Heat transfer radiation shape factors

If the emissivities of all surfaces are equal, a rather simple relation may be derived for the heat transfer when the surfaces may be considered as infinite parallel planes. Let the number of shields be n. Considering the radiation network for the system, all the surface resistances would be the same since the emissivities are equal. There would be two of these resistances for each shield and one for each heat-transfer surface. There would be n + I space resistances, and these would all be unity since the radiation shape factors are unity for the infinite parallel planes. The total resistance in the network would thus be... 

B. T. F. Chung and P. S. Sumitra, Radiation Shape Factors from Plane Point Sources, J. Heat Transfer, 94(3), pp. 328-330, August, 1972. 

A very important aspect of radiative heat transfer is the system geometry. This is accounted for by using radiation shape factors, also called view factors, angle factors, or configuration factors and defined as follows... 

In a series of papers, Derby and Brown (144, 149-152) developed a detailed TCM that included the calculation of the temperature field in the melt, crystal, and crucible the location of the melt-crystal and melt-ambient surfaces and the crystal shape. The analysis is based on a finite-ele-ment-Newton method, which has been described in detail (152). The heat-transfer model included conduction in each of the phases and an idealized model for radiation from the crystal, melt, and crucible surfaces without a systematic calculation of view factors and difiuse-gray radiative exchange (153). 

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. 

The radiation network is shown in the accompanying figure where surface 3 is the room and surface 2 is the insulated surface. Note that 7, = Eh, because the room is large and (I - ed/ejAi approaches zero. Because surface 2 is insulated it has zero heat transfer and J2 = J2 floats" in the network and is determined from the overall radiant balance. From Fig. 8-14 the shape factors are... 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

To account for the effects of orientation on radiation heat transfer between two surfaces, we define a new parameter called the vieu factor, which is a purely geometric quantity and is independent of the surface properties and temperature. It is also called the shape factor, configuration factor, and angle factor. The view factor based on the assumption that the surfaces are diffuse emitters and diffuse reflectors is called the diffitse view factor, and the view factor based on the assumption that the surfaces are diffuse emitters but specular reflectors is called the specular view factor. In lliis book, we consider radiation exchange between diffuse surfaces only, and ihu.s the term view factor simply means diffuse view factor. 

Consider two black surfaces of arbitrary shape maintained at uniform temperatures T and Ti, as shown in fig. 13 -18, Recognizing that radiation leaves a black surface at a rate of E/, - crT" per unit surface area and that the view factor i, 2 represents the fraction of radiation leaving surface 1 that strikes surface 2, the net rate of radiation heat transfer from surface 1 to surface 2 can be expressed as... 

U. Gross, K. Spindler, and E. Hahne, Shape Factor Equations for Radiation Heat Transfer between Plane Rectangular Surfaces of Arbitrary Position and Size with Rectangular Boundaries, Lett. Heat Mass Transfer, 8, pp. 219-227,1981. 

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