Net radiation between two plates with different emissivities

[shyta]

Hi guys, I need help on my concept on the steffan's law.

I understand that for a grey body sphere confined within a blackbody spherical shell, the general formula for the net power radiated from inner sphere should be

P_rad = kAe(T⁴-T_surr⁴)
where k=stefan Boltzmann's constant

However recently, I was given a question which made me really confused.
For the case where 2 plates are separated by a gap and the left plate has a higher temp, T₁ and emissivity, e₁ = 1, as compared to the right plate having T₂ and emissivity e₂. Then the net rate of power transferred from the left to the right is?

P_{rad by left plate} = kAe₁T₁⁴
P_{rad by right plate} = kAe₂T₂⁴
which all of it is absorbed by the left plate since emissivity is 1

then fraction absorbed by right plate is e₂
and fraction reflected by right plate is 1-e₂(all absorbed by the left plate)


P_{net flow from left to right} = kA (e₁T₁⁴e₂ - e₂T₂⁴)

But how??

 

[shyta]

anyone? :(

 

[cesiumfrog]

P_{net flow from left to right} = kA (e₁T₁⁴e₂ - e₂T₂⁴)

But how??

I agree, net power transfer is $Ake_2(T_1^4-T_2^4)$. What's the problem?

 

[shyta]

why is it e_2 and not e_1?

 

[cesiumfrog]

Why is which e_2? Why wouldn't it be e_2?
It would really help if you explained in detail how much you do understand. Which parts did you work out yourself, and which parts have you simply copied from somewhere else? Please show your own work more clearly. Explain with more detail exactly what it is that you don't understand. And is this part of a homework question?

 

[shyta]

No it's not part of a homework.

isn't the emissivity used supposed to be the one of the object, as in for the 2 plates situation, the emissivity of the left plate (e=1?)

what i am confused about is why there is a need to multiply by the fraction absorbed by the right plate when finding the net transfer?

 

[cesiumfrog]

If you want my help, it would be great if you helped me more to help you. Are you saying that you yourself figured out everything except the second last line of your original post, but that second last line you took from a book and do not know how to obtain yourself?

Some components to consider:
- the radiation from 2 to 1.
- the fraction of the radiation from 1 that is absorbed by 2.
Note these are in opposite directions. Do you know how to combine these into the "net"?

Alternatively:
- the initial radiation from 2, in one direction.
- all the radiation from 1, in the other direction.
- the part, of the radiation originally from 1, that reflects back from 2 to 1 again, in the first direction.
This should also give the same answer.

 

[shyta]

yes i did take it from something like a book.

Prad by left plate = kAe₁(T₁⁴-T₂⁴)


then we have to consider the fraction reflected by the right plate which got absorbed back by the left plate which is 1-e₂

which is the part I'm confused about

 

[cesiumfrog]

What is "like a book"? Please cite your source.

What does "Prad" mean?

Can you calculate exactly how much power, in Watts, the right plate absorbs? (And exactly how much power reflects back at the left plate?)

Is it easier to solve the problem if I choose e_2 = 0.25 ?