How Does Thermal Conductivity Affect Heat Distribution in Spherical Sources?

[magicuniverse]

Homework Statement

For a metal bar of cross sectional area A, the rate of flow of heat along the bar is given by the expression:

Q=-KA(dT/dx)

where K is the thermal conductivity of the material of the bar, and T and x refer to temperature and the distance measured from the high temperature end of the bar respectively. Use this information to solve the following problem.

A spherical heat source of radius a sits at the centre of a solid sphere of radius b>a.

The material of the sphere has thermal conductivity K. The source emits heat equally in all directions at the rate Q per second and the outside of the sphere is held at a constant temperature T0. Determine the temperature T at the surface of the source.


Some help needed please.

 

[cristo]

What work have you done thus far?

 

[tacman]

To solve this problem, we can use the equation given in the homework statement for the rate of flow of heat along a bar. In this case, we can consider the solid sphere as a bar with a very small cross-sectional area at the center, where the heat source is located.

First, let's define our variables. The radius of the heat source is given as a, the radius of the solid sphere is b, and the thermal conductivity is K. We also know that the heat is being emitted equally in all directions, so we can assume that the rate of heat flow is constant throughout the sphere. Let's also define the distance x as the distance from the center of the sphere to any point on its surface.

Using the equation Q=-KA(dT/dx), we can set up an integral to find the temperature at the surface of the heat source, T.

∫dQ = -K∫A(dT/dx)dx

Since the heat is being emitted equally in all directions, the total heat emitted from the heat source will be equal to the rate of heat flow multiplied by the surface area of the sphere, 4πa^2. Therefore, we can replace dQ with Q, and A with 4πa^2 in the integral.

Q = -K∫4πa^2(dT/dx)dx

Next, we need to find the limits of the integral. The distance x ranges from 0 (at the center of the sphere) to b (at the surface of the sphere). Therefore, our integral becomes:

Q = -K∫0b4πa^2(dT/dx)dx

Integrating, we get:

Q = -K4πa^2(T-T0)

Finally, we can solve for T by dividing both sides by 4πa^2 and adding T0 to both sides:

T = T0 - Q/(4πa^2K)

Therefore, the temperature at the surface of the heat source is given by T = T0 - Q/(4πa^2K).

I hope this helps! Let me know if you have any other questions.