Short Answer:

The net heat exchange between two diffuse gray surfaces is the net radiative power transferred from one surface to the other after accounting for emission, reflection and geometry. It can be written in a resistance-network form using black-body emissive powers , surface emissivities , areas  and the view factor . The general expression is

For special symmetric cases (equal areas, ) this reduces to familiar simpler forms such as for two large parallel plates:

Detailed Explanation :

Net heat exchange between two gray surfaces

When two diffuse gray surfaces (surface 1 and surface 2) face each other in an enclosure, each surface both emits radiation (according to its temperature and emissivity) and reflects part of the radiation falling on it. The net heat exchange is the algebraic difference between radiant power leaving surface 1 that is absorbed by surface 2 and the power leaving surface 2 that is absorbed by surface 1.

A convenient and widely used way to evaluate this exchange is the radiosity–irradiation method combined with a resistance network analogy. Key quantities:

•  : black-body emissive power of surface .
•  : emissivity of surface  (diffuse gray).
•  : area of surface .
•  : view (configuration) factor from 1 to 2 (geometry only). Reciprocity gives .
•  : radiosity of surface  = total radiant energy leaving surface  per unit area (emitted + reflected).
•  : irradiation on surface  = total incident radiation.

For diffuse gray surfaces the surface energy balance (per unit area) gives:

For the two-surface enclosure:

The net power leaving surface 1 and absorbed by surface 2 is

Solving the coupled radiosity equations for  and substituting yields the closed form expression.

Resistance network interpretation and final formula

Radiative heat exchange can be arranged as a series of three resistances between black-body emissive powers  and :

• Surface resistance of surface 1: .
• Space (geometric) resistance between surfaces: .
• Surface resistance of surface 2: .

Thus the net heat transfer from 1 to 2 is analogous to a current driven by the “potential” difference  across the total resistance:

This is the general two-surface formula for diffuse gray surfaces. It is exact under the assumptions of diffuse (Lambertian) surfaces and gray (wavelength-independent) properties.

Useful special cases

1. Large parallel plates, equal areas  and :
   The expression simplifies to

This is a common formula used for two infinite parallel plates.

1. One surface black () and arbitrary surface 1:
   Resistances reduce because ; the network is simpler:

For equal areas and : .

1. Highly emissive surfaces ():
   Surface resistances vanish and the net heat flux approaches the black-body exchange limited by geometry: .

Practical notes and assumptions

• The formula assumes surfaces are diffuse (radiation emitted and reflected equally in all directions) and gray (emissivity independent of wavelength). If surfaces are specular or spectral dependencies are important, more complex models are needed.
• The view factor  must be computed for the geometry; tables and formulas exist for many standard configurations.
• In enclosures with more than two surfaces, the radiosity method leads to a system of linear equations for the  values and the resistance analogy generalizes to a network that can be solved numerically.

Conclusion

The net radiative heat exchange between two diffuse gray surfaces is obtained by treating emission, reflection and geometry as resistances in series driven by the difference in black-body emissive powers . The compact resistance form shown above is convenient for hand calculations and design checks and reduces to simpler well-known formulas for common symmetric cases.