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 = qk/Ak radiative flux on surface k
Jk outgoing radiative flux from surface k (“radiosity”)
Fk-j fraction of diffuse radiation leaving surface k that is incident on surface j (the configuration or shape factor)
Gk incident radiative flux striking surface k (“irradiance”)
Tk absolute temperature of surface k, K
εk emissivity of surface k
αk 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
net heat flux = outgoing flux - incident flux
The outgoing flux Jk is made up of the emitted flux leaving surface k plus that portion of the incident flux that is reflected from that surface, or
outgoing flux = emitted flux + reflected flux
where Kirchhoff’s law is used for a gray diffuse surface to replace αk by εk.
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 Fj-k, so the energy reaching surface k from j is JjAjFj-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
Note that Fk-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 3N equations in 3N unknowns, with the unknowns being Gk, Jk, and either Tk or q''k (whichever is not specified as a boundary condition) for each surface. Because neither Gk nor Jk 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
Now, solve Eq. (2) for Gk to give
and substitute Eq. (5) into Eq. (1) to eliminate Gk, resulting in
Now, substitute Eq. (6) to eliminate Jk (and Jj) in Eq. (4). Gathering all T terms on the left and q'' terms on the right results in
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 Tk or q''k for each surface.
For computer applications, note that ΣNj=1Fk-j = 1 for an enclosure. This can be used in Eq. (7) to give the form
Here, δkj 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 T1 and emissivity ε1, and plate 2 has temperature T2 and emissivity ε2.
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, F1-1 = 0 and F1-2 = 1, and using q''1 = -q''2, this reduces to