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Archived Fourier's law heat conduction

  • Thread starter tweety1234
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1. Homework Statement

a) A slab of thickness L and constant thermal conductivity [tex] \lambda [/tex] 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

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

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'?
 
228
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There's a little error in that temperature profile, here's how it should look like
[tex]T = \frac{gL^2}{8 \lambda} \left[1 - \left(\frac{2x}{L}\right)^2 \right] + \frac{gL}{2h} + T_{f}[/tex]
Fourier's Law of heat conduction states
[tex]q'' = - \lambda \frac{dT}{dx}[/tex]
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.
[tex]\frac{dT}{dx} = - \frac{gx}{\lambda}[/tex]
So the expression for the heat flux as a function of x is
[tex]q'' = gx[/tex]
 

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