Coefficient of Performance and Figure of Merit

The coefficient of performance (COP) is used to describe the effectiveness of refrigerators, including those operating at cryogenic temperatures. The COP is defined as the amount of heat removed at the cryogenic operating temperature of the refrigerator divided by the amount of work that must be applied to remove the heat. If two refrigerators remove heat at the same temperature, the one with the larger COP will require less work (and ultimately less electrical power) to remove the same amount of heat. Generally speaking, with all other factors being equal, refrigerators with higher COPs have better performance and are more energy efficient. As will be seen below, however, comparison of COPs between refrigerators has to be done carefully and with a full understanding of the assumptions made in the COP calculation.

For most cryogenic refrigerators, the prediction of the COP is quite complicated and is dependent on both the specific thermodynamic cycle chosen and on the equipment used to implement that cycle in the refrigerator. However, for the ideal Carnot cycle, it can be shown that the COP is defined as Tc/(Th–Tc), where Tc is the cryogenic temperature at which the heat is removed and Th is the temperature at which the heat is rejected. Recall that the Carnot cycle is an ideal cycle and describes the most efficient cryogenic refrigeration cycle permitted by the laws of thermodynamics. The definition of COP for the Carnot cycle illustrates a very fundamental aspect of cryogenics. Assuming that This almost always at or near 300K, then the COP for the Carnot cycle will increase as the temperature Tc increases. Thus, in any cryogenic system, it is always more thermodynamically efficient to remove heat at a higher temperature. This fact drives a lot of design choices in cryogenics, including the use of intermediate heat sinks and actively cooled thermal radiation shields (Cold Facts Fall 2013).

The COP for a Carnot cycle refrigerator operating between 300K and 4.2K is given by 4.2/(300-4.2) or 0.0142. If we take the inverse of the COP (1/COP) we have a term that describes the number of Watts of work required to remove 1 Watt of heat at a given temperature. In the case of the Carnot cycle operating at 4.2K, the inverse of the COP is 1/0.0142 or 70 W/W. Thus, in the best possible case, it requires 70 W of work to remove 1 W of heat at 4.2K. Contrast this to a Carnot cycle removing heat at 77K, where 1/COP is 2.9 W/W.

The Carnot cycle is an ideal cycle that can’t be realized with practical cryogenic refrigerators. The question then becomes, how close can real cryogenic refrigerators get to the COP of the ideal Carnot cycle? This is indicated by the figure of merit (FOM). The FOM is defined as the COP of a real cryogenic refrigerator divided by the COP of a Carnot cycle refrigerator operating between the same temperatures (FOM = COPreal/COPcarnot). Modern large scale helium refrigerators operating at 4.2K generally have a FOM of approximately 0.25 – 0.32. In some usages, the FOM is referred to as a percent Carnot; thus, a FOM of 0.25 may be called “25 percent Carnot.

”Small cryocoolers can have very low FOMs, in some cases less than 0.1. This illustrates an important point: While the COP and FOM of cryogenic refrigerators are important performance parameters, they are not the only consideration in choosing a given refrigerator design. Other considerations such as availability, capital cost, size, weight, operating temperature and capacity may be more or equally important. In general, high COPs are more important for larger-capacity refrigerators that require more energy to operate.

Comparing COPs (either measured or calculated) between different refrigerators should be done carefully with a full understanding of the assumptions made in each case. For example, does the work required to remove the heat include items such as the power needed for cooling water circulation pumps or cooling tower fans? These can be significant energy requirements for large systems. Always ensure that you are comparing systems with the same set of assumptions. Additionally, keep in mind that COPs may be defined or measured at a specific operating point of the cryogenic refrigerator. However, many cryogenic refrigerators, particularly those in research institutions, operate over a wide range of capacity. Modern cryogenic plants can be designed to operate over a wide range of capacity reduction without significantly reducing the COP.

A very good description of COP, FOM and the thermodynamics associated with cryogenic plants may be found in Cryogenic Engineering by T. Flynn (Dekker 1997) and Cryogenic Systems by R. Barron (Oxford 1985). A discussion of COP and FOM for small cryocoolers can be seen in “Figures of Merit for Multi-Stage Cryocoolers,” J. Delmas et al. Adv. Cryo. Engr. Vol. 55A (2010).

Energy efficiency in cryogenic systems goes beyond COP and FOM values. Recent papers on this broader topic include “Energy Efficiency of Large Cryogenic Systems: The LHC Case and Beyond,” S. Claudet et al. Proc. ICEC 24-ICMC 2012 (2013), “Helium Refrigeration Considerations For Cryomodule Design,” V. Ganni and P. Knudsen, and “Waste Heat Recovery From The European Spallation Source Cryogenic Helium Plants—Implications for System Design,” J. Jurns et al., both in Adv. Cryo. Engr. Vol. 59 (2014).