# Temperature evolution of a Heated Sphere (high Biot number)

A homogeneous sphere is uniformly heated (or cooled) while the boundary is kept at constant temperature.

How does its temperature evolve in time and how is it distributed spatially?

Physics Asked on November 21, 2021

Newton's Law of heating or Fourier's Heat Equation?

$$text{Bi}=frac{Rh}{k}$$

(where $$R$$ is the radius, $$h$$ the convection coefficient and $$k$$ the thermal conductivity) internal temperature gradients $$frac{partial u}{partial r}$$ will be small and Newton's Law of heating (so-called 'lumped thermal analysis') can be used.

• But when $$text{Bi}$$ is high, spatial temperature distribution becomes uneven and we need to use Fourier's Law of heat conduction.

For a sphere with complete symmetry, we're looking for a function $$u(r,t)$$ that satisfies:

$$frac{partial u}{partial t}=frac{alpha}{r^2}frac{partial}{partial r}Big(r^2frac{partial u}{partial r}Big)+q$$ with $$q$$ the heat source. Boundary and initial condition: $$u(R,t)=0text{ and }u(r,0)=f(r)$$ (Attentive readers may now wonder about a 'missing' boundary condition) where $$alpha$$ is the thermal diffusivity: $$alpha=frac{k}{rho c_p}$$.

Developed and using shorthand: $$u_t=frac{2alpha}{r}u_r+alpha u_{rr}+qtag{1}$$ The problem now is that $$(1)$$ is not homogeneous, so separation of variables doesn't work here.

To try and homogenise it we define: $$u(r,t)=u_E(r)+v(r,t)$$ where $$u_E(x)$$ is the steady state temperature, for $$u_t=0$$: $$u_t=0 Rightarrow u_E(r)$$ From $$(1)$$: $$alpha ru''_E+2alpha u'_E+qr=0$$ Which solves to: $$u_E(r)=frac{c_1}{r}+c_2-frac{qr^2}{6alpha}$$ Note that: $$rto 0 Rightarrow u_E(0)to +infty Rightarrow c_1=0$$ (this was our 'hidden' boundary condition) $$r=Rrightarrow u_E(r)=frac{q}{6alpha}(R^2-r^2)$$ Now remember that: $$u(r,t)=u_E(r)+v(r,t)tag{2}$$ Let's calculate some derivatives: $$u_t=0+v_t$$ $$u_r=u'_E(r)+v_r$$ $$u_{rr}=u''_E(r)+v_{rr}$$ $$u'_E=-frac{qr}{3alpha}Rightarrow u''_E=-frac{q}{3alpha}$$ Insert it all into $$(2)$$: $$u_t=frac{2alpha}{r}(-frac{qr}{3alpha}+v_r)+alpha(-frac{q}{3alpha}+v_{rr})+q$$ $$Rightarrow v_t=frac{2alpha}{r}v_r+alpha v_{rr}$$ So the PDE in $$v(x,t)$$ is homogeneous. Checking also the boundary condition:

$$u(R,t)=u_E(R)+v(R,t)=0text{ with } u(R,t)=0 Rightarrow v(R,t)=0$$

So the boundary condition remain homogeneous.

Separation of variables can now be executed. Ansatz: $$u(r,t)=R(r)Theta(t)$$ $$frac{Theta'}{alpha Theta}=frac{R''}{R}+frac{R'}{rR}=-lambda^2$$ $$frac{Theta'}{alphaTheta}=-lambda^2$$ $$Theta(t)=exp(-alphalambda^2 t)$$ $$frac{R''}{R}+frac{R'}{rR}=-lambda^2$$ $$rR''(r)+R'(r)+lambda^2rR(r)=0$$ This solves to: $$R(r)=c_1J_0(lambda r)+c_2Y_0(lambda r)$$ Where $$J_0$$ and $$Y_0$$ are the Bessel functions.

Note that for: $$r to 0 Rightarrow Y_0 to -infty Rightarrow c_2=0$$ $$R(R)=0=J_0(lambda_n R)$$ $$lambda_n R=z_n$$ The roots $$z_n$$ of the first Bessel function are:

$$R(r)=c_1J_0(lambda_n R)$$ $$u_n(r,t)=C_nexp(-alphalambda_n^2 t)J_0(lambda_n R)$$ With the superposition Principle: $$u(r,t)=displaystylesum_{n=1}^{infty} C_nexp(-alphalambda_n^2 t)J_0(lambda_n R)$$ Initial condition: $$u(r,0)=u_E(r)+v(r,0) Rightarrow v(r,0)=f(r)-u_E(r)$$ $$v(r,0)=f(r)-u_E(r)=displaystylesum_{n=1}^{infty} C_nJ_0(lambda_n R)$$ So that: $$C_n=frac{2}{R}int_0^R[f(r)-u_E(r)]J_0(lambda_n R)text{d}r$$ Putting it all together: $$boxed{u(r,t)=frac{q}{6alpha}(R^2-r^2)+displaystylesum_{n=1}^{infty} C_nexp(-alphalambda_n^2 t)J_0(lambda_n R)}$$

Answered by Gert on November 21, 2021

## Related Questions

### Are Newton’s first and third laws contradictory?

2  Asked on November 29, 2020 by pyridine

### Yang-Mills massive ghosts

0  Asked on November 29, 2020

### Measuring the one-way speed of light?

2  Asked on November 29, 2020 by user276997

### Quantum information = quantum gravity?

1  Asked on November 29, 2020 by redhood

### Superfluids in areogel and porous media: why?

1  Asked on November 29, 2020

### Possible Error in Marion and Thornton’s Classical Dynamics of Particles and Systems

1  Asked on November 29, 2020 by elvis

### Binnig’s fractal evolution applied to multiple universes?

1  Asked on November 28, 2020

### Lorentz invariance of the Klein-Gordon equation action

1  Asked on November 28, 2020 by arthur

### What is meant by the component sinusoids being “correlated”?

1  Asked on November 28, 2020

### Why is an RG fixed point scale invariant?

1  Asked on November 28, 2020 by jiahao-mao

### Tilted screen in Young’s double slit experiment

1  Asked on November 28, 2020

### Plotting the magnetic susceptibility of the mean field Ising model

0  Asked on November 28, 2020

### Proving parallel rays converge to focus using Fermat’s principle for a concave mirror

0  Asked on November 28, 2020 by examination12345

### What is momentum really?

4  Asked on November 28, 2020

### Fermi energy in conduction band while valence band and conduction band have no overlap

1  Asked on November 28, 2020 by stratofortress

### How to produce a given entangled state of two quantum bits?

1  Asked on November 28, 2020 by suissidle

### Skyrmion number

1  Asked on November 28, 2020 by eric-z

### The travel of soundwaves

1  Asked on November 28, 2020 by 00xxqhxx00

Get help from others!