Efficiency of Heat Engines & Refrigerators: Is Impossible Possible?

[EngineeringStudent]

Homework Statement

An entrepreneur approaches you with an investment opportunity. This person has developed a new type of heat engine – refrigerator combination in which the net power output of the engine is used to drive the refrigerator. The heat engine, supplied with heat at the rate of Q(dot)_H= 1.75 kW from a reservoir at 727 °C, delivers W(dot) = 1 kW of mechanical work while rejecting heat to a sink at 60 °C.

The refrigeration unit has a cooling capacity Q(dot)_L = 3.5 kW and operates between a refrigerated region at – 20 °C and a high temperature heat sink at 90 °C.

Should you invest in this venture? Show appropriate calculations related to the laws of thermodynamics to justify your decision.

Relevant Equations

For the heat engine:
η_max=1-T_L/T_H
η_claim = W(dot)_out/Q(dot)_in

For the refrigerator:
COP_max=T_c/T_H-T_c
COP_claim=Q_c/w_in

For the heat engine:
First I converted all the temperatures to Kelvin,

η_max=1-(333)/(1000)=0.667

η_claim=(1*10^3)/(1.75*10^3)=0.5714

So the heat engine seems to be less efficient than a Carnot heat engine which means it can exist.

For the refrigerator:

COP_max=(253)/(363-253)=2.3

COP_claim=(3.5*10^3)/(1*10^3)=3.5

The refrigerators COP is greater than the theoretical maximum.

So does my work look all right?
I believe this system is impossible.

 

[Lnewqban]

The refrigeration section cannot deliver more work than the input work it receives from the engine to which is mechanically connected.