How Do You Convert Spectral Irradiance into Heat Flux?

[herpetology]

I have found an equation in an article which will give me the spectral irradiance due to sunlight as function of longitude, latitude, atmospheric pressure, surface albedo, and precipitable water vapor.

I haven't really worked with these sorts of problems before, so my question is how to turn spectral irradiance into heat flux--in other words, to convert W/m^3 to W/m^2. I'm thinking the best thing to do would be to integrate the function with respect to wavelength, and plug in the max and min wavelengths of interest. But what would those be for sunlight?

also, if anyone can think of a simpler way to model heat flux due to solar radiation as a function of time, I'd be interested in hearing it. the equation in this paper is extremely complex and i don't need anything too accurate--as long as its within a degree or two it will serve my purposes just fine.

thanks

 

[omoplata]

Why don't you integrate from 0 to infinity? Does it have to be a numerical integration?

The sun can be approximated to be a http://en.wikipedia.org/wiki/Black_body" with a surface temperature of 5800 K. A black body will radiate in all wavelengths from 0 to infinity.

May I suggest the following approach find upper and lower limits of wavelength? You can find $$\lambda_{max}$$ (wavelength at which the intensity of the radiation produced by a black body is at a maximum) using http://en.wikipedia.org/wiki/Wien%27s_displacement_law" and the Sun's surface temperature. Then try plugging in values on either side of that to your equation until you think the Spectral Irradiance value will be insignificant and will no longer contribute to the integration.

For an alternative method of modeling heat flux, Wikipedia the article on http://en.wikipedia.org/wiki/Insolation" might be helpful to you?

 

[SpectraCat]

I have found an equation in an article which will give me the spectral irradiance due to sunlight as function of longitude, latitude, atmospheric pressure, surface albedo, and precipitable water vapor.

I haven't really worked with these sorts of problems before, so my question is how to turn spectral irradiance into heat flux--in other words, to convert W/m^3 to W/m^2. I'm thinking the best thing to do would be to integrate the function with respect to wavelength, and plug in the max and min wavelengths of interest. But what would those be for sunlight?

also, if anyone can think of a simpler way to model heat flux due to solar radiation as a function of time, I'd be interested in hearing it. the equation in this paper is extremely complex and i don't need anything too accurate--as long as its within a degree or two it will serve my purposes just fine.

thanks

That integral you describe will only tell you the total irradiance (i.e. energy flux) .. if you assume every joule of radiant energy impingent on the surface will be converted to heat, then you integral should be equivalent to heat flux. On the other hand, I'm no expert on the topic but I don't think it can be directly converted to heat flux in a straightforward manner. First of all, much of the energy is reflected back into space without being absorbed and converted to heat .. the surface albedo gives you a quantitative measure of that reflection. Cloud cover and terrain type are likely also to be important. If you already know those factors are being properly accounted for in the spectral irradiance expression you want to integrate (and they might well be), then you might indeed get a reasonable result.

One final suggestion .. watch the units carefully .. spectral irradiance is sometimes reported in W/(m^2*nm), which will change your calculation rather drastically.

 

[herpetology]

That integral you describe will only tell you the total irradiance (i.e. energy flux) .. if you assume every joule of radiant energy impingent on the surface will be converted to heat, then you integral should be equivalent to heat flux. On the other hand, I'm no expert on the topic but I don't think it can be directly converted to heat flux in a straightforward manner. First of all, much of the energy is reflected back into space without being absorbed and converted to heat .. the surface albedo gives you a quantitative measure of that reflection. Cloud cover and terrain type are likely also to be important. If you already know those factors are being properly accounted for in the spectral irradiance expression you want to integrate (and they might well be), then you might indeed get a reasonable result.

One final suggestion .. watch the units carefully .. spectral irradiance is sometimes reported in W/(m^2*nm), which will change your calculation rather drastically.

Thanks guys! the equation is for cloudless days only--which isn't a problem , given the location of my study (phoenix , az). however, atmospheric pressure, precipitable water vapor in the atmosphere, and albedo are taken into account in the equation.

Regarding the use of empirical insolation data, instead of this more complicated model...I liked this idea, and so I looked into it. Apparently, Phoenix has an average solar insolation of 4.51 KWh/(day*m^2) = 187.9 W/m^2. I can't find any hourly data, and I want to approximate the heat flux as a function of time of day. I was thinking the average value would likely be reached 1/4 through the day, and then again 3/4 through the day, on a normal day, while there would be zero radiative heat at the beginning of the day (t=0) and the end of the day (t =1) . So, i might be able to construct a function which would allow me to predict heat flux as a function of time of day--either a parabolic or a trigonometric function. what do you think? would this be a decent approximation of heat flux?