Deriving Steady-State Heat Conduction Equation in a Sphere

[protonchain]

Homework Statement

Show that the equation governing the stead-state radial conduction of heat in a sphere or spherical shell with volumetric heat production is given by

$$\frac{k}{r^2}\frac{d}{dr}(r^2\frac{dT}{dr})+\rho H = 0$$

where k is thermal conductivity, H is the heat production per unit mass and $$\rho$$ is the density

Homework Equations

See above

The Attempt at a Solution

My first attempt was to start off with the 3-dimensional fourier's law equation with a steady-state case aka

$$\nabla^2 T = -\frac{A}{k}$$
$$\frac{d^2T}{dr^2} = -\frac{4\pi r^2}{k}$$
$$\frac{d}{dr}(\frac{dT}{dr}) = -\frac{4\pi r^2}{k}$$
$$\int \frac{d}{dr}(\frac{dT}{dr}) dr = \int -\frac{4\pi r^2}{k} dr$$
$$\frac{dT}{dr} = -\frac{4\pi r^3}{3k}$$

At this point I'm totally lost as to how to proceed. Have I even got the right equation to start with?

 

[jdwood983]

Doesn't look right to start there. Almost seems like you want to take the answer and find the starting point, which might not get you anywhere.

Consider the shell having inner radius of $$r$$ and total thickness $$\delta r$$. The heat flow out of the surface is given by
$$4\pi\left(r+\delta r\right)^2q_r(r+\delta r)$$

while the heat flow into the surface is
$$4\pi r^2q_r(r)$$

The total heat flow out will be the first minus the second equation. If you then Taylor expand the first equation about the variation $$\delta r$$, you should see the direction which you need to go to finish the problem

 

[gabbagabbahey]

$$\nabla^2 T = -\frac{A}{k}$$
$$\frac{d^2T}{dr^2} = -\frac{4\pi r^2}{k}$$

You seem to be claiming that $\nabla^2T(r)=\frac{d^2T}{dr^2}$...that's not really what you get for the Laplacian in spherical coordinates is it?

 

[jdwood983]

You seem to be claiming that $\nabla^2T(r)=\frac{d^2T}{dr^2}$...that's not really what you get for the Laplacian in spherical coordinates is it?

It is if there is no angular dependence, which he does seem to imply in the initial problem.

 

[gabbagabbahey]

It is if there is no angular dependence, which he does seem to imply in the initial problem.

No, it isn't...look up the Laplacian in spherical coordinates.

 

[jdwood983]

No, it isn't...look up the Laplacian in spherical coordinates.

I think we may be disagreeing on different aspects. I was talking about the use of $$d$$ instead of $$\partial$$ and not the form while you were talking about the form and not the use of $$d$$.

 

[gabbagabbahey]

I think we may be disagreeing on different aspects. I was talking about the use of $$d$$ instead of $$\partial$$ and not the form while you were talking about the form and not the use of $$d$$.

Okay, but that wasn't very clear from your last post.

 

[protonchain]

Sorry about the late response guys, I got held up at work.

So I followed your instructions jdwood, and I see how you had come up with the equation that you came up with. This is how far I've gotten.

$$4\pi\left((r^2+\delta r^2)\left(-k\frac{dT}{dr}+\delta r \left(-k\frac{d^2T}{dr^2}\right)\right) + r^2 k\frac{dT}{dr}\right) = q_{total}$$

I am unsure of how to proceed from this part though.

Is the $$q_{total}$$ = some constant value?

Sorry if I'm asking the stupidest questions.

 

[gabbagabbahey]

I don't see why you are deriving the heat conduction equation...surely you've already seen it done in your textbook/class? If so, just use it.

$$\rho H=-k\nabla^2T$$

Have you not seen the above form before?

 

[protonchain]

I have seen the above, but the problem is that because this is spherical, we need to start from the beginning and render the equation proper. At least that's how the prof wants to see it on the homework.

A HUGE thank you to jdwood for that massively helpful hint.

I just set your equation to what I should have realized it was right from the beginning

$$\rho H (4\pi r^2)\delta r$$

I was able to derive it using just basic algebra, taylor expansion, and assumption that $$\delta r^2 = 0$$.