Heat Radiation: Is Balance Fastest 2

[PentagonJohn]

Given two identical bodies with identical amounts of heat energy, both in a vacuum, radiating their heat away.

Body #1 is of non-uniform temperature (let's presume it's hot mostly on one side and cooler on the opposite side)

Body #2 is of uniform temperature.

Which one will radiate its heat away fastest and why? Does the Stefan-Boltzmann Law explain this?

 

[PentagonJohn]

The problem I have with concluding that the uneven disbursement loses its heat first is simply what was mentioned above, that the warmer temperatures first need to cool to the temperature of the cooler in order to "catch up."

I understand that the warmest sections of the uneven distribution will, at first, be radiating at a faster *RATE* than at any point on the completely evenly distributed case, but only for as long as it takes to cool to that initial temperature of the evenly distributed case, at which point it will be that much time *BEHIND*.

We could then focus on the *COOLEST* sections of the unevenly distributed case and say that they are "ahead" in the sense of losing their heat (while admitting that they are radiating at the SLOWEST rate of all).

It would seem to me that the coolest areas of the uneven distribution would ultimately lose all their heat first, followed by the entirety of the even distribution case, followed lastly by the originally warmest sections of the uneven distribution case.

Conclusion, the even distribution case loses its all its heat within a shorter time period than that of the uneven distribution case.

Am I missing anything?

 

[IRstuff]

You should do the math, then.

TTFN

FAQ731-376

 

[chicopee]

To keep in tune with 25362 for the unsteady state condition, and assuming that there is no temperature gradient in both bodies as they cool down, I think that you can determine your answer by the relationship of M*Cp*(Delta T/Delta t)=s*A*(T^4-Ta^4) where T is temp;Ta is ambient temp.;t is time;s=boltzeman constant;A=surface area; M is mass. The body with two temperature zones would be assumed to have half the mass of the entire body and would be separated by a non conductive material.

 

[electricpete]

I think the point made by Bribyk was important and I'm not sure if it was discussed much.

Let's say we have one copper cylinder at 50C uniform temperautre.

The other copper cylinder has non-uniform temperautre at 50C + 10C * cos(theta).

The above conditions are t=0. Clearly at t=0 the non-uniform cylinder radiates more total heat for reasons stated by Bribyk (I assume we are limited to radiation and convection is not relevant).

That is t=0. The temperature distributions will evolve in a complex manner after that but the non-uniform distribution initially has an advantage in heat radiation and I suspect it would be hard to contrive a scenario where it wouldn't win the race.

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[electricpete]

I guess I should mention that winning the race in my book means having a lower remaining total heat after a specified time. If our criteria for winning was something else like all regions below T = -20C, then that might be a factor giving advantage to the initially uniform cylinder. (don't know which advantage would be bigger).

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