Sol-air temperature equation's variables

[TerriKH]

Homework Statement

I’m trying to figure out the equation for sol-air temperature and don't undertand how to find ΔR.

Homework Equations

tsa=to+(αIt-εΔR)/ho

tsa - sol-air temperature
to – ambient air temperature (outside dry-bulb air temperature)
α – surface solar absorptance (inverse of the product’s listed solar reflectance)
It – total solar radiation incident on the surface (can get from solar radiation ground monitoring sites)
ε – surface emittance (can get from product data sheet)
ΔR – net radiation exchange (difference between the long-wave radiation received and emitted by the surface)
ho – external surface heat transfer coefficient (can get from several places on the web)

The Attempt at a Solution

I think I understand most of the variables (please correct me if I’m wrong) except, as stated above, how to find ΔR. Do I calculate the emissivity of the sky and then multiply the emittance of the product, then subtract the answer from the sky emissivity?

 

[LawrenceC]

Long wavelength radiation implies low temperatures. You have the flux from the sun as one component and the heat loss due to the fact the panel is radiating to the sky. That radiation is lost because nitrogen, oxygen, and helium gases are transparent to radiation. Thus the sky can be considered a black body at absolute zero temperature. Carbon dioxide, carbon monoxide, and water vapor are not transparent but can be neglected here.

So in conclusion the delta R term is the Stephan-Boltzmann constant multiplied by the absolute surface temperature of the solar device. Dividing by h changes the units to temperature.

 

[TerriKH]

Thank you for your reply.
However, I don't know the absolute surface temperature of the roof. I was under the impression that I could calculate the sol-air temperature and then put it into the heat balance equation q/a=H (Tsa-Tsurf) to get the temperature. It is ultimately the roof's surface temperature that I am looking for (without calculating in the effects of wind). Any suggestions on resources I can go to find the right equation to calculate the temperature?

 

[LawrenceC]

Can you draw a sketch of what is going on here or describe it more fully. I am confused about the geometric situation. I was assuming it's a solar panel sitting on a roof and that the roof beneath the panel is shaded from the sun by the panel.

 

[TerriKH]

I’m trying to calculate the temperature of the surface of a roof with no solar panels and not concerned about any mechanical equipment. Just the temperature of the roof itself. There isn’t a particular roof that I’m trying to get the answer for, but I picture a low-slope commercial roof with a standard membrane.

The question came up because I was curious as to how one would figure out the temperature if they didn’t have the software to calculate it for them. My background is in design, not engineering, so I thought it would be an easy thing to find out. However, after reading about short-wave and long-wave radiation, convective flux and so on, it is rather confusing.

 

[LawrenceC]

Why don't you get an infrared non-contact thermometer and actually measure it?

To calculate it you have several concerns.

1. Attic temperature
2. 90 degree incident heat flux is about 320 Btu/hr-ft^2 in east coast US area
3. Heat loss by radiation
4. Heat loss by natural convection
5. Heat loss by conduction through the plywood, paper, shingle to shingle surface

Mathematically, the steady state solution is a nonlinear conduction problem due to its boundary conditions of natural convection and radiation.

If I use my FE software to solve the problem assuming the plywood under the shingle poses essentially all the thermal resistance between the attic air temperature and the shingle surface, 90 F outside air temperature, 115 F attic temperature, and 320 BTU/hr-ft^2 incident heat flux, I get about 150-160 F for the shingle temperature.

 

[TerriKH]

There isn't an actual roof to measure. I was just curious how it would be done mathematically, but I think instead I will just be glad there is the software to do it for me.

Thank you for the time you have taken to answer my question. I do appreciate it.

 

[LawrenceC]

I have an infrared thermometer. If the sun is still out when it moves to a certain side of my house I'll measure the temperature of the shingles and post it. I just happen to have a ladder on that side of the house for gutter cleaning. One has to be fairly close with my thermometer else an average of a large area (not necessarily the shingles) will be measured. If it is not sunny, I'll post in a few days. It supposed to snow tomorrow.