Radiant heat transfer between gray surfaces forming an enclosure

Abstract

An analytical investigation was made of the radiant heat transfer between the surfaces of a gray enclosure filled with a non-reacting gas considering all multiple reflections between the surfaces. The problem was developed by writing two different heat transfer equations for each surface of the enclosure: Q[subscript i] = Σ[n above sigma, k=1 below sigma] A[subscript i]F[subscript i-k] (J[subscript i]-J[subscript k]) i = 1 ,2, 3, 4, ..., n (1) and Q[subscript i] = A[subscript i] (J[subscript i]-H[subscript i]) (2) where where Q[subscript i] is the net heat exchange between the ith surface and the rest of the enclosure; A[subscript i] is the area of the ith surface; F[subscript i-k] is the view factor between the ith and kth surfaces; J[subscript i] is the radiosity of the ith surface; H[subscript i] is the irradiation of the ith surface; and n is the total number of surfaces. The radiosity is composed of the emitted flux density, the reflected flux density, and the flu x density transmitted into the enclosure from the outside or J[subscript i] = σε[subscript i]T^4[subscript i] + ρ[subscript i]h[subscript i] + τ[subscript i]H'[subscript i] (3) where is the temperature; H'[subscript i] is the outside radiant flux density; ε is the emissivity; ρ is the reflectivity; and τ is the transmissivity of the surface. Eliminating H[subscript i] from Equations (2) and (3) gives: Q[subscript i] = A[subscript i] [(ε[subscript i] / ρ[subscript i] σT^4[subscript i]) + (τ[subscript i] / ρ[subscript i] H'[subscript i]) -- (1-ρ[subscript i] / ρ[subscript i]) J[subscript i]]. (4) For each surface of the enclosure there are three possible boundary conditions that can exist: (a) the temperature and the outside radiant flux density are known; (b) the net heat transfer is known; (c) neither of the above. Equations (1) and (4) are used, depending upon boundary conditions (a) or (b), to form a set of simultaneous linear equations with the radiosities as the unknowns. If boundary (c) exists, then both (a) and (b) must be known for another surface and the number of equations in the set is reduced by one. After the system of equations is written and the radiosities have been determined, then the net heat transfer by each surface of the enclosure can be determined. The other methods which can be applied to an enclosure problem require that either a generalized view factor or an absorption factor be defined. These factors are defined for particular pairs of surfaces in the enclosure and all factors must be determined to find the net heat transfer by each surface of the enclosure. There are, however, only (n)(n-1)(1/2) independent factors for each enclosure. The determination of the independent factors requires the solution of (n-1) different sets of simultaneous linear equations. Once all the necessary generalized view or absorption factors have been determined then the net heat transfer by each surface of the enclosure can be determined. The proposed Radiosity method requires the solution of only one set of n simultaneous linear equations to determine the net heat transfer by each surface of an enclosure, whereas, the other methods require the solution of (n-1) different sets. The Radiosity method also allows a greater flexibility in boundary conditions. The temperature of the adiabatic surfaces can be determined also using the Radiosity method. All methods were applied to specific problems. The numerical results of these problems show excellent agreement between the Radiosity method and other methods.