Friday, July 31, 2009

Fermi Problem Friday: How much energy do you save by having a white roof?

This Friday, in honor of my new facebook friend Steven Chu, I'd like to examine his "White Roofs" campaign through a Fermi problem.

Steven Chu has said that we could all save energy simply by painting our roofs white. The idea is the same one behind why I don't wear a black shirt on a sunny day (and white after Labor Day). Black objects absorb light, and with it, heat, while white objects reflect it. So if you're trying to keep your house cool in the summer, it makes sense to have a roof that reflects sunlight instead of absorbing it. Here's the schpiel:



But how much energy do you save, really?

The Fermi Problem:



The solar flux at Earth's atmosphere is about 1360 watts per square meter. About 0.3 of this is reflected back into space, while the rest is absorbed by the earth. Given the square footage of your roof, how much energy (assumed as heat) would a black roof absorb? How much energy would a white roof absorb?

Now, you could give Secretary Chu the benefit of the doubt and assume that black roofs absorb all of the sun's energy, while white roofs reflect it all. But we're talking getting people to actually paint their roofs white. For reals! So we're going above and beyond the Fermi Problem protocol for this week's problem.

How can we test the difference in absorption between light and dark materials? Let's try placing two thermometers in the sunlight. Procure two boxes. Cover one with white paper, and the other with black paper. Put a thermometer in each box. Go inside and reread a few chapters of Dialogue Concerning Two New Sciences. Then check to see which thermometer reads a higher temperature. Let us know what you find out. Now use this information to calculate the energy you will really save by painting your roof white.

On Monday we'll be posting the results from our own experiment. Send us a photo of your experiment to physicscentral@aps.org with your name and a mailing address, and we will mail you something fun for your scientific efforts!

Now, are you going to do with all the money you save from lower electric bills minus the cost of the white paint? Buy Swedish fish, of course.

4 comments:

  1. Should you repaint your roof black every winter? Then white again for the summer, etc.?

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  2. There are about 100 million housholds in the US, according to the 2000 census.

    Each house has a total roof area of about 50 square meters (I'm estimating this based on the houses in my neighborhood).

    So the total roof area of the US is about

    Roof area=(50 m^2/houshold)*(100,000,000 households)*(1 km^2/1,000,000 m^2) = 5,000 km^2

    The total land area of the US is about 10 million square kilometers, so the percentage of land area of the US covered by roofs is

    (5,000 km^2/10,000,000 km^2)*100%=0.05%

    If we went even further than painting our roofs white, and covered them with perfectly reflecting mirrors (which would presumably reflect the light back into space), we could reduce the amount of energy from the sun that warms the nation by no more than 0.05%.

    The benefit would be a little less than that, because the ground is not perfectly absorbing, and roofs will not be perfectly reflective, so I'd guess we could reduce the nation's solar heat load by 0.01% to 0.02% by painting roofs white.

    Seems like a TOTAL waste of white paint to me.

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  3. I'm in Florida. It's hard number to figure, but I know I can set a pot of water boiling within a minute by reflecting 8.5 square feet of sunlight from a parabolic reflector. Considering this is about as much time as it takes to boil a pot on the stove, the energy hitting my roof is the equivalent of leaving almost 200 stoves turned on all day. Pointing a bit parabola at a dark grey cooking pot is probably not the most efficient approach either. Disturbing, isn't it? We really should try capturing some of this.

    Of course heat rises and the roof already reflects a lot. And of course there's insulation. Some heat is not transmitted through the roof/attic either, but through walls from surrounding air (which is a scorcher today, mind you). Air conditioning is still by far the largest part of my consumption every month though, especially during the summer (actually it stays off for both months of winter we have, but otherwise...)

    Managing to reflect this relatively small amount of energy back out of the atmosphere would be interesting, but not huge. Reflecting this back out of your house would be significant, but not game changing. Capturing this energy and using it to power and cool your house? That would be a revolution.

    (sorry, couldn't manage to find numbers on BTU absorption of white paint vs gray roofing)

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  4. I dispute the anonymous poster who said that painting roofs would be a total waste of time based on 0.05% of land area being covered by roofs in the U.S. First of all, the percentage of land is probably even lower, at around 0.02%, since you have to take into account the large number of households in aggregate that live in apartments or multi-family houses. Beyond that however, instead of looking at the overall proportion of roofs to national landmass, you have to factor in the concentration of those roofs at specific points (cities). Painting all of the roofs white would probably go a fair way towards reducing the heat island effects noted around U.S. cities, making for a noticeable impact on climate for many residents.

    ReplyDelete