Heat transfer with nonlinear radiation

Example 3.2.7. Heat Transfer with Nonlinear Radiation Boundary Conditions... 

Physical situations that involve radiation with other modes of heat transfer are fairly common. If conduction enters the problem, the Fourier conduction law states that the heat flow depends upon the temperature gradient, thus introducing derivatives of the first power of the temperature. If convection matters, the heat flow depends roughly on the first power of the temperatures, the exact power depends on the type of flow. For instance, natural convection depends on a temperature difference between the 1.25 and l. 4 power. Physical properties that are temperature dependent introduce more temperature dependencies. This all means that the governing equations are highly nonlinear. 

The radiation boundary condition involves the fourth power of temperature, and thus it is a nonlinear condition. As a result, the application of this boundary condition results in powers of the unknown coefficients, which makes it difficult to determine them. Therefore, it is tempting to ignore radiation exchange at a surface during a heat transfer analysis in order to avoid the complications associated with nonlinearity. This is especially the case when heat transfer at the surface is dominated by convection, and tlie role of radiation is niinor. 

Consider heat transfer in a slab with a nonlinear fourth order radiation boundary condition at the surface.[16] (Schiesser, 1991). The governing equation in dimensionless form is... 

The present section deals with a number of examples combining radiation with conduction and/or convection. Most problems involving more than one mode of heat transfer are relatively involved, as they yield nonlinear differential equations and/or boundary conditions whenever radiation is included. They are usually solved after a linearization of the Stefan-Boltzmann law. During this process, however, the quantitative nature of a problem gets lost. 

Items 1-4 determine the degree to which the radiation source and material load are thermally coupled and can be addressed with the heat transfer analysis methods outlined in Chap. 7 of this handbook. Items 5 and 6 may be quantified with an analysis, which takes into account the multimode heat transfer effects discussed elsewhere in this handbook. Because of the nonlinear nature of radiative heat transfer, few correlations exist that can be applied to relevant materials processing situations. 

Nandeppanavar, M. M. Vajravelu, K. Abel, M. S. (2011). Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink. Comm. Nonlinear. Sci and Num. Simulation, 16, pp. 3578-3590, ISSN 1007-5704. 

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