Final answer:
The minimum heat capacity is used in determining the heat transfer rate in a heat exchanger because the fluid with the lower heat capacity limits the heat transfer. This is based on the principles of the Second Law of Thermodynamics, which ensure that some heat is always lost and cannot be perfectly converted to work.
Step-by-step explanation:
The reason why the minimum heat capacity is applied in equations to determine the maximum heat transfer rate of a heat exchanger is because the fluid with the lower heat capacity will limit the actual heat transfer. During the heat exchange process, the amount of heat transferred depends on both the temperature difference between the fluids and their respective heat capacities. If one fluid has a lower heat capacity, it will be unable to absorb or give away as much heat as the fluid with a higher heat capacity, given the same temperature difference.
This constraint is governed by the Second Law of Thermodynamics, which states that the efficiency of a heat engine cannot be 100% due to unavoidable increases in entropy. Thus, there's always some heat (Qc) that must be rejected to the cold reservoir, and this loss is minimized when a heat exchanger is designed based on the fluid with the lower heat capacity, to optimize the amount of useful heat energy (Qh) available for work.
Efficiency is improved when the ratio Tc/Th is minimized, meaning a high temperature of the hot reservoir and a low temperature of the cold reservoir favor better heat transfer to work. Nonetheless, the fluid with the lower heat capacity will still be the limiting factor in the heat transfer rate in the heat exchanger.