Contains second law of thermodynamics

1. Prepared by:
Pradeep Kumar Gupta
Assistant Professor
Department of Mechanical Engineering

2. Thermal reservoir
Thermal reservoir is that part of the system which has a large heat
capacity i.e., it is a body which is capable of absorbing or rejecting
any amount of heat without affecting its temperature.
Heat source: The reservoir which is at high temperature and
supplies heat is known as heat source or source. Such as furnace,
combustion chamber etc.
Heat sink: The reservoir which is at low temperature and receives
heat is known as heat sink or sink. Such as atmosphere, sea, river
etc.

3. Second law of thermodynamics
Second law of thermodynamics may be stated in various ways given as under:
(a) Clausius statement
(b) Kelvin Plank statement
(a) Clausius Statement: According to Clausius statement, “it is impossible for
self acting device (or machine), while operating in a cyclic process, to transfer
heat from a reservoir at a lower temperature to a reservoir at a higher
temperature without any external work being done on it.
Or
In other words, “heat cannot naturally flow from a colder body to a hotter
body.”
(b) Kelvin- Planck statement: According to Kelvin- Planck statement, “it is
impossible to construct a device (or heat engine) working on a cyclic process,
whose only aim is to convert heat energy from a single thermal reservoir into
an equivalent amount of work.”

4. Heat engine
A heat engine is a system that converts heat (thermal energy) or
chemical energy of working substance (any type of fuel) to
mechanical energy, which can then be used to do mechanical work.
Some facts about heat engines:
1. Heat engine receives heat form a higher temperature body
(source: such as nuclear reactor, solar energy etc.)
2. Heat engine converts part of this heat into useful work.
3. Heat engine rejects the remaining waste heat to the low
temperature body (sink: such as rivers, atmosphere etc.).
4. Heat engine operate in a cycle.

5. Efficiency of heat engine
Efficiency of a heat engine may be defined as the ratio of net work done by the
engine to the heat supplied (or in simple words it is defined as the ratio of
output to input). If as per the fig. WHE is the work done by the engine and Q1 is
the heat supplied.
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒
𝐻𝑒𝑎𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑
Since,𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝐻𝑒𝑎𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑 − ℎ𝑒𝑎𝑡 𝑟𝑒𝑗𝑒𝑐𝑡𝑒𝑑
𝑊𝐻𝐸= 𝑄1 − 𝑄2
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =
𝐻𝑒𝑎𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑 − ℎ𝑒𝑎𝑡 𝑟𝑒𝑗𝑒𝑐𝑡𝑒𝑑
𝐻𝑒𝑎𝑡 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑
𝜂 =
𝑄1 − 𝑄2
𝑄1
𝜂 = 1 −
𝑄2
𝑄1

6. Coefficient of performance: It is defined as the ratio of heat
extracted from the cold body to the amount of work done on the
refrigerant.
𝐶. 𝑂. 𝑃. =
𝑄
𝑊
Where,
Q= amount of heat extracted from the cold body
W= amount of work done on the refrigerant

7. Refrigerator: Refrigerator is a device which is used to maintain the temperature
lower to its surroundings.
Or
Refrigerator is a reversed heat engine or a heat pump which extract heat from a cold
body and delivers it to a hot body.
Refrigerator is a reversed heat engine which either cool or maintain the temperature
of the body (T1) lower than the atmospheric temperature (Ta). This is obtained by
extracting heat (Q1) from a cold body and delivering it to a hot body (Q2). In doing so,
work Wr is required to be done on the system. According to the first law of
thermodynamics,
𝑊𝑟 = 𝑄2 − 𝑄1
The performance of a refrigerator is defined by the ratio of heat extracted from the
cold body (Q1) to the amount of work done on the system (Wr). It is known as
coefficient of performance. Mathematically it is written as,

8. (𝐶. 𝑂. 𝑃. ) 𝑟=
𝑄1
𝑊𝑟
=
𝑄1
𝑄2 − 𝑄1

9. Heat Pump: Heat pump is also a reversed heat engine which extracts heat (Q1) from a
cold body and delivers it to a hot body.
According to the first law of thermodynamics,
𝑊𝑝 = 𝑄2 − 𝑄1
The performance of a pump is defined by the ratio of heat delivered to the hot body
(Q2) to the amount of work done on the system (Wp). It is known as coefficient of
performance. Mathematically it is written as,
(𝐶. 𝑂. 𝑃. ) 𝑝 𝑜𝑟 𝐸𝑃𝑅 =
𝑄2
𝑊𝑝
=
𝑄2
𝑄2−𝑄1
Where EPR is known as Engine performance ratio.
(𝐶. 𝑂. 𝑃. ) 𝑝 𝑜𝑟 𝐸𝑃𝑅 =
𝑄1
𝑄2 − 𝑄1
+ 1
(𝐶. 𝑂. 𝑃. ) 𝑝= (𝐶. 𝑂. 𝑃. ) 𝑟+1

10. Carnot cycle
This cycle was given by Nicolas Leonard Sadi Carnot, a French engineer.
Let us consider some elements for making analysis of Carnot’s cycle:
i. A working substance is assumed to be a perfect gas.
ii. Two heat reservoirs at different temperatures one is at high temperature and other is at
lower temperature.
iii. Cylinder walls are perfectly insulated.
(a) P-v diagram (b) T-s diagram

11. Isothermal expansion (Process 1-2): The working substance (air) is expanded isothermally (i.e. at
constant temperature T1= T2) as shown by the curve 1-2 on P-v and T-s diagrams. During the process
the pressure decreases from P1 to P2 and volume increases from v1 to v2. Heat supplied on the air
during the isothermal expansion process is given by
Q1-2= Area 1-2 -s2-s1
Q1-2= T1(s2-s1)
Adiabatic expansion (Process 2-3): During this process, the air is allowed to expand adiabatically.
The reversible adiabatic expansion process is represented by the curve 2-3 on P-v and T-s diagram.
The temperature of working substance falls from T2 to T3 at constant entropy. During adiabatic
process no heat is absorbed or rejected by the air.
Isothermal compression (Process 3-4): The air is compressed isothermally (i.e. at constant
temperature T3= T4) as shown by the curve 3-4 on P-v and T-s diagrams. During the process the
pressure increases from P3 to P4 and volume decreases from v3 to v4. Heat rejected by the air during
the isothermal expansion process is given by
Q3-4= Area 4-3-s2 –s1
Q3-4= T3(s2-s1)
Adiabatic compression (Process 4-1): During this process, the working substance is allowed to
compress adiabatically. The reversible adiabatic compression process is represented by the curve 4-1
on P-v and T-s diagram. The temperature of working substance increases from T4 to T1 at constant
entropy. During adiabatic process no heat is absorbed or rejected by the working substance.

12. The net work done= Heat supplied – Heat rejected
Work done= Q1-2 – Q3-4
Work done= T1(s2-s1) - T3(s2-s1)= (T1-T3) (s2-s1)
𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =
𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒
𝐻𝑒𝑎𝑡 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑
𝜂 =
𝑇1 − 𝑇3 𝑠2 − 𝑠1
𝑇1 𝑠2 − 𝑠1
𝜂 =
𝑇1 − 𝑇3
𝑇1
= 1 −
𝑇3
𝑇1

13. Carnot’s Theorem
It states that all heat engines operating between a given constant
temperature source and a given constant temperature sink, none
has a higher efficiency than a reversible engine.
Let two engines E1 and E2 operate between the given source
at temperature T1 and the given sink at temperature T2 as shown in
figure.

14. Let E1 be the heat engine and E2 be a reversible heat engine. It has to
prove that the efficiency of E2 is more than that of E1. Let us assume
that it is not true and 𝜂1 > 𝜂2. Let the rates of working of the
engines be such that
𝑄1
′
= 𝑄1
′′
= 𝑄1
Since 𝜂1 > 𝜂2
𝑊1
𝑄1
>
𝑊2
𝑄2
So that,
𝑊1 > 𝑊2

15. Now, let E2 be reversed. Since E2 is assumed as a reversible heat engine, the magnitudes of heat and work
quantities will remain the same, their directions will be reversed as shown in figure.
Since 𝑊1 > 𝑊2 , some part of W1 may be fed to drive the reversed heat engine ∃2.
Since 𝑄1′ = 𝑄1′′ = 𝑄1, the heat discharged by ∃2 may be supplied to E1. The source may, therefore, be
eliminated. The net result E1 and ∃2 together constitute a heat engine which, operating in a cycle,
produces net work W1-W2 while exchanging with a single reservoir at T2. This violates the statement of
second law(Kelvin- Planck). Thus the assumption that η_1>η_2 is wrong.
Therefore,

16. 𝜂2 ≥ 𝜂1

17. Corollary of Carnot’s theorem
The efficiency of all reversible heat engines operating between the same
temperatures is equal.
Let, both the heat engines E1 and E2 be reversible and 𝜂1 > 𝜂2. If E2 is
reversed i.e. it works as a heat pump using some part of work output of
engine E1. We obtained that combined system of heat engine E1 and heat
pump E2, becomes a perpetual motion machine (PMM-II). So, 𝜂1 cannot
be greater than 𝜂2. Similarly if we assume that 𝜂2 > 𝜂1and reverse the
heat engine E1, we observe that 𝜂2 cannot be greater than 𝜂1. Therefore
𝜂1 = 𝜂2
Since, efficiencies of all reversible heat engines operating between the
thermal reservoirs are same. The efficiency of reversible heat engine is
independent pf the nature or amount of the working substance taking
part in the cycle.

18. Equivalence of Kelvin- Planck statement and Clausius statement
Though Kelvin- Planck statement and Clausius statement of second law of
thermodynamics appear two different interpretations of the same basic fact,
but both these statements are equivalent in all aspects. For establishing
equivalence of the two statements, it has to be proved that violation of Kelvin-
Planck statement implies the violation of Clausius statement and vice-versa.
This is explained as under:

19. Let us consider a system as shown by Fig. 4.4 (a). In this system, a heat engine of 100% thermal
efficiency (i.e. PMM-2) is violating the Kelvin –Planck statement as it converts the heat energy
(Q1) from a single high temperature body at T1, into an equivalent amount of work (W= Q1).
This work output can be used to drive the heat pump which receives an amount of heat Q2 from
a low temperature body at T2 and rejects heat Q1+Q2 to a high temperature body at T1. If the
combination of heat engine and a heat pump is taken as a single system (Fig.4.4(a)), then the
result is a device that operates in a cycle and has no effect other than the transfer of heat Q2
from a low temperature body to a high temperature body, thus it is a violation of Clausius
statement. Hence it proves that violation of Kelvin-Planck statement leads to violation of
Clausius statement.
Now let us consider another system as shown by fig. 4.4 (b). In this system, a heat pump (i.e.
PMM-2) is violating the Clausius statement as it transfers heat from a low temperature body at
T2 to a high temperature body at T1without any disbursement of work. Let a heat engine
operating between the same heat bodies that receives an amount of heat Q1 from a high
temperature body at temperature T1 and does work (WHE= Q1-Q2) and rejects an amount of heat
Q2 to a low temperature body at temperature T2. If the combination of heat pump and heat
engine is taken as a single system (Fig.4.4(b)), then the result is a device that operates in a cycle
whose entire effect is to remove heat at the rate of (Q1-Q2) and convert it completely into an
equivalent amount of work, thus it is violation of Kelvin- Planck statement. Hence it proves that
violation of Clausius statement leads a violation of Kelvin-Planck statement.
Thus from the above we can say that both the statements of second law of thermodynamics
are complimentary to each other.

20. Perpetual motion machine of the first kind (PMM-1)
A device or a machine which violates the first law of thermodynamics (i.e.
the energy can neither be created nor be destroyed it can only be
transformed from one form to another) is termed as perpetual motion
machine of the first kind (PMM-1).
Perpetual motion machine of second kind (PMM-2)
A heat engine that violates the second law of thermodynamics (i.e. a heat
engine which converts entire of the heat energy in mechanical work) is
termed as perpetual motion machine of second kind (PMM-2).

21. Concept of reversibility
A process is reversible if the system passes through a continuous series
of equilibrium states.
Process 1-2 followed by 1-a-2 and process 2-1 followed by 2-a-1, is a
reversible process.
Process 1-2 followed by 1-a-2 and process 2-1 followed by 2-b-1, is an
irreversible process.

22. Clausius inequality
It was first given by R.J.E. Clausius (1822-1888), a German Physicist
and is expressed as,
𝛿𝑄
𝑇
≤ 0
According to above equation, “the cyclic integral of δQ/T is always
less than or equal to zero. This inequality is holds good for all
reversible and irreversible cycles.”

23. Entropy
Entropy may be defined as, “a thermodynamic quantity representing
the unavailability of a system's thermal energy for conversion into
mechanical work, often defined as the degree of randomness or
disorder in the system.”
Or
“The cyclic integral of the quantity
𝛿𝑄
𝑇
for a reversible cycle being
equal to zero indicate that
𝛿𝑄
𝑇
is a point function and is therefore
property of a system. This property is termed as entropy. It is
expressed by ‘S’. Mathematically it is written as:
1
2
𝑑𝑆 = 1
2 𝛿𝑄
𝑇
in kJ/K
𝑑𝑆 = (𝑆2 − 𝑆1) = 1
2 𝛿𝑄
𝑇
in kJ/K