This past weekend, I was at Boskone, where I appeared on a few science-y panels. One of these was on the possibility of beaming power down from space:

Energy From Space

Beam me down some juice, Scotty? Let’s talk about the possibilities — and practicalities — of really long-distance power transmission.

Tom Easton (M), Jordin T. Kare, Chad Orzel, Jeff Hecht, Joan Slonczewski

This was a little odd, as Jordin does this for a living– he’s been working on a proposal to NASA for a solar power generating satellite that would use lasers to beam power down to photovoltaic panels on the ground– and I know basically nothing about the subject. I’m not even much of a space buff by SF fan standards, so I didn’t have a whole lot to contribute, other than shutting down one line of speculation. As it’s something that comes up a fair bit in discussions of power generating things, I figure I might as well type it out here for future reference.

The question that was asked was what effect having all this power beamed down from space would have on the local weather, which started off into a big thing about how adding a lot of heat to the Boston area would suck in moist air from the ocean, and create a local micro-climate. Just as that was really getting going, though, I pointed out that this couldn’t possibly be a real issue, or else we’d already have weird little tropical zones around every coal-burning plant in the country. The power-beaming schemes being discussed would produce somewhere between a megawatt and a gigawatt of electricity from arrays of panels between 100m and 1000m in diameter. The efficiency of either photovoltaics or fossil fuel generation is around 50% (close enough for back-of-the-envelope purposes), so they’ll be dumping about as much heat into the local environment as they produce electricity. If a gigawatt capacity conventional power plant (on the high end for fossil fuel powered power plants) doesn’t warp the local climate, a gigawatt of waste heat from solar panels isn’t going to change anything, either.

It’s a tempting path to go down, though, because a gigawatt sound really big. I mean, that’s 1,000,000,000 joules of heat every second. That’s a lot of zeroes, and it’s hard to believe that that isn’t significant. But when you’re talking about the energy scale involved with the climate, it’s completely insignificant.

You can get a better sense of the proper scale from a sophomore-level modern physics problem, having to do with black-body radiation. Students in those classes learn that the total power radiated by an object at some temperature T is given by the Stefan-Boltzmann Law:

P=A \sigma T^4

where A is the surface area of the object, and σ is standing in for a bunch of constants with a total value of 5.67×10-8 W/m2/K4 (the temperature is in Kelvin, because this is physics– a temperature change of 1K is the same as one Celsius degree, and 0C = 273K). This is a really steep function of temperature– if you double the temperature, the power radiated goes up by a factor of 16.

One of the classic homework problems to assign with this is to ask students to find the temperature of the Earth. In theory, anyway, the Earth ought to be radiating away as much energy as it receives from space (otherwise the temperature would be increasing even more dramatically than it is), so you can come up with an estimate of the temperature of the Earth by balancing the energy received from the sun with the energy radiated away by the Earth. The total energy flux from the Sun is about 1.7×1017 W (that’s the energy contained in all the light from the Sun passing through a disc the radius of the Earth at a distance of Earth’s average orbital radius). If you plug that into the Stefan-Boltzmann equation along with the Earth’s radius (6400 km, give or take), and solve for the temperature, you get:

T_{estimate} = 276 K

that corresponds to about 3 Celsius or 37 Fahrenheit, if you’re an American. That seems reasonable enough during a Northern winter, but the actual average temperature of the Earth is a good deal higher,

T_{actual} = 288 K

What accounts for the difference? Well, the simple calculation ignores the Earth’s atmosphere. In fact, as anybody who’s paid any attention to science news over the last couple of decades knows, the Earth’s atmosphere absorbs some of the energy radiated away from the surface, and radiates part of what it absorbs back down to the surface. The estimate above is accurate for a rocky, airless ball, but the Earth is manifestly not one of those, and as a result the actual temperature is higher.

(The general idea that the Earth as a whole is in thermal equilibrium with is surroundings is still accurate, though– it’s just more complicated to work out the details. The extra energy that needs to be radiated away comes in the form of reflected sunlight, and thermal radiation from the upper atmosphere, which is cooler than the surface, but has more surface area.)

If you want to know the proper energy scale for thinking about affecting climate, this gives you an upper bound. If you take the actual surface temperature of the Earth and plug it into the Stefan-Boltzmann equation, you get that the real power radiated should be something a hair over 2×1017 W, about 17% more than the actual energy coming in. That’s about 29,000,000 gigawatts, a few thousand times the total energy consumption of human civilization. So the waste heat of a few power plants isn’t going to move anything.

Another way to look at it is in terms of the mass of air you’re trying to move. If you’re going to change the local weather, that would require changing the temperature of the air in the vicinity of the power station. That doesn’t seem like much of a problem– after all, if you’re burning stuff, you’re producing hot air– but if you’re talking about climate and weather you’re talking huge volumes of air.

The size of the power generating facilities being considered here is something around a kilometer on a side (again, we’re doing back-of-the-envelope estimates here). If you want to change the weather by dumping heat into the air, you need to raise the temperature of all the air above the station; as a very rough estimate, let’s call it a cube a kilometer on a side. That’s a billion cubic meters of air, or about 1.2×109kg. Air doesn’t have much heat capacity, but that much of it will soak up a lot of heat without much of a temperature change– the 109 joules dumped by a gigawatt power plant in one second would raise the temperature of that volume of air by a bit less than 1K. And while there are a lot of seconds in a day, that air doesn’t stick around for all that long before it moves off into the rest of the atmosphere– conduction and convection will carry that heat off very quickly (if you only considered radiation, like we did above, the power radiated away by a 1km-a-side cube at the average surface temperature of the Earth is about 2 gigawatts anyway; increasing that by a gigawatt would require a significant temperature increase, but that’s about the least efficient heat transfer mechanism available). The actual effect of a power plant on local temperatures is pretty minimal, and if you can’t affect the temperature, you’re not going to change the weather.

So, while a gigawatt sounds like a huge number, it’s a piddling amount when we’re talking about climate and weather. We’re a long, long way from needing to worry about the waste heat generated by our power plants having a significant effect on the climate.

Why do we worry about power plant emissions, then? Because the carbon dioxide emitted by a power plant doesn’t just carry heat from the plant, it helps trap heat from the Sun. The atmosphere we already have helps effectively trap an additional 17% of the energy coming in from the Sun. Boosting the CO2 content of the atmosphere increases that percentage by some amount– a tiny amount, granted, but that’s a small percentage of a whopping huge number. To get an extra gigawatt, we only need to increase the percentage of energy kept in by about one one-millionth of a percent. So it’s not the energy dumped into the system by the power plants that matters, but how their exhaust affects the energy retained from the Sun.

Comments

  1. #1 Eric Lund
    February 19, 2013

    Last month there was a case of a snow band downwind of a nuclear power plant in western Pennsylvania (technical discussion here). But that, too, was due to the exhaust products (steam, in this case) from the plant. The band was quite narrow and produced only about an inch of snow at most, which is minor compared to the lake effect snows that are commonly seen in that part of the US.

    There is also the urban heat island effect, but that is due more to land use patterns (roofs, roads and parking lots absorb more solar radiation than does ground covered with grass or trees) than power plant siting.

  2. #2 Anthony
    February 19, 2013

    Power plants do create local climate zones; a gigawatt over a square kilometer is about 80% of the brightness of sunlight and would probably be hot enough to cause heatstroke if a human walked through the area. They also create indirect greenhouse effects by increasing the rate of water evaporation. The thing is, these power plants don’t cover a very large part of the surface of the planet, and if you turn the power plant off, temperature will revert to normal over a period of a few days, whereas it seems to take around 10,000 years for climate to revert to normal after a period of elevated CO2.

  3. #3 wereatheist
    Berlin, Germany
    February 19, 2013

    Think you forgot to mention albedo.
    And then, heat capacity of gases is (you know it, of course) ~2J/kg, not J/g.

  4. #4 wereatheist
    February 19, 2013

    heat capacity of gases is (you know it, of course) ~2J/kg, not J/g.

    this is the opposite of true :)
    Should not drink…

  5. #5 wereatheist
    February 19, 2013

    Blockquote fail

  6. #6 Hamish
    February 22, 2013

    One significant effect is the heating of lakes etc by cooling water. When I was a lad we used to go fishing in a channel that took cooling water from the Lakeview coal generator near Toronto into Lake Ontario. In the winter it was teaming with trout that gorged on the baitfish that seemed to be attracted to the warmer water. Lakeview has since shutdown — good news for local health, but as Anthony points out the effect vanished, and I assume so did the fisherman!

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