Page 255 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

247
CONVECTION IN POROUS MEDIA
Energy equation
2
σ ∂T ∗ + u ∗ i ∂T ∗ = k ∗ ∂ T ∗ ! (8.30)
∂t ∗ ∂x ∗ ∂x ∗2
i i
Other alternative scales are possible and the appropriate references should be consulted
to learn more about scaling. In the above formulation, the buoyancy effects are incorpo-
rated by invoking the Boussinesq approximation as discussed in Chapter 7. The kinematic
viscosity ν, used in the above scales, is defined as
µ
ν = (8.31)
ρ
and α is the thermal diffusivity, given as
k f
α f = (8.32)
(ρc p ) f
It may be observed that the scales and non-dimensional parameters are defined by
using the fluid properties. Often, a quantity called the Darcy–Rayleigh number is used in
the literature as a governing non-dimensional parameter for Darcy flow. This is the product
of the Darcy (Da) and fluid Rayleigh (Ra) numbers as defined previously.

8.2.2 Limiting cases

The equations discussed above represent a porous medium, which tends to a solid as the
porosity, → 0. Thus, a conjugate problem, in which part of the domain is completely
solid, can be dealt with by using the above equations.
Another limiting case of these equations is that they approach the incompressible
Navier–Stokes equations as → 1. Again, a very general problem in which the porous
medium and a single-phase fluid are part of the domain (porous-fluid interface (Massarotti
et al. 2001)) can be solved by using the above equations. Thus, many applications such as
alloy solidification (Sinha et al. 1992) and heat exchanger design can be analysed via these
equations.

8.3 Discretization Procedure

The CBS scheme will be employed to solve the porous medium flow equations. In this
context, the same four steps, with minor modifications, will be utilized as discussed in the
previous chapter.
In the following subsections, the temporal and spatial discretization schemes are given,
which will then be employed to solve the porous medium equations. Use will be made
only of simple, linear triangular elements to study porous medium flow problems.

8.3.1 Temporal discretization

Before going into the details of the CBS split, let us first consider the temporal discretization
of the governing equations. The momentum equation is subjected to the characteristic