What is the Correct Way to Calculate Emissivity and Absorptivity?

[Kqwert]

Homework Statement

An impenetrable flat horizontal plate is radiated with 3000 W/m^2, where 500 W/m^2 is reflected. The surface temperature of the plate is 200 °C and it emits 500 W/m^2. Air with temperature 25 °C flows over the plate, and the heat transfer coefficient due to this is 20 W/m^2*K.

Calculate the (i) emissivity, (ii) absorptivity

Homework Equations

The Attempt at a Solution

(i) A black body absorbs all incoming radiation, and all absorbed radiation is emitted. Therefore, I would say that a black plate would emit 3000 W/m^2, and the emissivity of my plate would be 500/3000 = 0.166. This is wrong, could anyone please explain me what is wrong?(ii) Using Kirchoff's law I would assume that emissivity = absorptivity, but this is also wrong according to the solutions manual. Could anyone correct my thinking?

 

[tnich]

Homework Statement

An impenetrable flat horizontal plate is radiated with 3000 W/m^2, where 500 W/m^2 is reflected. The surface temperature of the plate is 200 °C and it emits 500 W/m^2. Air with temperature 25 °C flows over the plate, and the heat transfer coefficient due to this is 20 W/m^2*K.

Calculate the (i) emissivity, (ii) absorptivity

Homework Equations

The Attempt at a Solution

(i) A black body absorbs all incoming radiation, and all absorbed radiation is emitted. Therefore, I would say that a black plate would emit 3000 W/m^2, and the emissivity of my plate would be 500/3000 = 0.166. This is wrong, could anyone please explain me what is wrong?(ii) Using Kirchoff's law I would assume that emissivity = absorptivity, but this is also wrong according to the solutions manual. Could anyone correct my thinking?

I can see several problems in your argument.
The first is that power per area radiated by an object depends on its temperature and emissivity. Since the plate is not a blackbody you cannot apply Kirchoff's law to it.
The second is that you have not considered the heat that the plate is losing to the air around it.
The third is that you have not considered the reflected radiation.
The problem does not say that the system is in thermal equilibrium, but I think it would be difficult to determine the emissivity without knowing the rate of change of temperature. If it is in equilibrium, then you should be able to write an energy balance equation with the heat gains on one side and the heat losses on the other, and then solve for the emissivity and absorptivity. When I try to do that, I end up with more heat losses than heat gains, even without considering radiative emission, so perhaps I have made a mistake somewhere. I hope you have better luck when you try it.

 

[Kqwert]

I can see several problems in your argument.
The first is that power per area radiated by an object depends on its temperature and emissivity. Since the plate is not a blackbody you cannot apply Kirchoff's law to it.
The second is that you have not considered the heat that the plate is losing to the air around it.
The third is that you have not considered the reflected radiation.
The problem does not say that the system is in thermal equilibrium, but I think it would be difficult to determine the emissivity without knowing the rate of change of temperature. If it is in equilibrium, then you should be able to write an energy balance equation with the heat gains on one side and the heat losses on the other, and then solve for the emissivity and absorptivity. When I try to do that, I end up with more heat losses than heat gains, even without considering radiative emission, so perhaps I have made a mistake somewhere. I hope you have better luck when you try it.

But doesn't Kirchoff´s law apply to non-black bodies as well? And will the heat lost through convection influence the emissivity/absorbtivity?

 

[tnich]

But doesn't Kirchoff´s law apply to non-black bodies as well? And will the heat lost through convection influence the emissivity/absorbtivity?

The simple answer is no. As I understand it, Kirchoff's law says that absorptivity is equal to emissivity for a blackbody in equilibrium. For other objects, they are not necessarily equal.
Convective heat loss should not influence emissivity or absorptivity, but in this problem it should help you determine how much heat is lost by radiative emission, from which you can determine the emissivity.
Are you sure that the values of the constants are correct in the problem statement?

 

[Kqwert]

m which you can determine the emissivity.
Are you sure that the values of the constants are correct in the problem statement?

Yeah, everything is correct. I am wondering if the emitted radiation from the plate is "independent" of the incoming radiation, could that be an explanation?

 

[tnich]

Yeah, everything is correct. I am wondering if the emitted radiation from the plate is "independent" of the incoming radiation, could that be an explanation?

It depends only on temperature and emissivity, so yes, in that sense it is independent of incoming radiation.

 

[haruspex]

As I understand it, Kirchoff's law says that absorptivity is equal to emissivity for a blackbody in equilibrium.

My reading is that if you know the absorption coefficient of the body at a given wavelength and the body's temperature then you can determine the emission power at that wavelength. See https://en.m.wikipedia.org/wiki/Kirchhoff's_law_of_thermal_radiation.

 

[haruspex]

Yeah, everything is correct. I am wondering if the emitted radiation from the plate is "independent" of the incoming radiation, could that be an explanation?

In principle, the solution should consist of:
Assuming thermal equilibrium (because there is not enough info otherwise)
Calculating the losses from conduction and reflectance
Deducing from that the heat lost by radiation
Comparing that with the radiation power of a black body at the same temperature.

However, something seems to be wrong with the data. What would you calculate for the heat lost by conduction to the air?

 

[tnich]

My reading is that if you know the absorption coefficient of the body at a given wavelength and the body's temperature then you can determine the emission power at that wavelength. See https://en.m.wikipedia.org/wiki/Kirchhoff's_law_of_thermal_radiation.

Yes, I think you are right. Unfortunately, that will still not help in this problem since we don't know the wavelength distribution of incident radiation or how the absorptivity/emissivity varies with wavelength.