# Chapter 4: Second Law of Thermodynamics (Solution)

**Problem 4.1: **Calculate the minimum work required to produce 10 kg ice cubes from water initially at 273 K. Assume that the surroundings are at 300 K. The latent heat of fusion of water at 273 K = 333.5 kJ/kg.

**Problem 4.2: **A 800 MW thermal power plant uses steam at 600 K and discards heat to a river at 295 K. Determine the heat discarded to the river if the thermal efficiency of the plant is 70 percent of the maximum possible value.

**Problem 4.3: **The overhead vapor from a distillation column is taken to a condenser where it is condensed by heat exchange with cooling water that enters at 300 K and leaves at 310 K. The vapor can be treated as pure water vapor at 101.3 kPa. Its flow rate is 5000 kg/h. It is condensed and subcooled to 345 K. If the condenser were replaced by a reversible engine calculate the following: (a) The work output of the engine (b) The per cent reduction in the cooling water circulation rate when the condenser is replaced with Carnot engine.

**Problem 4.4: **A reversible refrigerator absorbs heat from water at 273 K in order to produce ice at the same temperature and rejects heat to the surroundings at 300 K. The work requirement of the refrigerator is to be met by a reversible heat engine operating between a heat source at 425 K and surroundings at 300 K. For each kilo joule of heat received by the engine, calculate: (a) The heat removed from water (b) The heat rejected to the surroundings.

**Problem 4.5: **An inventor claims to have developed a heat pump with a COP of 6 which maintains the cold space at 250 K when operating in a surrounding temperature of 310 K. Would you agree with his claim?

**Problem 4.6: **An engine using hot spring water at 330 K as the heat source receives 420 kJ and is said to reject 335 kJ to melting ice at 273 K. Comment on the engine efficiency.

**Problem 4.7: **An inventor claims to have developed a refrigerating machine which maintains the refrigerated space at 267 K while operating in a room at 300 K. If the COP claimed is 8.5, how would you evaluate his claim?

**Problem 4.8: **One mole of an ideal gas is compressed isothermally at 400 K from an initial pressure of 1 bar to a final pressure of 10 bar. The work required for this irreversible process is 20 percent more than that required for a reversible compression. The heat liberated during compression is absorbed by a heat reservoir at 300 K. Determine: (a) The entropy change of the gas (b) The entropy change of the reservoir (c) The total entropy change.

**Problem 4.9: **Three moles of hydrogen and 1 mol nitrogen, both at 10 bar and 373 K are separately admitted, mixed, and then heated to 773 K in a heat exchanger with a corresponding increase in pressure. Calculate the entropy change in the process. Assume C_{V} = 21 J/mol K for the mixture.

**Problem 4.10: **Ten kilograms of water (the system) is heated from 290 K to 340 K under the following conditions: (a) Using saturated steam at 10 bar (b) Using saturated steam at 5 bar (c) Using superheated steam at 5 bar and 513 K (d) Using 10 kg water at 340 K in a counter-current exchanger. Assume that the condensate is not subcooled. In each case calculate the entropy change of the system, the entropy change of the surroundings, and the total entropy change and compare them.

**Problem 4.11: **An ideal gas at 300 K and 10 bar enters an apparatus, which is thermally and mechanically insulated from the surroundings. The gas leaves the apparatus in two streams in equal quantities. One stream is at 360 K and 1 bar and the other is at 240 K and 1 bar. The heat capacity of the gas is 30 kJ/kmol K. Determine the total entropy change. Is the process thermodynamically possible?

**Problem 4.12: **An ideal gas at 420 K and 5 bar enters a steady-flow system and leaves at 298 K and 1 bar. Heat is exchanged with the surroundings at 298 K. The mean specific heat of the gas is 30 kJ/kmol K. Assuming reversible operation, determine the work extracted from each kilo mole of gas flowing through the system.

**Problem 4.13: **Oil of specific heat 3.2 kJ/kg K is cooled from 495 K to 315 K at a rate of 5000 kg/h by exchanging heat with a large thermal reservoir at a constant temperature of 300 K. What is the lost work in the process?

**Problem 4.14: **It has been suggested that a building could be heated by a refrigeration engine operating in a Carnot cycle. The engine takes heat from the outside at 280 K and delivers heat to the building at 295 K. How much work would have to be done for every kilo joule heat delivered to the building?

**Problem 4.15: **An inventor claims to have developed an engine that produces 1200 kJ of work while receiving 1000 kJ of heat from a single heat reservoir. Such an engine would violate both the first and second laws of thermodynamics. Do you agree? Why?

**Problem 4.16: **Calculate the molal entropy of fusion and vaporization of ethyl alcohol which melts at 158.8 K and boils at 351.5 K. The latent heat of fusion and vaporization are 108 kJ/kg and 855 kJ/kg respectively.

**Problem 4.17: **Air at an initial condition of 1.4 bar, 295 K, and 0.3 m^{3} is compressed to a pressure of 4.5 bar whereby its volume is reduced to 0.12 m^{3}. Calculate the change in the molal entropy of the gas in this process. Take the specific heat at a constant volume of air to be 20.52 kJ/kmol K.

**Problem 4.18: **A block of copper at a temperature of 825 K and weighing 5 kg is dropped into 50 kg water at 300 K. If there are no heat losses what is the change in entropy of (a) copper, (b) water, and (c) copper and water both considered together? C_{P} of copper is 0.4 kJ/kg K and that of water is 4.2 kJ/kg K.

**Problem 4.19: **A refrigerator is to be used to maintain the temperature of a certain storage space at 250 K. The compressor is rated at 250 W and the cooling duty is estimated to be 5×10^{4} kJ per day. Determine the fraction of the time the compressor runs if the refrigerator is used in (a) a cold country where the ambient temperature is 290 K and (b) a tropical country where the ambient temperature 310 K.

**Problem 4.20: **How much work must be done in order to cool the air in an otherwise empty room of dimensions 5.0 m×5.0 m×3.0 m from 303 K to 295 K when the ambient temperature is (a) 293 K (b) 303 K? Take specific heat of air to be 29 kJ/kmol K and density to be 1.2 kg/m^{3}.

**Problem 4.21: **Liquid helium is produced from its saturated vapor at 4.2 K. The latent heat of vaporization of helium at this temperature is 83.3 kJ/kmol. Calculate the minimum work required to produce liquid helium if the ambient temperature is 305 K.

**Problem 4.22: **A heat engine absorbs 260 kJ of heat from a source at 325 K and yields a work output of 72 kJ rejecting 100 kJ of heat to a reservoir at 300 K and 88 kJ of heat to another reservoir at 275 K. Does this engine violate the laws of thermodynamics?

**Problem 4.23: **The heat capacity of chloroform in the range 240 K to 330 K is given by

where C_{P} is in J/mol K and T in K. Calculate the change in molar entropy of chloroform when it is heated from 273 K to 300 K.

**Problem 4.24: **One kilogram of saturated liquid water at 100 kPa is mixed with 5 kg of superheated steam at 100 kPa and 573 K in an insulated container. Assuming that the pressure remains constant during mixing, calculate the entropy generation due to the mixing process.

**Problem 4.25: **In a heat exchanger, air is heated from 293 K to 353 K by means of a second air stream which enters the exchanger at 423 K. The molar flow rates of both the streams are equal and the specific heat of air is 29.3 kJ/kmol K. (a) Calculate the entropy change of both the streams and the total entropy change. (b) Repeat the above calculations if the cooler air temperature is raised to 393 K. Why could this be achieved only in a counter-current system?

**Problem 4.26: **A cylinder of free volume 0.283 m^{3} is divided into two compartments by a piston. Initially, the piston is at one end and the cylinder contains 1 kg steam at 423 K. Air from a supply main at 1400 kPa and 288 K is then admitted slowly via a throttle valve to the other side of the piston. Estimate the temperature of the air when pressure equilibrium is attained. Neglect any heat loss from the cylinder and assume no heat transfer across the piston. State clearly the assumptions made. C_{V} = 21 J/mol K for air.

**Problem 4.27: **One kilo mole of water is heated at the constant pressure of 1400 kPa from a temperature of 294 K to the boiling point and then completely vaporized at this pressure. What portion of the heat transferred is unavailable for transformation to work in a heat engine assuming that the temperature of the heat sink is 283 K? Boiling point of water at 1400 kPa = 468 K. Latent heat of vaporization = 1960 kJ/kg.

**Problem 4.28: **What is the increase in unavailable energy for the conditions in Exercise 4.27, if the heat is transferred from a furnace at 1480 K?

**Problem 4.29: **Determine the entropy of the mixture of two ideal gases A and B in the proportion 1:2. If this mixture is expanded through a throttle from pressure 500 kPa to 100 kPa what is the entropy change of the universe?

**Problem 4.30: **Oil with a heat capacity of 3.2 kJ/kg K is to be cooled from 495 K to 315 K at a rate of 5000 kg/h. An unlimited supply of cooling water at a constant temperature of 303 K is available. Determine the lost work in the process and the thermodynamic efficiency of the process.

**Problem 4.31: **A hot hydrocarbon oil (C_{P} = 2.512 kJ/kg K) is cooled from 422 K to 339 K in a heat exchanger at the rate of 2500 kg/h. Cooling water at the rate of 5000 kg/h enters the exchanger at 294 K. Assume that there is no heat loss in the exchanger. (a) What is the change in entropy of the oil? (b) What is the total change in entropy? (c) How much work could be obtained if the cooling of the oil were carried out by a reversible Carnot engine rejecting heat to a sink at 294 K?

**Problem 4.32: **(a) Nitrogen gas (C_{P} = 30 kJ/kmol K) at 1 bar and 300 K is to be compressed to 10 bar and 450 K at a rate of 5000 kg/h. Cooling water (C_{P} = 4.1868 kJ/kg K) enters the compressor at 300 K at the rate of 7500 kg/h and leaves at 320 K. Determine the power required by the compressor. (b) For the same conditions of nitrogen as in the previous part, determine the minimum power required by the compressor. The rate of circulation and the inlet temperature of cooling water remain the same as above.

**Problem 4.33: **Calculate the change in entropy when one gram of ice at 273 K is converted into steam at 373 K. Latent heat of fusion of ice = 336 J/g, latent heat of vaporization = 2268 J/g and the mean specific heat of water between 273 K and 373 K = 4.2 J/g K.