EXPLAIN THE CARNOT’S THEOREM AND PROOF IT

We were discussing various basic concepts of thermodynamics such as thermodynamic state, path, process and cycles in our previous post. We have also discussed the concept of reversible and irreversible process in our recent post.

Today we will see here the very important theorem in thermal engineering i.e. Carnot’s theorem and after discussing the Carnot’s theorem we will see the Carnot cycle in our upcoming posts.

Let us see first Carnot’s theorem

There will not be any heat engine that will have more efficiency as compared to a reversible cyclic process or reversible heat engine working between same temperature limits and all reversible heat engines working between same temperature limits will have the same efficiency.

Let us see here what is temperature limits?

Reversible heat engine absorbs heat at one temperature and rejects heat at another temperature and these two temperatures are termed as temperature limits. If we are operating a reversible engine and irreversible engine between the same temperature limits, reversible heat engine will have always higher efficiency as compared to irreversible heat engine.

Lets we have number of reversible heat engines working between same temperature limits, all reversible heat engines will have same efficiency even reversible heat engines might be different in their specifications or design but their efficiency will surely be same if these reversible heat engines are working between same temperature limits.

However if we are considering the case of irreversible heat engines, if we have number of irreversible heat engines working between the same temperature limits, all irreversible heat engines will have different efficiency. We must note it here that if temperature limits are same, an irreversible heat engine can never secure the efficiency of a reversible heat engine.

Therefore, we have noted here that if temperature limits are same for both types of engines i.e. a reversible heat engine and an irreversible heat engine are working between same temperature limits then reversible heat engine will have higher efficiency as compared to irreversible heat engine.

Let us proof the Carnot’s theorem

Let we have two heat engines, HE_A and RHE_B as displayed in following figure, let us see the various nomenclature used here.

HE_A is a heat engine i.e. natural engine or irreversible heat engine and RHE_B is a reversible heat engine. Let higher temperature thermal reservoir i.e. source has temperature T₁ and lower temperature thermal reservoir i.e. sink has temperature T₂.

Now according to Carnot’s theorem, we need to show that efficiency of reversible heat engine (η_B) will be higher than the efficiency of irreversible heat engine (η_A). We can also say that we will have to show that

η_B > η_A

Let us think that above statement about the efficiency is not true and we have assumed that

η_A > η_B

Let us think that heat supplied in both engines is same, therefore we will have following information

W_A > W_B

Let we have reversed the reversible heat engine and now heat engine will absorb the heat from lower temperature reservoir or sink and deliver the heat at source or at higher temperature thermal reservoir after securing some amount of work as displayed in following figure.

If we will conclude the net effect, what you will see here? We will conclude here that irreversible heat engine and reversible heat engine will provide one heat engine which is operating with single thermal reservoir T₂ and delivering the net work energy by taking heat energy from single thermal reservoir T₂.

This will be the violation of Kelvin plank statement of second law of thermodynamics and therefore our assumption (η_A > η_B) will never be true and hence η_B > η_A will be the correct statement.

It is proved that

η_B > η_A

We will see "What is Carnot cycle and its efficiency?" in our upcoming posts.

Do you have suggestions? Please write in comment box

Reference:

Engineering thermodynamics by P.K. Nag

Engineering thermodynamics by Prof S K Som

Image courtesy: Google

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