Page 248 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

8

Convection in Porous Media

8.1 Introduction

The phenomenon of fluid flow and heat transfer in porous media has been recognized
as a separate engineering topic for the last three decades. Several books have been pub-
lished on this topic (Kaviany 1991; Lewis and Schrefler 1998; Nield and Bejan 1992;
Zienkiewicz et al. 1999). Convective heat transfer in porous media occurs in many engi-
neering applications including packed beds, thermal insulation, metal solidification and
geothermal problems. Advanced applications such as petroleum reservoirs, multi-phase
flows and drying have also been studied using finite elements (Lewis and Ferguson 1990;
Lewis et al. 1984, 1983, 1989; Lewis and Sukirman 1993; Murugesan et al. 2001; Pao
et al. 2001). A wide variety of solution methodologies, both analytical and numerical, are
available for solving porous media flow and heat transfer. Analytical methods are limited
by many factors and the solution of realistic field problems is normally intractable by such
techniques. With the advent of computing power in the last three decades, solutions to
many practical porous medium problems are feasible using numerical methods (Lewis and
Schrefler 1998; Zienkiewicz et al. 1999). Such numerical solution procedures have their
own limitations, for example, accuracy, implementation difficulties and so forth. However,
with a proper combination of algorithms and discretization techniques, it is possible to
obtain reasonably accurate solutions for complex problems, in which analytical approaches
would not be feasible. In this chapter, the finite element modelling of incompressible flow
and heat transfer through porous media will be outlined in detail.
The flow of fluid in a saturated porous media was quantified by a simple, phenomeno-
logical, linear relation by Darcy in the nineteenth century (Darcy 1856). Darcy’s law relates
the pressure drop (head) to the flow rate across a porous column. The following relation
can be written from such observations:

κ ∂p
u i =− (8.1)
µ ∂x i

Fundamentals of the Finite Element Method for Heat and Fluid Flow R. W. Lewis, P. Nithiarasu and K. N. Seetharamu
 2004 John Wiley & Sons, Ltd ISBNs: 0-470-84788-3 (HB); 0-470-84789-1 (PB)