Cooling of a Uniform Sphere with Convective heat loss at the boundary

Question

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?

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

Answer 1

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:

$$-mc_p\frac{\text{d}T(t)}{\text{d}t}=hA[T(t)-T_e]\tag{1}$$

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.

2. 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:

$$v_r(0,t)=0$$

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 $u(r)$.