## Heat conduction through a slab with internal heat generation

1. The problem statement, all variables and given/known data
There's a slab of a material with temperature T1 on the left and the T2 on the right. The thickness of the material is l with area A. In the centre, there is heat generation Qvol in the centre, which is a thin rod.

Find the heat transfer Q through the material.
2. Relevant equations

3. The attempt at a solution
My issue is with modeling the Qvol. If it was heat generation through the material, I'd just use the DE
D2T/dt2 = -(1/k)Qvol
with BCs T(0) = T1
T(l) = T2
which is easily solved,

But it's a 'point' source of heat, and unless I'm mistaken, I can't use this. The lecturer seemed to hint that it could be solved with looking at each side of the slab seperately.

Any help?
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 Recognitions: Gold Member Homework Help Science Advisor I think the lecturer gave a good hint. Assume that the heat flux from Qvol is divided up between the left and right sides. For each side, you have a known temperature on one side and an unknown fraction of Qvol entering from the other side. Also, the fractions must add up to one. And finally, the temperature at the middle is the same in each equation. That should give you an equal number of equations and unknowns to solve the problem.
 So basically treat it as a series question? Defining the temps as TL, TM and TR, (left, mid, right), Qin as the Q coming in from the left, Qmid as the heat through the middle, Qout as the heat from the right, a*Qvol as the fraction going into the left, a*Qvol as the fraction going into the right, L a the length with each side = to L/2: Qin - a*Qvol = Qmid Qout = Qmid + b*Qvol Qin = (Tm - T1)*(2A*k)/L so that Qmid = (Tm - T1)*(2A*k)/L - a*Qvol and (Fourier's Equation for the right side): Qmid = (T2 - Tm)*(2A*k)/L Solving the above two equations to get rid of Tm: Qmid = (T2 - T1)*(A*k)/L - (a/2)*Qvol So, Qout = (T2 - T1)*(A*k)/L + (b - a/2)*Qvol Qout = (T2 - T1)*(A*k)/L + (1 - 3a/2)*Qvol

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