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Standard XIII

Physics

Carnot Cycle

Two Carnot en...

Question

Two Carnot engines $A$ and $B$ are operated in succession. The first one, $A$ receives heat from a source at $T_{1} = 800 K$ and rejects to a sink at $T_{2} K$ . The second engine $B$ receives heat rejected by the first engine and rejects to another sink at $T_{3} = 300 K$ . Find the value of $T_{2}$ if
(i) the work outputs of the two engines are equal.
(ii) the efficiencies of the two engines are equal.

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Solution

(i)Given work done are equal
$W_{1} = W_{2}$
where, $W_{1} = Q_{1} - Q_{2}$ and $W_{2} = Q_{2} - Q_{3}$
In Carnot's engine, $\frac{Q_{i n}}{Q_{o u t}} = \frac{T_{s o u r c e}}{T_{s i n k}}$
$∴ \frac{W_{1}}{W_{2}} = \frac{Q_{1} - Q_{2}}{Q_{2} - Q_{3}} = \frac{T_{1} - T_{2}}{T_{2} - T_{3}} = 1$
$T_{2} = 550 K$

(ii) Here the efficiencies are equal.
$η_{A} = η_{B}$
where, $η_{A} = 1 - \frac{T_{2}}{T_{1}}$ and $η_{A} = 1 - \frac{T_{3}}{T_{2}}$
$∴ \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{2}} \Rightarrow T_{2} = \sqrt{T_{3} T_{1}} = 200 \sqrt{6}$
$T_{2} = 489.89 ≃ 490 K$

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(i)Given work done are equal W1=W2 where, W1=Q1−Q2 and W2=Q2−Q3 In Carnot's engine, QinQout=TsourceTsink ∴W1W2=Q1−Q2Q2−Q3=T1−T2T2−T3=1 T2=550 K (ii) Here the efficiencies are equal. ηA=ηB where, ηA=1−T2T1 and ηA=1−T3T2 ∴T2T1=T3T2⇒T2=√T3T1=200√6 T2=489.89≃490 K