# The cost of global warming, from thermo 101

There’s a formula in thermodynamics 101 called Carnot’s theorem that goes like this:

This is a famous equation because it defines the absolute upper limit of efficiency achievable by a heat engine, irrespective of how much its performance is optimised. ηth is the thermal efficiency; Tc is the temperature of the surroundings into which the engine releases its exhaust heat; Th is the temperature at which heat enters the engine.

Say an engine combusts its fuel at 1,000 K and the ambient temperature is 298.15 K (25º C). Then the engine will have a thermal efficiency of 70.18% at best. Effectively, the engine needs to work at a hotter temperature and release heat into a cooler environment.

This is why, to make better engines, engineers are trying to build materials that can withstand a higher operating temperature (e.g. using ceramics). In doing so, they can increase the value of Th in the denominator, and reduce the value of the Tc/Th term.

Sadi Carnot, ‘the father of thermodynamics’ for whom the Carnot theorem is named, propounded his studies of thermodynamics in 1824. Making the reasonable assumption that the world didn’t warm significantly between 1824 and 1870, when the historical record for making common climate change measurements begins, the average global surface temperature in his time was 0.1º C cooler than the average between 1901 and 2000. In 2017 – in the post-industrial period – it was 0.8º C warmer.

Now, let’s extrapolate Carnot’s equation to a global context encompassing all the heat engines in the world – to the point where we’re effectively treating all of them as one big heat engine. Let’s also assume for simplicity’s sake that the average operating temperature of this engine is 1,500 K (1,227º C). Going by the numbers above, the thermal efficiency ceiling of this engine has fallen by 0.1% just because of global warming.

This may not seem like much until you couple it to the amount of power this mega-heat-engine contributes. For example, if a nuclear power plant generates 3,000 MW to move a turbine that produces 1,000 MW of electrical power, then a 0.1% drop in thermal efficiency means it will have to generate 344 MW more to keep producing 1,000 MW of electrical power. This in turn translates to higher resource consumption: uranium, coal, sunlight, wind, whatever.

Now compare this to the fact that the world’s total energy consumption in 2014 was an est. 109,613 TWh, which is 960 trillion MW over the course of the year.

And while some of the resources are renewable, they are all founded on the availability of one very important finite resource: land.

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Any engineer (myself included) will be able to tell you that my calculations are based on grossly oversimplified assumptions. Possibly the grossest of all requires an acknowledgment that the typical steam-powered engines in Carnot’s time had an efficiency of 3%.

But I think my overall point still stands: as the world warms, heat engines are going to become less efficient than they would’ve been if the world was cooler, regardless of whether the consequences are trivial at a given scale. This efficiency could either be measured as an efficiency of the engine itself or one that takes into account resource requirements across the entire value chain.

The reason I write about this now is because of an article that appeared in the Bulletin of the Atomic Scientists on June 10, discussing the economic costs of adapting to climate change and contrasting them to the amount we’ll have to spend fixing things that a warming world will break. And one way things will break is capture in the following lines from the article:

… a 2012 paper in the American Economic Journal [found] that higher temperatures reduce economic growth rates, particularly in poorer countries. A 2015 paper by Stanford scientists published in Nature Climate Change built on this work, similarly finding that global warming will particularly hurt economic growth in poorer countries, and that “Optimal climate policy in this model stabilizes global temperature change below 2 degrees C.” This finding is consistent with the target set by the Paris climate accords.

It’s possible that one way these effects will be perceived on ground is by forcing thermal efficiency down, in turn forcing innovation that draws funds away from other activities. I acknowledge that these effects will be almost immeasurable, if only because it’s not a useful way to start working towards a solution.

On the other hand, I find it quite fascinating that a very simple equation first worked out almost two centuries ago provides a glimpse of the costs anthropogenic global warming has forced us to confront.