The interaction of thermal nonequilibrium and heterogeneous conductivity effects in forced convection in layered porous channels

International Journal of Heat and Mass Transfer 44 (2001) 4369±4373

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Technical Note

The interaction of thermal nonequilibrium and heterogeneous conductivity e€ects in forced convection in layered porous channels D.A. Nield a, A.V. Kuznetsov b,* b

a Department of Engineering Science, University of Auckland, Auckland, New Zealand Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, NC 27695-7910, USA

Received 19 September 2000; received in revised form 16 February 2001

Abstract Forced convection in a parallel plate channel ®lled with a porous medium, consisting of two layers with the same porosity and permeability but with di€erent solid conductivity, and saturated by a single ¯uid, is analyzed using a twotemperature model. It is found that the e€ect of local thermal nonequilibrium is particularly signi®cant when the solid conductivity in each layer is greater than the ¯uid conductivity, and in these circumstances the e€ect is to reduce the Nusselt number de®ned in terms of the overall e€ective thermal conductivity. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Owing to the use of hyperporous media in the cooling of electronic equipment, there has recently been renewed interest in the classical problem of forced convection in a porous medium channel or duct. Global heterogeneity e€ects due to the variation of permeability and/or thermal conductivity, with the medium layered in the transverse direction, have been analyzed in a number of papers by the authors [1±4]. These papers have been concerned with the usual situation in which thermal equilibrium between the solid and ¯uid phases can be assumed. However, there are several industrial applications where high-speed ¯ow leads to a signi®cant degree of local thermal nonequilibrium (LTNE), even in the case of steady forced convection, and this situation for a homogeneous porous medium has also been analyzed by the authors [5,6]. In the present paper the interaction

between heterogeneity e€ects and LTNE e€ects is investigated. 2. Analysis We consider the case (see Fig. 1(a)) of a parallel plate channel with walls at y  ˆ H, divided into core and sheath layers occupied, respectively, by two porous media, of the same porosity / and permeability K, and saturated by the same ¯uid of thermal conductivity kf , but with the solid thermal conductivity ks being given by ks ˆ ks1 ks ˆ ks2

for 0 < jy  j < nH ; for nH < jy j < H :

Corresponding author. Tel.: +1-919-515-5292; fax: +1-919515-7968. E-mail address: [email protected] (A.V. Kuznetsov).

…1b†

Asterisks are used to denote dimensional variables. Thus the mean solid conductivity is ks ˆ nks1 ‡ …1

*

…1a†



n†ks2 :

…2†

We de®ne k~s1 ˆ ks1 =ks ;

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 1 ) 0 0 0 7 8 - 3

k~s2 ˆ ks2 =ks ;

krel ˆ ks =kf ;

…3a†

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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Mass Transfer 44 (2001) 4369±4373

Fig. 1. De®nition sketch, and plots of Nusselt number Nu versus e€ective conductivity ratio keffratio for various values of the solid-to¯uid heat transfer number g (n ˆ 0:5; / ˆ 0:5).

keff ˆ /kf ‡ …1 keffratio

/†ks ;

/kf ‡ …1 ˆ /kf ‡ …1

…3b†

/†ks2 : /†ks1

The thermal energy equations for the solid and ¯uid phases are taken to be /†r
…ks r Ts † ‡ hfs …Tf

…1

/r
…kf r Tf † ‡ hfs …Ts

Ts † ˆ 0;

…4a†

Tf † ˆ …qcP †f v
r Tf :

…4b†

The ¯uid±solid heat exchange coecient hfs is assumed to be uniform. The case where the wall temperature Tw is uniform is considered. For the ¯uid ¯ow, Darcy's law is assumed. Since the permeability is uniform, so is the axial velocity U. We introduce dimensionless variables and parameters by xˆ

x ; H

Pe ˆ

yˆ

y ; H

UH …qcP †f ; kf

hf ˆ

gˆ

Tf Tw ; Tref

hfs H 2 : keff

hs ˆ

Ts Tw ; Tref

…5†

…6†

 o2 Nh hsj ‡ Nf 2 oy

Nh

 o hfj ˆ 0: ox

…8b†

These equations constitute an eighth-order system, which is to be solved subject to the following eight symmetry/matching/boundary conditions: ohf1 ohs1 ˆ ˆ 0 at y ˆ 0; oy oy hf1 ˆ hf2 ;

hs1 ˆ hs2 ;

ohs1 ~ ohs2 ˆ ks2 k~s1 oy oy

…9†

ohf1 ohf2 ˆ ; oy oy

…10†

at y ˆ n;

hf2 ˆ hs2 ˆ 0 at y ˆ 1:

…11†

The individual equations in Eq. (10) express the matching of temperature and heat ¯ux at the interface between the two layers, for the ¯uid and solid phases separately. (This matching is based on the uniformity principle discussed by Nield and Kuznetsov [6], that requires that the matching conditions are independent of the porosity.) The variables are now separated by letting

Thus Pe is the Peclet number based on ¯uid properties, and g is the dimensionless exchange parameter introduced by Nield and Kuznetsov [6]. It is convenient to carry out the analysis in terms of the following parameters:

hfj ˆ Hfj ekx ;

With D denoting d/dy, we then obtain h i Ns k~sj D2 Nh Hsj ‡ Nh Hfj ˆ 0;

…13a†

/ …1 /†krel ; ; Ns ˆ Pe Pe g‰/ ‡ …1 /†krel Š Nh ˆ : Pe

Nh Hsj ‡ ‰Nf D2

…13b†

Nf ˆ

…7†

With subscripts 1, 2 referring to the subdomains 0 < y < n, n < y < 1, respectively, then, for j ˆ 1, 2, we have   o2 …8a† Ns k~sj 2 Nh hsj ‡ Nh hfj ˆ 0; oy

hsj ˆ Hsj ekx :

Nh

…12†

kŠHfj ˆ 0:

Elimination of Hsj leads to n  Ns k~sj D2 Nh …Nf D2 Nh

k†

o Nh2 Hfj ˆ 0:

…14†

The solution of this equation subject to the conditions in Eqs. (9) and (11) is Hf1 ˆ A cos sa1 y ‡ B cosh sb1 y; Hf2 ˆ C sin sa2 …1

y† ‡ D sinh sb2 …1

…15a† y†:

…15b†

D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Mass Transfer 44 (2001) 4369±4373

2

Here, for j ˆ 1, 2, ("   Nh Nf ‡ Nh Ns k~sj ‡ kNs k~sj saj ˆ ‡



Nh Nf ‡ Nh Ns k~sj ‡ kNs k~sj

4kNh Nf Ns k~sj

1=2 #,

D2 ˆ 4 2 D3 ˆ 4

2

2

)1=2 2Nf Ns k~sj

;

…16a†

("

‡



Nh Nf ‡ Nh Ns k~sj ‡ kNs k~sj

4kNh Nf Ns k~sj

2

)1=2 1=2 #, ~ : 2Nf Ns ksj

…16b† Hs2 ˆ

The matching conditions in Eq. (10) then imply that Mc ˆ 0;

…17†

where M

2 6 6 6 ˆ6 6 4

C2 sb1 S2 s2b1 C2

S3 sa2 C3 s2a2 S3

S4 sb2 C4 s2b2 S4

k~s1 sb1 S2 Ns k~s1 s2b1 ‡Nh

k~s2 sa2 C3 Ns k~s2 s2a2 ‡Nh

k~s2 sb2 C4 Ns k~s2 s2b2 ‡Nh

C1 sa1 S1 s2a1 C1 k~s1 sa1 S1 Ns k~s1 s2a1 ‡Nh

2

A

3

6B7 6 7 and c ˆ 6 7; 4C5

3 7 7 7 7; 7 5

…18†

where in turn C1 ˆ cos sa1 n;

S1 ˆ sin sa1 n; n†;

C4 ˆ cosh sb2 …1

n†;

S3 ˆ sin sa2 …1 S4 ˆ sinh sb2 …1

C2 D1 ˆ 4 sb1 S2 s2b1 C2

S3 sa2 C3 s2a2 S3

n  Nf s2a2 ‡ Nh ‡ k C sin sa2 …1 y† h i ‡ Nf s2b2 ‡ Nh ‡ k D sinh sb2 …1

y†

o.

Nh :

/†ks2 sb2 …24†

n†;

0

 ‰/hf2 ‡ …1 /†hs2 Šdy n     Nf 2 k kx AS1 / ‡ …1 /† 1 ‡ sa1 ‡ ˆ Tref e Nh sa1 Nh    BS2 Nf 2 k / ‡ …1 /† 1 s ‡ ‡ sb1 Nh b1 Nh    C…1 C3 † Nf k / ‡ …1 /† 1 ‡ s2a2 ‡ ‡ Nh sa2 Nh    D…C4 1† Nf 2 k : / ‡ …1 /† 1 sb2 ‡ ‡ Nh sb2 Nh ‡

…19† n†:

With k found, the corresponding eigenvector is given by

where 2

C2 sb1 S2 s2b1 C2

The heat ¯ux at the boundary is given by     oTf oTs ‡ …1 /†ks q00 ˆ /kf oy y  ˆH oy y  ˆH  Tref kx ˆ /kf sa2 ‡ …1 /†ks2 sa2 e H    Nf k C ‡ /kf sb2 ‡ …1  1 ‡ s2a2 ‡ Nh Nh    Nf 2 k D :  1 sb2 ‡ Nh Nh

…20†

A B C D ˆ ˆ ˆ ; D1 D2 D3 D4

C1 sa1 S1 s2a1 C1

…22†

…23b†

The eigenvalue equation (regarded as an equation for k) is det M ˆ 0:

C2 sb1 S2 s2b1 C2

Z

S2 ˆ sinh sb1 n;

C3 ˆ cos sa2 …1

C1 sa1 S1 s2a1 C1

S4 sb2 C4 5; s2b2 S4 3 S4 sb2 C4 5; s2b2 S4 3 S3 sa2 C3 5: s2a2 S3

The di€erence between the e€ective bulk temperature and the wall temperature is Z n Tbeff Tw ˆ Tref ‰/hf1 ‡ …1 /†hs1 Šdy

D

C2 ˆ cosh sb1 n;

S3 sa2 C3 s2a2 S3

(Without loss of generality one can put D ˆ 1.) Then Eqs. (13b), (15a) and (15b) yield n h i  Nf s2b1 ‡ Nh ‡ k Hs1 ˆ Nf s2a1 ‡ Nh ‡ k A cos sa1 y ‡ o. B cosh sb1 y Nh ; …23a†

Nh Nf ‡ Nh Ns k~sj ‡ kNs k~sj

sbj ˆ

D4 ˆ 4

C1 sa1 S1 s2a1 C1

4371

3

…21†

1

…25† 3 S4 sb2 C4 5; s2b2 S4

We de®ne the Nusselt number as Nu ˆ

2Hq00 : keff …Tw Tb eff †

…26†

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D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Mass Transfer 44 (2001) 4369±4373

It is now just a matter of substituting from Eqs. (24) and (25) into (26). The factor Tref ekx cancels. The expression may be simpli®ed by using the quadratic equations satis®ed by s2a1 , s2a2 , s2b1 and s2b2 . Further simpli®cations can be made using the de®nitions in Eq. (7). In this manner one obtains the expression ( " # k~s2 Ns Nh 2 Csa2 Nf ‡ Nu ˆ …Nf ‡ Ns † Nh ‡ k~s2 Ns s2a2 " #)!, ( k~s2 Ns Nh AS1 ‡ Dsb2 Nf ‡ 2 ~ sa1 Nh ks2 Ns sb2 # # " " …1 /†Nh BS2 …1 /†Nh ‡ /‡  /‡ sb1 Nh ‡ k~s1 Ns s2a1 Nh k~s1 Ns s2b1 # " C…1 C3 † …1 /†Nh /‡ ‡ sa2 Nh ‡ k~s2 Ns s2a2 #)! " D…C4 1† …1 /†Nh : …27† /‡ ‡ sb2 Nh k~s2 Ns s2b2

This expression can be readily checked for a special case, namely that for a homogeneous medium, for which k~s1 ˆ k~s2 ˆ 1;

B ˆ D ˆ 0;

sa1 ˆ sa2 ˆ p=2;

C ˆ A;

sb1 ˆ sb2 ˆ 1:

For this case, Eq. (27) reduces to      p2 2…Nf ‡ Nu ˆ p2 Nf Nh ‡ Ns ‡ Ns Nh 4     p2 Ns † / Nh ‡ Ns ‡ …1 /†Nh ; 4

…28†

…29†

an expression which is independent of the value of n, and which agrees with Eq. (36) of Nield and Kuznetsov [6] in the limit as Lf and s2 both tend to in®nity. In the particular case of large g (and hence of large Nh ), as is normal in a practical situation, Eq. (29) leads to ( ) p2 p2 /…1 /†2 ks …kf ks † 1‡ Nu ˆ : …30† 4hfs H 2 keff 2

Fig. 2. Plots of Nusselt number versus solid-to-¯uid conductivity ratio ks2 =kf for various values of the solid-to-¯uid heat transfer number g for: (a) ks1 =kf ˆ 0:3; (b) ks1 =kf ˆ 1; (c) ks1 =kf ˆ 3; (d) ks1 =kf ˆ 10 (n ˆ 0:5; / ˆ 0:5).

D.A. Nield, A.V. Kuznetsov / International Journal of Heat and Mass Transfer 44 (2001) 4369±4373

This illustrates the immediate fact that the e€ect of LTNE is negligible in the case where the porosity tends to either zero or unity or when the solid and ¯uid conductivities are closely the same. Further, it is seen that the e€ect of LTNE is to increase or decrease Nu (as we have de®ned it) according as ks is less than or greater than kf , respectively. 3. Results and discussion We have calculated Nu from Eq. (27) for various parameter values. We con®rmed that Nu is independent of Pe as expected. We checked that we could recover the results for the homogeneous case expressed in Fig. 3 of [6] in the limiting case of very large Biot number (corresponding to the isothermal boundary conditions that we have adopted in the present work). We then checked that we could obtain results in agreement with the middle curve in Fig. 4(b) of [1] for the case of local thermal equilibrium. This was a severe test of the correctness of the present analysis and its numerical implementation, because the formulation in the present paper is distinctly di€erent from that in [1]. In the present paper an eigenvalue equation is solved for the separation constant (decay parameter) k and the result is fed into an expression for the Nusselt number Nu, whereas in [1] the value of Nu is obtained directly from another eigenvalue equation. Our results are presented in Figs. 1 and 2, for the case where the porosity is 0.5 and each layer occupies half of the channel. Fig. 2 has been designed for comparison with Fig. 4(b) of [1]. As we have already reported, the results for a large value of g (i.e., for negligible LTNE e€ects) match those in our earlier paper [1], and are in fact independent of the value chosen for ks1 =kf . Fig. 1(b)

4373

shows that the chief e€ect of LTNE is to reduce the Nusselt number when keffratio is greater than unity (and has little e€ect when it is less than unity). Fig. 2 presents results for various values of ks1 =kf and ks2 =kf varied separately. As ks1 =kf increases the curves become more peaked, with the position of the peaks occurring at increasing values of ks2 =kf and the e€ect of LTNE is to reduce the magnitude of the peaks. When ks1 =kf is less than unity the e€ect of LTNE is small. (We have not presented results for ks1 =kf ˆ 0:1 because they closely resemble those for ks1 =kf ˆ 0:3.) The existence of the peaks was explained in [1] as a result of two opposing e€ects of conductivity variation. References [1] D.A. Nield, A.V. Kuznetsov, E€ects of heterogeneity on forced convection in a porous medium: parallel plate channel or circular duct, Int. J. Heat Mass Transfer 43 (2000) 4119±4134. [2] D.A. Nield, A.V. Kuznetsov, E€ects of heterogeneity on forced convection in a porous medium: parallel plate channel, Brinkman mode, J. Porous Media (in press). [3] D.A. Nield, A.V. Kuznetsov, E€ects of heterogeneity on forced convection in a porous medium: parallel plate channel, asymmetric property variation and asymmetric heating, J. Porous Media (in press). [4] A.V. Kuznetsov, D.A. Nield, E€ects of heterogeneity on forced convection in a porous medium: triple layer or conjugate problem, Numer. Heat Transfer Part A (in press). [5] D.A. Nield, E€ects of local thermal nonequilibrium in steady convective processes in a saturated porous medium: forced convection in a channel, J. Porous Media 1 (1998) 181±186. [6] D.A. Nield, A.V. Kuznetsov, Local thermal nonequilibrium e€ects in forced convection in a porous medium channel: a conjugate problem, Int. J. Heat Mass Transfer 42 (1999) 3245±3252.