Physics Asked by user239504 on February 26, 2021

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

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

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

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):

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

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

- 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:

$$U_t=T_ttext{ and } U_r=T_rtext{ and }U_{rr}=T_{rr}$$

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

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

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

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.

Correct answer by Gert on February 26, 2021

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