Radiosity

W = total leaving-flux density (also radiosity). 

Evaluation of the AS" s that charac terize an enclosure involves solution of a system of radiation balances on the surfaces. If the assumption is made that all the zones of the enclosure a re gray and emit and reflec t diffusely, then the direct-exchange area ij, as evaluated for the black-siirface pair A and Aj, applies to emission and reflections between them. If at a surface the total leaving-flnx density, emitted phis reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form ... 

If the temperature of a grey surface is known, then the net heat transfer to or from the surface may be determined from the value of the radiosity qo- With regard to signs, the usual convention is that a positive value of g, indicates heat transfer from grey surfaces. 

Radiation arrives at a grey surface of emissivity 0.75 al a constant temperature of 400 K, at the rate of 3 kW/m2. What is the radiosity and the net rate of radiation transfer to the surface What coefficient of heat transfer is required to maintain the surface temperature at 300 K if the rear of the surface is perfectly insulated and the front surface is cooled by convective heat transfer to air at 295 K ... 

For the more complex case of a multi-sided enclosure formed from n surfaces, the radiosities may be obtained from an energy balance for each surface in turn in the enclosure. Thus the energy falling on a typical surface i in an enclosure formed from... 

Equations similar to equation 9.158 may be obtained for each of the surfaces in an enclosure, 1 = 1,1 = 2, 1 = 3, 1 = n and the resulting set of simultaneous equations may then be solved for the unknown radiosities, qoi,qm- qun The radiation heat transfer is then obtained from equation 9.140. This approach requires data on the areas and view factors for all pairs of surfaces in the enclosure and the emissivity, reflectivity and the black body emissive power for each surface. Should any surface be well insulated, then, in this case, Qj — 0 and ... 

Irradiation G Total thermal radiation energy incident on a surface per unit time per unit area Irradiation (G), and Radiosity J) are all energy fluxes (i.e., rate... 

Radiosity J Total thermal radiation energy leaving a surface (emitted and reflected) per unit time per unit area of energy transfer per unit area). The three terms, Absorptivity (a), Reflectivity (p), and Transmissivity (x), are all surface properties... 

The equation for the radiosity is given below, where p is replaced by (1-e) according to assumption (7). 

In addition to the assumptions stated above, we shall also assume that the radiosity and irradiation are uniform over each surface. This assumption is not strictly correct, even for ideal gray diffuse surfaces, but the problems become exceedingly complex when this analytical restriction is not imposed. Sparrow and Cess [10] give a discussion of such problems. The radiosity is the sum of... 

The net energy leaving the surface is the difference between the radiosity and the irradiation ... 

Taking the resistance (1 — e,)/ejAi as zero, we have the network as shown. To calculate the heat flows at each surface we must determine the radiosities J, and J2. The network is solved by setting the sum of the heat currents entering nodes J, and J2 to zero ... 

We define the diffuse radiosity Jn as the total diffuse energy leaving the surface per unit area and per unit time, or... 

Equation (8-77) expresses the diffuse radiation leaving 1 which arrives at 2 and which may contribute to a diffuse radiosity of surface 2. The factor 1 - ps represents the fraction absorbed plus the fraction reflected diffusely. The inclusion of this factor is most important because we are considering only diffuse direct exchange, and thus must leave out the specular-reflection contribution... 

Now consider the diffuse exchange between surfaces 1 and 3. Of the energy leaving 1, the amount which contributes to the diffuse radiosity of surface 3 is... 

The diffuse radiosity <5f a particular surface of the medium is defined by... 

The diffuse radiosity is still defined as in Eq. (8-91), and the net energy exchange with a transmitting surface is given by Eq. (8-93). The analysis of transmitted energy exchange with other surfaces must be handled somewhat differently, however. 

Making use of Eqs. (8-99) and (8-102) gives the complete network for the system as shown in Fig. 8-61. Of course, all the radiation shape factors in the above network are unity, but they have been included for the sake of generality. In this network Jw refers to the diffuse radiosity on the left side of 2, while J 2D is the diffuse radiosity on the right side of this surface. 

The heat transfer at each surface is then evaluated in terms of the radiosities Jj. These parameters are obtained by recalling that the heat transfer can also be expressed as... 

The nodal equations for the radiosities may also be derived from the nodes in the network formulation. At each J, node an energy balance... 

The solution for the radiosities is found by obtaining the inverse to [/4] such that... 

To obtain the heat transfers the set of equations is first solved for the radiosities. For set (c),... 

To illustrate the radiation formulation for numerical solution we consider the circular hole 2 cm in diameter and 3 cm deep, as shown in the accompanying figure. The hole is machined in a large block of metal, which is maintained at l000oC and has a surface emissivity of 0.6. The temperature of the large surrounding room is 20°C. A simple approach to this problem would assume the radiosity uniform over the entire heated internal surface. In reality, the radiosity varies over the suiface, and we break it into segments 1 (bottom of the hole), 2. 3, and 4 (sides of the hole) for analysis. 

The large room acts like a blackbody at 20°C, so for analysis purposes we can assume the hole is covered by an imaginary black surface S at 20°C. We shall set the problem up for a numerical solution for the radiosities and then calculate the heat-transfer rates. After that, we shall examine an insulated-surface case for this same geometry. 

The equations for the radiosities are now written in the form of Eq. (8-112), noting that... 

It is of interest to compare this heat transfer with the value we would obtain by assuming uniform radiosity on the hot surface. We would then have a two-body problem with... 

Thus, the simple assumption of uniform radiosity gives a heat transfer which is 3.9 percent above the value obtained by breaking the hot surface into four parts for the calculation. This indicates that the uniform-radiosity assumption we have been using is a rather good one for engineering calculations. 

It is of interest to compare the heat transfer calculated above with that obtained by assuming surfaces 2, 3, and 4 uniform in temperature and radiosity. Equation (8-41) applies for this case ... 

In this case the assumption of uniform radiosity at the insulated surface gives an overall heat transfer with surface 1 (bottom of hole) that is 18.7 percent too high. 

With such a small difference between the solutions we may conclude that the extra complexity of choosing each surface at a different radiosity is probably not worth the effort, particularly when one recognizes the uncertainties which are present in the surface emissivities. This points out that our assumptions of uniform irradiation and radiosity, though strictly not correct, give answers which are quite satisfactory. 

---