Temperature as a function of time in black body radiation

[Daniel Sellers]

Homework Statement

2. Consider a metal sphere of radius R and heat capacity C, initially at a temperature To which is much hotter than the background temperature.

a) Derive an analytical result for the temperature of such a sphere as a function of time. Clearly state any simplifying assumptions.

b) Early models to calculate the age of the Earth presumed that the Earth had the heat capacity of iron3 and started out at a temperature of T ∼ 4000 K4. Using these assumptions, how long would it take Earth to cool to its current surface temperature of Tsurf ≃ 280K? Make a reasonable speculation about why this estimate fails so spectacularly.

c) For a sufficiently hot object, obeying the same assumptions as part (a) and the initial problem statement, what is the time scale of cooling to a background temperature T1, independent of the initial temperature?

I am only asking for help with part a)

Homework Equations

dU/dt = σA(T_background⁴ - T_sphere⁴)
C = dU/dT
dT/dt = (σA/C)(T_background⁴ - T_sphere⁴)

The Attempt at a Solution

This appears to be a separable differential equation, but not an easy one to solve. It is fairly straight forward to non-dimensionalize this equation and by assuming T_sphere/T_background >> 1.

But this leads to results which seem like they can't possibly be correct;

T_sphere = T_background²(σA/3tC)^1/3

I suppose there should also be an integration constant as well, but ignoring that for now;

I can't see how this result could be used to solve part b), attempting to do so results in a much too small time.

This equation is also undefined at infinity, possibly a result of the assumption I mentioned above?

I would appreciate it if someone could suggest a coherent approach to this problem as I have been struggling with it for a while now

 

[haruspex]

but not an easy one to solve.

Doesn't look hard to me. Use partial fractions.
By the way, you have already made a simplifying assumption to get that ODE. What is it?

 

[Daniel Sellers]

I'm not sure what you mean. The only assumption I'm aware I made is the one I stated.

Separating the equation is not difficult but rather integrating 1/(1-T_sphere⁴) leads to a result which I can't seem to solve for T_sphere. Is this where you're suggesting I use partial fractions?

Using my stated assumption to remove the 1 and integrating 1/(T_sphere⁴) leads to a result that is apparently incorrect.

 

[haruspex]

The only assumption I'm aware I made is the one I stated.

Think conduction.

Is this where you're suggesting I use partial fractions?

Yes. But on reflection, I doubt this is necessary. Ignoring the background temperature should be fine. However, that should mean the background temperature disappears from your equation, so I don't know how you got an equation with it still present. Did you mean T_final rather than T_background?

 

[Daniel Sellers]

Oh indeed you're correct, the original ODE is contigent on the assumption that conduction is much faster than radiation, so the sphere cools pretty much homogenously.

Perhaps I made a mistake, I'll check again. The Tbackground came from turning the Tsphere into a dimensionless temperature and then changing it back at the end of the calculation.

Either way, having an expression in terms of 1/t seems problematic, since this would make the temperature undefined at t = 0. Hence my confusion

 

[Daniel Sellers]

No, you're correct again, non-dimensionalizing the equation is unnecessary, by ignoring T_background completely I get the same equation I stated in the OP, without the T_background² term, still seems problematic

 

[haruspex]

No, you're correct again, non-dimensionalizing the equation is unnecessary, by ignoring T_background completely I get the same equation I stated in the OP, without the T_background² term, still seems problematic

What answer do you get now?

 

[Daniel Sellers]

At first the same answer without any term TSUB]background[/SUB].

On a different forum I found a totally new approach, which seems to work.

This is to write T(t) = a(t+b)ⁿ and then solving for the new constants. This gives an expression which seems much more reasonable.

Applying that expression to part b) of this problem gives a ridiculously small value for the age of the Earth (60,000 years) but then it is a very oversimplified estimate

 

[haruspex]

At first the same answer without any term [TSUB]background[/SUB].

So that would be Tsphere = (σA/3tC)^1/3?
Shouldn't it be (3σAt/C)^-1/3? That gave me 45000 years.

You are told that the calculation produces much too small a value.

 

[vela]

Either way, having an expression in terms of 1/t seems problematic, since this would make the temperature undefined at t = 0. Hence my confusion

That's because you've ignored the constant of integration.

 

[Daniel Sellers]

Using the method I mentioned above I arrive at the expression,

T(t) = (C/3σA)^1/3(t + [C/3σA][1/T_i³])^-1/3

Which gives 60,000 years unless I've made a calculation error.

vela, I could not see how the integration constant would stop the 1/t term from becoming undefined, unless it ended up as 1/(t+k).

I suppose that's equivalent to the round about method I ended up taking but I was getting lost in the algebra and calculus

 

[Daniel Sellers]

Think conduction.

Thanks for that note, I probably would have forgotten about that one and the problem specifically asks for all assumptions being used.

So that would be Tsphere = (σA/3tC)^1/3?
Shouldn't it be (3σAt/C)^-1/3? That gave me 45000 years.

Yes, that is the expression I got at first, I must have entered it wrong when I posted. But would also be undefined at t = 0

 

[vela]

vela, I could not see how the integration constant would stop the 1/t term from becoming undefined, unless it ended up as 1/(t+k).

I'm guessing you're solving for $T$ and then adding in the integration constant. You can't do that. You have to introduce the constant immediately upon integrating, and then solve for $T$.

Alternately, you could just use definite integrals:
$$\int_{T_0}^T \frac{dT}{T^4} = -\int_0^t \frac{\sigma A}{c}\,dt.$$

 

[haruspex]

But would also be undefined at t = 0

You are maybe misinterpreting t in that equation. It represents the period of time you are trying to find. It would not make sense to consider t=0.

 

[Daniel Sellers]

Wow, you are absolutely correct. That is an embarrassing rookie calculus mistake. There is no undefined expression, the sky is not falling.

Thanks very much to you both.