A carnot engine receives heat at ‘Ta’ from source and rejects heat to a sink at ‘Tb’. This engine drives a carnot refrigerator which removes heat at ‘Tc’ and rejects heat at ‘Tb’. Determine:

i) The ratio of $[\frac{Q_c}{Q_a} ]$ where $Q_c$ is the heat removed at ‘Tc’ and $Q_a$ is heat received at ‘Ta’.

ii) If Ta =500K and Tc =250K determine ‘Tb’ if $Q_a=Q_c$

Determine ideal efficiency of carnot engine and ideal COP of carnot refrigerator.

Mumbai University > Mechanical Engineering > Sem8 > Refrigeration and Air Conditioning

Marks: 12M

Year: May 2013

Let W1 be the work produced by Carnot engine which drives the Carnot refrigerator.

We know,efficiency of carnot engine $η=\frac{W1}{Q_a} =\frac{(T_a-T_b)}{T_a}$ … (1)

Also COP=$\frac{Q_c}{W_1} =\frac{T_C}{(T_b-T_c )}$… (2)

From (1) and (2),

$$ηQ_a=\frac{Q_c}{COP} \\ ∴\frac{Q_c}{Q_a }=η×COP=\frac{(T_a-T_b)}{(T_a )}{\frac{T_C}{(T_b-T_c )}} $$

For, Ta=500K and Tc = 250 K and $Q_a=Q_c$

$$1=\frac{(500-T_b)}{500}\frac{250}{(T_b-250)} \\ ∴T_b=333.33 K $$

Ideal efficiency of Carnot engine=$\frac{W_1}{Q_a} =\frac{(Q_a-Q_1)}{Q_a }=\frac{(T_a-T_b)}{T_a}$ =0.3333

Ideal COP of carnot refrigerator=$\frac{Q_c}{W_1} =\frac{T_C}{(T_b-T_c )}=3$