Heat and Mass Transfer 33 (1998) 377±382 Ó Springer-Verlag 1998

Non-Darcy mixed convection from a vertical plate in saturated porous media-variable surface heat flux Z. H. Kodah, A. M. Al-Gasem

Abstract Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous medium are presented for variable surface heat ¯ux (VHF) of the power-law form. The entire mixed convection region is divided into two regimes. One region covers the forced convection dominated regime and the other one covers the natural convection dominated regime. The governing equations are ®rst transformed into a dimensionless form by the nonsimilar transformation and then solved by a ®nite-difference scheme. Computations are based on Keller Box method and a tolerance of iteration of 10ÿ5 as a criterion for convergence. Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Rax =Pe3=2 characterizes the effect of buoyancy x forces on the forced convection; while the parameter Pex =Ra2=3 characterizes the effect of forced ¯ow on the x natural convection. The second aspect represents the effect of the inertial resistance where the parameter K 0 U1 =m is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter …K 0 U1 =m†…Ra2=3 x =Pex † characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition. Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity pro®les for both Darcy and non-Darcy models are presented. List of symbols F; F1 G hx h

Dimensionless stream functions Parameter of the inertial effect on natural con2=3 vection dominated regime …K 0 U1 =m†R ax =Pex Local heat transfer coef®cient Average R L heat transfer coef®cient, …1=L† 0 h…x† dx

Received on 26 May 1997

Z. H. Kodah A. M. Al-Gasem Mechanical Engineering Department Jordan University of Science and Technology P.O. Box 3030 Irbid ± Jordan Correspondence to: Z. H. Kodah

k K K0 L Nux Nu Pex qw R Rax Re T T1 Tw u; v U1 x; y

Thermal conductivity Permeability coef®cient of the porous medium Coef®cient of Forchheimer's equation Length of the plate Local Nusselt number, hx=k Average Nusselt number, hL=k Local Peclet number, U1 x=a Local surface heat ¯ux Parameter of the inertial effect on forced convection dominated regime, K 0 U1 =m Modi®ed local Rayleigh number, gbqw …x†Kx2 =…kma† Reynolds number, U1 x=v Temperature Free stream temperature Wall temperature Velocity components in x- and y-directions Free stream velocity Axial and normal coordinates

Greek symbols a Effective thermal diffusivity of saturated porous medium b Volumetric coef®cient of thermal expansion d Boundary layer thickness H; H1 Dimensionless temperatures l Dynamic viscosity m Kinematic viscosity nf Nonsimilarity parameter for the forced convection dominated regime nn Nonsimilarity parameter for the free convection dominated regime e Porosity q Fluid density sw Local wall shear stress w Stream function Subscripts f Forced convection dominated condition n Natural convection dominated condition 1 Free stream condition w Wall condition

1 Introduction In most of the previous studies of mixed convection, the boundary-layer formulation of Darcy's law and energy equation were used. However, the inertial effect is

377

378

expected to become very signi®cant when the pore Reynolds number is large. This is especially true for the case of either the high Rayleigh number or for high-porosity media. In spite of its importance in many application, the non-Darcy ¯ow has not received much attention. Vafai and Tien [1] investigated the inertia and boundary effects in forced convection over a horizontal, heated plate based on volume-average technique. Vafai [2] studied the inertia effect in a similar problem but also considered the variable-porosity effect. For natural convection in saturated, porous media, Plumb and Huenefeld [3] studied the inertia effect along an isothermal, vertical plate based on the Ergun model using the boundary-layer approximations and obtained the similarity solutions. Aldoss et al. [4, 5] analyzed mixed convection over horizontal surface neglecting boundary and inertia terms, nonsimilar solutions were obtained. Velocity and temperature ®elds for different values of mixed convection parameters were obtained. Hsieh et al. [6] solved the Darcy model problem of mixed convection from a vertical ¯at plate in a porous medium under variable surface heat ¯ux and variable wall temperature conditions for entire regime of convection by employing two different transformation. Also, Hsieh et al. [7] solved the same problem but used a single nonsimilarity parameter which covers the entire regime of mixed convection. Lai and Kulacki used both Darcy and non-Darcy models from horizontal [8] and vertical [9] surfaces embedded in saturated porous media. Chen and Ho [10] studied the effects of ¯ow inertia on vertical, natural convection in saturated, porous media. Duwairi [11] studied non-Darcian mixed convection along a horizontal wall in a saturated porous-medium. Kodah and Duwairi studied the inertia effects on mixed convection for vertical plates with variable wall temperature in saturated porous media [12]. The purpose of this study is to examine the inertia effect on vertical plate subjected to a power-law variation in surface heat ¯ux and embedded in a saturated porous medium. This study is speci®cally for a class of nonsimilar problems in terms of the inertia parameter.

2 Analysis Consider the problem of steady, two-dimensional convection in a porous medium adjacent to a vertical semiin®nite plate. Under the assumptions that constant ¯uid and medium (isotopic) properties and local thermodynamic equilibrium between ¯uid and solid phases, the governing equations are given by: ou ov ‡ ˆ0 ox oy

  K 0 2 ÿK op ‡ qg u‡ u ˆ m l ox   K 0 2 ÿK oP v‡ v ˆ m l oy  2  oT oT o T o2 T ‡v ˆa ‡ u ox oy ox2 oy2

…1† …2† …3† …4†

The Cartesian coordinates x and y are measured along and perpendicular directions of the free stream velocity, respectively. The gravitational acceleration g is acting downward in direction opposite to the x-direction. The continuity equation is automatically satis®ed by introducing the stream function w as

uˆ

ow oy …5†

and

ow ox When the Boussinesq and boundary-layer approximation are invoked, and eliminating P from Eqs. (2) and (3) by cross differentiation, the resulting governing equations in term of stream function, w, are:  2 ! o2 w K 0 o ow K oT …6† ‡ ˆ gb 2 m oy oy oy m oy        2  ow oT ow oT oT ÿ ˆa …7† oy ox ox oy oy2 vˆÿ

With the power law variation in the surface heat ¯ux, the boundary conditions can be written as:

y ˆ 0; u ˆ 0; v ˆ 0; qw …x† ˆ bxm ou ˆ0 …8† at y ˆ 1; u ˆ U1 ; T ˆ T1 ; oy where b and m are prescribed constants. Note that m ˆ 0 corresponds to the case of constant heat ¯ux. at

2.1 Forced convection The suitable similarity variables for solving the non-Darcy forced convection problem are y Y ˆ Pe1=2 …9† x x …10† nf ˆ nf …x† w ˆ aPe1=2 x F…nf ; Y† H…nf ; Y† ˆ

…T ÿ T1 †Pe1=2 x qw …x†x=k

…11† …12†

After transformation, the governing equations and boundary conditions are

  K 0 0 00 F F ˆ nf H0 F ‡ 2U1 m 00

…13†

where nf ˆ Rax =Pe3=2 x

R ˆ U1

K0 m

  1 1 0 0 H ‡ FH ÿ m ‡ F H 2 2 #   " 1 0 oF 0 oH ÿH ˆ m ‡ nf F 2 onf onf 00

…14†

F…nf ; 0† ˆ 0:0

nn ˆ

F 0 …nf ; 1† ˆ 1:0 H…nf ; 1† ˆ 0:0

U1 K 0 mnn     m‡2 2m ‡ 1 00 0 F1 H1 ÿ H1 F10 H1 ‡ 3 3     2m ‡ 1 oF1 oH1 ˆ ÿ F10 nn H01 onn onn 3 G ˆ

H0 …nf ; 0† ˆ ÿ1:0

…15† Rax

2

where Pex ˆ U1 x=a and ˆ gbqw …x†Kx =…kma†, and the primes denote partial differentiations with respect to Y. In the above system of equations, the parameter nf represents the buoyancy force effect on forced convection. The case of nf ˆ 0 corresponds to pure forced convection and the limiting case of nf ˆ 1 corresponds to pure natural convection. The above system of equations is solved for the region covered by nf ˆ 0 to nf ˆ 1 to provide the ®rst half of the total solution of the mixed convection regime. In terms of the new variables, it can be shown that the velocity components are given by

u ˆ U1 F 0 …nf ; Y† " #   a 1 1 1 oF Pe1=2 vˆÿ F ÿ YF 0 ‡ m ‡ nf x x 2 2 2 onf

…16† …17†

the wall shear stress sw de®ned as sw ˆ l…ou=oy†yˆ0 , and the local Nusselt number Nux ˆ hx=k, where h ˆ qw =…Tw …x† ÿ T1 † are given as

  lU1 00 Pe1=2 sw ˆ x F …nf ; 0† x

…18†

Pe1=2 x Nux ˆ H…nf ; 0†

…19†

The average Nusselt number, Nu, can be expressed as

Nu ˆ

Pax Ra2=3 x

2 1=2 ÿ1=…2m‡1† Pe n 2m ‡ 1 L fL Z nf nÿ2m=…2m‡1† L f dnf  H…nf ; 0† 0

…20†

where PeL and nfL are values of Pex and nf at x ˆ L.

…26† 379

F1 …nn ; 0† ˆ 0:0 F10 …nn ; 1† ˆ nn H1 …nn ; 1† ˆ 0:0 H01 …nn ; 0† ˆ ÿ1:0

…27†

The primes denote partial differentiations with respect to Y1. Note that the nn parameter here is a measure of forced ¯ow effect on natural convection. The case of nn ˆ 0 corresponds to pure natural convection and the limiting case nn ˆ 1 corresponds to pure forced convection. The above system of equations is solved over the region covered by nn ˆ 0 to nn ˆ 1 to provide the other half of the solution for the entire mixed convection regime. The velocity components u and v, the wall shear stress, and the local Nusselt number are given, respectively, by:

a u ˆ Ra2=3 F10 …nn ; Y1 † x x  a 1 1 Ra1=3 …m ‡ 2†F1 ‡ …m ÿ 1†Y1 F10 vˆÿ x x 3 3  1 oF1 ÿ …2m ‡ 1†nn onn 3 la  00 sw ˆ 2 Rax F1 …nn ; 0† x Ra1=3 x Nux ˆ H1 …nn ; 0†

…28†

…29† …30† …31†

The average Nusselt number Nu is given by

3 1=3 2.2 Nu ˆ ÿ Ra nn…m‡2†=…2m‡1† L Natural convection 2m ‡ 1 L Z nnL ÿ…3m‡3†=…2m‡1† The suitable similarity variables for solving the non-Darcy nn dnn  …32† natural convection problem are: H1 …nn ; 0† 0 y 1=3 Y1 ˆ Rax …21† where Ra and n are values of Ra and n at x ˆ L. nL n L x x Note that the relation between the different dimen…22† nn ˆ nn …x† sionless groups in forced convection dominated regime 1=3 …23† and natural convection dominated regime can be given as: w ˆ aRax F1 …nn ; Y1 † H1 …nn ; Y1 † ˆ

…T ÿ T1 †Ra1=3 x qw …x†x=k

…24†

The governing equations and boundary conditions can then be transformed into:

  K0 00 F 0 F 00 ˆ H01 F1 ‡ 2U1 nn m 1 1 where

…25†

ÿ2=3

nn ˆ nf

G ˆ R=nn : 3 Results and discussions The range of m values for which the present problem has physically realistic values is based on the requirement that both U and d, the streamwise velocity component and the

boundary layer thickness, must increase or at least remain constant with respect to x. From Eq. (28) one ®nds that U varies like x…2m‡1†=3 . Also, from Eq. (21) the boundary layer thickness d, which

is of the order of y, varies like x…1ÿm†=3 . Thus, the above condition can be satis®ed if ÿ0:5  m  1. Based on the above argument, the numerical computations were carried out for values of m within the above range.

380

Fig. 1a±c. Forced convection. a Effects of nf on temperature Fig. 2a±c. Forced convection. a Effects of nf on velocity pro®les; gradient; b effect of variable m on temperature gradient; c inertia b effects of variable m on velocity pro®les; c inertia effects on effects on temperature gradient velocity pro®les

3.1 Forced convection dominated regime The results for H…nf ; Y†=H…nf ; 0† and F 0 …nf ; Y† are presented in Fig. 1 and Fig. 2 for different values of nf , m, R . Figure 1a shows that as the buoyancy parameter nf increases, the temperature gradient at the wall increases. This means that higher heat transfer rate is expected at higher values of nf . Also, Fig. 1b shows that as m increases the thermal boundary layer thickness decreases and that for a particular value of nf , the temperature gradient at the wall increases as m increases resulting in higher heat transfer rate at higher values of m. Figure 1c shows that increasing inertia parameter R decreases the rate of heat transfer as a result of increasing the thermal boundary layer thickness. The effect of increasing nf value on the velocity pro®les is evident from Fig. 2a where it is seen that as nf increases, the slip velocity increases as a result of buoyancy-induced favorable pressure gradient. Figure 2b shows that as m increases the momentum boundary layer thickness decreases, and for a particular value of nf , the velocity gradient decreases with increasing value of m, resulting in lower wall shear stresses. Figure 2c shows that increasing the inertia parameter decreases the velocity within the boundary layer, pushing it to the pure natural convection limit.

381

3.2 Natural convection dominated regime Results for H1 …nn ; Y†=H1 …nn ; 0† and F10 …nn ; Y† for the natural convection dominated regime are presented in Fig. 3 and Fig. 4 respectively. Figure 3a shows that increasing forced parameter nn , increases the temperature gradient at the wall which means higher heat transfer rate at higher values of nn . Figure 3b indicates that higher heat transfer rate is expected at higher values of m, because increasing m value reduces the thermal boundary layer thickness. Figure 3c illustrates the effect of inertia parameter G on the temperature pro®les and shows that increasing G increases the thermal boundary layer thickness, and consequently reduces the rate of heat transfer. The effect of increasing nn value on the velocity pro®les is evident from Fig. 4a, where as nn increases the momentum boundary layer thickness and the velocity gradient at the wall increases. Figure 4b indicates that increasing the value of the exponent m increases the slip velocity and decreases the momentum boundary layer thickness. As before Figure 4c shows that increasing the value of G decreases the velocity within the boundary layer. 4 Conclusion The problem of mixed convection along a vertical wall subjected to variable surface heat ¯ux embedded in a saturated porous medium was investigated. It is found that increasing mixing parameters increases the rate of heat transfer and increases the velocity within the boundary layer due to induced-favourable pressure gradient. Also, increasing the exponent m of the power-law

Fig. 3a±c. Free convection. a Effects of nn on temperature gradient; b effects of variable m on temperature gradient; c inertia effects on temperature gradient

form heating boundary conditions will decrease the velocity within the boundary layer, and increase the rate of heat transfer. The non-Darcian effects tend to decrease the velocity within the boundary layer, and broaden the tem-

References 1.

2. 3. 4.

382

5.

6.

7.

8. 9. 10. 11. 12.

Fig. 4a±c. Free convection. a Effects of nn on velocity pro®les; b effects of variable m on velocity pro®les; c inertia effects on velocity gradient

perature distributions, which result in lower heat transfer rate at higher value of this effect (inertia forced effect).

Vafai K; Tien CL (1988) Boundary and inertia effects on ¯ow and heat transfer in porous media. Int J Heat Mass Trans 24: 195±203 Vafai K (1984) Convective ¯ow and heat transfer in variableporosity media. J Fluid Mech 147: 233±259 Plumb OA; Huenefeld JC (1981) Non-Darcy natural convection from heated surfaces in saturated porous media. Int J Heat Mass Trans 24: 765±768 Aldoss TK; Chen TS; Armaly BF (1993) Non-similarity solutions for mixed convection from horizontal surfaces in a porous medium-variable surface heat ¯ux. Int J Heat Mass Trans 36: 463±470 Aldoss TK; Chen TS; Armaly BF (1993) Non-similarity solutions for mixed convection from horizontal surfaces in a porous medium-variable wall temperature. Int J Heat Mass Trans 36: 471±477 Hsieh JC; Chen TS; Armaly BF (1993) Nonsimilarity solutions for mixed convection from vertical surfaces in porous media: variable surface temperature or heat ¯ux. Int J Heat Mass Trans 36: 1485±1493 Hsieh JC; Chen TS; Armaly BF (1993) Mixed convection along a nonisothermal vertical plate embedded in a porous medium: The entire regime. Int J Heat Mass Trans 36: 1819± 1825 Lai FC; Kulacki FA (1987) Non-Darcy convection from horizontal impermeable surface in saturated porous medium. Int J Heat Mass Trans 30: 2189±2192 Lai FC; Kulacki FA (1991) Non-Darcy mixed convection along a vertical wall in a saturated porous medium. Transactions of the ASME 113: 252±255 Chen KS; Ho JR (1988) Effects of ¯ow inertia on vertical natural convection in saturated, porous, media. Int J Heat Mass Trans 29: 753±759 Duwairi H: Non-Darcian mixed convection along a horizontal wall in a saturated porous medium. Master Thesis, Jordan University of Science and Technology, 1992 Kodah ZH; Duwairi HM (1996) Inertia effects on mixed convection for vertical plates with variable wall temperature in saturated porous media. Heat and Mass Trans 31: 333± 338