Carnot's Theorem-Engineering Thermodynamics

1. Carnot’s Theorem
Presented by
Pulagala Venkateswara Rao (PVRao)
ME (IISc)
Associate Professor
Department of Mechanical Engineering
Andhra University, Visakhapatnam, A.P., INDIA

2. Carnot’s Theorem
Here, two heat engines operating between
same thermal energy reservoirs: A high
temperature reservoir at ‘TH’ and a low
temperature reservoir at ‘TL’
Assume; Heat engine ‘HEa' is a generalized
heat engine (which means ‘HEa’ is reversible or
irreversible). Heat supplied to ‘HEa’ is ‘Qsa’ and heat
rejected by ‘HEa’ is ‘Qra’, and the work output is
‘Wa’.
Another heat engine ‘HEb’ is a reversible
heat engine. (A reversible heat engine can be reversed
without changing their values of energy interactions). Heat
and work interactions are: ‘Qsb’, ‘Qrb’ and ‘Wb’.

3. Assume,
1. Both heat engines are supplied with same
amount of heat; Qsa = Qsb
2. Efficiency of a generalized heat engine
(HEa) is greater than a reversible heat engine (HEb);
ηHEa > ηHEb, which means Wa > Wb
Efficiency of a
reversible heat engine=
Work Output/Heat Input
ηHEb=Wb/Qsb
Efficiency of a
generalized heat engine =
Work Output/ Heat Input
ηHEa = Wa/Qsa

4. Since a reversible heat
engine ‘RHE’/‘HEb’ can be
reversed without changing
the values of heat and work
interactions; this reversible
engine can be turned into
heat pump (HP), which
transfers heat from a low
temperature reservoir ‘TL’ to
a high temperature reservoir
‘TH’ by consuming ‘Wb’
amount of work (as input).
Since Wa>Wb (# assumption
2), some fraction of the work
(=Wb) can be used to drive
the HP. Therefore, there is an
availability of excess amount
of work (Wc = Wa-Wb).

5. Since Qsa=Qsb (# assumption 1), the
reservoir at a temperature ‘TH’ can be
eliminated. Therefore, the combined heat
engine (Here, the generalised heat engine ‘HEa’ is driving
a reversible heat engine ‘HEb/HP’) is operating with
a single reservoir at a temperature TL, and
producing a network output ‘Wc’, which
means the combined system is violating
the Kelvin-Planck statement of SLT. Which
means, ‘Wa>Wb’(# assumption 2) is wrong (or)
the assumption: ηHEa > ηHEb is wrong.
Therefore, ηHEb ≥ ηHEa
It implies that, “No heat engine
can be more efficient than a reversible
heat engine operating between the
same thermal energy reservoirs”
ηHEb = ηHEa (if ‘HEa’ is reversible)
ηHEb > ηHEa (if ‘HEa’ is irreversible)

6. Corollary of Carnot's Theorem
Corollary of Carnot's Theorem: The efficiency of all the reversible heat engines operating between
the same reservoirs (Source and Sink) is the same; which means the efficiency of a heat engine is
independent of the nature (or the amount) of the working substance (gas/steam) undergoing the
thermodynamic cycle. The efficiency of a Reversible Heat Engine purely depends on temperature
levels of the source and the sink.