FOM is a ratio of actual COP to the ideal COP and Exergy efficiency is a ratio of product exergy (output Exergy) to the fuel exergy (input exergy). The rationale behind the use of FOM and Exergy efficiency is not clear.
It is all about the perspective. Sometimes it is more interesting to show how a process perform in relationship with the ideal process, thus the comparison is made for a same technology/process. For instance: if the theoretical efficiency is 50% (e.g. a power system) and your process efficiency is 45%, then you are actually very close to the maximum although the numbers are not even close to 100%.
In the other hand, this type of analysis can lead to false conclusions. For example: there are two technologies, the previously described (50%/45%) and a new technology that can reach 80% efficiency, but right now we can only get 30%. One may say that the first option is more efficient (right now), but there is not a lot space to increase the efficiency further.
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