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Can we upgrade the atmosphere to 1000 ppm of CO2 for $1 trillion?

Heating limestone won't do the job because Worrall missed a factor of 3,000.

On Anthony Watts' influential blog, Eric Worrall proposes to make a "perfect U-turn" in the carbon dioxide policy:

Let's Cook Limestone to Raise Atmospheric CO2 to 1,000ppm
He claims that by heating limestone, the energy we need to consume would cost just $100 billion and perhaps $300 billion with additional expenses. Because there is a missing factor of three in the calculation of the mass of CO2 in the atmosphere, a more realistic price would be $1 trillion.

Limestone blocks in Malta. To make it more confusing, "malta" means mortar (the material to glue bricks) in Czech.

I would still find it extremely cheap because the benefits would be immense. But is the calculation right?

The current CO2 concentration in the air is 400 ppm. 1,000 ppm would be 2.5 times higher. Take a square root of that and you get a good estimate of the improvements of the agricultural yields. They would jump by a factor of 1.5 or so i.e. by 50 percent. Because agriculture produces some 4% of the global GDP now, or over 3 trillion dollars, the investment of $1 trillion could be returned in 4 months or so.

Imagine the happy Africans with lots of food etc.

OK, what is the algorithm to get to this quasi-paradise? Worrall tells us to take limestone (Czech: vápenec) and heat it. The reaction is\[

{\rm CaCO}_3 + {\rm heat} \to {\rm CaO} + {\rm CO}_2.

\] The product CaO is quicklime (Czech: nehašené vápno). Burning of 1 kilogram of limestone produces\[

\frac{12+16+16}{40+12+16+16+16} = \frac{44}{100} = 0.44

\] kilograms of CO2. The inverse of this ratio is 2.3. How much carbon dioxide we need? The current mass of this gas in the atmosphere is some 3,000 gigatons. Worrall got a result smaller by a factor of 3 because he didn't appreciate that "ppm" refers to "ppmv", parts per million of the volume, and he probably made other mistakes, too.

OK, 3,000 gigatons is 400 ppm. We need to add 600 ppm so we need to add some 4,500 gigatons. Multiply it by the factor of 2.3 mentioned above and you get 10,350 gigatons of limestone that you need to heat up. Worrall points out that you only need 3.6 MJ/kg to get the carbon dioxide from the limestone. We need to do that with \(10.35\times 10^{15}\,{\rm kg}\) of limestone, as I mentioned, so you need \(37\times 10^{15}\) megajoules of energy. Or \(3.7\times 10^{22}\,{\rm J}\).

If you hear the car horn while you're shopping, it's probably because your puppies are already losing patience with you.

Because one hour is 3,600 seconds, \(37\times 10^{15}\) megajoules is about \(10\times 10^{12}\) megawatthours. One megawatthour may be produced for $30 so it costs $300 trillion (one trillion is ten to the twelfth power). Worrall announced $100 billion only. So I am sorry, he was too optimistic by a factor of 3,000.

It is more expensive than he thinks to dramatically increase the CO2. I previously said that agriculture produces some $3 trillion per year so even if you neglect the fact that its greater output would lower the food prices, the expenses of $300 trillion would only be repaid in 100 years.

Moreover, you would need to keep on heating additional limestone to keep the CO2 high. Right now, we are emitting the equivalent of 4 ppm of CO2 every year. About 2 ppm of it is absorbed by Mother Nature – the remaining 2 ppm is the observed increase of the CO2 concentration every year.

I think that the absorption by Nature (oceans and biosphere) is approximately proportional to the excess of CO2 above the equilibrium value of 280 ppm. Right now, the excess is 400-280 = 120 ppm. At 1,000 ppm, it would be 720 ppm, i.e. 6 times higher than now. So without any CO2 emissions, Nature would be reducing the CO2 concentration by some 12 ppm every year.

So in one century, you would need to heat additional limestone equivalent to 1200 ppm of extra CO2. That's 2 times higher than the 600 ppm increase we fabricated before. So in a century, the total expenses would be higher by another factor of 3. So to bring CO2 to 1,000 ppm in this way and keep it there for a century would require some $1,000 trillion or a quadrillion dollars, over one decade of the world GDP.

At this level, I feel that the proposed solution is no longer economic. Once you know this result – with my fix of the factor of 3,000 – you might say that it should have been obvious that the global re-engineering of the whole atmosphere won't be economic. The atmosphere is just too large. Efforts to increase the concentration in the whole atmosphere are the opposite project to the plans to reduce the carbon dioxide – but in absolute value, it's pretty much equally stupid. At least, Worrall got the correct sign, unlike the IPCC types.

But I do believe that there are more focused or regional or local projects that would make sense. I believe that for vastly lower amounts of money, much of Sahara could be made fertile. Bring water over there, create big greenhouses, restore a big lake over there, and so on. The wealthy nations should think about such projects these days because it's plausible that energy will continue to be rather cheap for quite some time.

By the way, there's a more obvious way to get the CO2. Just burn the fossil fuels. Coal is cheap but arguably too dirty (or it's too hard to remove the real pollutants). And oil production can't be increased too much and most of it is already burned, anyway. ;-) But it's still interesting to know how much the carbon dioxide would cost if you just got it from burning oil.

One barrel of oil produces some 0.3 tons of carbon dioxide when it burns. We needed to add 4,500 gigatons of CO2, if you remember, so we need 4,500/0.3 = 15,000 gigabarrels (billions of barrels) – without the tripling for the "preservation" of the elevated concentration. At the price $50 per barrel, it is 15,000 times billion times $50 = $750 trillion dollars. For the same problem, we got $300 trillion using limestone. So limestone is indeed by a factor of 2.5 cheaper than just burning oil. It's probably cheaper than burning oil but not spectacularly cheaper.

If you think that you can find a method to increase CO2 to 1,000 ppm for a price much lower than "hundreds of trillions of dollars", I am surely interested.

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