How would you treat a radiation heat transfer boundary condition for a one-dimensional steady problem? Develop the difference equation for a control volume near the boundary and explain how to solve the entire system of difference equations. Assume that the heat flux at the surface is q = ? ? (Ts4 – Te4) where Ts is the surface temperature and Te is the temperature of an enclosure surrounding the surface.
GIVEN
• Radiation boundary condition
• One-dimensional steady conduction
FIND
(a) Difference equation for control volume near surface (b) Solution method
SKETCH
An energy balance on the half control volume surrounding the surface node is
The right side of the above equation can be written
The difference equation can be written in the tridiagonal form like as follows
The coefficients for 1 < i < N are given just before For i = 1, the coefficients will
depend on the boundary condition at the left boundary. For i = N, the coefficients are
To solve the set of difference equations, an initial temperature distribution guess will be made. This
will allow a determination of all of the coefficients. The tridiagonal matrix can then be solved to get an
updated temperature distribution. This distribution will be used to update the coefficients, and the
procedure will be repeated to convergence.
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