Skip to main content
Physics

Return to Answer

Bounty Ended with 25 reputation awarded by CommunityBot
added 689 characters in body
Source Link
Ilja
Ilja
  • 2.5k
  • 15
  • 25

Why do these two methods give different results and which one is correct?

I think, bothBoth are incorrect; but especially the second more, I'd say. But asAs H.Tofaili pointed out, both often are very similar, since every of them makes some approximation which is notonly always allowed in some limit.

  • In your second approach you assume, that there is no heat conducted across the (imagined) cut between the cylinder and the hemispheres. But it will. It may be very little in most cases, but you can see it's there in an extreme case:

take a thick spherical shell and insert a very thin cylinder (great $R_f-R_i$, small $\ell$). It is in practice still a shell and will transport heat radially. Then with $\frac{R_i}{R_f} \rightarrow 0$ arbitrarily little of the heat starting in the cylinder will end up in it, it will all cross the border.

  • In your first approach you assume, that the surfaces of equal temperature are all of the same form and equally spaced. I don't see an obvious counterexampleThis example is not as convincing as the above (I mean, here the assumption doesn't get arbitrarily wrong), but... why should they? consinder:

Take now a thin ($R_f\approx R_i$) figure. The central parts of the cylinder and the hemispheres are far from the other form respectively, so the heat conduction in them is decoupled from each other and behaves just as in a cylinder or hemisphere. The mean temperature will be at the geometrical mean of the radii in the cylinder and at the harmonical mean of the radii in the hemisperes.

So every approximation has a limit in which it is bad. But I think, the first approximation is "better", it looks so "second-order", and as I said, I don't see how to make it arbitrarily wrong. But this get sheer intuition now...

Why do these two methods give different results and which one is correct?

I think, both are incorrect; but especially the second. But as H.Tofaili pointed out, both often are very similar, since every of them makes some approximation which is not always allowed.

  • In your second approach you assume, that there is no heat conducted across the (imagined) cut between the cylinder and the hemispheres. But it will. It may be very little in most cases, but you can see it's there in an extreme case:

take a thick spherical shell and insert a very thin cylinder (great $R_f-R_i$, small $\ell$). It is in practice still a shell and will transport heat radially. Then with $\frac{R_i}{R_f} \rightarrow 0$ arbitrarily little of the heat starting in the cylinder will end up in it, it will all cross the border.

  • In your first approach you assume, that the surfaces of equal temperature are all of the same form and equally spaced. I don't see an obvious counterexample as above, but... why should they?

Why do these two methods give different results and which one is correct?

Both are incorrect; but the second more, I'd say. As H.Tofaili pointed out, both often are very similar, since every of them makes some approximation which is only always allowed in some limit.

  • In your second approach you assume, that there is no heat conducted across the (imagined) cut between the cylinder and the hemispheres. But it will. It may be very little in most cases, but you can see it's there in an extreme case:

take a thick spherical shell and insert a very thin cylinder (great $R_f-R_i$, small $\ell$). It is in practice still a shell and will transport heat radially. Then with $\frac{R_i}{R_f} \rightarrow 0$ arbitrarily little of the heat starting in the cylinder will end up in it, it will all cross the border.

  • In your first approach you assume, that the surfaces of equal temperature are all of the same form and equally spaced. This example is not as convincing as the above (I mean, here the assumption doesn't get arbitrarily wrong), but consinder:

Take now a thin ($R_f\approx R_i$) figure. The central parts of the cylinder and the hemispheres are far from the other form respectively, so the heat conduction in them is decoupled from each other and behaves just as in a cylinder or hemisphere. The mean temperature will be at the geometrical mean of the radii in the cylinder and at the harmonical mean of the radii in the hemisperes.

So every approximation has a limit in which it is bad. But I think, the first approximation is "better", it looks so "second-order", and as I said, I don't see how to make it arbitrarily wrong. But this get sheer intuition now...

Source Link
Ilja
Ilja
  • 2.5k
  • 15
  • 25

Why do these two methods give different results and which one is correct?

I think, both are incorrect; but especially the second. But as H.Tofaili pointed out, both often are very similar, since every of them makes some approximation which is not always allowed.

  • In your second approach you assume, that there is no heat conducted across the (imagined) cut between the cylinder and the hemispheres. But it will. It may be very little in most cases, but you can see it's there in an extreme case:

take a thick spherical shell and insert a very thin cylinder (great $R_f-R_i$, small $\ell$). It is in practice still a shell and will transport heat radially. Then with $\frac{R_i}{R_f} \rightarrow 0$ arbitrarily little of the heat starting in the cylinder will end up in it, it will all cross the border.

  • In your first approach you assume, that the surfaces of equal temperature are all of the same form and equally spaced. I don't see an obvious counterexample as above, but... why should they?