WHAT ARE THE COROLLARIES OF CARNOT THEOREM?

We were discussing “Carnot cycle and its efficiency” and also we have seen the “Carnot theorem and its explanation” in our previous post. Today we will see here the corollary of Carnot’s theorem.

So let us see here the corollary of Carnot’s theorem

There are following two corollaries of Carnot’s theorem and these are as mentioned here

“All reversible heat engines working between same temperature limits will have the same efficiency”.

“Efficiency of any reversible heat engine working between two thermal energy reservoirs will never depend over the nature, type or quantity of working fluid but also it will be only dependent on the temperature of the both thermal reservoir”.

We can also say that efficiency of any reversible heat engine will only depend on the temperature limits of thermal energy reservoirs and it will be independent of the nature, type or quantity of working fluid.

Let we have two heat engines i.e. HE_A and HE_B and both are reversible heat engines working between two temperature limits. Hot thermal energy reservoir i.e. source has temperature T1 and cold thermal energy reservoir i.e. sink has temperature T2 as shown in figure.

Two reversible heat engines HE_A and HE_B are working between same temperature limits

Let us assume that statement of corollary of Carnot ‘theorem i.e. “All reversible heat engines working between same temperature limits will have the same efficiency” is not true.

And let us assume that, η_A> η_B

η_A> η_B

W_A/Q₁ >W_B/Q₁

W_A >W_B

As we are dealing here with reversible heat engines and therefore we can say that now we have reversed the heat engine HE_B and now all flow directions will be reversed and therefore heat engine HE_Bwill now act as heat pump.

Hence it will now absorb heat energy from cold thermal energy reservoir and will deliver the heat energy to higher thermal energy reservoir and for that it will require work to be done on the system.

Now let us see the combined system of heat engine HE_A and heat pump EH_B. We can easily conclude here that combine system is now working as “PMM2” because combined system is now exchanging heat energy (Q_2B-Q_2A) from single thermal energy reservoir and providing the work energy (W_A-W_B) as output work.

We can see here that "Kelvin plank statement of second law of thermodynamics" is violated and therefore it is concluded here that our assumption η_A> η_B is wrong or we can say that η_A could not be greater than η_B.

In similar way we can assume that η_B> η_A and reverse the heat engine HE_A to work as heat pump. And similarly we will show that η_B could not be greater than η_A or we can say that

η_A= η_B

As the efficiency of all reversible heat engines working between the same temperature limits is same.

Therefore we can say that efficiency of any reversible heat engine working between two thermal energy reservoirs will never depend over the nature, type or quantity of working fluid but also it will be only dependent on the temperature of the both thermal energy reservoirs.

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We will see another topic in our next post in the category of thermal engineering.