Question

A Carnot heat engine receives heat from a reservoir at 900°C at a rate of 800 kJ/min and rejects the waste heat to the ambient air at 27°C. The entire work output of the heat engine is used to drive a refrigerator that removes heat from the refrigerated space at 25°C and transfers it to the same ambient air at 27°C. Determine (a) the maximum rate of heat removal from the refrigerated space and (b) the total rate of heat rejection to the ambient air.

Answer 1

Heat engine

$T_L=27°c=27+273=300K$

$T_H=900°c=900+273=1173K$

$Q_H=800KJ/min$

The power output is obtained from the different efficiency relation

$W=Q_H(1-\frac{T_L}{TH})$

Put value

$$W=800(1-\frac{300}{1173})KJ/min$$

W=595kJ/min 

The rate of heat rejected to the ambient air $Q_L=Q_H-W$

Put value

$Q_L=800-595=205 KJ/min$

Refrigerator 

$W=595kJ/min$

$T_H=27°c=300K$

$T_L=25°c=273+25=298k$

The rate of heat removed from the refrigerator is obtained by manipulating the different efficiency relation

$Q_L=\frac{W}{\frac{T_H}{T_L}-1,}$

Put values

$Q_L=\frac{595}{\frac{300}{298}-1}$ =88655KJ/min

The rate of heat related of the ambient air is

$Q_H=W+Q_L$

$Q_H=(88655 +595)KJ/min$

$Q_H=89250KJ/min$

(b)

Total rate of heat rejected to the ambient air is

$Q_{amb}=Q_H+Q_L$

$Q=(89250+205)KJ/min$

$Q=89455KJ/min$