Take the spot cooler as a control volume
shown on the left. If the spot cooler satisfies both the first and the
second law of thermodynamics, it will work. To verify if it satisfies the first law,
check the energy balance of the system. To
verify if it satisfies the second law, check the entropy change of the entire process.
With all the assumptions, the mass and energy balance of the control volume are:
The mass balance is satisfied since
The constant-specific-heat assumption is also used. So with constant
specific heats, the energy balance becomes
Substituting the temperatures into both sides of the above equation
right side = 0.25(24 + 273) +0.75(40 + 273) = 309
left side = 36 + 273 = 309 = right side
The result above shows the energy balance is satisfied.
To check if the spot cooler satisfies the second law, calculate
the entropy change of the entire process. If the entropy change is not negative, the
process can happen. With constant specific heats, the entropy change equals
The result shows that the spot cooler cannot work at atmospheric
pressure. The air at the inlet needs to be compressed to a
higher pressure to make the cooler work, which requires energy input.