Problems
3.1 A battery gets accidentally short circuited, and in the process its charge reduces from 3000 Coulombs to 1000 Coulombs. During this process the heat lost to the surroundings is 5 kJ. Determine the entropy change in the process. Assume, the average battery voltage during this process is 12V, and its average temperature is 370C.
3.2 In Example 3.3, show the equivalence of Caratheodory’s formulation with Kelvin-Planck and Clausius formulations for the case when the state 2, taken to the left of isentrope through state 1, is at a higher temperature than state 1 (Fig 3.23).
3.3 A 2.5W resistor is connected to a battery such that a current of 5 Amps passes through it for 20 seconds. If the temperature of the resistor is maintained constant at 200C, determine the entropy change of the resistor, the environment (taken as a thermal reservoir at 200C) and that of the ‘universe’? What is the entropy generation in the resistor, and the environment?
3.4 During a process, an object receives 100kJ of heat from a thermal energy reservoir and produces 200 kJ of work output. Can we return the object to the original state through an adiabatic process?
3.5 Show that any violation of the Clausius statement of the second law would result in violation of the Kelvin-Planck and the Caratheodory’s statements too.
3.6 Producing and maintaining very low temperatures is an extremely difficult process. Estimate, the power requirements of a reversible refrigeration machine needed to provide 1W of cooling at .0001K when the condensing temperature is 500C. Compare it with the power requirement for an evaporator temperature of -200C (~ domestic refrigerator).
3.7 A window air conditioner produces 5kW of cooling effect maintaining the room at 250C when the outdoor temperature is 450C. Its evaporator temperature is 100C and the condenser temperature is 550C, and it consumes 2kW of power. Estimate the entropy generation rate due to (a) internal irreversibilities, and (b) external irreversibilities.
3.8 The air conditioner of problem 3.7 is to be operated as a heat pump in winter when the outdoor temperature drops to -150C and the indoor temperature is being maintained at 200C. Keeping the temperature differences to enable heat transfer at both the indoor and the outdoor coils to be the same as before, and assuming the same amount of internal irreversibility, estimate the heat transfer to the room for the same power consumption.
3.9 An inventor claims to have developed a new method of producing refrigeration wherein following typical operating conditions are achieved:
Heat transfer from a high temperature source at 1500C = 10 kW
Heat rejection to a sink at 300C = 20.1 kW
Power input to the pump = 0.1 kW
Heat transfer from a low temperature space at 200C = 10 kW.
The temperature difference at each heat exchanger is 100C
Is this method feasible thermodynamically?
3.10 Estimate the “energy loss” involved in heating water by an electric heater of 1 kW capacity, rather than using the thermal energy directly. Assume electricity production to be done in an engine operating at 70% of the reversible engine efficiency, coupled to an alternator of 90% efficiency. The electric transmission losses are 10%. The two operating temperatures of the engine are 6000C and 400C respectively.
3.11 In problem 3.10, if heat pump were used instead, how would the result alter? Assume its operating temperatures to be 00C and 600C, and COP 70% of the reversible heat pump operating under same condition.
3.12 Two reversible refrigeration systems are cascaded by having the heat rejected from the low temperature system input to the evaporator of the high temperature system. The three cardinal temperatures are:
Evap temperature of low temperature system = -600C
Condenser temperature of low temperature system =
Evaporator temperature of high temperature system= -100C
Condenser temperature of high temperature system= 400C
What is the COP of the cascade system? How does it compare with the COP of a single reversible refrigeration system operating between -600C and 400C?
3.13 In problem 3.12, if the refrigeration systems are not reversible, but have a COP which is 60% of the corresponding reversible system, how would the results be altered?
3.14 The mini cold store in a students’ hostel in being cooled down after a long shut down from the ambient temperature of 350C to the operating temperature of 50C, with the help of a refrigeration machine. The heat ingress into the cold store is 100 W per 0C temperature difference between the outdoor and the indoor temperatures. The evaporator coil of the refrigerator machine is operated steadily at 00C and its heat removal rate is 0.4(T) kW where T is the temperature of air in the indoor space in 0C. Determine the time required for pulling down the cold store temperature and the total power consumed in this process. Assume the COP of the refrigeration machine as 60% of the COP of a reversible system, and the energy of air inside the cold store to be given by the equation E = 4T kJ, where T is the temperature in 0C.
3.15 In an air-conditioned library, it is proposed to replace the 40W fluorescent tubes by energy efficient 9W-CFL lamps. The total power consumption of each fluorescent tube is about 60W and that of CFL lamps of same illumination is 10W, but it costs about 50 Rs more. If 200 of such fluorescent tubes are replaced estimate the yearly saving in running cost and the payback period, i.e. the number of years in which the additional initial cost will be recovered through this saving. Given the overall COP of the air conditioning system is 3.0 and energy costs Rs 4/kWh.
3.16 The energy content of an object is related to its temperature by the equation E = CT where T is the temperature is Kelvin. The object initially at temperature Ti is brought in contact with a thermal energy reservoir at temperature Tf (Tf>Ti) and allowed to come to equilibrium. Show that the total entropy change of the object and the reservoir amount to
C [ln (Tf/Ti) – 1 + Ti/Tf]
3.17 Find the entropy change in the quenching process of example 2.11
3.18 Two identical objects, whose energy content is related to the temperature T (in K) by equation E = CT, are initially at temperatures T1 and T2 (T1>T2) show that the maximum possible work that can be obtained by using a reversible engine operating between the two objects is
W = C (ÖT1 - ÖT2)2
What is the final temperature of both the objects?
3.19 The energy content of a given mass of water is related to its temperature T (in K) as
E = 2T kJ
This has to be heated from 200C to 1000C. Examine the following possibilities from the second-law perspective, (i.e. the entropy increase of universe):
(a) Bringing it in contact with a TER at 2000C.
(b) Bringing it in contact with a TER at 1200C.
(c) Heating it upto 500C by bringing it in contact with a TER at 700C, followed by heating upto 1000C by bringing it in contact with a TER at 1200C.
How could the heating be done without any increase in entropy?