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Heat current through a spherical shell

  • Thread starter Swatch
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A spherical shell has inner radii a and outer radii b. The temperatures at the inner and outer surfaces are T2 and T1. The thermal conductivity of the shell material is k. I have to derive an equation for the total heat current through the shell.


The equation for heat current through a rod is H=k*A*DeltaT/L where L is the length of the rod.

For this sperical shell the area which is perpendicular to the flow of the heat is changing with the radius. So I have to integrate the area. Am I right here?
I have tried to integrate the area by doing:
A=integrate(4*pi*r^2) from a to b
But I end up with something far from the right answer, don't want to get into that.
Could anyone please give me a hint to this problem?
Thanks

The right answer is H=4*pi*k*a*b*DeltaT/(b-a)
 

Answers and Replies

Astronuc
Staff Emeritus
Science Advisor
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1,685
Along the lines of the example H=k*A*DeltaT/L

the current is the area integral of the heat flux.

In general, heat flux is given by -k*[itex]\nabla T[/itex], and [itex]\nabla T[/itex] can be approximated by [itex]\frac{\Delta T}{\Delta x}[/itex], where x is the generalized length dimension.

In the case of the sphere, one would apply [itex]\frac{\Delta T}{\Delta r}[/itex].

Also the area midway between the inner and outer surfaces may be given the the 4[itex]\pi[/itex]*R2, where R = [itex]\sqrt{r_i r_o}[/itex], i.e. the geometric mean radius.

Properly integrating the problem should also give the same answer, but the limits of integration are not (a, b), since one is considering a spherical surface.
 
89
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Thank you. :smile:

Just one question. How do you derive the mean of the radius?
 
Last edited:

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