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Research Papers: Porous Media

# Restrictions on the Validity of the Thermal Conditions at the Porous-Fluid Interface—An Exact Solution

[+] Author and Article Information
Kun Yang

School of Energy and Power Engineering,  Huazhong University of Science and Technology, Wuhan 430074, P. R. China; Department of Mechanical Engineering, University of California, Riverside, Riverside, CA 92521-0425

Kambiz Vafai1

Department of Mechanical Engineering,  University of California, Riverside, Riverside, CA 92521-0425vafai@engr.ucr.edu

1

Corresponding author.

J. Heat Transfer 133(11), 112601 (Sep 16, 2011) (12 pages) doi:10.1115/1.4004350 History: Received January 10, 2011; Revised May 22, 2011; Published September 16, 2011; Online September 16, 2011

## Abstract

Thermal conditions at the porous-fluid interface under local thermal nonequilibrium (LTNE) conditions are analyzed in this work. Exact solutions are derived for both the fluid and solid temperature distributions for five of the most fundamental forms of thermal conditions at the interface between a porous medium and a fluid under LTNE conditions and the relationships between these solutions are discussed. This work concentrates on restrictions, based on the physical attributes of the system, which must be placed for validity of the thermal interface conditions. The analytical results clearly point out the range of validity for each model for the first time in the literature. Furthermore, the range of validity of the local thermal equilibrium (LTE) condition is discussed based on the introduction of a critical parameter. The Nusselt number for the fluid at the wall of a channel that contains the fluid and porous medium is also obtained. The effects of the pertinent parameters such as Darcy number, Biot number, $Bi$, Interface Biot number, $Biint$, and fluid to solid thermal conductivity ratio are discussed.

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## Figures

Figure 1

Schematic diagram for flow through a channel partially filled with a porous medium and the corresponding coordinate system

Figure 2

βcr distributions for different parameters Bi and k

Figure 3

Dimensionless temperature distributions for Model A for α*=0.78, Da=1×10-5, and ɛ=0.9

Figure 4

Dimensionless heat flux distributions at the interface for α*=0.78

Figure 5

Dimensionless temperature distributions for Model C for α*=0.78, Da=1×10-5, and ɛ=0.9

Figure 6

%Δθ variations as a function of η1 for α*=0.78 and ɛ=0.9

Figure 7

η1,cr variations as a function of pertinent parameters k, Bi, Biint, and Da for α*=0.78 and ɛ=0.9

Figure 8

Nusselt number variations as a function of pertinent parameters k, Bi, and Da for Model A for α*=0.78 and ɛ=0.9

Figure 9

Dimensionless velocity distributions as a function of η1 for α*=0.78

Figure 10

The variation of the maximum velocity at the open region for pertinent parameters η1 and Da for α*=0.78

Figure 11

Nusselt number variations as a function of pertinent parameters k,Bi, and β for Model B for α*=0.78 and ɛ=0.9

Figure 12

Nusselt number variations as a function of pertinent parameters k,Bi, and Biint for Model C for α*=0.78 and ɛ=0.9

## Errata

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