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Maximum density effects on natural convection in a vertical annulus filled with a non-Darcy porous medium

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Summary

This article numerically studies the problem of natural convection in a porous medium saturated with cold water, under density inversion, within a vertical annulus. In modeling the flow in the porous medium the non-Darcy effects, which include the Forchheimer inertia and Brinkman viscous effects are taken into account. The governing equations are solved numerically by the finite difference method using the modified strongly implicit procedure. The effects of the inversion parameterΘ m , radius ratioR *, aspect ratioAR, Forchheimer inertia parameter Fc/Pr, and Darcy number parameter Da on the heat transfer and fluid flow characteristics are discussed in detail. Results show that both the inversion parameter and radius ratio have a significant influence on the flow structure and heat transfer rate in the annulus. It is also found that the mean Nusselt number decreases as the Forchheimer inertia parameter or the Darcy number increases. Moreover, the results obtained here are also compared and favorably agree with numerical results and with experimental data.

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Abbreviations

AR :

aspect ratio,H/L

d :

particle diameter

Da:

Darcy number,K/(ϕL 2)

Fc:

Forchheimer number,K′/L

g :

gravitational acceleration

H :

annulus height

K :

permeability

K′:

transport property defined in Eq. (5)

L :

gap width,r o −r i

Nu i , Nu o :

local Nusselt number of the inner and outer cylinders, respectively

\(\overline {Nu} _i ,\overline {Nu} _o \) :

mean Nusselt number of the inner and outer cylinders, respectively

p :

pressure

Pr:

Prandtl number, ν/α

q :

constant in Eq. (9)

r :

radial coordinate

R :

dimensionless radial coordinate,(r−r i )/L

r i :

radius of inner cylinder

r o :

radius of outer cylinder

R * :

radius ratio,(r o −r i )/r i

Ra:

Rayleigh number,Kϱ m gβL(T h −T c )q/μα

T :

dimensional temperature

T c :

dimensional temperature of inner cylinder

T h :

dimensional temperature of outer cylinder

T m :

temperature corresponding to the density maximum, 4.029325°C

u, v :

Darcian velocity components inr andz directions, respectively

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  1. M. -I. Char & G. -C. Lee

    Present address: Department of Applied Mathematics, National Chung Hsing University, 402, Taichung, Taiwan, Republic of China

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    1. M. -I. Char
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    2. G. -C. Lee
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    Char, M.I., Lee, G.C. Maximum density effects on natural convection in a vertical annulus filled with a non-Darcy porous medium. Acta Mechanica 128, 217–231 (1998). https://doi.org/10.1007/BF01251892

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    • DOI: https://doi.org/10.1007/BF01251892

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