CHAPTER 04
THE SECOND LAW OF
THERMODYNAMICS
4-1 EFFICIENCY OF A HEAT ENGINE Example 4-1
A diesel engine performs 2200 Jof mechanical work and discards 4300 Jof heat each cycle.
(a) How much heat must be supplied to the engine in each cycle?
(b) What is the thermal efficiency of the engine? Solution (a) QIN =QOUT +W =4300+2200=6500 J (b) 0.34 34 % 6500 2200 or Q W IN = = = η Example 4-2
An engine with 20 %efficiency does 100 J of work in each cycle.
(a) How much heat is absorbed in each cycle? (b) How much heat is rejected in each cycle? Solution
(a) The efficiency of a heat engine is given by IN Q W = η J Q Q IN IN 500 20 . 0 100 = = = η (b) QOUT =QIN −W =500−100=400 J
Example 4-3
An engine absorbs 100 Jand rejects 60 J in each cycle. (a) What is its efficiency?
(b) If each cycle takes 0.5 s, find the power output of this engine in watts.
Solution
(a) The efficiency of the heat engine is defined as % 40 40 . 0 100 60 100 or Q Q Q Q W IN OUT IN IN = − = − = = η (b) W t Q Q t W P IN OUT 80 5 . 0 60 100 = − = − = = Example 4-4
Find the efficiency of a Carnot engine working between the temperatures 2250C and250C. B.U. B.Sc. (Hons.) 1983A Solution
The efficiency of a Carnot engine is given by
1 ×100 % − = H L T T η % 40 % 100 273 225 273 25 1 × = + + − = η Example 4-5
What is the maximum efficiency of a steam engine that utilizes steam from a boiler at 480 Kand exhausts at
K
373 . K.U. B.Sc. 2003 Solution
The efficiency of a Carnot engine is given by
1 ×100 % − = H L T T η % 3 . 22 % 100 480 373 1 × = − = η
Example 4-6
A heat engine performs 200 joules of work and at the same time rejects 3000 joules of heat to the cold body. Calculate the efficiency. B.U. B.Sc. (Hons.) 1989A Solution
The efficiency of a Carnot engine is given by
1 ×100 % − = IN OUT Q Q η % 25 . 6 % 100 200 3000 3000 1 × = + − = η Example 4-7
A Carnot engine is operated between two heat reservoirs at temperatures of 400 K and300 K. If the engine receives
1200 joules of energy from the reservoir at 400 K in each cycle, how many joules of energy does it reject to the reservoir at300 K? B.U. B.Sc. (Hons.) 1986A Solution
The efficiency of a Carnot engine is given by
− = − = IN OUT H L Q Q T T 1 1 η IN OUT H L Q Q T T = J Q T T Q OUT H L OUT (1200) 900 400 300 = = = Example 4-8
A steam engine, operating between 1150Cand 600C, develops 2500 hp. How much heat must be supplied if its overall efficiency is 10 percent? What is the maximum efficiency that an ideal heat engine can have when it operates between the stated temperatures? (1 hp = 746 W). B.U. B.Sc. 1999A
Solution
Let QINbe the supplied heat, then
W =10 % of QIN =0.1 QIN 2500×746=0.1 QIN J QIN 7 10 865 . 1 1 . 0 746 2500 × = × =
The efficiency of ideal heat engine is given by
% 2 . 14 142 . 0 273 115 273 60 1 1 or T T H L = + + − = − = η Example 4-9
A heat engine absorbs 52.4 kJof heat and exhausts 36.2 kJ
of heat per cycle. Calculate the efficiency and the work done by the engine per cycle. P.U. B.Sc. 2003, 2009 Solution
The efficiency of a Carnot engine is given by
1 ×100 % − = IN OUT Q Q η % 9 . 30 % 100 4 . 52 2 . 36 1 × = − = η
The work done by the engine per cycle is
kJ Q
Q
W = IN − OUT =52.4−36.2=16.2
Example 4-10
A Carnot’s engine takes 3.36×106joules of heat from a reservoir at 5270Cand gives some heat to sink at270C. What is the efficiency of engine? How much work does it perform in joules? K.U. B.Sc.2006 Solution
The efficiency of heat engine is given by
− = H L T T 1 η
% 5 . 62 625 . 0 273 527 273 27 1 = or + + − = η
The work done by the engine is
J Q
W =η IN =(0.625)(3.36×106)=2.1×106
Example 4-11
A Carnot engine extracts 240 Jof heat from a high temperature reservoir during each cycle. It rejects 100 J of heat at reservoir at150C.
(i) How much work does the engine do in one cycle? (ii) What is its efficiency?
(iii) What is the temperature of the hot reservoir?
K.U. B.Sc. 2001 Solution
(i) The work done by the engine is given by J Q
Q
W = IN − OUT =240−100=140
(ii) The efficiency of the given Carnot engine is
% 3 . 58 % 100 240 100 1 % 100 1 × = − = × − = IN OUT Q Q η
(iii) The efficiency of Carnot engine in terms of temperatures of reservoirs is given by
1 ×100 % − = H L T T η 58.3 1 15 273×100 + − = H T H T 288 1 583 . 0 100 3 . 58 − = = 288 =1−0.583=0.418 H T C or K TH 0 418 691 418 . 0 288 = =
Example 4-12
A heat engine operates between two reservoirs at 300 K
and500 K. During each cycle it absorbs 200 calof heat from the hot reservoir.
(a) What is maximum efficiency of the engine?
(b) Determine the maximum work the engine performs during each cycle. B.U. B.Sc. 2008A Solution
(a) The efficiency of heat engine is given by − = H L T T 1 η % 40 40 . 0 500 300 1− = or = η
(b) The heat absorbed by the engine from hot reservoir in each cycle is
J J
cal
QIN =200 =(200)(4.186) =837.2 The work done by the engine in each cycle is J Q
W =η IN =(0.40)(837.2)=334.88 Example 4-13
A Carnot engine absorbs heat from a reservoir at a temperature of 10000Cand rejects heat to a cold reservoir at a temperature of 00C. If the engine absorbs 1000joules from the high temperature reservoir, find heat rejected and efficiency of heat engine. B.U. B.Sc.(Hons.) 1988A, 1989A Solution Now H L IN OUT T T Q Q = J Q T T Q IN H L OUT 214 ) 273 1000 ( ) 1000 )( 273 0 ( = + + = =
1 ×100 % − = H L T T η % 6 . 78 % 100 273 1000 273 0 1 × = + + − = η Example 4-14
In a Carnot cycle, the isothermal expansion of an ideal gas takes at 1400Cand isothermal compression at250C. During the compression, 2090 Jof heat energy is transferred to the gas. Calculate
(i) The work performed by the gas during isothermal expansion.
(ii) Heat rejected from the gas during isothermal compression.
(iii) Work done on the gas during isothermal compression. K.U. B.Sc. 2002, 2005 Solution
(a) W =−QIN =−2090 J The negative sign indicates that the work is done by the gas.
(b) The efficiency of Carnot engine is given by
1 ×100 % − = H L T T η % 8 . 27 % 100 273 140 373 25 1 × = + + − = η Now 1 ×100 % − = IN OUT Q Q η (100) 2090 1 8 . 27 − = QOUT − = = 2090 1 278 . 0 100 8 . 27 QOUT 1 0.278 0.722 2090 = − = OUT Q
QOUT =(0.722)(2090)=1509 J
(c) Work done on the gas during isothermal compression W =QOUT =1509 J
Example 4-15
A Carnot cycle has efficiency 75 % working between highest and lowest temperatures. What is the lowest temperature of its high temperature is 9270C?
B.U. B.Sc. 2007S Solution
The efficiency of a Carnot engine is given by
− = H L T T 1 η 273 927 1 100 75 + − = TL 1200 1 75 . 0 TL − = 1 0.75 0.25 1200 = − = L T C K TL 0 27 300 ) 1200 )( 25 . 0 ( = = = Example 4-16
A Carnot engine has an efficiency of 22 %. It operates between heat reservoirs differing in temperature by750C. Find the temperature of the reservoirs.
K.U. B.Sc. 2004, 2008 Solution
Let TLand TH be the temperatures of cold and hot reservoirs (i.e. sink and source) respectively, then the efficiency of Carnot engine is given by H L T T − = 1 η Now η=22 %=0.22
TH =(TL+75) K Hence 75 75 75 1 22 . 0 + − + = + − = L L L L L T T T T T 75 75 22 . 0 + = L T 0.22 TL +16.5=75 0.22 TL =75−16.5=58.5 TL K or C 0 7 266 22 . 0 5 . 58 − = = TH =TL+75=266+75=341 K or 680C Example 4-17
A Carnot heat engine with an efficiency of 0.20 operates between two temperatures that differ from each other by100 K. What are the temperatures between which the cycle operates?
Solution
The efficiency of a Carnot engine is given by
H L T T − = 1 η Now η=0.20 TH =(TL +100) K Hence 100 100 100 1 20 . 0 + − + = + − = L L L L L T T T T T 100 100 20 . 0 + = L T 0.20 TL +20=100 0.20 TL =100−20=80 TL 400 K 20 . 0 80 = =
TH =TL +100=400+100=500 K The given engine operates between 400 K and 500 K.
Example 4-18
The exhaust temperature of a heat engine is 2200C. What must be the high temperature if the Carnot efficiency is to be 36 %? Solution Now H L T T − = 1 η H T 273 220 1 100 36 + − = H T 493 1 36 . 0 = − 493 =1−0.36=0.64 H T C or K TH 770 4970 64 . 0 493 = = Example 4-19
In one cycle of operation a heat engine takes in 2200 Jof heat and performs 620 J of work.
(a) What is the efficiency of this engine? (b) How much heat is exhausted in each cycle?
(c) If the engine completes a cycle each 0.033 s, then at what rate is heat added, heat exhausted, work done? Solution
(a) The efficiency of heat engine is defined as % 28 28 . 0 2200 620 or Q W IN = = = η
(b) The heat exhausted in each cycle is
J W Q QOUT = IN − =2200−620=1580 (c) W kW t Q P IN IN 6.67 10 66.7 033 . 0 2200 4 = × = = =
W kW t Q POUT OUT 4.79 10 47.9 033 . 0 1580 4 = × = = = W kW t W P 1.88 10 18.8 033 . 0 620 4 = × = = = Example 4-20
The low temperature reservoir of a Carnot engine is at
C
0
10 and has an efficiency of50 %. How much the temperature of the high temperature reservoir is increased to measure the efficiency to50 %? K.U. B.Sc. 2007 Solution
The efficiency of a Carnot engine is given by
− = H L T T 1 η H T 273 10 1 50 . 0 = − + 50 . 0 50 . 0 1 283 = − = H T TH 566 K 50 . 0 283 = =
Let η1be the new efficiency of heat engine with new source temperature as TH*, then * 1 1 H L T T − = η * 283 1 60 . 0 H T − = 283* =1−0.60=0.40 H T TH 707.5 K 40 . 0 283 * = =
The desired increase in temperature of hot reservoir is K T
T
T = H* − H =707.5−566=141.5 ∆
Example 4-21
A heat engine utilizes a heat source at 5800Cand has a Carnot efficiency of 29percent. To increase the efficiency to
35percent, what must be the temperature of the heat source?
Solution
The temperature of the cold reservoir i.e. sink is calculated as under H L T T − = 1 η 273 580 1 29 . 0 + − = TL 1 0.29 0.71 853 = − = L T K TL =(0.71)(853)=606
Let η1be the new efficiency of heat engine with new source temperature as * H T , then * 1 1 H L T T − = η * 606 1 35 . 0 H T − = 606* =1−0.35=0.65 H T C or K TH* 932 6590 65 . 0 606 = =
Example 4-22
A Carnot heat engine operates between a reservoir at
K
950 and has heat transfer of 400 Jexhausted to another reservoir at 300 Kduring each cycle. What is the amount of work done by the heat engine during each cycle?
Solution Now IN OUT H L Q Q T T = J Q T T Q OUT L H IN (400) 1267 300 950 = = = W =QIN −QOUT =1267−400=867 J
4-2 COEFFICIENT OF PERFORMANCE OF A REFRIGERATOR ENGINE
Example 4-23
The low temperature of a freezer cooling coil is −150Cand the discharge temperature is 300C. What is the maximum theoretical coefficient of performance?
Solution
The coefficient of performance is defined as
L H L T T T K − = 733 . 5 ) 273 15 ( ) 273 30 ( ) 273 15 ( = + − − + + − = K Example 4-24
A heat pump acting as a refrigerator is used to heat a house. The temperature outside the house is −100Cand the interior to be kept at220C. Find the maximum coefficient of performance of heat pump. P.U. B.Sc. 2005, 2006 Solution
The coefficient of performance of heat pump acting as a refrigerator is given by L H L T T T K − = Now TL =−100C =(−10+273) K =263 K K K C TH =220 =(22+273) =295 TH −TL =295−263=32 K Hence 8.218 32 263 = = K Example 4-25
What is the coefficient of performance of a refrigerator that operates with Carnot efficiency between temperatures
C
0
3
The coefficient of performance of heat pump acting as a refrigerator is given by L H L T T T K − = Now TL =−30C=(−3+273) K =270 K K K C TH =270 =(27+273) =300 TH −TL =300−270=30 K Hence 9 30 270 = = K Example 4-26
A household refrigerator, whose coefficient of performance is 4.7, extracts heat from the cooling chamber at rate of 250
J per second. How much work per cycle is required to
operate the refrigerator? P.U. B.Sc. 2002 Solution
The coefficient of performance K is defined as
W Q Q Q Q K L L H L = − = cycle J K Q W L 53.2 / 7 . 4 250 = = = Example 4-27
A refrigerator does 153 J of work to transfer 568 J of heat from its cold compartment.
(a) Calculate the refrigerator’s coefficient of performance. (b) How much heat is exhausted to the kitchen?
Solution
(a) The coefficient of performance K is defined as 71 . 3 153 568 = = = W Q K L
(b) Heat exhausted to kitchen will be J W
Example 4-28
Apparatus that liquefies helium is in a laboratory at296 K. The helium in the apparatus is at4 K. If 150 mJof heat is transferred from helium, find the minimum amount of heat delivered to the laboratory.
Solution
The coefficient of performance is defined as
L H L T T T K − = 2 10 37 . 1 4 296 4 − × = + = K Now K Q Q W Q QH = L + = L + L J QL 11.1 10 37 . 1 1 1 ) 10 150 ( 3 2 = × + × = − −
ADDITIONAL PROBLEMS
(1) A Carnot engine is operated between two reservoirs at temperatures 2270Cand1270C. Evaluate the efficiency of the engine. B.U. B.Sc.(Hons.) 1988S (2) A heat engine performs 1000 joules of work and at
the same time rejects 4000joules of heat energy to the cold reservoir. What is the efficiency of the engine? B.U. B.Sc.(Hons.) 1991A (3) A heat engine receives heat 120 joules and the work done by it is 90 joules. Calculate the efficiency in percentage. B.U. B.Sc. 1997A (4) A heat engine performs 2000 joules of work and at the same time rejects 6000joules of heat energy to the cold reservoir. What is the efficiency of the engine? B.U. B.Sc. 1998A (5) A Carnot engine whose heat source is at 1270C takes
100calories of heat at this temperature in each cycle and gives up 80calories to the heat sink. Find the temperature of heat sink. B.U. B.Sc. 1988S (6) A Carnot engine absorbs heat from a reservoir at a
temperature of 1000Cand rejects heat to a cold reservoir at a temperature of 00C. If the engine absorbs 1000 joules from the high temperature reservoir, find heat rejected and efficiency of heat engine. B.U. B.Sc. 1986S (7) What is the maximum efficiency of a steam engine
that utilizes steam from a boiler at 480 K and exhausts at 373 K? K.U. B.Sc. 2003 (8) A heat engine performs 200 joules of work and at the same time rejects 300 joules of heat to the cold body. Calculate its efficiency. B.U. B.Sc.(Hons.) 1989A (9) Calculate the efficiency of a Carnot engine working
between the temperatures 9270Cand270C.
(10) A Carnot engine’s working substance is water. It uses steam at 3000Cand condenses to water at 400C. What is the maximum theoretical efficiency of this engine? B.P.S.C. 1995 (11) A Carnot engine operates between the temperatures
K
850 and300 K. The engine performs 1200 J of work each cycle, which takes0.25 s. Calculate its efficiency and its average power. What are the rates of heat input and heat exhaust per cycle? F.P.S.C. 2009
Answers
(1) 20 % (2) 20 % (3) 75 % (4) 25 % (5) 320 K (6) QOUT =732 J , η=26.8 % (7) 22.3 % (8) 6.25 % (9) 75 % (10) 37 %
