Nonlinear Radiation Boundary Condition

Consider the following boundary value problem with a nonlinear radiation boundary condition at y = 0. 

6 Method of Lines for Elliptic Partial Differential Equations 

This equation is solved below in Maple using the program developed for example 6.3. The semianalytical method developed earlier is valid for nonlinear boundary conditions also. This is true because the vector equation (6.6) is linear as both the governing equation and the boundary conditions in x are linear. The nonlinear boundary condition comes into the picture only for solving the constants. This is illustrated in the following program. 

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

If the surface temperature does not differ greatly from the surrounding temperature, the highly nonlinear surface boundary condition may be simplified by linearizing the expression for the radiation flux and the Clausius-Clapeyron equation to yield the approximation... 

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. 

Tien attempting to gel an analytical solution to a physical problem, there is always the tendency to oversimplify the problem to make the mathematical model sufficiently simple to warrant an analytical solution. Therefore, it is common practice to ignore any effects that cause mathematical complications such as nonlincarities in the differential equation or the boundary conditions. So it comes as no surprise that nonlinearities such as temperature dependence of tliernial conductivity and tlie radiation boundary conditions aie seldom considered in analytical solutions. A maihematical model intended for a numerical solution is likely to represent the actual problem belter. Therefore, the numerical solution of engineering problems has now become the norm rather than the exception even when analytical solutions are available. 

Note that thermodynamic temperatures must be used in radiation heat transfer calculations, and ail temperatures should be expressed in K or R when a boundary condition involves radiation to avoid mistakes. We usually try to avoid the radiation boundary condition even in numerical solutions since it causes the finite difference equations to be nonlinear, wlu ch are more difficult to solve. 

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... 

A fiill solution of tlie nonlinear radiation follows from the Maxwell equations. The general case of radiation from a second-order nonlinear material of finite thickness was solved by Bloembergen and Pershan in 1962 [40]. That problem reduces to the present one if we let the interfacial thickness approach zero. Other equivalent solutions involved tlie application of the boundary conditions for a polarization sheet [14] or the... 

Yet another boundary condition encountered in polymer processing is prescribed heat flux. Surface-heat generation via solid-solid friction, as in frictional welding and conveying of solids in screw extmders, is an example. Moreover, certain types of intensive radiation or convective heating that are weak functions of surface temperature can also be treated as a prescribed surface heat-flux boundary condition. Finally, we occasionally encounter the highly nonlinear boundary condition of prescribed surface radiation. The exposure of the surface of an opaque substance to a radiation source at temperature 7 ,-leads to the following heat flux ... 

Fortunately, numerical modeling despite its many limitations associated with grid resolution, choice of turbulence model, or assignment of boundary conditions is not intrinsically limited by similitude or scale constraints. Thus, in principle, it should be possible to numerically simulate all aspects of fires within canopies for which realistic models exist for combustion, radiation, fluid properties, ignition sources, pyrolysis, etc. In addition it should be possible to examine all interactions of fire properties individually, sequentially and combined to evaluate nonlinear effects. Thus, computational fluid dynamics may well provide a greater understanding of the behavior of small, medium, and mass fires in the future. 

At the harmonic wavelength, the source to the radiated fields is the sheet of polarization. However, the total field radiated at location f in the outer medium cannot be taken as the simple superposition of the spherical waves generated by each point f at the surface of the particle because the metal particle is also highly polarizable at the harmonic frequency. The total field in the particle must be the superposition of the radiated field and the polarization field at the harmonic wavelength and the total field at the detector in the outer medium is obtained by solving the boundary conditions at the surface of the particle. In spherical coordinates, the harmonic field at location r radiated by a nonlinear polarization source at location E is now also an expansion of the parameter x = a/X of the form ... 

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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