Diffusion heat on an egg

Question

Let suppose that I have a egg at $T=20ºC$ and I assume it's almost an sphere of radius $R$, let's call it surface $S$. I put the egg inside a bath with water at $T_w=100ºC$. I want to know the temperature of the egg as a function of time and position. I already know that have to solve diffusion equation for the heat in spherical coordinates

$$ \left( \frac{\partial}{\partial t}-\chi\cdot \nabla^2 \right) T(t,r,\theta,\varphi)=0 $$

The problem is that I'm not sure about the initial conditions, because when I put the egg inside the bath, in the boundary are at two temperatures, witch one should I use? $T(t=0,\vec{r} \in S)=20 ºC$, $T(t=0,\vec{r}\in S)=100 ºC$ or some superposition?

From the point of view of physics the egg surface initially have to be at $T(t=0,\vec{r} \in S)=20 ºC$, but I'm not sure from the point of view of mathematics and PDE theory

Answer 1

Firstly, due to symmetry you're looking for a function:

$$T(t,r)$$

because the geometry is radially symmetric. So the angles $\theta$ and $\varphi$ do not matter because:

$$\partial_{\varphi}T(t,r,\theta,\varphi)=\partial_{\theta}T(t,r,\theta,\varphi)=0$$

Fourier's heat equation then reduces to (in PDE shorthand):

$$T_t=\alpha\Big(\frac{2}{r}T_r+T_{rr}\Big)$$

As regards the boundary conditions (and NOT the initial condition, see below), you have some choices to make:

1. elect to have the outer layer of the egg at $100 ºC$, at all times:

$$T(t,R)=100$$

where $R$ is the radius of the egg.

This is very convenient because with a small transformation of the dependent variable $T$:

$$U(t,R)=T(t,R)-100$$

So that:

$$U_r(t,R)=0$$

So you have a homogeneous boundary condition. Solving the Fourier PDE then becomes an eigenvalue problem.

The partials remain the same:

$$U_t=T_t\text{ and } U_r=T_r\text{ and }U_{rr}=T_{rr}$$

This BC is quite realistic: an infinitesimally thin outer layer of egg would quickly reach $100 ºC$ and then stay there.

2. assume convective heat flow between the egg's shell and the water:

$$k\Big(\partial_{r}T(t,r)\Big)_{r=R}=-h[T(t,R)-T_{water}]$$

with $h$ a convective heat transfer coefficient and $k$ the thermal diffusivity.

This BC is a little more demanding, mathematically. But here too the transformation of $T$ as above is helpful.

As regards the initial condition ($t=0$), it is simply:

$$T(0,r)=20 ºC$$

Note: the problem of the conductive heating or cooling of a uniform sphere has been solved numerous times and googling will find those derivations.