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 T_{1}and lower temperature thermal reservoir i.e. sink has temperature T_{2}.
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 T

_{2}and delivering the net work energy by taking heat energy from single thermal reservoir T_{2}.
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}

_{B}

_{A}

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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