Conduction through a spherical shell.

[kristen32123]

How do I solve for the temperature at a certain time for conduction through a spherical shell? I have a shell 5 cm thick with a radius of 0.5m. I am starting with Q=KA(T2-T1)/L and I am completely lost as to where to go with it.

Thanks!

 

[Mentz114]

Imagine the sphere has lines of latitude and longitude drawn on it. The (differential element of ) area of a rectangle defined by these lines is ( working in polar coords $r, \theta, \phi$)

$$dA = rd\theta*rsin\theta d\phi$$

so, using your formula

$$dQ = rd\theta*rsin\theta d\phi*K(T_1-T_2)/0.05$$

Now integrate out $d\theta, d\phi$ between 0 and 2*pi, and that gives the formula for the whole shell.

Of course you don't actually need to integrate - just think about it and the answer is obvious.

This solution is not exact because we're ignoring the difference in area btween the outer and inner surfaces ...

To solve exactly one needs a differential dL of thickness. Not a trivial problem.

 

[baba89]

To solve for the temperature at a certain time for conduction through a spherical shell, you can use the heat equation, which is derived from the Fourier's Law of Heat Conduction. The heat equation is given by:

Q = -kA (dT/dr)

Where Q is the heat flow, k is the thermal conductivity of the material, A is the surface area, and dT/dr is the temperature gradient.

To solve for the temperature at a certain time, you will need to use the initial and boundary conditions of the problem. In this case, the initial condition would be the temperature at time t=0 and the boundary condition would be the temperature at the inner and outer surfaces of the shell.

To apply the heat equation to a spherical shell, you will need to use the spherical coordinate system. The heat equation in spherical coordinates is given by:

Q = -kA (1/r^2)(d/dr)(r^2dT/dr)

Where r is the radial distance from the center of the sphere.

To solve for the temperature at a certain time, you can use numerical methods such as finite difference or finite element methods. Alternatively, you can use analytical solutions for simplified cases such as steady-state or one-dimensional heat transfer.

In your specific case, you have a spherical shell with a thickness of 5 cm and a radius of 0.5m. You will need to determine the thermal conductivity of the material and the initial and boundary conditions to solve for the temperature at a certain time. It is recommended to consult a heat transfer textbook or consult with a heat transfer expert for more specific guidance on solving this problem.