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 ?


1 Answer 1


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? $\endgroup$ Sep 22, 2015 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 $\endgroup$ Sep 22, 2015 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, 2015 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, 2015 at 16:45
  • $\begingroup$ @AbanobEbrahim: I've added a link with full derivation. $\endgroup$
    – Gert
    Sep 22, 2015 at 17:04

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