Pump and Circumstance

My parents are in the process of installing a set of heat pumps in my childhood home, so they don't have to rely on the old wood stove, which requires buying, stacking, and splitting large quantities of lumber every year. I was interested to learn more about these real-world pumps, since physicists are introduced to the idea of heat pumps and heat engines in any intro thermodynamics class.

The idea with both these devices is that we have some working gas that we move to different levels of temperature (T), pressure (p), volume (V), and entropy (S). The reservoir where we keep this gas is attached to a piston/shaft. The piston controls the pressure and volume, and we can either turn the shaft to create a temperature difference (heat pump), or apply a temperature difference to make the shaft turn (heat engine).

Usually these different stages the pump goes through are shown on a p-V diagram, which shows the relationship between pressure and volume at each step. Here's a simple example:

The steps are

1. At a low, constant pressure, expand the gas to a larger volume
2. Keeping that volume fixed, increase the pressure to a high value
3. Decrease the volume again, but at the new pressure
4. Bring the pressure back to the start, at the original volume

The reason physicists like to represent this in terms of pressure and volume is that it's easy to find the energy:

This says that the energy produced by the system is equal to the area under each curve in the plot above. This is where the arrows come in: pointing to the right means the volume is increasing, and dV is positive, while to the left is negative. How does this relate to temperature though? If we're using an ideal gas (air is close), then these factors are related by

so in the plot above, the upper right corner will be the hottest, while the lower left will be the coolest. Since the arrows are to the left at the higher pressure, we put in more energy than we get out, and this is a heat pump that will make that temperature difference.

What's really interesting about heat pumps is their efficiency: The amount of energy you get out (through heating/cooling), divided by the energy you put in (turning the shaft). In the case of an old fashioned electric heater, you run current through a big piece of metal, which heats up, converting 100% of the input energy into heat. It seems odd that you could do better than 100%, but heat pumps manage it.

The key is that we use ambient energy, which doesn't count against our input. In my parents' case, the pump is taking energy from the cold outside, and moving it to the warm inside, so the efficiency is

where Q is the energy exchanged with each location.

When physicists talk about efficiency, we like to bring in the Carnot cycle, which is the theoretical maximum efficiency. The p-V diagram for this case is a bit funny looking:

For this type of pump, we can replace the energy in the efficiency equation with the indoor and outdoor temperatures, in Kelvin. According to Steve's last NOAA report in 2012, the lowest recorded temperature in Ashfield was 249 K. For an indoor temperature of 293 K (brisk for my taste, but I'm sure he would prefer cooler), the efficiency comes to 666%! Clearly these heat pumps are the work of the devil...