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2.3.4 Radiation Exchange Between Surfaces

When studying the radiant energy exchanged between two surfaces separated by a non-absorbing medium, one should consider not only the temperature of the surfaces and their characteristics but also their geometric orientation with respect to each other. The effects of the geometry of radiant energy exchange can be analyzed conveniently by defining the term view factor, F12, to be the fraction of radiation leaving surface A1 that reaches surface A2. If both surfaces are black, the radiation leaving surface A1 and arriving at surface A2 is Eb1A1F12, and the radiation leaving surface A2 and arriving at surface A1 is Eb2A2F21. If both surfaces are black and absorb all incident radiation, the net radiation exchange is given by

If both surfaces are of the same temperature, Eb1 = Eb2 and Q12 = 0. Therefore,

It should be noted that Eq. (2.59) is strictly geometric in nature and valid for all diffuse emitters, irrespective of their temperatures. Therefore, the net radiation exchange between two black surfaces is given by

From Eq. (2.36), Eb = ctT4, Eq. (2.60) can be written as

where T1 and T2 are the temperatures of surfaces A1 and A2, respectively. As the term (Eb1 - Eb2) in Eq. (2.60) is the energy potential difference that causes the transfer of heat, in a network of electric circuit analogy, the term 1/A1F12 = 1/A2F21 represents the resistance due to the geometric configuration of the two surfaces.

When surfaces other than black are involved in radiation exchange, the situation is much more complex, because multiple reflections from each surface must be taken into consideration. For the simple case of opaque gray surfaces, for which e = a, the reflectivity p = 1 — a = 1 — e. From Eq. (2.42), the radiosity of each surface is given by

The net radiant energy leaving the surface is the difference between the radiosity, J, leaving the surface and the irradiation, H, incident on the surface; that is,

Combining Eqs. (2.62) and (2.63) and eliminating irradiation H results in

Therefore, the net radiant energy leaving a gray surface can be regarded as the current in an equivalent electrical network when a potential difference (Eb - J) is overcome across a resistance (1 - e)/Ae. This resistance is due to the imperfection of the surface as an emitter and absorber of radiation as compared to a black surface.

By considering the radiant energy exchange between two gray surfaces, Aj and A2, the radiation leaving surface A1 and arriving at surface A2 is J1A1F12, where J is the radiosity, given by Eq. (2.42). Similarly, the radiation leaving surface A2 and arriving surface A1 is J2A2F21. The net radiation exchange between the two surfaces is given by

Q12 = J1AF12 - J2AF21 = A1F12(J1 - J2) = A2F21(J1 - J2) (2.65)

Therefore, due to the geometric orientation that applies between the two potentials, J1 and J2, when two gray surfaces exchange radiant energy, the resistance 1/A1F12 = 1/A2F21.

An equivalent electric network for two the gray surfaces is illustrated in Figure 2.24. By combining the surface resistance, (1 - e)/Ae for each surface and the geometric resistance, 1/AjF12 = 1/A2F21, between the surfaces, as

Ae1 A1F12 A2F21 A2e2

FiGuRE 2.24 Equivalent electrical network for radiation exchange between two gray surfaces.

Ae1 A1F12 A2F21 A2e2

FiGuRE 2.24 Equivalent electrical network for radiation exchange between two gray surfaces.

shown in Figure 2.24, the net rate of radiation exchange between the two surfaces is equal to the overall potential difference divided by the sum of resistances, given by

Eb1 Eb2

M AiFi

A2e2

A1e1 A!F1

A2e2

In solar energy applications, the following geometric orientations between two surfaces are of particular interest.

A. For two infinite parallel surfaces, A1 = A2 = A and F12 = 1, Eq. (2.66) becomes

B. For two concentric cylinders, F12 = 1 and Eq. (2.66) becomes