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Analytical study of flow and heat transfer in an annular porous medium subject to asymmetrical heat fluxes

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Abstract

Fully developed forced convective heat transfer in an annulus filled with a porous medium subject to asymmetrical heating is investigated analytically with different models in this work. The classic Darcy and Brinkman models were employed for the fluid flow, while the local thermal equilibrium (LTE) and the local thermal non-equilibrium (LTNE) models were employed to describe the heat transfer process in porous media. An analytical model based on fin theory was also employed for analyzing this problem. Exact solutions with Darcy-LTNE, Darcy-LTE, Brinkman-LTNE, Brinkman-LTE, and the fin models were obtained. Among these solutions, the Brinkman-LTNE solution can be treated as the benchmark, as it is a complete model, which covers the effect of viscous force near the solid wall and the temperature difference between the solid and fluid phases. The basic parameters that affect the velocity and temperature fields were analyzed in depth. The velocity and temperature profiles with these different models were also presented. The effects of some critical parameters on thermal performance of asymmetrically heated annulus fitted with a porous medium were discussed. The cited different analytical models were compared in detail with each other. The critical heat flux (HF) ratios for the inner and outer walls were presented in terms of a Nu–ξ curve for the five models. These solutions were developed for an asymmetrically heated annular channel filled with a porous medium, which can predict the thermal performance within a wide range of radii and HF ratios.

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Abbreviations

a sf :

Specific surface area (m−1)

A :

Area (m2)

c p :

Specific heat (J kg−1 K−1)

Da :

Darcy number

f :

Friction factor

h :

Heat transfer coefficient (W m−2 K−1)

h sf :

Local convective heat transfer coefficient (W m−2 K−1)

K :

Permeability (m2)

k :

Thermal conductivity (W m−1 K−1)

k r :

Thermal conductivity ratio (k r = k f/k s)

M :

Viscosity ratio

Nu :

Nusselt number

p :

Pressure (N m−2)

P :

Dimensionless pressure drop

Pr :

Prandtl number

q :

Heat flux (W m−2)

r :

Radius (m)

r 1 :

Inner radius (m)

r 2 :

Outer radius (m)

R :

Dimensionless radius

R 2 :

Radius ratio

Re :

Reynolds number

s :

Shape factor

t :

Dimensionless factor

T :

Temperature (K)

u :

Velocity (m s−1)

u m :

Mean velocity (m s−1)

U :

Dimensionless velocity

x :

Axial position (m)

ε :

Porosity

θ :

Dimensionless temperature

μ :

Dynamic viscosity (kg m−1 s−1)

ξ :

Heat flux ratio

ρ :

Density (kg m−3)

φ :

Polar angle (rad)

1:

Inner wall

2:

Outer wall

b:

Bulk

e:

Effective

f:

Fluid

fe:

Effective value of fluid

i:

Interface

m:

Mean

p:

Porous

r:

Ratio

s:

Solid

se:

Effective value of solid

w:

Wall

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (51406238).

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Authors and Affiliations

  1. Department of Energy and Power Engineering, College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, 266580, China

    Huijin Xu

  2. Institute of Engineering Thermophysics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Huijin Xu & Changying Zhao

  3. Department of Mechanical Engineering, University of California, Riverside, CA, 92521, USA

    Huijin Xu & Kambiz Vafai

Authors
  1. Huijin Xu
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  2. Changying Zhao
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  3. Kambiz Vafai
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Correspondence to Kambiz Vafai.

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Xu, H., Zhao, C. & Vafai, K. Analytical study of flow and heat transfer in an annular porous medium subject to asymmetrical heat fluxes. Heat Mass Transfer 53, 2663–2676 (2017). https://doi.org/10.1007/s00231-017-2011-x

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  • DOI: https://doi.org/10.1007/s00231-017-2011-x

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