Net thermal radiation from an object outside at night

[sian130]

I am trying to model the cooling of an object (for example, a sheet of glass) placed outside at night. At the moment I am only considering heat loss by radiation.

I know that the net radiation from the object will be:

R_net = R_obj - R_sky

where:
R_net = the net radiation from the object
R_obj = the total thermal radiation from the object
R_sky = the thermal downwelling radiation from the sky

I have come across a formula which does the above, but also takes into consideration the absorptivity of the sheet of glass at the 8-14μm wavelength (I have only considered emissivity of the glass in this wavelength as I'm only concerned with thermal radiation heat loss to the sky).

The formula is:

R_net = A((ε_obj²/α_obj²)σT_obj⁴ - ε_skyσT_amb⁴)

where:
ε_obj = the emissivity of the object at 8-14μm
α_obj = the absorptivity of the object at 8-14μm
ε_sky = the emissivity of the sky
T_obj = the temperature of the object
T_amb⁴ = the ambient air temperature
A = the area of the object

What I would like to know is where does the ε_obj²/α_obj² bit come from? I know if I wasn't considering absorptivity then it would just be ε_obj, but why are ε_obj and α_obj now squared? I have since lost where I saw it, and I'm pretty sure there wasn't an explanation there anyway. I have searched all over for a derivation of it but have had no luck.

Can anyone help?

 

[elegysix]

I am not familiar with this, but I would agree that something is off here:

Since the eqn is Rnet= Robj - Rsky

these terms should have equal forms.

But as you pointed out there is a squared ratio only on one term.
My guess is that it may be simplified, but who knows.

So I can't really answer your question, but I can say that I agree with it.
Perhaps try to find a different formula / approach to use?

I did find this though - a program for modelling thermal radiation. Looks like it is free to use, and mentions finding radiation from any given objects of given temperature. Maybe it will help. Here's the link http://www.fire-engineering-software.com/tra.html

good luck

 

[JeffKoch]

Without knowing how the terms are defined, and what units they have, it's impossible to say what is going on - "I found it on the internet somewhere" isn't very helpful. What units does each term in this equation have, and does the result ("net radiation" is not precise either) make sense?

 

[sian130]

Thanks for the link, elegysix.

JeffKoch - my problem in finding the source of the equation stems from the fact that it was found in a Solar Energy journal somewhere but I can no longer gain access to Solar Energy journals via my university account. Hence why I'm trying to find it by other means. I thought perhaps it may have been used elsewhere but now I'm beginning to realize that perhaps the assumptions and simplifications made in the particular journal article are more important than I'd first thought!

As for the units, they are all SI units, i.e. T in Kelvin, Stefan-Boltzmann in W(m^−2)(K−4). Emissivity and absorptivity are dimensionless. Net radiation is therefore in units of W. So the equation is dimensionally correct as far as I can tell.

What do you mean about "net radiation" not being precise? By net radiation, I mean the net thermal radiation (i.e. primarily in the 8-14micro m wavelength band) leaving the surface of the glass, as opposed to the thermal radiation leaving the glass before one has taken into account the downwelling thermal radiation from the sky.

 

[256bits]

The formula is:

Rnet = A((εobj2/αobj2)σTobj4 - εskyσTamb4)

where:
εobj = the emissivity of the object at 8-14μm
αobj = the absorptivity of the object at 8-14μm
εsky = the emissivity of the sky
Tobj = the temperature of the object
Tamb4 = the ambient air temperature
A = the area of the object

Your equation is most likely derived for a particluar situation since it has a ratio of emissivity/ absortivity, which is what I have no idea, and the ambiant air temperature rather than the temperature of the sky ( which is different than the sky temperature ),

 

[JeffKoch]

What do you mean about "net radiation" not being precise? By net radiation, I mean the net thermal radiation (i.e. primarily in the 8-14micro m wavelength band) leaving the surface of the glass, as opposed to the thermal radiation leaving the glass before one has taken into account the downwelling thermal radiation from the sky.

Think, man. Net thermal radiation per unit time? Per unit area? Per Hz? Per Sr? Integrated over any of these quantities? Photons or joules? What units do each of the terms have, and does the result make sense? Find a book on this stuff, there are many - for example the very useful one by Rybicki and Lightman.