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.
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.