I am trying to find a way with which I can find out the temperature of a certain point inside a body undergoing thermal conduction. I understand that this might not be very simple to calculate since the temperature difference changes with time and so the rate of heat transfer changes too.

To make this clear, I am trying to find the temperature variation with time at the center of a steel sphere that is 10 centimeters in radius that starts at t=0 with surface temperature of 500 °K, knowing that the heat capacity of steel is 500 J/kg.K and its thermal conductivity is 55 W/(m.K). So is there a way that is simple enough to do this ? Or are there any simulators on the internet that can do the job ?


Because of the high degree of symmetry, this is actually a relatively simple problem.

Use the 3 dimensional heat propagation equation and transform to spherical coordinates.

At the boundary steel-air, calculate the overall heat loss as:

$\frac{dQ}{dt}=-hA(u_R(t)-u_0)$ where $h$ is the heat transfer coefficient (estimates of which you'll find for simple problems on the Internet), $A$ the outside surface area of the sphere, $u_R(t)$ the temperature of the surface of the sphere (in time $t$) and $u_0$ the ambient temperature.

You'll also have to define what really means:

with surface temperature of 500 °K

Was the ball isothermal at the start of cooling or was the centre hotter than the edge?

You can also find the full derivation for the quenching of a sphere here - *pdf (point 4.3 and 4.4). For heating, reverse the relevant signs.

  • $\begingroup$ I am assuming that the entire sphere was at room temperature initially, and then the sphere is transferred inside say an oven with temperature of 500 °K. But in this case, what would $u_R(t)$ and $u_0$ be? $\endgroup$ – Abanob Ebrahim Sep 22 '15 at 16:33
  • $\begingroup$ And by the way to clear this more, I am trying to see how the center will heat when the outside is higher in temperature and not vice versa. Sorry if I didn't explain that well in the main question $\endgroup$ – Abanob Ebrahim Sep 22 '15 at 16:41
  • $\begingroup$ @AbanobEbrahim: in that case $u_0=500 \text{ K}$. Let's call $r$ the distance from the centre and $R$ the total radius of the sphere. Solving the differential equation then gives a function $u(r,t)$. At $t=0$, $u_R(t)$ would be ambient temperature. $\endgroup$ – Gert Sep 22 '15 at 16:44
  • $\begingroup$ @AbanobEbrahim: there is a much more simple but more approximate approach by assuming the whole of the ball's temperature is homogeneous (but increasing). $\endgroup$ – Gert Sep 22 '15 at 16:45
  • $\begingroup$ @AbanobEbrahim: I've added a link with full derivation. $\endgroup$ – Gert Sep 22 '15 at 17:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.