What mathematical techniques can be used to find the evolution of the temperature of a uniform sphere if it is subject to convective cooling at its surface? Are these techniques different from the case of conductive cooling?

Cooling sphere

For concreteness, assume the initial, uniform temperature to be $T_0$ and the surrounding temperature to be $T_e$. Radius is $R$.

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    $\begingroup$ I have attempted to edit the question to focus it more on the mathematical techniques involved, rather than seeming like a homework problem (in the hopes that it can then be reopened.) Feel free to roll the edits back if this conflicts with your intent. $\endgroup$ – Michael Seifert Aug 20 '20 at 14:05
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    $\begingroup$ Thank you, Michael. $\endgroup$ – Gert Aug 20 '20 at 15:00

What mathematical approach is preferred depends primarily on the Biot number $\text{Bi}$ of the system:

$$\text{Bi}=\frac{hR}{k}$$ where $h$ is the convective heat transfer coefficient, $k$ the thermal conductivity of the sphere's material and $R$ the sphere's radius.

  1. High $\text{Bi}$ number:

This means the heat loss proceeds primarily by convection and that temperature gradients inside the sphere are small:

$$\frac{\partial T}{\partial r} \approx 0$$

It also means that lumped thermal analysis can be applied here. We can use Newton's Law of Cooling:


with $T_e$ the environment's temperature (far away from the sphere's surface)

$(1)$ is an ODE with separation of vaiables and is easy to solve.

  1. Low $\text{Bi}$ number:

Here conduction prevails over convection and significant temperature gradients inside the sphere exist:

$$\frac{\partial T}{\partial r} < 0$$

Lumped thermal analysis is likely to cause significant error here and we need to use Fourier's Heat Equation.

Define $u(r,t)=T(r,t)-T_e$, so that with Fourier for a sphere:

$$\frac{\partial u}{\partial t}=\frac{\alpha}{r^2}\frac{\partial}{\partial r}\Big(r^2\frac{\partial u}{\partial r}\Big)$$ Make a subsitution: $$v(r,t)=u(r,t)r$$ We obtain: $$v_t=\alpha v_{rr}\tag{1}$$ BCs:


BC from convection: $$-ku_r(R,t)=hu(R,t)$$ Transcribed: $$-kv_r(R,t)=\Big(h-\frac{k}{R}\Big)v(R,t)\tag{2}$$

Initial condition: $$v(r,0)=(T_0-T_e)r$$ Going back to $(1)$:

$$v_t=\alpha v_{rr}\tag{1}$$

Using Ansatz and separation:

$$v(x,t)=R(r)\Theta(t)$$ allow determining $v(r)$ and thus $u(r)$.


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