Heat transfer with nonlinear radiation boundary conditions

Example 3.2.7. Heat Transfer with Nonlinear Radiation Boundary Conditions 

Example 3.2.4 is solved here using Maple s dsolve command. The boundary condition at the surface, x = 1 is taken as  

(in dsolve/numeric/bvp) unable to store Limit(0.+1.0000000000000 1, 

Maple is not able to solve this problem directly. An approximate solution can be provided to arrive at the exact solution. The approximate solution can be found using the linear boundary condition at x = 1. Note that the approximate solution has to be evaluated for at least eight node points. 

The numerical solution of the original boundary value problem is found as  

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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. 

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