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It's probably academic, but if natural gas is that cheap, then I wonder about more thermodynamically-efficient uses of it.

The classic thing is cogeneration. It's a thermodynamic sin to create a large temperature difference (say, furnace combustion chamber vs. house) without running that heat through a heat engine. In the Nordic countries, in parts of Russia, and on some university campuses in the US, that's done with a "neighborhood power plant" and district heating (steam pipes). Heat engines get less efficient as they get smaller, but possibly a house-scale cogeneration setup could still make sense?

But I'm also now curious about something more interesting: In the same way that you can have, say, a propane powered refrigerator, is it also possible to have a natural gas powered heat pump? Suppose it's a cold winter day, and you're burning natural gas in a combustion chamber in an appliance in your basement. There are three temperature reservoirs: The outdoors, the house, and the combustion chamber, at temperatures T1 < T2 < T3, respectively. By harnessing heat flow (Q32) from the combustion chamber into the house, can additional heat (Q12) be pumped from the outdoors into the house? Then the house will get a total flow Q = Q32 + Q12.

From Q32, work

W = (1 - T2/T3) * Q32

is available. That can then be used to pump

Q12 = 1 / (1 - T1/T2) * W

= ((1 - T2/T3) / (1 - T1/T2)) * Q32

additional heat. And thus

Q = Q32 + Q12

= (1 + (1 - T2/T3) / (1 - T1/T2)) * Q32

= B(T1, T2, T3) * Q32 .

Here, B (a function of T1, T2, T3) is the coefficient by which the natural gas' energy is effectively multiplied, for heating purposes.



Belatedly, let's plug in some numbers for the Carnot heat-pump furnace.

T1 = -9.4 F = -23 C = 250.15 K (Outdoor temperature in winter on cold day.)

T2 = 71.6 F = 22 C = 295.15 K (Indoor room temperature.)

T3 = 2236 K (Adiabatic flame temperature of methane at constant volume.)

Then we have B(T1, T2, T3) = 6.69.

That's very substantial!

Of course, this assumes Carnot efficiencies for everything, so it's an upper bound.

Also, my assumption of T3 being the adiabatic flame temperature may be too optimistic. Google says

> Today's commercial jet engines can reach temperatures as high as 1,700 degrees Celsius (that's 3,092 degrees Fahrenheit)

https://engineering.virginia.edu/news/2018/11/generating-cur....

so letting T3 = 1,700 C = 1973.15 K, we get the slightly lower B(T1, T2, T3) = 6.58.

This is still fantastic.

This is also assuming a very cold day, which is when the system will have lower efficiency. If we instead assume T1 = 0 C = 273.15 K, and use the more conservative T3 = 1973.15 K assumption, then we get B(T1, T2, T3) = 12.4. That's huge.

So, assuming my math/modeling is right, a hypothetical heat-pump furnace could (using Carnot bounds) use around 1/6th - 1/12th the natural gas as a conventional one, even in a very cold place, if you keep all the other properties of the house (insulation, air exchange) constant.




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