Thermal radiation in an enclosure made up of gray-diffuse surfaces is a problem of solving a set of linear equations if some simplifying assumptions are made. The equations involve radiative heat flux, absolute temperatures, geometrv specifications, and surface properties. The necessary assumptions are as follows:

All properties are temperature independent. If this is not the case, the solution becomes iterative unless the temperatures of all enclosure surfaces are specified. Otherwise, properties must be reevaluated after each iteration and new temperatures found and this process repeated until convergence.

All surfaces are diffuse and gray. The assumption of diffuse surfaces allows use of configuration (shape) factors between surfaces, and the gray-diffuse assumption allows substitution of total emissivity for total absorptivity (Kirchhoff’s law).

With these assumptions, we can derive the relations for radiative exchange in an enclosure of N diffuse-gray surfaces. Each surface has either temperature or heat flux prescribed. The quantities needed in the analysis are as follows:

*q'' _{k} = q_{k}/A_{k}* radiative flux on surface

*k*

*J _{k}* outgoing radiative flux from surface

*k*(“radiosity”)

*F _{k-j}* fraction of diffuse radiation leaving surface

*k*that is incident on surface

*j*(the configuration or shape factor)

*G _{k}* incident radiative flux striking surface

*k*(“irradiance”)

*T _{k}* absolute temperature of surface

*k*, K

ε_{k} emissivity of surface *k*

α_{k} absorptivity of surface *k*

The radiative energy balance on surface *k* indicates that the net heat flux at the
surface (the energy added to surface *k*) is the difference between the outgoing and
incident radiation as

(1) |

net heat flux = outgoing flux - incident flux

The outgoing flux *J _{k}* is made up of the emitted flux leaving surface

*k*plus that portion of the incident flux that is reflected from that surface, or

(2) |

outgoing flux = emitted flux + reflected flux

where Kirchhoff’s law is used for a gray diffuse surface to replace α_{k} by ε_{k}.

Finally, the incident flux (irradiance) on surface *k* can be found. It is the sum of
the radiation leaving each other surface j in the enclosure that is incident on surface
*k*. The fraction of the total radiant energy (radiosity) leaving surface *j* that arrives at
surface *k* is, by definition, the configuration (shape) factor F_{j-k}, so the energy
reaching surface *k* from *j* is *J _{j}A_{j}F_{j-k}*. The total energy reaching

*k*from all surfaces is then

where the last term is found by using reciprocity on the configuration (shape) factor, resulting in

(3) |

Note that *F _{k-k}* in the summation may have a nonzero value if a particular surface

*k*is concave so that some radiation from

*k*may impinge on itself.

Equations (1)-(3) can be written for each surface, giving 3*N* equations in 3*N*
unknowns, with the unknowns being *G _{k}, J_{k}*, and either

*T*or

_{k}*q''*(whichever is not specified as a boundary condition) for each surface. Because neither

_{k}*G*nor

_{k}*J*are of interest in most engineering problems, it makes sense to eliminate them from the equation set. This is done by substituting Eq. (3) into Eq. (1) to get

_{k}(4) |

Now, solve Eq. (2) for *G _{k}* to give

(5) |

and substitute Eq. (5) into Eq. (1) to eliminate *G _{k}*, resulting in

(6) |

Now, substitute Eq. (6) to eliminate *J _{k}* (and

*J*) in Eq. (4). Gathering all

_{j}*T*terms on the left and

*q''*terms on the right results in

(7) |

Equation (7) can be written for each of the *N* surfaces in the enclosure,
giving a set of *N* equations in the *N* unknowns of either *T _{k}* or

*q''*for each surface.

_{k} For computer applications, note that **Σ*** ^{N}_{j=1}F_{k-j}* = 1 for an enclosure. This can
be used in Eq. (7) to give the form

(8) |

Here, δ_{kj} is the Kronecker delta, which has a value of 0 when *k* ≠ *j*, and 1 when
*k = j*. These equations differ from those found in numerical solutions of convection or
conduction problems because each equation may contain all of the unknowns. In a
convection or conduction problem, each equation normally only contains the
unknown for a given node plus the surrounding nodes, resulting in many fewer terms
in each equation. The matrix of coefficients in these cases is “sparse.” where for
radiation it may be full (i.e., all elements are nonzero). This is because the radiation
equations are essentially finite difference forms of integral equations, while
finite difference conduction and convection equations describe differential
equations.

EXAMPLE: Use Eq. (7) to derive a relation for radiative heat flux *q''* between
two infinite parallel plates. Plate 1 has temperature *T*_{1} and emissivity ε_{1}, and plate 2
has temperature *T*_{2} and emissivity ε_{2}.

SOLUTION: The heat flux for this 1D geometry will be the same for each plate,
so *q''*_{1} = -*q''*_{2}. Write Eq. (7) for this case for plate 1:

Observing that for infinite parallel plates, *F*_{1-1} = 0 and *F*_{1-2} = 1, and using
*q''*_{1} = -*q''*_{2}, this reduces to

or