Heat transfer by radiation between two plates

[Karol]

Homework Statement

Two large plates are parallel and close to each other, vacuum is between them. they are held at 200⁰K and 300⁰K.
What is the energy transfer rate between them.

Homework Equations

The energy emission rate per unit area-the Stefan-Boltzmann Law: $$R=\varepsilon\sigma T^4$$
$\varepsilon$ is the emissivity of the surface
$\sigma=5.672E-8[w/sec]$

The Attempt at a Solution

The solution is 368[w/m²], it's the answer if the emission is of a black body:
$1\cdot 5.672E-8\cdot\left(300^4-200^4\right)=368$
But why don't the emissivities of the surfaces count? what if one surface emits more or less than the other?
In the Stefan-Boltzmann formula the outer temperature exists, not the nature of the surrounding surfaces.
I guess it has something to do with the fact that the rest of the radiation, which isn't absorbed, is reflected and comes into account

 

[mfb]

But why don't the emissivities of the surfaces count?

They do count, and the problem statement should specify that ideal black bodies are meant. The sign of the energy transfer cannot change, but the overall magnitude of exchanged energy can go down.

 

[Karol]

The sign of the energy transfer cannot change, but the overall magnitude of exchanged energy can go down.

What do you mean by "sign"?

 

[haruspex]

What do you mean by "sign"?

Meaning which way the net flow goes.
If one body has lower emissivity than that of a black body then it both emits less and absorbs less. Different emissivities cannot result in the net flow being from the cooler to the hotter.

 

[Karol]

Of course not from the cooler to the hotter, but different emissivities must mean smaller energy flow. it doesn't matter which plate's emissivity is smaller, right?

 

[haruspex]

Of course not from the cooler to the hotter

You say 'of course', but if the emissivity affected only the emission rate and not the absorption rate then you could have the paradoxical situation that a hot body with low emissivity could gain energy from a cooler body.

but different emissivities must mean smaller energy flow. it doesn't matter which plate's emissivity is smaller, right?

It's not that they're different, just that one, the other or both are lower than a black body's.

 

[Karol]

Thanks

 

[Karol]

Why in an insulated chamber the emission of a body and the emission of the walls is black body? why don't we take into account the emissivity of the walls?

 

[mfb]

Why in an insulated chamber the emission of a body and the emission of the walls is black body?

It is not.
The energy density in the cavity does not depend on the emissivity, but the energy exchange rate with the objects does.

 

[Karol]

The energy density in the cavity does not depend on the emissivity

Why? if the walls inside emit low radiation, because of small emissivity, then the energy density in the opening will also be low, no?

 

[haruspex]

Why? if the walls inside emit low radiation, because of small emissivity, then the energy density in the opening will also be low, no?

No. This is exactly the same point as for the two bodies at different temperatures. If the emissivity is low then the absorption from the cavity is correspondingly low. Equilibrium will still be achieved when the energy density corresponds to the temperature of the enclosure.