Heat conduction through a slab with internal heat generation

rafehi

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?

Mapes

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.

rafehi

So basically treat it as a series question?

Defining the temps as
T_L, T_M and T_R, (left, mid, right),
Q_in as the Q coming in from the left,
Q_mid as the heat through the middle,
Q_out as the heat from the right,
a*Q_vol as the fraction going into the left,
a*Q_vol as the fraction going into the right,
L a the length with each side = to L/2:

Q_in - a*Q_vol = Q_mid
Q_out = Q_mid + b*Q_vol

Q_in = (T_m - T₁)*(2A*k)/L
so that
Q_mid = (T_m - T₁)*(2A*k)/L - a*Q_vol

and (Fourier's Equation for the right side):
Q_mid = (T₂ - T_m)*(2A*k)/L

Solving the above two equations to get rid of T_m:
Q_mid = (T₂ - T₁)*(A*k)/L - (a/2)*Q_vol

So,
Q_out = (T₂ - T₁)*(A*k)/L + (b - a/2)*Q_vol

Q_out = (T₂ - T₁)*(A*k)/L + (1 - 3a/2)*Q_vol

Mapes

Why 2A instead of A?

rafehi

2A because each side has length L/2 - so Ak/(L/2) = 2Ak/L.

Is the reasoning right?

Mapes

Ah, got it. That part's OK, then. It's just that your last line looks a little off and I'm looking for problems in the calculations.

Why do you have Qin = (Tm - T1)*(2A*k)/L when Qin is defined as the heat transfer coming in? If Tm>T1, this predicts positive flux coming in from the left, which doesn't make sense.

Then you have Qmid = (T2 - Tm)*(2A*k)/L; shouldn't this be Qout? (And again there seems to be a minus sign missing.)