# Assignment on Second Law and Entropy

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```					                                                                                     Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra
ITER
FLUID MECHANICS AND THERMODYNAMICS
ASSIGNMENT – 5
SECOND LAW OF THERMODYNAMICS AND ENTROPY

Introductory Notes
Thermal Reservoir is a large system to which a finite amount of heat can be added or extracted
without changing its temperature. A source is a thermal reservoir at higher temperature from which
energy in the form of heat is withdrawn isothermally. A sink is a thermal reservoir at lower
temperature to which energy as heat can be added isothermally.
Thermal Efficiency of a Heat Engine:
Performance of a heat engine is called its thermal efficiency. It is a measure of the degree of useful
W net , out Q i n−Q out     Q out
utilization of the heat input into a heat engine. So, th =            =            =1−
Qi n            Qi n       Qi n
Second Law of Thermodynamics : Kelvin-Planck Statement
“It is impossible for any device that operates on a cycle to receive heat from a single reservoir and
produce a net amount of work.”
Clausius Statement – “It is impossible to have a device which working cyclically will produce no
other effect than to transfer heat from a low temperature body to a high temperature body.”
The two statements of the Second Law are equivalent and violation of one leads/implies the violation
of the other statement.

Factors that cause irreversibility:
–     Friction (solid friction or viscosity)
–     heat transfer across a finite temperature difference
–     electric resistance
–     expansion and compression due to finite pressure difference
–     spontaneous mixing of two fluids
–     unrestrained expansion of a gas
–     elastic deformation of solids

Carnot Cycle:

P-v diagram of Carnot Cycle
T-s diagram of Carnot Cycle
Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra

Process 1-2 → Reversible isothermal expansion at TH
Process 2-3 → Reversible adiabatic expansion where temperature changes from TH to TL
Process 3-4 → Reversible isothermal compression at TL
Process 4-1 → Reversible adiabatic compression where temperature changes from TL to TH
TL
Thermal efficiency of a Carnot cycle rev =1−           This formula is applicable for a reversible cycle
TH
only.
QL
Note: The efficiency of a heat engine (reversible or irreversible) is 1−        . But for a reversible
QH
QL       T                                     QL T L
heat engine rev =1−       =1− L , since for reversible engines,          =        . Here, temperatures
QH       TH                                    QH T H
must be taken in Kelvin. Remember that efficiency of a reversible heat engine (say Carnot Engine) is
never 100%.
Carnot Principle:
I.     No heat engine operating between the two thermal reservoirs, with fixed temperatures, can be
more efficient than a reversible engine operating between the same two thermal reservoirs.
II.    The efficiencies of all reversible engines operating between the same two reservoirs are the
same.

Clausius Inequality:    ∮ Q ≤0
T
The limiting case of equality is for reversible processes i.e

∮    
Q
T i nt. rev.
=0
Entropy:
dS = Q
T i nt.rev.
Here, dS is the infinitesimal change in entropy, δQ is the infinitesimal amount of
heat interaction across the boundary where the temperature is T in Kelvin.
 S =S 2−S 1=∫1    2 Q

T i nt.rev.
 kJ /K  Entropy is an extensive property.
Change in Entropy during an internally reversible isothermal heat transfer :
1 2
 S = ∫1 Qi nt.rev. =1 Q 2 / T  kJ / K  This expression is useful to determine the entropy changes
T
of a thermal reservoir that can absorb or supply heat indefinitely at a constant temperature.
For an isentropic process, ΔS = 0. The performance of devices like pumps, turbines, nozzles,
compressors and diffusers etc. are compared with an idealized isentropic version of their operations.
[Note: A reversible, adiabatic process is necessarily isentropic but an isentropic process is not
necessarily a reversible, adiabatic process. The entropy increase of a substance during a process as a
result of irreversibilities may be compensated by a decrease in entropy as a result of heat losses.]
1Q2 int.rev. = T ΔS
Combining the First and Second law for a closed system: (for both reversible and irreversible process)
TdS =dU + PdV. Using H = U + PV, we can also express this as TdS = dH – VdP
Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra

du      dv
ds=      P
T       T
So,
dh     dP
ds= −v
T       T
Entropy Change of Solids and Liquids:
Approximating liquids and solids as incompressible substances, Cp = Cv = C. If we also assume that

the heat capacity of the substance does not change much with temperature, s 2−s1=C ln
T2
T1  .

For an isentropic process with incompressible substances, ΔS = 0 and hence, T2 = T1. Therefore, for a
truly incompressible substance, an isentropic process is also isothermal.
Entropy change for Ideal Gases: (taking constant specific heats)
Using du = Cv dT, dh = Cp dT and Pv = RT:

  
T           v
s 2−s1=C v ln 2 R ln 2 kJ /kg.K
T1          v1

=C p ln
   
T2
T1
P
− R ln 2 kJ /kg.K
P1
Principle of Entropy Increase:
“For an isolated system, the entropy never decreases. It increases for any irreversible process and
remains the same for reversible processes only.”
S 2−S 1= S system=∫1 
2 Q

T
S gen

Assignment Problems
1. An inventor claims that he has built an engine working between temperature limits of 1000 K
and 400 K and having an efficiency of 70%. Is his claim valid?
2. Among the following processes, pick those which are internally reversible:
• A process with friction
• Throttling process
• A process involving mixing
• Frictionless constant pressure process
• Free expansion process
3. A reversible engine working between temperature limits of 800 K and 300 K receives 1000 kW
of heat from the high temperature reservoir. What power is developed by this engine?
(625kW)
4. A Carnot refrigerator takes heat at the rate of 150 kW from a space maintained at 300 K. If the
refrigerator rejects heat at 320 K, what is the power required to
drive the refrigerator?                                    (10 kW)
5. A cycle is constructed with three frictionless processes and one                             Q

process with friction. What will be the value of   ∮ Q
T
for this
W
cycle – positive, negative or zero ?
6. Prove that the cyclic device shown(Figure 1) can not be reversible.
7. An inventor claims that he has developed an engine that works
between temperature limits of 500 K and 1000 K and develops                      Figure 1
Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra

a power of 20 kW while consuming heat at the rate of 50 KW. Is this engine possible? If
possible, is it reversible or irreversible?
8. Working between the same temperature limits, show that coefficient of performance of a heat
pump is greater than that of a refrigerator by unity.
9. Two Carnot engines A and B are connected in series between two thermal reservoirs
maintained at 1000 K and 100 K respectively. Engine A receives 1680 kJ of heat from the high
temperature reservoir and rejects heat to the Carnot engine B. Engine B takes in heat rejected
by engine A and rejects heat to the low temperature reservoir. If engines A and B have equal
thermal efficiencies, determine : (i ) the heat rejected by engine B (ii ) the temperature at
which engine A rejects heat (iii ) the work done.
(168 kJ, 316.23 K, 1512 kJ)
10. Two reversible engines are arranged in series as detailed below (Figure 2). The first engine
receives energy from a reservoir at TH and rejects energy to a reservoir at temperature T. The
second engine receives the energy rejected by the first engine from the reservoir at T and
rejects energy to a reservoir at temperature TL . Here, TH>T > TL . Derive an expression for the
temperature T in terms of TH and TL, if the net work of the two engines are equal.

Figure 2
11. A reversible engine operates as shown in Figure 3. It has two sources and a sink. One source is
at 1200 K and supplies heat at the rate of 800 kW, while the other source is at 1000 K and
supplies heat at the rate of 500 kW. The engine rejects heat at the rate of 400 kW to a sink at
an unknown temperature. Determine the sink temperature, the power developed and the
engine efficiency.                                                 (343 K, 900 kW, 0.6923)
Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra

Figure 3
12. A heat engine operating between two reservoirs at 1000 K and 300 K is used to drive a heat
pump which extracts heat from a reservoir at 250 K at a rate twice that at which the engine
rejects heat. If the efficiency of the engine is 80% of the maximum possible and COP of the
heat pump is 1/3 of the maximum possible, what is the temperature of the reservoir to which
the heat pump rejects heat? What is the rate of heat rejection from the heat pump if the rate of
heat supply to the engine is 50 kW?                                            (303 K, 72 kW)
13. A compressor takes in 500 kg/minute of air at 0.98 bar and 68oC. The diameters of inlet and
delivery pipes are 450 mm and 200 mm. The power input is 1000 kW. Determine the rate and
direction of heat flow. (Cp = 1.005kJ/kg OC)                                      (-580.3 kW)
14. An inventor claims that he has developed a heat engine which absorbs 1200 kJ and 800kJ
from reservoir at 800 K and 600 K respectively and rejects 600 kJ and 200 kJ as heat to
reservoir at 400 K and 300 K respectively. It delivers 1200 kJ work. Determine whether the
heat engine is theoretically possible.
15. A system at 27OC is compressed reversibly and isothermally such that it receives 30 kJ of work.
Calculate its entropy change.                                                       (-0.1 kJ/K)
16. During a reversible isothermal process occurring at 400 K, the entropy change was found to be
0.2 kJ/K. Find the work done during this process.                                      (80 kJ)
17. A system undergoes a process during which its entropy changes by -0.2 kJ/K. The system also
transfers 90 kJ of heat to the surroundings at 300 K during this process. What is the change in
entropy of the universe due to this process?                                       (0.1 kJ/K )
18. A system performs an irreversible process at 200 K and receives 200 J of heat from the
surroundings. Then which of the following relations is correct for the entropy changes (ΔS) of
the system
(a ) ΔS = 1 J/K (b ) ΔS > 1 J/K (c ) ΔS < 1 J/K (d ) ΔS = 0 J/ K
19. If an ideal gas undergoes a reversible adiabatic process and changes from (P1, T1) to (P2, T2),
show that the change in entropy is equal to zero.
20. Two identical bodies of constant heat capacity (Cp) are at the same initial temperature T1. A
refrigerator operates between these two bodies until one body is cooled to temperature T2. If
the bodies remain at constant pressure and undergo no change of phase, show that the

minimum amount of work needed to do this is Wmin =       Cp
[   T2
1
T2
T 2−2 T 1
]
21. A system maintained at constant volume is initially at temperature T1 and a heat reservoir at
the lower temperature T0 is available. Show that the maximum recoverable work as the system
Mechanical Engg. Dept
D.N.Thatoi, S.N.Das, A.K.Mishra

is cooled to T0 is
[
W max =C v T 1 −T 0 −T 0 ln
 ]
T1
T0
.

22. A rigid vessel contains air at 2 bar, 100oC while the surroundings are at 30oC. The air in the
vessel gets cooled by heat transfer to the surroundings and reaches thermal equilibrium with
the surroundings. For 5 kg of this air, determine the :
(i ) Entropy change of air.                                                      (-0.746 kJ/K)
(ii ) Entropy change of surroundings.                                            (0.829 kJ/K )
(iii ) Entropy change of the universe.                                           (0.083 kJ/K )
23. One kg of superheated steam at 0.2 M Pa and 200oC contained in a piston-cylinder assembly
is kept at ambient condition of 300 K till steam is condensed to saturated liquid at constant
pressure. Calculate the change in the entropy of the universe associated with this process.
(1.9089 kJ/K)
24. Steam at 0.8 MPa, 2500C and flowing at the rate of 1 kg/s passes into a pipe carrying wet
steam at 0.8 MPa, 0.95 dryness fraction. After adiabatic mixing, the flow rate is 2.3 kg/s.
Determine the condition of steam after mixing. The mixture is expanded in a frictionless nozzle
isentropically to the pressure of 0.4 MPa. Determine the quality of steam leaving the nozzle.
(178.965oC, 6.709 kJ/kg-K; 0.96)

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