Example Problem with Complete Solution

6C-2 : Is This a Perpetual Motion Machine ? 3 pts
Air is compressed by a compressor that is driven by an electric motor. The effluent air is warmer than the air at the inlet of the compressor. An entrepreneur plans to use this system to heat his home. 
He claims that this system is 10% more efficient than an eletrical resistance heater. Is this possible or is this another perpetual motion machine? If so, what kind? Explain your reasoning.
Read : Short and sweet.
Find: Is this a PMM ?  Explain
Assumptions: None.
Equations / Data / Solve:
Efficiency for a heating system is really the coefficient of performance and it is defined by :
Eqn 1
The COP for the resistance heater is 1 because it converts all of the electrical energy input into heat rejected into the hot reservoir…your home.
If the COP of the compressor/heater device is 10% greater, then : COP = 1.1
Eqn 2
But if you consider a system in which the air is recycled as the working fluid, then the 1st Law for the compressor/heater system is:
Eqn 3
Eqns 2 and 3 cannot both be true.
Therefore, the inventor's device violates the 1st Law !
Verify: No assumptions to verify !
Answers : It is a PMM of the 1st Kind !