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?

 

[IRstuff]

This sounds like homework. Student posting is not allowed.

TTFN

FAQ731-376

 

[PentagonJohn]

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?

 

[IRstuff]

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

TTFN

FAQ731-376

 

[Bribyk]

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.

 

[25362]

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.

 

[MintJulep]

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.

 

[davefitz]

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.

 

[Bribyk]

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.

 

[PentagonJohn]

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.

 

[IRstuff]

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

FAQ731-376

 

[Prestz]

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

 

[Bribyk]

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.

 

[PentagonJohn]

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

 

[IRstuff]

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

FAQ731-376

 

[25362]

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 ?

 

[IRstuff]

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

FAQ731-376

 

[25362]

QED.

 

[SomptingGuy]

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

- Steve

 

[IRstuff]

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

FAQ731-376