Encyclopaedia Index

1. The nature of the radiation-simulation problem

1.1 The geometrical difficulty

Radiative heat transfer differs in character from conductive and convective heat transfer in that it involves "action at a distance".

Heat conduction to a point is influenced by the temperatures of the materials at the immediately-surrounding locations.

Heat convection to a point is influenced by the temperature on the immediately-upstream side.

Heat radiation, by contrast, depends on the temperatures at all surrounding points, no matter how far away they are.

Admittedly, the more remote points usually have less influence than the nearer ones; but the temperature of the sun affects the heat flux to the Earth; and that is remote enough by terrestrial standards.

The mean-free-path ( = MFP) explanation

One way of expressing the difference between radiative heat transfer and the other kinds is to state that the "mean free path of radiation" is often much larger than the dimensions of the domain of study.

The "MFP of conduction" and the "MFP of convection", on the other hand, are usually much smaller than those dimensions.

Their MFPs are indeed of the order of the distance between molecules, or (in turbulent flow) of the size of the smallest eddies.

The MFP of radiation varies inversely with the amount of radiation-absorbing material per unit path length; which is why it is so large in "outer space", where there is no such material.

The conduction approximation to radiative heat transfer

Where much radiation-absorbing material intervenes, e.g. within a furnace, where pulverized-coal particles and finely-divided soot absorb, scatter and re-emit radiation, the MFP of radiation can be smaller than the apparatus dimensions.

Then radiative transfer can be regarded as similar to conduction, but with an increased thermal conductivity.

The magnitude of the "radiative conductivity" is of the order of:

sigma * T**3 / ( a + s)

      where: sigma = Stefan-Boltzmann Const = 5.6678E-8 W/(m^2 K^4)
                 T = absolute temperature, K
                 a = absorption coefficient, 1/m
                 s = scattering coefficient, 1/m

The mathematical viewpoint

Because the mathematical expression for the radiative heat transfer to a point involves adding up (ie integrating) the contributions from an infinite number of nearer and more-remote locations, the equation describng it is called "integro-differential".

This contrasts with the more modest "differential-equation" label which is used for the conduction and convection processes.

Differential equations are much easier to solve than integro- differential ones.

For this reason, nearly all practical simulations of radiative heat transfer in CFD codes employ differential-equation approximations for radiation.

The approximations differ mainly in their formulations of the radiative conductivity.

1.2 Other difficulties

The problem of simulating radiation mathematically is complicated further by the need to account for the facts that:

(1) rays of radiation passing through any point differ in angle, corresponding with their differing locations of origin;

(2) they differ also in wave-length (ie "colour"); and

(3) both angle and wave-length influence their interactions with the materials on which they impinge.

The task of accounting correctly for these facts is so large that they are either totally neglected or greatly simplified.

The simplifications usually involve coarsely discretising the variations with wave-length and/or angle.