Radiation between black surfaces

A black surface will not only absorb all of this radiation but will also emit radiation, and the net rate at which radiation is exchanged between the gas and the surface at temperature Ts is given by ... 

Problem A black element 0.5 cm by 0.5 cm, is at a temperature of 800°C and is near a tube of 2 cm diameter. The opening of the tube may be approximated as a black surface, and is at 400°C. Calculate the net radiation exchange along the connecting path R between the square element and the tube opening. 

Consider two black surfaces Ai and A2, as shown in Fig. 8-8. We wish to obtain a general expression for the energy exchange between these surfaces when they are maintained at different temperatures. The problem becomes essentially one of determining the amount of energy which leaves one surface and reaches the other. To solve this problem the radiation shape factors are defined as... 

The calculation of the radiation heat transfer between black surfaces is relatively easy because all the radiant energy which strikes a surface is absorbed. The main problem is one of determining the geometric shape factor, but once this is accomplished, the calculation of the heat exchange is very simple. When nonblackbodies are involved, the situation is much more complex, for all the energy striking a surface will not be absorbed part will be reflected back to another heat-transfer surface, and part may be reflected out of the system entirely. The problem can become complicated because the radiant energy can be reflected back and forth between the heat-transfer surfaces several times. The analysis of the problem must take into consideration these multiple reflections if correct conclusions are to be drawn. 

Evaluation of the A9 s that characterize 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 are gray and emit and reflect diffusely, then the direct-exchange area ij, as evaluated for the black-surface pair A and Ap applies to emission and reflections between them. If at a surface the total leaving-flux density, emitted plus reflected, is denoted by W (and called by some the radiosity and by others the exitance), radiation balances take the form ... 

When a surface of emissivity and surface area at a thermodynamic tem-peratwe T, is completely enclosed by a much larger (or black) surface at thermodynamic temperature T separated by a gas (such as air) that does not intervene with radiation, the net rate of radiation heat transfer between these two surfaces is given by (Fig. 1-37)... 

Consider the 5-m x 5-m X S-m cubical furnace shown in Fig. 13-19, whose surfaces closely approximate black surfaces. The base, top, and side surfaces of the furnace are maintained at uniform temperatures of 800 K, 1500 K, and 500 K, respectively. Delermine (a) the net rale of radiation heat transfer be-tvreen the base and the side surfaces, (b) the net rate of radiation heat transfer between the base and the top surface, and (c) the net radiation heat transfer from the base surface. 

If the bounding surface is black at temperature the surface will emit radiation to the gas at a rate of A,crT/ without reflecting any, and the gas will absorb this radiation at a rale of UgA aTj, where is the absorptivity of the gas. Then the net rate of radiation heal transfer between the gas and a black surface suriouiidiiig it becomes... 

The rate of net radiation heal transfer between two black surfaces is determined from... 

Now reconsider two bodies each with a flat black surface at absolute temperatures T and Tz separated l distance apart in a vacuum, as shown in Fig. 1.15(b). Surface 1 has emissive power Ebi, surface 2 has emissive power Ebz, and the radiation heat transfer between these bodies is... 

Figure US (a) The Stefan-Boltzm aim law, (b) radiation heat transfer between two parallel black surfaces.

The simplest type of radiation between two surfaces is where each surface can see only the other, e.g., where the surfaces are very large parallel planes, as shown in Fig. 143a, and where both surfaces are black. The energy emitted per unit area by the first plane is crTj that emitted by the second plane is cTf. Assume that Ti > T2. All the radiation from each of the surfaces falls on the other surface and is completely absorbed. Since the areas of the two surfaces are equal, the net... 

The equation of radiative transfer will not be solved here since solutions to some approximations of the equation are well known. In photon radiation, it has served as the framework for photon radiative transfer. It is well known that in the optically thin or ballistic photon limit, one gets the heat flux as q = g T[ - T ) from this equation for radiation between two black surfaces [13]. For the case of phonons, this is known as the Casimir limit. In the optically thick or diffusive limit, the equation reduces to q = -kpVT where kp is the photon thermal conductivity. The same results can be derived for phonon radiative transfer [14,15]. 

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