An irreversible heat engine extracts heat from a high temperature

An irreversible heat engine extracts heat from a high temperature source at a rate of 100 kW and rejects heat to a sink at a rate of 50 kW. The entire work output of the heat engine is used to drive a reversible heat pump operating between a set of independent isothermal heat reservoirs at 17°C and 75°C. The rate (in kW) at which the heat pump delivers heat to its high temperature sink is

W = Q₁ – Q₂
= 100 – 50 = 50 kW

C.O.P =	348	=	Q₁
	348 - 290		W

Q₁ = 300 kW
Alternately
Work output from irreversible heat engine,
W = Q₁ – Q₂
= 100 –50 = 50 kW

Also for irreversible heat engine
Q₁ + W = Q₂
∴ T₁ ∆S + 50 = T₂ ∆S
or (T₂ – T₁) ∆S = 50
or (348 – 290) ∆S = 50

or ∆S =	50
	58

∴ Rate of heat rejection,
Q₂ = T₂ ∆S

= 348 ×	50	= 300 kW
	58