Answer to Question #282737 in Mechanics | Relativity for Nour Aldeen

Question

A Carnot heat engine receives heat at T1 and rejects the waste heat to the environment at T2. The

entire work output of the heat engine is used to drive a Carnot refrigerator that removes heat

from the cooled space at 258 K at a rate of 400 kJ/min and rejects it to the same environment at

T2. Determine

(a) The rate of heat supplied to the heat engine and

(b) The total rate of heat rejection to the environment.

Answer 1

a)

$T_1=750\ K,T_2=T_4=300\ K,T_3=258\ K,Q_3=400\ kJ/min$

coefficient of performance of refrigerator:

$COP=\frac{Q_3}{W_R}=\frac{T_3}{T_4-T_3}=\frac{258}{300-258}=6.143$

work input:

$W_R=\frac{Q_3}{COP}=\frac{400}{6.143}=65.115$ kJ/min

work output W_E of heat engine is equal to work input W_R

efficiency of heat engine:

$\eta_E=\frac{W_E}{Q_1}=1-\frac{T_2}{T_1}=1-\frac{300}{750}=0.6$

rate of heat supplied to the heat engine:

$Q_1=\frac{W_E}{\eta_E}=\frac{65.115}{0.6}=108.525$ kJ/min

b)

heat rejected by engine:

$Q_2=Q_1-W_E=108.525-65.115=43.41$ kJ/min

heat rejected by refrigerator:

$Q_4=Q_3+W_R=400+65.115=465.115$ kJ/min

total rate of heat rejection to the environment:

$Q_2+W_R=43.41+465.115=508.525$ kJ/min