How Does Fourier's Law Apply to Heat Flux in a Uniformly Heated Slab?

[tweety1234]

Homework Statement

a) A slab of thickness L and constant thermal conductivity $$\lambda$$ generates heat at a constant rate throughout of g W m–3. The heat is dissipated from both sides of the slab by convection into the ambient air at a temperature Tf with a heat transfer coefficient h. The expression for the steady state temperature profile throughout the slab is given by

$$T(x) = \frac{g}{8 \lambda} L^{2}( 1- (\frac{2x^{2}}{L})) + \frac{gL}{2h} + T_{f}$$

where symbols have their usual meaning in this context.

(i) Derive an expression for the heat flux as a function of position x. Should I differentiate with respect to 'x'?

 

[MexChemE]

There's a little error in that temperature profile, here's how it should look like
$$T = \frac{gL^2}{8 \lambda} \left[1 - \left(\frac{2x}{L}\right)^2 \right] + \frac{gL}{2h} + T_{f}$$
Fourier's Law of heat conduction states
$$q'' = - \lambda \frac{dT}{dx}$$
Where q'' is the heat flux in the x direction. So the way to go is to differentiate the temperature profile wrt x and multiply it by -λ.
$$\frac{dT}{dx} = - \frac{gx}{\lambda}$$
So the expression for the heat flux as a function of x is
$$q'' = gx$$