Two ideal Carnot engines operate in $$cascade(all$$ heat given up by one engine is used by the other engine to produce $$work)$$ between temperatures, $$T1$$ and $$T2$$. The temperature of the hot reservoir of the first engine is $$T1$$ and the temperature of the cold reservoir of the second engine is $$T2$$. T is temperature of the sink of first engine which is also the source for the second engine. How is T related to $$T1$$ and $$T2$$, if both the engines perform equal amount of work?

Question

Two ideal Carnot engines operate in $$cascade(all$$ heat given up by one engine is  used by the other engine to  produce $$work)$$ between temperatures, $$T1$$ and $$T2$$. The temperature of the hot reservoir of the first engine is $$T1$$ and the temperature of the cold reservoir of the second engine is $$T2$$. T is temperature of the sink of first engine which is also the source for the second engine. How is T related to $$T1$$ and $$T2$$, if both the engines perform equal amount of work?

Infinity Learn · Accepted Answer

QH:  Heat input to $$1st$$ engineQ : Heat rejected from $$1st$$ engineQL: Heat rejected from IInd engine η$$=1-TsinkT$$ $$source---(1)$$η$$=WQ1=Qabsorbed-QrejectedQabsorbed=1-QrejectedQabsorbed---(2)equation(1)$$ $$=Equation(2)$$ $$TsinkTsource=QrejectedQabsorbedAs$$ per first engine, QQH $$=TT1$$ ⇒$$QH=QT1TAs$$ per second engine, $$QLQ=T2T$$     ⇒$$QL=QT2TNow$$, work done by $$1st$$ engine $$=$$ work done by $$2nd$$ engine⇒QH−$$Q=Q$$−QL        ⇒$$2Q=QH+QL$$           ⇒ $$2=T1T+T2T$$ ⇒T $$=$$ $$T1+T22$$

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