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I am modeling a situation in which an object that is a small object that is spherical in shape is receiving heat at a constant rate and then conducting it away into a medium that can be model as infinite in size relative to it (large glacier) such that the bulk temperature at infinity of the glacier is unchanged by the conduction. In a steady state, there should be some sort of temperature gradient set up in which all of the energy coming into the sphere is going out into the ice. in order to evaluate this Fourier's law of q = -k* dT/dr (assuming spherical symmetry such that all variations are only caused by variations in r and this becomes a 1D problem) should be integrated across the surface of the sphere, yielding dQ/dt = -kA* dT/dr. In order to evaluate dT/dr, the most common approach is to approximate it as (T1-T2)/L where L is the distance between the two different temperature sources. Unfortunately in the case where these two objects are "infinitely" far apart as L approaches a large number at which the bulk temp is unchanged relative to object, the quantity of (T1-T2)/L approaches 0, yielding that no heat would flow between the object and surrounding medium, which is unphysical.

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  • $\begingroup$ What is your goal? Are you trying to find the temperature everywhere inside the sphere? $\endgroup$ Jul 22, 2014 at 18:51
  • $\begingroup$ @user3814483 yes. I am attempting to find the equilibrium temp difference between the object and the medium at infinity $\endgroup$
    – Patrick
    Jul 22, 2014 at 19:11
  • $\begingroup$ Without doing the math explicitly, I believe there is no "steady state" solution when you have a heat sink at infinity. Instead you need to think of this as a diffusion equation which is solved (for arbitrary time t) with Fourier methods - or you have to set a (spherical) boundary condition at a sensible distance. $\endgroup$
    – Floris
    Jul 22, 2014 at 19:59

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To solve for the steady-state temperature everywhere, you need to solve the heat equation:

$$\frac{\partial T}{\partial t} - \chi \nabla^2 T = \frac{S}{c_p \rho}$$

Here, $\chi$ is the thermal diffusivity of the sphere in question, which is defined as the ratio $\kappa / c_p \rho$ (ratio of thermal conductivity to product of specific heat capacity and density). $S$ are sources and sinks.

Since you only care about steady state, the partial time derivative can be tossed out. In your case, I imagine you have some source that's providing the heat for your sphere, so you can plug that in. The sink is best treated using a Neumann boundary condition. Since you're at steady state, the heat in must equal the heat out. Fourier's Law provides you with the boundary condition:

$$\frac{dQ}{dt}_{in} = \frac{dQ}{dt}_{out} = 4\pi a^2 \kappa \frac{dT}{dr}$$

The last term on the right is evaluated at the boundary of your sphere. So the derivative you point out in your question is actually your boundary condition. You have some knowledge about the heat source, that must equal the heat that leaves the sphere per unit time (you have $dQ/dt$, so you have a constraint on $dT/dr$ which you'll apply when solving the heat equation).

You can find the Laplacian for spherical coordinates here. For a constant source (in space and time) the solution should be straightforward integration.

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