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.

Question

GIVEN

• Radiation boundary condition

• One-dimensional steady conduction

FIND

(a) Difference equation for control volume near surface (b) Solution method

SKETCH

Answer 1

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.