Compute temperature rise from solar irradiance

[donbock]

Homework Statement

(This is not a homework problem; it comes from a real-world problem I'm trying to solve.)

Sunlight is falling on an absorptive object. What will be equilibrium temperature of the object given the solar irradiance (I) and the ambient temperature?
I = solar irradiance = 1120 W/m^{2} (from MIL-STD-810G).

Homework Equations

My guess is that thermal equilibrium is reached when the energy flowing into the object from solar irradiance is balanced by the gray-body energy radiated from the object.

Radiated energy is given by the Stefan-Boltzmann Law:
P/A = e * \sigma * (T^{4} - T_{a}^{4})

P/A = energy radiated per second per unit area (W/m^{2}). Set this equal to I.
e = emissivity of the object = 0.93.
\sigma = the Stefan-Boltzmann Constant (5.67 x 10^{-8} W/m^{2}/K^{4})
T = temperature of the object (degrees Kelvin)
T_{a} = ambient temperature (degrees Kelvin)

To use this formula I will also have to be given the emissivity (e) of the object.

The Attempt at a Solution

This assumes the object absorbs all infalling solar irradiance (no reflection, no transmission).
This ignores convective cooling.
This ignores conductive cooling.

Solving for T as a function of T_{a} yields the following results. (Notice I converted the temperatures to Celsius in the following list.)
T_{a} (degC), T (degC):
0, 132
20, 138
40, 146
60, 155
80, 165
100, 176

These results don't feel plausible: I've never seen an object heated above the boiling point simply by letting the sun shine on it.
Is my approach all wrong?
Are the counterintuitive results solely due to the effects of my three assumptions?

 

[RTW69]

If you are tying to find the surface temperature of the component given I, e, Ta your answer is correct. But I don't know what you mean by "How much will the temperature of the object increase?" Do you mean, what is the surface temperature? The difference between the surface temperature and the ambient temperature? I am not sure what you are trying to find.

 

[donbock]

Prior to exposure to the sun, the object can be expected to be at the ambient temperature. When I said "how much will the temperature increase", I really meant "what is the object's equilibrium temperature" and, oh by the way, how much hotter is that than the ambient.
( I revised the original question to eliminate this source of confusion.)

I don't trust my results because, for example, with ambient temperature of 0 degC, my result is that the object heats up to 132 degC just by letting the sun shine on it! That is outside my experience.

 

[RTW69]

Your analysis is correct given the assumptions you have imposed on the problem but remember the surface has no mass nor is it loosing heat by convection, conduction or re-radiation. If you stick a bucket of water in the sun at 1200 w/m^2 with ta=0 c its surface temperature will not be 132 C. You intuition is correct. Some of the heat will raise the temperature of the water and container, heat will be lost through the bottom of the container and by convection from the sides and top of the container. An interesting exercise is to add those elements to the problem to find the new T-surface.

 

[donbock]

If I knew how much energy was conducted away from the surface to heat the bulk of the object, then I could use the specific heat capacity to compute the temperature change -- but I don't see how to predict how much energy is conducted.

Even if I knew how to do this, I'm not sure it is very relevant. The sun will keep pumping energy into my object until the entire thing reaches that gray-body temperature ... unless I figure out how to model the energy that leaves the object and flows into the surroundings. I'm at a loss as to how to do this.

I'm looking into this because we have plastic lenses that occassionally melt [or at least deform] due to heat. The ambient temperature is high, but not high enough. The first lens in the stack shows this damage. It is located immediately beneath, and almost touching, a glass filter that transmits only a narrow range of frequencies. I want to compute how hot the filter is likely to get if the sun shines directly on it for several hours. Perhaps solar irradiation raises the filter temperature high enough to melt the nearby plastic lens.

We are instrumenting the assembly, but that won't tell us much more than that it is hot; and we already suspect that. The question is why is it so hot. If we understand that then we may be able to find a mitigation strategy.

 

[RTW69]

OK, you may want to move this request to the general or mechanical engineering thread. Your problem is more involved than a general physics homework problem. This is a heat transfer problem and can get complicated fast. It looks similar to a solar collector where you have a glass cover over a metal surface. There has been a lot of analysis of those systems. Try "Solar Engineering of Thermal Systems" by Duffie and Beckman. You have radiation and convection effects between surfaces and convective effects off the outer surface. This is really an engineering question. Good luck.

 

[JerryKops]

I am looking at a similar problem, guessing the temperature of train wheel set sitting outside in the Pilbara sun.
I think you need to eliminate the ambient temperature factor in you radiation equation.
Conduction and convection are relative to ambient T but not radiation.
Think that will solve your problem

cheers

Jerry