Derive expression for x-dependence of steady-state heat flux

Consider a one-dimensional plane wall with constant properties and uniform internal generation qdot. The left face is insulated, and the right face is held at a uniform temperature Tc. The diagram depicts a vertical wall, with the origin in the bottom right corner indicating that positive x is to the right. Epsilon (the greek symbol) represents a portion of the wall's width across, which will be used in part b.

(a) Using the appropriate form of the heat equation, derive an expression for the x-dependence of the steady-state heat flux qdoubleprime(x)

(b) Using a finite volume spanning the range 0<=x<=epsilon, derive an expression for qdoubleprime(epsilon) and compare the expression to your result for part (a)

 

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Mechanical Engineering: Derive expression for x-dependence of steady-state heat flux
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