Thermal Conduction Through a Sphere

[Brooks]

I have come across a problem where I must find the rate of heat flow to the surface of a sphere from the center. The sphere has a constant coefficient of thermal conduction. The problem also supplies the temperature difference between the center/surface and the radius of the sphere. My question is whether it would be correct to simply apply the conduction formula Q/t=kA(T2-T1)/L even though the area of contact between two adjacent surfaces will be changing at different radii. If applying this equation is incorrect, what method should be used for this problem? Will calculus be required due to the varying area of contact? Thank you for your time.

 

[Astronuc]

Q/t=kA(T2-T1)/L would apply in one dimension in rectilinear coordinate system.

See this for a spherical system, and yes, calculus is involved.

http://rpaulsingh.com/teaching/LecturesIFE/CondSphere/condsph.htm

or

http://www.rh.edu/~ernesto/C_S2002/CHT/notes/s08/s08.html

 

[Brooks]

Thanks for the reply, Though I can't seem to find any elementary techniques for finding the rate of heat transfer through a sphere with a constant thermal coefficient k.

 

[Astronuc]

Oops! Sorry about that. Let me see what I can find.

Also, is this steady-state, i.e. does $$\frac{\partial{T(r)}}{\partial{t}}$$ = 0?

 

[Brooks]

Yes it is steady state

 

[Integral]

To solve this you need to solve the heat equation wit h appropriate boundary conditions on the sphere. You may be able to use a simple Newtons law of cooling for the heat loss from the surface of the sphere. Another condition will be 0 heat loss from the center and perhaps a initial heat distribution. Due to the symmetries this problem can be reduced to a single variable, r, greatly simpling the solution.

 

[Shooting Star]

I don't quite feel that Newton's law of cooling would be any substitute for the treatment required by the OP. That is more appropriate when the inside temperature is almost constant, unlike here. But of course, without any calculus, this can't be solved at all. The best is http://rpaulsingh.com/teaching/LecturesIFE/CondSphere/condsph.htm , which Astronuc has already sited. It requires the minimum of calculus, and the diagram is good.

Could the OP tell us under what chapter he came across this?

 

[coomast]

For a sphere no formulas are available to my knowledge because of diverging problems at the center. They are available, as can be read in the given links, for a hollow sphere, not a complete solid one. One can check that the formula is indeed diverging for an inner radius going towards 0.

However, there might be an approximate solution to the problem. Consider therefore the formula for the heat flow derived in the link:

$$q=-4 \pi k r^2\frac{dT}{dr}$$

This can be approximated in stating the following:

outer radius sphere: $$r=R$$
temperature difference: $$dT=\Delta T$$, from the original post
"radius difference": $$dr=0-R=-R$$

giving thus:

$$q=4 \pi k R \Delta T$$

Can someone confirm this before using it...

 

[Shooting Star]

So, q/t = kA(T2-T1)/L becomes q/t = k(T2-T1)*4pi*r^2/r = k(T2-T1)*surface area/radius. I don't know...

 

[Shooting Star]

As has been pointed out, no exact solution exists even using calculus, because of the div at 0. Some sort of mean of the max area and min area (which is 0) has to be taken. The one given by coomast seems as good as any. It's useless to ponder on this any further without knowing at what level the problem was presented.

 

[HVOF]

Heat conduction equation in spherical coordinates and with transient surface temperature is not an easy problem to solve. There is no analytical solution but only approximations that some times are not accurate. If you are familiar with numerical methods and discretization have a look to my publication:

http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TWM-4SBRTK1-1&_user=153063&_coverDate=10%2F31%2F2008&_alid=1290075656&_rdoc=2&_fmt=high&_orig=search&_cdi=5566&_sort=r&_docanchor=&view=c&_ct=8&_acct=C000012698&_version=1&_urlVersion=0&_userid=153063&md5=6d0c4eddd4872a90ae0a0d1c5a698946