We were discussing “

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*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}/Q_{1}>W_{B}/Q_{1}
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_{B }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 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.

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.

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**Reference:**

Engineering thermodynamics by P. K. Nag

Image Courtesy: Google

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