Notes on the Volume averaged Energy Equation for Porous Medium Flows

Porous medium by definition is made of at least two materials, one solid and stationary with respect to a reference frame and the other a fluid that could move (flow). Fluid flow in such a configuration has been discussed earlier and is governed by the momentum conservation statement also known as the Darcy Law or one of its extensions.

How to formulate an energy conservation statement that governs the temperature distribution under such porous medium convection situation?

The concept of volume averaging [briefly discussed here] is invoked in writing this energy equation. This equation is nothing but the first law of thermodynamics, applied on an open system where mass and energy are allowed to "cross" the boundaries that separate the system from its surroundings. [see also [[this note|First Law and Fourier Law]] on how the energy equation for a heat conduction situation is simply the application of First Law for a closed system]

In the following figure, the sample parallel plate bounded porous medium convection configuration is modeled as a homogeneous one dimensional heat and flow configuration as pictured. The Darcy Law in its global and differential form (text inside the green box in the figure) is taken to govern the momentum conservation in this particular case. The surface porosity is invariant in the x direction and hence equal to the volumetric Porosity?.

[click on image for bigger picture] Figure 1: Schematic of the porous medium convection configuration considered for formulating the energy conservation equation

As one can expect, two energy conservation equations, one for the solid part and one for the fluid part could be written for this convection configuration as follows.

Solid side

Fluid Side

Observe that the volumetric internal heat generation is present only in the solid side (the last term in Eq. (1)) and the viscous dissipation term is present only in the fluid side (the last term in Eq. (2)). Also, the viscous dissipation is modeled as the power required for the fluid to "extrude" (as used in reference [1]) itself over the porous solid structure. This is equivalent to the pressure drop in the Darcy form of the momentum conservation times the seepage speed, resulting in an expression of the form given as the last term in Eq. (2).

The concept of a Representative Elemental Volume (REV) [briefly discussed here] is the basic step for performing a volume averaging in a porous medium. If one were to solve these equations separately, an interfacial closure term involving a local convection heat transfer coefficient is required that account for the local heat transfer between the solid and the fluid flow. Obviously, this term is neglected in the above two equations. This is possible with an important but pertinent assumption that the temperature of the solid and the fluid inside an REV is identical. In other words, local thermal equilibrium between the solid and the fluid is assumed while writing the above two equations. Hence the temperature T that is present in both the equations (1) and (2) is neither the temperature of the solid nor that of the fluid but is that of the porous medium.

From the figure above it is obvious that the one-dimensional porous medium convection configuration is a "parallel" arrangement at the REV level. Invoking the volume averaging concept and combining the above two equations to obtain a single energy equation for the porous medium results in

Generalizing the above one-dimension equation for three-dimensions using vector notations, it would look like

Many simplifications are consciously swept under the rug in arriving at the above equation but let me explain the important ones alone and await discussion on any specific points that arise. First, a note on the effective properties viz. k_e, and . Owing to the parallel arrangement of the solid and the fluid flow in the REV level, the effective conductivity can be defined as

Other combinations of are possible depending on the modeling of the REV but that warrants a separate essay. Proceeding to the sigma, we note that it is the porosity weighted ratio of the (density times heat capacity) of the solid and the fluid.

It can be expressed for the parallel model of homogenization of the porous medium considered in Figure 1 as

In the absence of heat generation within the porous medium and for negligible viscous dissipation effects, the above equation simplifies to

Which further simplifies for steady state to

where is the effective thermal diffusivity of the porous medium ( of fluid).

Some of the major assumptions in writing all of the above volume averaged energy conservation statements are as follows

1. homogeneous porous medium
2. local thermal equilibrium between the solid and the fluid of the porous medium exists in the REV level itself.
3. effective properties used in the volume averaged energy equation are accurate in predicting the effects in the REV level.
4. the rest of the primitive variables, u, P and T are volume averaged quantities defined on a porous continuum.

Reference

• Convection Heat Transfer by A. Bejan, [Amazon Link]

Related posts:

1. Flow Transition in Porous Media
2. Flow Through Porous Media Summary
3. Predicting Flow Transition in Porous Media
4. Variable Viscosity Effects Explained
5. Free Convection and the Rayleigh Number