Example Problem with Complete Solution

6C-2 : Is This a Perpetual Motion Machine ? 3 pts
It is common knowledge that the temperature of air rises as it is compressed.  An inventor thought about using this high-temperature air to heat buildings.  He used a compressor driven by an electric motor.  The inventor claims that the compressed hot-air system is 25% more efficient than a resistance heating system that provides an equivalent amount of heating.  Is this claim valid, or is this just another perpetual motion machine ?  Explain.
               
Read: Short and sweet.
Given: Figure 1 Find : Is this a PMM ?  Explain
Assumptions: None.
Diagram :   Figure 2 Equation 1
Equations / Data / Solve :
Efficiency for a heating system is really the coefficient of performance and it is defined by :
Equation 2 Eqn 1
The COP for the resistance heater is 1 because it converts all of the electrical energy inout into heat rejected into the hot reservoir…your home.
If the COP of the compressor/heater device is 25% greater, then :
COP = 1.25
And: Equation 3 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:
Equation 4 Eqn 3
Eqns 2 and 3 cannot both be true.
Therefore, the inventor's device violates the 1st Law !
Verify : No assumptions to verify !
Answer : It is a PMM of the 1st Kind !