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Heat Radiation: Is Balance Fastest
2

Heat Radiation: Is Balance Fastest

Heat Radiation: Is Balance Fastest

(OP)
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?

RE: Heat Radiation: Is Balance Fastest

(OP)
No, I'm not a student, though I wish I were, because then I could simply ask my professor.  No, my college days are long gone, but the answer to this question is of importance to me.  

Do you know the answer?  Do you know of an easy way to find out the answer?

RE: Heat Radiation: Is Balance Fastest

Sure, how must the temperature be manifested to achieve the same heat energy as the uniform temperature object?

TTFN

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RE: Heat Radiation: Is Balance Fastest

Radiation energy emitted per unit of time and unit of surface area is proportional to temperature^4 (fourth power).  If one has a non-uniform temperature it must has areas that are hotter than the uniform object (and, obviously, some colder) .  The fourth power on temperature means that the higher temp areas on the non-uniform object out-weigh the lower temp areas on the same object and, thus, this object radiates heat away faster.  This is the Stephan-Boltzmann law.

RE: Heat Radiation: Is Balance Fastest


This is a tricky unsteady state heat transfer situation, in which temperature is a function of both space and time.

A basic assumption would be that the heat content to be lost by both bodies is equal and radiates towards a black body absorbing all the incoming radiation.

It is generally assumed that a hot body cools down to a certain heat content level in a longer time period that the same body starting at a lower temperature.

Considering the unequally heated body, part of the heat of the hotter side would most probably be internally transmitted by conduction to its colder part.

I may be wrong, but it seems to me that when the hotter surface cools down to a certain temperature level, the total cooling effect by radiation from both of its surfaces may slow down to a pace similar to that of the original homogeneously heated body.

Thus, qualitatively speaking, the answer to PentagonJohn's question seems to be that the unequally heated body may take equal or longer periods to cool down to a given heat content level.

Please correct me if I'm wrong.

 



 

RE: Heat Radiation: Is Balance Fastest

Without more details I would say "It depends".

As noted by Bribyk, body 1 will have a higher radiant heat loss per unit area of its hot portions than body 2.    So what are the respective areas?  How hot are they wrt each other and the surrounding environment?

Both bodies will transfer heat internally by conduction.  How fast is conduction compared to the radiant loss?  How big are the bodies?

So without knowing the initial conditions, the question can't be answered.


 

RE: Heat Radiation: Is Balance Fastest

The case you cite is exactly the typical case of a space vessel, one side facing the sun, and the other facing space. The temperature of the body in steady state can be controlled to roughly earth temperatures by selectively specifying the absorptivity of each side, as a function of wavelength.  

RE: Heat Radiation: Is Balance Fastest

He doesn't mention incident radiation on the bodies ("radiating their heat away") so I was assuming internal heat generation or an initial temperature distribution.  Unless the material is infinitely conductive and the uneven distribution instantaneously assumes an even distribution my answer stands (and you don't really have a question, you have two identical bodies under identical conditions - no difference).  It's a yes or no question, he doesn't provide enough info for, nor appear to need a value.

RE: Heat Radiation: Is Balance Fastest

(OP)
I apologize for the omission of said information.  I am not an expert in heat radiation so it did not occur to me cover other details.

For clarification, I was simply interested in heat radiating away from bodies #1 and #2.  

Which one would radiate all its heat away first, as in heat bleeding off into space, or into the vacuum.

Assuming no other heat generation.  The bodies start with the same amount of heat energy (one with the heat perfectly evenly distributed and everything the exact same temperature, the other with the heat unevenly distrbuted)

Assume the bodies are spherical, like planets, or soccer balls, or does it even make a difference?

I just want to know if the evenness of the distribution of the heat energy will affect the loss of heat radiated into space overall such that one radiates all its heat faster than the other.

RE: Heat Radiation: Is Balance Fastest

No, but in the abstract problem where the heat is nonuniform, and nonmixing, and the material is thermal conductive otherwise, then the nonuniformly distributed heat will radiate more heat as posted earlier.

TTFN

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RE: Heat Radiation: Is Balance Fastest

Good Morning. Yes, i am gree with 25362, you have for this case an unsteady state heat transfer situation on body 1. Part of its energy will transfer from hot side to cold side, because no uniform temperataure distribution in the body 1. In other hand the body 2 will emits energy to the body 1 and the body 1 emits to the body 2 and make exchanger energy.  the thermal balance reach when the energy emited in both bodies are the same, AE=0. so  i think the same time.
Please correct me if I'm wrong.
thanks

RE: Heat Radiation: Is Balance Fastest

I don't think PentagonJohn is saying the objects are in the same space at the same time.  He's just looking for the individual cases.

RE: Heat Radiation: Is Balance Fastest

(OP)
Further clarification:

The bodies are NOT together.

Which case (case 1 or case 2) will radiate away all its heat first?

I realize the case 1 (nonuniform heat distribution) will have points that radiate heat at the fastest rate (the warmer points) but it will also have points that radiate heat away very slowly.  

I'm wondering if the uniform heat distribution will lose all its heat BEFORE any non-uniformly distributed case, or if it will lose all its heat AFTER those non-uniformly distributions.

- JCP

RE: Heat Radiation: Is Balance Fastest

Again, assuming no mixing between the halves, the unbalanced one will win.  

Analytically, the rate expression is proportional to:

{(heat - delta)4 + (heat + delta)4}/2

compared with just heat4

When you do the expansions, the terms with even powers of delta will all be positive, so the overall rate for the unbalanced case is always higher.

TTFN

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RE: Heat Radiation: Is Balance Fastest


I remember having read in this forum some time ago about the comparison of cooling times between a hot cup of tea and a less hot one, both equal in all other respects.

Discarding any natural air convection effect, it seems the warmer one takes longer to cool because along the way it has to reach the temperature level of the cooler analog whose cooling time should then be added.

Any comment ?

RE: Heat Radiation: Is Balance Fastest

The usual hot cup and cooler coffee problem is when to add cream to the originally hot coffee, immediately, or at your destination.  A crude numerial analysis says that the uncreamed coffee winds up being about about 0.25°C warmer at the end.

TTFN

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RE: Heat Radiation: Is Balance Fastest


QED.

RE: Heat Radiation: Is Balance Fastest

... doesn't it depend whether the cream is ambient or refrigerated at T0?

- Steve

RE: Heat Radiation: Is Balance Fastest

Marginally so, but the cream is assumed to be a smaller quantity in a smaller container.  Its heat transfer rates are all much slower.

TTFN

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RE: Heat Radiation: Is Balance Fastest

(OP)
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?

RE: Heat Radiation: Is Balance Fastest

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.

RE: Heat Radiation: Is Balance Fastest

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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RE: Heat Radiation: Is Balance Fastest

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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