Problem Sheet 4 Second Law of Thermodynamics

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Mechanical Engineering Department
Jadavpur University
Engineering Thermodynamics
Problem Sheet 4 Second Law of Thermodynamics
1. Prove that the COP of a reversible refrigerator operating between to given temperatures is
the maximum. [PKN 141]
2. Prove that the COP of a reversible heat pump operating between to given temperatures is the
maximum. 
3. An inventor claims to have a heat engine that is capable of developing 9 kW while working
between the temperature limits of 20oC and 40oC. It receives only 1047 kJ/min of heat.
Discuss the possibility of the claim.
4. An inventor claims to have developed a refrigeration unit which maintains the refrigerated
space at –10oC while operating in a room where the temperature is 25oC, and which has a
COP of 8.5. How do you evaluate his claim? How would you evaluate his claim of a COP of
7.5? [VWS 217]
5. A heat pump is to heat a house in the winter and then reversed to cool the house in the
summer. The interior temperature of the house is to be maintained at 20oC. Heat transferred
through the walls and roof is estimated to be 2400 kJ per hour per degree Celcius
temperature difference between the inside and outside.
a. If the outside temperature in the winter is 0oC, what is the minimum power required
to drive the heat pump?
b. If the power input is the same as that in part a., what is the maximum outside
temperature for which the inside can be maintained at 20oC? [VWS 218-219]
6. Consider an engine in the outer space that operates on the Carnot cycle. The only way in
which heat can be transferred from the engine is by radiation. The rate at which heat is
radiated is proportional to the fourth power of the absolute temperature and to the area of the
radiating surface. Show that for a given power output and a given Th, the area of the radiator
will be a minimum if Tl/Th = ¾, where Tl and Th are the lower and higher temperatures
respectively. [VWS 218]
7. Two bodies of equal mass m and heat capacity C are at temperatures T1 and T2 respectively
(T1 > T2). If the first body is used as the source of heat for a reversible engine and the second
as sink, show that the maximum work obtainable from such an arrangement is
mC(√T1-√T2)2.
8. A reversible heat engine operates between three heat reservoirs A, B and C. The heat
received by the engine from each of the reservoirs A and B is same and at temperatures TA
and TB respectively. The engine rejects heat to the reservoir C at temperature TC. If the
engine efficiency is K times the efficiency of a reversible engine operating between the two
reservoirs A and C only, show that
TA/TB = {2(1-K)TA/TC + 2K –1} [PKN 140]
9. A heat engine receives half of its heat supply at 1000 K and half at 500 K while rejecting
heat to a sink at 300 K. What is the maximum thermal efficiency of the heat engine? [PKN
141] Ans. 55%
10. A reversible heat engine operates between temperatures T1 and T (T1 > T). The energy
rejected from this engine is received by a second reversible engine at the same temperature
T. The second engine rejects energy at temperature T2 (T2 < T1). Show that (a) temperature T
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is the arithmetic mean of the temperatures T1 and T2 if the engines produce the same amount
of work output, and (b) temperature T is the geometric mean of temperatures T1 and T2 if the
engines have the same cycle efficiencies. [PKN 140]
11. A solar powered refrigerator receives heat from a solar collector at Th, rejects heat to the
atmosphere a Ta, and pumps heat from a cold space at Tc. The three heat transfer rates are
Qh, Qa, and Qc respectively. Derive an expression for the minimum ratio Qh/Qc, in terms of
the three temperatures.
If Th = 400 K, Ta = 300 K, Tc = 200 K, Qc = 12 kW, what is the minimum Qh? If the collector
captures 0.2 kW/m2, what is the minimum collector area required? [PKN 141] 
Ans. 24 kW, 120 m2
12. A reversible power cycle is used to drive a reversible refrigeration cycle. The power cycle
takes in Q1 heat units at T 1 and rejects Q2 at T 2. The refrigerator extracts
Q4 from the sink at T 4 and discharges Q3 at T 3 . Develop an expression for the
ratio 
Q 4
Q1
 in terms of the four temperatures. [PKN 141] Ans. 
T 4T 1 – T 2
T 1T 3 – T 4
13. Lord Kelvin was the first to point out the thermodynamic wastefulness of burning fuel for
the direct heating of a house. It is much more economical to use the high temperature heat
produced by combustion in a heat engine and then to use the work so developed to pump
heat from the outdoors up to the temperature desired in the house. In the following figure the
boiler furnishes heat Q1 at the high temperature T 1 . The heat is absorbed by a heat
engine, which extracts work W and rejects the waste heat Q2 into the house at T 2 .
Work W is in turn used to operate a mechanical heat pump, which extracts Q3 from
outdoors at temperature T 3 and rejects Q ' 2 into the house. As a result of this cycle of
operations, a total quantity of heat Q2Q' 2 is liberated in the house, against Q1
which would be provided directly by the ordinary combustion of the fuel. Thus the ratio
Q2Q ' 2/Q1 represents the heat multiplication factor of this method. Determine this
multiplication factor if T 1 = 473 K, T 2 = 293 K, and T 3 = 273 K. [PKN 134]
 Ans. 6.2
Figure Q.13
Boiler
T1
 Q1
House
T2
 Q2
 Q'2
Outdoors at T3
 Q3
W
W
H.E.
H.P.
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14. Two kg of water at 80oC are mixed adiabatically with 3 kg of water at 30oC in a constant
pressure process at 1 atmosphere. Find the increase in entropy of the total mass of water
due to the mixing process. [PKN 183] Ans. 0.0576 kJ/K
15. One kg of air initially at 0.7 Mpa, 20oC changes to 0.35 Mpa, 60oC by the three
reversible non-flow processes as shown in the following figure. Process 1-a-2 consists of
a constant pressure expansion followed by a constant volume cooling, process 1-b-2 an
isothermal expansion followed by a constant pressure expansion, and process 1-c-2 an
adiabatic expansion followed by a constant volume heating. Determine the change of
internal energy, enthalpy, and entropy of each process, and find the work transfer and
heat transfer for each process. Take Cp = 1.005 kJ/(kg.kK), Cv = 0.718 kJ/(kg. K), and
assume the specific heats to be constants. Also assume for air Pv = 0.287T, where P is
the pressure in kPa, v is the specific volume in m3/kg, and T is the temperature in K.
[PKN 184]
 P Reversible adiabatic
Reversible isothermal
 a
 b
 c
 v
Figure Q.15
16. A reversible engine, as shown in the following figure, during a cycle of operations draws
5 MJ from the 400 K reservoir and does 840 kJ of work. Find the amount and direction
of heat interaction with the other reservoirs. [PKN 186]
 Ans Q2 = -4.98 MJ, Q3 = +0.82 MJ
 Q3 Q2 Q1 = 5 MJ
W = 840 kJ
Figure Q. 16
17. For a fluid for which Pv/T is a constant quantity equal to R, show that the change in
specific entropy between two states A and B is given by
sB – s A=∫
T A
T B C pT dt – R ln PBPA
18. A rigid tank contains an ideal gas at 40oC that is being stirred by a paddle wheel. The
paddle wheel does 200 kJ of work on the ideal gas. It is observed that the temperature of
the ideal gas remains constant during this process as a result of heat transfer between the
system and the surroundings at 25oC. Determine (a) the entropy change of the ideal gas
and (b) the total entropy generation. [PKN 190] Ans. (a) 0, (b) 0.671 kJ/K
19. Consider a Carnot cycle heat engine with water as the working fluid. The heat transfer to
200 K 300 K 400 K
E
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the water occurs at 300oC, during which process the water changes from saturated liquid
to saturated vapour. The heat is rejected from water at 40oC. (a) Show the cycle on T-s
diagram. (b) Find the quality of the water at the beginning and end of the heat rejection
process. (c)Determine the net work output per kilogram of water and the cycle thermal
efficiency. [VWS 273]
20. Consider a Carnot cycle heat pump with R-22 as the working fluid. Heat is rejected from
R-22 at 40oC, during which process the R-22 changes from saturated vapour to saturated
liquid. The heat is transferred to the R-22 at 0oC. (a) Show the cycle on a T-s diagram.
(b) Find the quality of R-22 at the beginning and end of the isothermal heat addition
process at 0oC. (C) Determine the coefficient of performance of the cycle. [VWS 273]
21. One kilogram of ammonia in a piston/cylinder at 50oC, 100 kPa is expanded in a (a)
reversible isothermal process to 100 kPa, (b) reversible isobaric process to 140oC, and
(c) reversible adiabatic process to -30oC. Find the work and heat transfer for each
process. Draw each process on Pv and Ts planes. [VWS 273]
22. A heavily insulated cylinder/piston contains ammonia at 1200 kPa, 260oC. The piston is
moved, expanding the ammonia in a reversible process until the temperature is -20oC.
During the process 600 kJ of work is given out by the ammonia. What was the initial
volume of the cylinder? [VWS 274]
23. Insulated cylinder/piston has an initial volume of 0.15 m3 and contains steam at 400 kPa,
200oC. The steam is expanded adiabatically, and the work output is measured very
carefully to be 30 kJ. It is claimed that the final state of the water is in the two-phase
(liquid and vapour) region. What is your evaluation of the claim? [VWS 276]
24. Two tanks contain steam, and they are both connected to a piston/cylinder as shown
below. Initially the piston is at the bottom and the mass of the piston is such that a
pressure of 1.4 MPa below it will be able to lift it. Steam in A is 4 kg at 7 Mpa, 700oC
and B has 2 kg at 3 Mpa, 350oC. The two valves are opened, and the water comes to a
uniform state. Find the final temperature and the total entropy generation, assuming no
heat transfer.
 g
Figure Q.24
A B