 ðnpÞ2 þ k2 Hence, the approximate solution predicts less accurately near the xBorC; n ¼ 2 ð43Þ entrance, but it still can predict accurate results in the thermally ðnpÞ þ Bi developing region. 2ðb þ b Þ 2b =ð1 þ jÞ d ¼ f s s ð44Þ The corresponding exact solution for thermally fully developed h;n 2 jrn ðnpÞ þ Bi flow is given by Yang and Vafai [15]: 8 hi jbs bf 2bs 2 dD;n ¼ ð45Þ > h ¼ n þ ð1gÞ 1=3 þ 1 1 tanhðkÞ j½ðnpÞ2 þ k2 ðnpÞ2 þ Bi > a;ed j 2ð1þjÞ Bi 1 2 k > ð þjÞ < X1 2 ð1gÞ21=3 j The exact solutions for Models B and C can be expressed as ¼ n þ þ m;0 ð36Þ > j 2ð1þjÞ jcm 8 > m¼0 > hi > bf þ bs bf þ bs 2 bs > > h ¼ n þ ½ð1 gÞ 1=3þ :> coshðkð1gÞÞ > a D 1 1 > j 21ðÞþ j ð1 þ jÞBi ed ¼ Bið1þjÞ coshðkÞ > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > X Fully developed component <> 1 The fully developed component of Eq. (31) is quite close to the exact dh;n expðxBorC; nnÞ cosðnpgÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n¼1 > expressions given by Eq. (36) but misses few terms in the fully > Developing component > developed component part of the expression for h . > X a > bf þ bs jbs bf cosh ðÞkðÞ1 g 1 > D ¼ þ dD;n exp ðÞxBorC; nn cos ðÞnpg > ðÞ1 þ j Bi jk sinh ðÞk |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}n¼1 :> |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Developing component 3.3. Exact solution for Models B and C Fully developed component ð46Þ Eqs. (9) and (10) for Models B and C can be expressed as The fully developed component in Eq. (46) is the same as the corre- @hf @hs sponding exact solution for thermally fully developed flow given by j ¼ bf ; ¼ bs at g ¼ 0 ð37Þ @g @g Yang and Vafai [15]. X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775 773

3.4. Exact solution for the LTE model (a) 50 Approximate results The governing equation for the LTE model can be obtained by 40 Numerical results assuming that the temperatures of the ﬂuid and solid phases are for Model A the same in Eq. (7). That is 30 @h @2h j ¼ðj þ 1Þ ð47Þ @n @g2

Nu 20 Bi=10 The boundary conditions for the LTE model is represented by 8 @h <> ðj þ 1Þ @ ¼ 1atg ¼ 0 g Bi=1 @h ð48Þ > @g ¼ 0atg ¼ 1 : Bi=0.1 h ¼ 0atn ¼ 0 10 The exact solution can be obtained as 0.1 1

2 X ξ n ð1 gÞ 1=3 1 2 h ¼ þ expðrnnÞ cosðnpgÞ ð49Þ n¼1 j 21ðÞþ j jrn (b) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Approximate results Fully developed component Developing component Numerical results Within all these analytical results, it should be noted that there are for Model A two aspects that are consistent with all of the solutions when 100 Bi ? 1: First, the ha’s based on LTNE model for Models A, B and C Bi=10 in Eqs. (31) and (46) approach the h based on LTE model in Eq.

(49). Second, the D’s in Eqs. (31) and (46) approach 0. Thus, all these Nu analytical solutions for the LTNE model are consistent with the Bi=1 solution for the LTE model.

4. Validation and comparisons Bi=0.1 10 4.1. Nussuelt number results 0.01 0.1 ξ The local Nusselt number for Model A is defined as Fig. 2. Local Nusselt number distributions for Model A for: (a) j = 1, (b) j = 0.1. qwð4HÞ 4 NuðnÞ¼ R ¼ ð50Þ k T þk T H jh j n f ;eff f s;eff s 1 a g¼0 kf ;eff k þk H 0 Tf dy f ;eff s;eff y¼0 porous media [15–17]. As shown in Fig. 4(a), for lower values of As expected and seen in Fig. 2, Nu decreases along with the axial the Biot number such as Bi = 0.1, hs is much higher than hf at every coordinate at first and then approaches a constant value. The tran- point in the channel for Model B. As a result, hs is not influenced by the thermal boundary effects of the fluid phase. Thus, the local sition point is n = nc where the flow can be recognized as thermally fully developed. Nusselt number does not show the thermally developing effects Fig. 2 shows the local Nusselt number comparisons between the for such a case. Fig. 4(b) shows another case with Bi = 10 for Model approximate solution and the numerical solution for Model A. The B where hs is of the same order of magnitude as hf. Thus, the ther- approximate solution diverges slightly from the numerical solution mally developing effects are significant. However, for Model A even for lower values of j and Bi. The approximate solution agrees very for low values of Bi, the imposed condition of hf = hs ensures that hs well with the numerical solution in the range of j and Bi given by and hf have the same order of magnitude near the heated wall, j > 0.01 and Bi > 0.1. where the thermal boundary effects can be important, as shown For Models B and C, the local Nusselt number is defined as in Fig. 4(c). Therefore, the local Nusselt number will always display the thermal developing effects for Model A. 4ðb þ b Þ NuðnÞ¼ f s ð51Þ jhajg¼0 nðbf þ bsÞ 4.2. Predictions of the thermal entry length Fig. 3 displays the Nusselt number distributions for Models B and C for e = 0.8 based on the obtained exact solution and the numerical The classical definition for thermal entry length is based on the solution. As it can be seen there is perfect agreement between the local Nusselt number. The dimensionless thermal entry length nc is two solutions. In contrast to Model A, the Nusselt number distribu- defined as tion curves for Models B and C become substantially flatter as Bi de- NuðnÞNued creases. For example, for Model B with j = 0.1 and Bi = 0.1, the ¼ 1%; at n ¼ nc ð52Þ Nued Nusselt number at the inlet is not much different from that in the fully developed region, as seen in Fig. 3. This demonstrates that where Nued is the Nusselt number for the fully developed section. the thermally developing effects have a negligible influence on That is the local Nussult number for Models B and C for lower values of Nued ¼ NuðnÞ; at n !1 ð53Þ the Biot number.

The primary reason for this effect is that Models B and C utilize Combining the definitions for nc and Nu, i. e., Eqs. (52), (50) and (51), a constant heat flux on the solid phase boundary. It should be and utilizing the presented analytical solutions, an explicit expres- noted that for Models B and C the heat received by the solid phase sion for nc can be derived for Models A, B and C, as follows: has to transfer to the fluid phase by the internal convection in For Model A: 774 X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775

(a) (a) Model B Model C 12 100 Exact solution Numerical solution 10 Solid phase 8 Bi=10 ξ =0 κ ξ

)/ =0.01

f 6 β ξ

+ =0.1 s β 4 ( Nu 10 ξ

Bi=1 − θ 2 Fluid phase 0 Bi=0.1 -2 1 0.0 0.2 0.4 0.6 0.8 1.0 0.01 0.1 1 η ξ (b) (b) 1.4 For Model B For Model C 1.2 ξ =0 Exact solution ξ 1.0 =0.01 Numerical solution ξ =0.1 0.8 Bi=10 κ )/ f 0.6 β

100 + s β 0.4 Solid phase ( ξ

− 0.2 Nu θ Bi=1 0.0 -0.2 Fluid phase Bi=0.1 -0.4 10 0.0 0.2 0.4 0.6 0.8 1.0 η 0.01 0.1 ξ (c) 3.5 Fig. 3. Local Nusselt number distributions for Model B and Model C with e = 0.8 for: 3.0 ξ =0 (a) j = 1, (b) j = 0.1. ξ =0.01 2.5 Solid phase ξ =0.1 2.0 P 1 2 ðD1 þ D3Þ expðxA0n ÞþðD2 þ D4Þ exp ðÞþxA1n expðrnn Þ 1.5 1 c c n¼2 jrn c κ ¼ 2 2 / Nued 1 j0;0 j1;0 ξ 1 þ Nuðn ÞNu c ed 3ð1þjÞ þ jc þ jc 1.0 0 1 − 6 θ f expðx n Þ p2 A A0 c 0.5 ð54Þ 0.0 And for Models B and C: P -0.5 Fluid phase 1 1 n¼1dh;n expðxBorC; nncÞ -1.0 Nu ¼ b þb 1 þ ed f s þ bs 0.0 0.2 0.4 0.6 0.8 1.0 Nuðnc ÞNued 3ð1þjÞ ð1þjÞBi η 6 n 2 fBorCexpðxBorC; 1 cÞð55Þ p Fig. 4. Dimensionless temperature distributions at different cross sections of the while for the LTE model: channel for: (a) Model B for j = 0.1 and Bi = 0.1, (b) Model B for j = 0.1 and Bi = 10, P (c) Model A for j = 0.1 and Bi = 0.1. 1 2 expðrnn Þ 1 n¼1 jrn c 6 ¼ fLTE expðr1ncÞð56Þ 8 Nued 1 2 1 þ 3ð1þjÞ p > v ¼ r1=xA0 Nuðnc ÞNued < A v ¼ ¼ Bi þ j =ð1 þ jÞ ð58Þ It is found that only the largest term dominates the series in Eqs. i > vBorC Biþp2 :> (54)–(56). As such the approximate form of nc can be found from vLTE ¼ 1 Eqs. (54)–(56) as follows: 8 , > 2 2 j lnð61:4f Þ > 2 j j n i 57 > p 1 0;0 1;0 c ¼ 2 vi ð Þ > fA ¼ 6 ðD1 þ D3Þ 31ðÞþj þ jc þ jc p ð1 þ jÞ > 0 1 < , hihi fi ¼ ð59Þ where vi and fi are the correction functions for the LTNE model pre- > 2 2b =ðb þb Þ b =ðb þb Þ > f p 2 s s f 1 s s f sented below, and the subscript i denotes different models that > BorC¼ 6 p2 þ p2þBi 3 þ Bi > were considered before, that is i = A, B, C,orLTE: :> fLTE ¼ 1 X.-L. Ouyang et al. / International Journal of Heat and Mass Transfer 67 (2013) 768–775 775

(a) for different fundamental models. A highly accurate approximate Prediction by Eq. (57) solution for Model A, and exact solution for Models B and C are 100 Numerical results for Model A established. These analytical solutions are in excellent agreement with the numerical results. For Models B and C, the thermal devel- κ oping effects are found to be negligible for lower values of the Biot -1 =1 10 numbers.

c The dimensionless thermal entry length nc is established based ξ κ =0.1 on the analytical solutions and it is found to be a function of the 10-2 Biot Number, Bi, the ratio of ﬂuid to solid conductivity, j as well as a function of models A, B and C. For Model A, nc increases as j κ =0.01 increases or Bi decreases. But for Models B and C, nc is non-mono- -3 tonic with Bi. 10 10-1 100 101 102 The present work paves the way for understanding and designing the thermally developing ﬂows in porous media while Bi incorporating the LTNE condition.

(b) Acknowledgements 0 Prediction by Eq. (57) 10 Numerical results for Model B This project was supported by the National Natural Science Foundation of China (Grant No. 51276094) and the Key Project κ =1 10-1 Fund of the National Natural Science Foundation of China (No. 50736003). We would also like to acknowledge the financial c ξ κ =0.1 support from China Scholarship Council. 10-2 References κ =0.01 [1] A. Bejan, Convection Heat Transfer, Wiley, New York, 2004. 10-3 [2] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in 10-1 100 101 102 porous media, Int. J. Heat Mass Transfer 24 (2) (1981) 195–203. [3] A. Haji-Sheikh, K. Vafai, Analysis of flow and heat transfer in porous media Bi imbedded inside various-shaped ducts, Int. J. Heat Mass Transfer 47 (8) (2004) 1889–1905. [4] A. Haji-Sheikh, D.A. Nield, K. Hooman, Heat transfer in the thermal entrance (c) Prediction by Eq. (57) region for flow through rectangular porous passages, Int. J. Heat Mass Transfer 49 (17) (2006) 3004–3015. 0 Numerical results for Model C 10 [5] A. Amiri, K. Vafai, Analysis of dispersion effects and non-thermal equilibrium, ε for =0.8 non-Darcian, variable porosity incompressible flow through porous media, Int. J. Heat Mass Transfer 37 (6) (1994) 939–954. κ [6] P.X. Jiang, Z.P. Ren, B.X. Wang, Numerical simulation of forced convection heat -1 =1 10 transfer in porous plate channels using thermal equilibrium and nonthermal equilibrium models, Numer. Heat Transfer: Part A: Appl. 35 (1) (1999) 99–113. c ξ κ =0.1 [7] P.X. Jiang, M. Li, Y.C. Ma, Z.P. Ren, Boundary conditions and wall effect for forced convection heat transfer in sintered porous plate channels, Int. J. Heat 10-2 Mass Transfer 47 (10) (2004) 2073–2083. [8] P.X. Jiang, Z. Wang, Z.P. Ren, B.X. Wang, Experimental research of fluid flow and convection heat transfer in plate channels filled with glass or metallic κ =0.01 particles, Exp. Thermal Fluid Sci. 20 (1) (1999) 45–54. 10-3 [9] P.X. Jiang, M. Li, T.J. Lu, L. Yu, Z.P. Ren, Experimental research on convection 10-1 100 101 102 heat transfer in sintered porous plate channels, Int. J. Heat Mass Transfer 47 (10) (2004) 2085–2096. Bi [10] D.A. Nield, A.V. Kuznetsov, Forced convection in porous media: transverse heterogeneity effects and thermal development, in: Handbook of Porous Media, second ed., Taylor and Francis, New York, 2005, pp. 143–193. Fig. 5. Dimensionless thermal entry length and its comparison with the numerical [11] K. Yang, K. Vafai, Transient aspects of heat flux bifurcation in porous media: an results for: (a) Model A, (b) Model B, (c) Model C for e = 0.8. exact solution, J. Heat Transfer 133 (5) (2011) 052602. [12] B. Alazmi, K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 45 (15) (2002) 3071–3087. As shown in Eqs. (57)–(59), nc is a function of Bi, j and the boundary [13] X.L. Ouyang, P.X. Jiang, R.N. Xu, Thermal boundary conditions of local thermal condition models. It should be noted that if Bi ? 1, then v ? 1 and i non-equilibrium model for convection heat transfer in porous media, Int. J. fi ? 1, because Bi ? 1 represents the LTE condition. Heat Mass Transfer 60 (2013) 31–40. The comparisons between the analytical expressions for nc gi- [14] D.Y. Lee, K. Vafai, Analytical characterization and conceptual assessment of ven by Eq. (57) and the numerical results are shown in Fig. 5.As solid and fluid temperature differentials in porous media, Int. J. Heat Mass Transfer 42 (3) (1999) 423–435. can be seen the analytical and numerical results are in excellent [15] K. Yang, K. Vafai, Analysis of temperature gradient bifurcation in porous media agreement. It can be seen in Fig. 5(a), for Model A that nc increases – An exact solution, Int. J. Heat Mass Transfer 53 (19) (2010) 4316–4325. [16] K. Yang, K. Vafai, Restrictions on the validity of the thermal conditions at the as j increases or Bi decreases. However, for Models B and C, nc is porous-fluid interface – An exact solution, J. Heat Transfer 133 (2011) 112601. non-monotonic with Bi, as seen in Fig. 5(b) and (c). The reason [17] K. Yang, K. Vafai, Analysis of heat flux bifurcation inside porous media for this behavior has been discussed earlier in Section 4.1. incorporating inertial and dispersion effects – An exact solution, Int. J. Heat Mass Transfer 54 (2011) 5286–5297. [18] A. Amiri, K. Vafai, T.M. Kuzay, Effects of boundary-conditions on non-Darcian 5. Conclusions heat transfer through porous media and experimental comparisons, Numer. Heat Transfer A Appl. 27 (6) (1995) 651–664. In the present work, analytical solutions for thermally developing flow in porous media incorporating LTNE condition are derived