International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 www.elsevier.com/locate/ijhmt Analysis of ¯uid ¯ow and heat transfer interfacial conditions between a porous medium and a ¯uid layer B. Alazmi, K. Vafai *
Department of Mechanical Engineering, The Ohio State University, 206 West 18th Avenue, Columbus, OH 43210-1107, USA Received 21 April 2000; received in revised form 15 June 2000
Abstract Dierent types of interfacial conditions between a porous medium and a ¯uid layer are analyzed in detail. Five primary categories of interface conditions were found in the literature for the ¯uid ¯ow at the interface region. Likewise, four primary categories of interface conditions were found in the literature for the heat transfer at the interface region. These interface conditions can be classi®ed into two main categories, slip and no-slip interface conditions. The eects of the pertinent parameters such as Darcy number, inertia parameter, Reynolds number, porosity and slip coecients, on dierent types of interface conditions are analyzed while ¯uid ¯ow and heat transfer in the neighborhood of an interface region are properly characterized. A systematic analysis of the variances among dierent boundary conditions estab- lishes the convergence or divergence among competing models. It is shown that in general, the variances have a more pronounced eect on the velocity ®eld and a substantially smaller eect on the temperature ®eld and even a smaller eect on the Nusselt number distributions. For heat transfer interface conditions, all four categories generate results, which are quite close to each other for most practical applications. However, small discrepancies could appear for applications dealing with large values of Reynolds number and/or large values of Darcy number. Finally, a set of correlations is given for interchanging the interface velocity and temperature as well as the average Nusselt number among various models. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Nader [2] presented one of the earlier attempts regard- ing this type of boundary condition in porous medium. Fluid ¯ow and heat transfer characteristics at the In this study, the authors proposed a continuity in both interface region in systems which consist of a ¯uid- the velocity and the velocity gradient at the interface by saturated porous medium and an adjacent horizontal introducing the Brinkman term in the momentum ¯uid layer have received considerable attention. This equation for the porous side. Vafai and Kim [3] pre- attention stems from the wide range of engineering sented an exact solution for the ¯uid ¯ow at the inter- applications such as electronic cooling, transpiration face between a porous medium and a ¯uid layer cooling, drying processes, thermal insulation, porous including the inertia and boundary eects. In this study, bearing, solar collectors, heat pipes, nuclear reactors, the shear stress in the ¯uid and the porous medium crude oil extraction and geothermal engineering. The were taken to be equal at the interface region. Vafai and work of Beavers and Joseph [1] was one of the ®rst Thiyagaraja [4] analytically studied the ¯uid ¯ow and attempts to study the ¯uid ¯ow boundary conditions at heat transfer for three types of interfaces, namely, the the interface region. They performed experiments and interface between two dierent porous media, the in- detected a slip in the velocity at the interface. Neale and terface separating a porous medium from a ¯uid region and the interface between a porous medium and an impermeable medium. Continuity of shear stress and * Corresponding author. Tel.: +1-614-292-6560; fax: +1-614- heat ¯ux were taken into account in their study while 292-3163. employing the Forchheimer-Extended Darcy equation E-mail address: [email protected] (K. Vafai). in their analysis. Other studies consider the same set of
0017-9310/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 7 - 9 3 1 0 ( 0 0 ) 0 0 217-9 1736 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749
Nomenclature Greek symbols a thermal diusivity (m2 s 1) 1 1 Ã cp speci®c heat at constant pressure (J kg K ) a velocity slip coecient in Table 1 2 Da Darcy number, K/H aT temperature slip coecient in Table 2 F geometric function de®ned in Eq. (4) e porosity h heat transfer coecient (W m 2 K 1) / parameter in Table 2 H height of the ¯uid layer (m) K inertia parameter, eFH/K1=2 k thermal conductivity (W m 1 K 1) l kinematics viscosity (kg m 1 s 1) K permeability (m2) m dynamic viscosity (m2 s 1) Nu local Nusselt number q density (kg m 3) Nu average Nusselt number H dimensionless temperature, 2 P pressure (N/m ) Tw T = Tw T1 Re Reynolds number, u1H/mf T temperature (K) Subscripts
Tm mean temperature (K) e eective property u velocity in x-direction (m s 1) f ¯uid 1 uint interface velocity (m s ) int interfacial U non-dimensional velocity, u/u1 w wall Uint non-dimensional interface velocity 1 free stream x, y Cartesian coordinates (m) ) plain medium side X, Y non-dimensional coordinates, x/H and y/H + porous medium side
boundary conditions for the ¯uid ¯ow and heat transfer have also presented another hybrid interface condition used in [4] such as [5±8]. for the heat transfer part in which they introduce a jump Ochoa-Tapia and Whitaker [9,10] have proposed a condition to account for a possible excess in the heat ¯ux hybrid interface condition (a hybrid between models 2 at the interface. Sahraoui and Kaviany [18] have pro- and 5 discussed later on) in which a jump in the shear posed yet another hybrid interface condition for the heat stress at the interface region is assumed. In their study, transfer part. They used the continuity of the heat ¯ux at the shear stress jump is inversely proportional to the the interface along with a slip in the temperature at the permeability of the porous medium. This proposed set of interface. The main focus of the present study is to interface conditions was used in [11±15]. More recently, critically examine the dierences in the ¯uid ¯ow and Ochoa-Tapia and Whitaker [16] have presented another heat transfer characteristics due to dierent interface shear stress jump boundary condition where the inertia conditions, including all the aforementioned models. eects become important. The same investigators [17] The current study complements a prior work by Alazmi
Table 1 Primary categories of ¯uid ¯ow interface conditions between a porous medium and a ¯uid layer Model Velocity Velocity gradient Refs.
1 u u du du [2,3,5,23] dy dy
du du 2 u u [4,6,7] leff l dy dy
l du du l 3 u u [9±15] l b1 p u e dy dy K
l du du l 4 u u 2 [16] l b1 p u b2qu e dy dy K
a à 5 du a [1,21] p uint u1 dy K a A Forchheimer term is added to the momentum equation in the porous side for the purpose of comparison. B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 1737
Table 2 Primary categories of heat transfer interface conditions between a porous medium and a ¯uid layer Model Temperature Temperature gradient Refs. oT oT I T T k k [2,4±8,13±15,23] eff oy f oy
oT oT II T T [17] / kf keff oy oy
III dT aT oT oT [18], using ¯uid ¯ow of model 1 T T k k eff oy f oy dy k
IV dT a T T T oT oT [18], using ¯uid ¯ow of model 3 keff kf dy k oy oy and Vafai [19] in which they presented a comprehensive lr2hV irhPi; 4 analysis of variants within the transport models in po- rous media. In their study, four major categories 2 hV iÁrhT iaf r hT i; 5 namely, constant porosity, variable porosity, thermal dispersion and local thermal non-equilibrium were where considered in great detail. kf af : 6 qf cpf 2. Analysis The pertinent dimensionless parameters for this problem are [3]: A comprehensive synthesis of literature revealed ®ve primary categories for interface conditions for the ¯uid K ¯ow and four primary forms of interface conditions for Da ; 7a the heat transfer between a porous medium and a ¯uid H 2 layer. Table 1 summarizes the models for the ¯uid ¯ow part while Table 2 summarizes the models for the heat transfer part. The fundamental con®guration used by Vafai and Kim [3], representing the interface region between a porous medium and a ¯uid layer, is used in the current study. This con®guration consists of a ¯uid layer sandwiched between a porous medium from above and a solid boundary from below. The physical con®g- uration and the coordinate system are shown in Fig. 1 (a). For the porous region, the governing equations are [20]: q l q F e f h V ÁrV i hiV pf hV iÁhV i J e K K l r2hV i rhPi; 1 e
a hV iÁrhT i eff r2hT i; 2 e where
keff aeff ; 3a qf cpf keff ekf 1 eks: 3b Fig. 1. (a) Schematic of the physical model and the coordinate The governing equations for the ¯uid region can be system. (b) Comparison between the exact solution of Vafai and written as [3,4]: Kim and the present numerical results. 1738 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749
Fig. 2. Eect of changing the eective viscosity on: (a) velocity ®eld; (b) temperature ®eld and Nusselt number distribution for the interface between a porous medium and a ¯uid layer, e 0:7; K 1:0; Da 10 3; Re 1:0.
u H Re 1 ; 7b an adjacent porous layer can be found in [4]. Therefore, mf the current study concentrates on analyzing and syn- F eH thesizing the eects of dierent interface conditions. The K p : 7c presentation of the results in the current study is given in K terms of the velocity ®elds for models given in Table 1 Primary categories of interface conditions utilized in the and in terms of temperature and Nusselt number dis- literature for the ¯uid ¯ow are given in Table 1, while tributions for models given in Table 2. In presenting the those for the heat transfer are given in Table 2. results it is useful to introduce the following dimen- sionless variables [3]: x y 3. Results and discussion X ; Y ; L H u T T A comprehensive analysis of ¯uid ¯ow and heat U ; H w 8 transfer for the interface region between a ¯uid layer and u1 T T1 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 1739
The local Nusselt number for the lower wall is de®ned Fig. 1(b). Considering the ®rst two models in Table 1, it as: can be seen that model 1 becomes identical to model 2 hH when the eective viscosity of the porous medium equals Nu ; 9 the viscosity of the ¯uid. For model 2, when there is a k f signi®cant dierence between the viscosity of the ¯uid where hx is the local heat transfer coecient at the wall, and the eective viscosity of the porous medium, the which is de®ned as slope of the velocity pro®le in the porous medium is not the same as the slope on the other side of the interface. H oT h : 10 On the other hand, the velocity gradients for model 1 are T T oy w m y0 equal on both sides. It should be noted that in the cur- rent study, based on the analysis presented in [20], the 3.1. Fluid ¯ow eective viscosity of the porous medium for model 2 is taken to be lf /e. Therefore, for a higher porosity medi- Comparison between the present numerical results um, the value of the eective viscosity is close to that of and the exact solution of Vafai and Kim [3] is shown in the ¯uidÕs. Models 3 and 4 present a jump in the shear
Fig. 3. (a) Velocity ®eld; (b) temperature ®eld and Nusselt number distribution for the interface between a porous medium and a ¯uid 3 layer, e 0:7; K 1:0; Da 10 ; Re 1:0; b1 1:0; b2 1:0; aT 10:0; / 10. 1740 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 stress while model 5 introduces a slip in the velocity at the porous medium. The velocity slip coecient aà for the interface region. model 5 is determined numerically by matching the When the inertia eects are negligible, model 4 be- velocity gradient of model 5 for each individual case comes identical to model 3. Therefore, model 3 is a according to the physical parameters for this case [21]. special case of model 4. Neale and Nader [2] proposed a According to the experimental study of Gilver and Al- mathematical representation for the slip coecient for tobelli [22], the ratio of the eective viscosity, l , to the p eff à model 5, they predicted that a leff =l. Model 5 ¯uid viscosity, lf , is in the range of 5:1 < leff =lf < 10:9: indicates that the velocity gradient in the ¯uid side is A mean value of 7.5 was recommended by this study. proportional to the dierence between the interfacial Physically this range of values appears to be quite high. velocity and the free stream velocity in the porous me- Nevertheless, a side study for the eect of choosing dium while inversely proportional to the permeability of the eective viscosity is performed in this study and
Fig. 4. Eect of porosity variation on the velocity ®eld for the interface between a porous medium and a ¯uid layer, 3 K 1:0; Da 10 ; Re 1:0; b1 1:0; b2 1:0. (a) e 0:5, (b) e 0:9. B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 1741
Fig. 5. Eect of inertia parameter variation on the velocity ®eld for the interface between a porous medium and a ¯uid layer, 3 e 0:7; Da 10 ; Re 1:0; b1 1:0; b2 1:0. (a) K 0:1, (b) K 10:0. presented in Fig. 2. Fig. 2(a) shows that changing the The comparative representation of velocity, tem- eective viscosity from lf to 7.5lf has a relatively minor perature and Nusselt number given in Fig. 3 will be used eect on the velocity pro®les considering such a wide as a baseline for studying the eects of dierent pertinent range of variations for the eective viscosity. The parameters. Comparisons between the ®ve primary ¯uid smaller velocity peak and interfacial velocity are asso- interface conditions are given in Figs. 4±9. Fig. 3(a) is ciated with the highest eective viscosity. The eect of used as a baseline in which moderate values of the per- changing the eective viscosity on the temperature and tinent parameters are used. Fig. 4 shows the eect of Nusselt number distributions is shown in Fig. 2(b). It is porosity variation on the velocity ®eld. It is clear that clear that changing the eective viscosity even within models 1 and 5 are unaected by changing the porosity, such a wide range has an insigni®cant eect on the this is expected as the boundary conditions in these thermal ®eld. models do not include a porosity term. Model 2 is the 1742 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749
Fig. 6. Eect of Darcy number variation on the velocity ®eld for the interface between a porous medium and a ¯uid layer, 4 3 e 0:7; K 1:0; Re 1:0; b1 1:0; b2 1:0. (a) Da 10 , (b) Da 10 . most aected by changing the porosity. An increase in terfacial velocity for the ®rst four models and a slightly the porosity causes model 2 to approach models 1 and 5 larger interfacial velocity for the ®fth model. The slight and a decrease in the porosity drives it in the opposite increase in the interfacial velocity for model 5 may be direction, i.e., toward models 3 and 4. In other words, ascribed to the fact that this is the only model that does not higher porosity creates a higher peak velocity and account for the boundary eect in the momentum equa- smaller porosity creates a smaller peak velocity for tion within the porous medium. The eect of Darcy model 2 in the plain medium. number variation on the velocity ®eld is shown in Fig. 6. It The eect of variations in the inertia parameter is can be seen that for smaller Darcy numbers, the velocity shown in Fig. 5. It is clear that an increase in the inertia pro®les of all the ®ve primary models are very close to each parameter results in a slightly larger peak velocity for other, while large Darcy numbers induces a discrepancy model 5 as compared to model 1. At the same time, an among the models. This is because higher Darcy numbers increase in the inertia parameter results in a smaller in- translate into higher permeabilities in the porous side of B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749 1743
Fig. 7. Eect of Reynolds number variation on the velocity Fig. 8. Eect of b variation on the velocity ®eld for the in- ®eld for the interface between a porous medium and a ¯uid 1 terface between a porous medium and a ¯uid layer, layer, e 0:7; K 10:0; Da 10 3; b 1:0; b 1:0. (a) 1 2 e 0:7; K 1:0; Da 10 3; Re 1:0; b 1:0. (a) b 0:1, Re 10:0, (b) Re 100:0. 2 1 (b) b1 10:0. the interface, which in turn results in a smaller ratio be- Large values of b1 may cause a sharp change in the tween the average velocity in the plain medium and the velocity gradient on both sides of the interface as shown Darcian velocity. As a result of this smaller velocity ratio, in Fig. 8(b). The eect of variations in parameter b2 on the discrepancy between these models becomes more the velocity ®eld is shown in Fig. 9. Smaller values of b2 pronounced. However, it should be noted that for most causes the velocity pro®le for model 4 to collapse on that practical applications Da >10 4 and as such the variation for model 3 while larger values of b2 increases the di- among the ®ve models becomes less signi®cant. vergence between these two models and creates a The eect of variations in Reynolds number is shown sharper slope for the velocity pro®le for model 4. in Fig. 7. For larger Reynolds numbers the deviation In general, models 1 and 5 are closer to each other between the velocity pro®les for models 3 and 4 in- while models 3 and 4 are closer to each other. Model 2 creases due to the presence of u2 term in the interface usually falls in between the results for models 1, 5 and condition for model 4. Reynolds number has a similar models 3, 4. Interfacial velocities for models 2±5 in terms eect as the inertia parameter had on models 1 and 5. A of the interfacial velocity for model 1 and pertinent similar result was found in [3]. Fig. 8 shows the eect of parameters such as the Darcy number, the Reynolds number, the inertia parameter and the porosity are given variations in the coecient b1. It indicates that for smaller values of b the velocity pro®les for models 2 below: 1 h and 3 collapse on each other. On the other hand, for 0:8762 Uint2 0:057 1:0194e 0:00082544K larger b1, velocity pro®les for models 3 and 4 approach each other. This is because the linear term in the inter- 0:0363 1000Da0:3893 face condition for model 4 becomes more dominant for i 0:0010228Re Uint1; 11 larger values of b1. 1744 B. Alazmi, K. Vafai / International Journal of Heat and Mass Transfer 44 (2001) 1735±1749
lf . In this ®gure, the temperature and Nusselt number distributions for the entire wide range of variations in
leff /lf are quite close to each other. The ®rst type of heat transfer interface condition, model I is based on a con- tinuity of temperature as well as heat ¯ux at the interface region. The second model includes a jump in the heat ¯ux at the interface region. As a logical extension the ¯uid ¯ow model 1 is in conjunction with the heat transfer models I and II. The third (III) and the fourth (IV) heat transfer interface condition models have a jump in the temperature at the interface. However, the heat transfer interface model III utilizes the ¯uid ¯ow model 1 while thermal model IV utilizes the ¯uid ¯ow model 3. The eects of the pertinent parameters on the tem- perature and Nusselt number distributions are shown in Figs. 10±13. Fig. 3(b) is used as the datum for tem- perature and Nusselt number comparisons. Fig. 10(a) shows the eect of porosity. Higher porosity results in a better agreement between models I and II while it de- creases the discrepancies between the interfacial tem- peratures for models III and IV. In other words, higher porosity causes the results for ¯uid ¯ow models 1 and 3 to be closer to each other, which in return coalesces the results for models III and IV. The eects of inertia parameter variation are shown in Fig. 10(b). It has a relatively insigni®cant eect on the convergence or divergence of the temperature and Nusselt number pro®les for the four models shown in Table 2. However, a careful examination indicates an Fig. 9. Eect of b2 variation on the velocity ®eld for the in- terface between a porous medium and a ¯uid layer, increase in the inertia parameter produces convergence 3 in the interfacial temperature for all the four models. e 0:7; K 1:0; Da 10 ; Re 1:0; b1 1:0. (a) b2 0:1, (b) b2 10:0. This is due to the higher values of the inertia parameter,  which results in larger convective heat transfer, which in 11:7265 Uint3 0:7077e 0:0202K 54:9355Da turn decreases the eect of dierent interface conditions à on the heat transfer results. 0:0082205Re 0:053b Uint1; 12 1 The eects of Darcy number variation, Da, are shown in Fig. 11(a). It is clear that for more practical U 14:9258 0:4185e 0:0012816K int4 values of Darcy number, all the four models displayed in 15:1505 1000Da0:000643 0:0117Re Table 2 converge. For very large Darcy numbers, the discrepancy in the interfacial temperatures for the four 0:0848b 0:0050702b2 1 1 models becomes more pronounced. Model IV has the