I was reading up on the physics of refrigeration for … reasons … and while working out some of the math, stumbled on something interesting. It’s almost certainly not novel, but I thought I’d write it out anyway. (It might not even be correct, if I’ve made a careless mistake somewhere. I’m cautiously confident it’s right, though.)

**Overview**

The gist of my “discovery” is that the (non-standard^{1}) efficiency of refrigeration using a gas cycle is a simple ratio of two temperatures.

I’ll define a gas cycle here for convenience. In a gas cycle, a volume of cold gas is placed in thermal contact with a system in order to extract heat from the system. The gas is then separated, compressed to a high temperature, then placed in contact with a heat sink, rejecting the heat into the sink. The gas is then allowed to expand, cooling down in the process, and the cycle starts again.

More technically, a gas starts off at temperature ( for minimum), and is placed in thermal contact with the system to be cooled, its temperature falling isochorically until equilibrium to ( for cold reservoir). The gas is then compressed adiabatically to a new temperature of ( for maximum), where it is then placed in contact with the heat sink and it cools isochorically until equilibrium to temperature ( for hot reservoir). The gas is then allowed to expand adiabatically, reaching the original temperature .

The quantity I’m interested in is :

Obviously, not all four temperatures are free variables, otherwise the efficiency could be anything. So, we need to find the relationship between the four temperatures, and plug it into the expression for efficiency.

**The Four Temperatures**

The core of the relationship lies in the adiabatic transitions from to and to .

We look first at and .

We start with the equation for adiabatic processes:

and we know that

and because the transition between these states is isochoric.

So, from the adiabatic equation, we proceed to isolate the terms so that we can later eliminate them with the other equations:

Similarly, for and we obtain

Substituting into (isochoric transition):

But since (isochoric), this simplifies down to:

which is the desired relationship between the four temperatures.

**Efficiency**

We pick a variable at random and substitute into the expression for efficiency:

This is the same expression from the earlier section!!

*what sorcery is this??*

**Conclusion**

So somehow the efficiency turns out to be this simple ratio.

It’s not completely out of the blue. The ratio structure of the relationship between the four temperatures hinted that it was describing some underlying parameter, but I didn’t expect the constant to be the efficiency of the transfer.

The expression for efficiency also makes some intuitive sense. You would expect that the efficiency goes down if you pump the gas up to higher temperatures. I didn’t expect it to be such a simple relationship, though.

Maybe there’s actually something really simple going on, but a intuitive physical understanding of this system continues to elude me. But whatever the case, the expression for efficiency is certainly one of the most elegant relationships I’ve seen.

- Usually, efficiency is defined as the ratio of useful work extracted (in this case, the energy removed from the system to be cooled) to the work input. However, the value I am interested in is the ratio of heat extracted from the system to be cooled to the heat output to the heat sink. There’s no name for this that I know of, so I just use “efficiency” here. ↩