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. HEA and HEB 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 HEA and HEB 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, ηA> ηB
ηA> ηB
WA/Q1 >WB/Q1
WA >WB
As we are dealing here with reversible heat engines and therefore we can say that now we have reversed the heat engine HEB and now all flow directions will be reversed and therefore heat engine HEB will 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 HEA and heat pump EHB. We can easily conclude here that combine system is now working as “PMM2 because combined system is now exchanging heat energy (Q2B-Q2A) from single thermal energy reservoir and providing the work energy (WA-WB) 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 ηA> ηB is wrong or we can say that ηA could not be greater than ηB.
In similar way we can assume that ηB> ηA and reverse the heat engine HEA to work as heat pump. And similarly we will show that ηB could not be greater than ηA or we can say that
ηA= ηB
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.

Do you have any suggestions? Please write in comment box
We will see another topic in our next post in the category of thermal engineering.

#### Reference:

Engineering thermodynamics by P. K. Nag