Why not heat the inside of the house directly? Having the heat go through a pump just results in more loss.
If the "waste heat" allows the intake not to freeze up on a heat pump, it would raise the efficiency of the heat pump, and it's quite possible that 1 BTU worth of natural gas could do the work of more than 5 if burned through a high-efficiency furnace.
If the outside of the house is colder than the inside, there is obviously no heat gradient to take advantage of. Thus a heat pump is a good choice.
If you're using what ever this micro fuel cell thing the article tasks about which apparently produces incredibly high temperature heat, you just heat the house with that. It's already hotter than the ambient temperature of the house You can't just slap a heat pump into such a system and have it suddenly gain efficiency. Where would the energy be coming from?
I was assuming the heat required was far more than the waste heat output of the device.
Of course if your heat pump's efficiency doesn't scale with temperature ratio there might a way to use the heat to move it towards carnot efficiency.
The thing is, gas is so dirt cheap that we burn the stuff in furnaces, so I wouldn’t be surprised that this hasn’t taken off. Even electric source heat pumps in homes are a fairly new thing from the last 5-10 years.
Outside of the peltier junction, there is no such thing as an electrical heat pump. They're all mechanical systems, usually with a working gas of some kind. So "mechanical heat pump" is a redundant term.
You can't put x units of heat that is already hotter than the hot well into the cold well of an idealized heat pump and come out ahead of just putting it into the hot well.
You could run it through a heat engine (TEG or stirling engine or similar) and extract some work with your heat pump's 'hot well' as your 'cold well' then use the work to move other heat inside but this is extra complexity.
There might also be eg. some temperature of your cold well where your heat pump's efficiency diverges because viscosity changes or you're not in a situation where modelling it as a single phase change in your working fluid is good or something. Then it might help to waste some free enthalpy (but not discard the energy into the environment entirely) to warm it up slightly.
Presumably none of these things are worth the added hassle especially if x + y is already close to the heating energy needed and y is close to the electrical energy needed inside. The benefit of whatever convoluted scheme you come up with would only be COP * second_heat_engine_efficiency * x which is likely around the same size as x or a bit smaller. Maybe a 30-50% reduction in overall energy input for more than doubling the complexity.