# Chapter 5: Some Applications of the Laws of Thermodynamics (Solution)

**Problem 5.1: **A 50-mm diameter horizontal jet of water strikes a vertical plane. If the horizontal force needed to support the plane is 360 N, what is the velocity of the jet?

**Problem 5.2: **Wet steam containing 5 % by weight of liquid at a pressure of 500 kPa is mixed at a rate of 1 kg/s with superheated steam at 500 kPa and 473 K (H = 2855 kJ/kg) to obtain dry saturated steam at 500 kPa (H_{l} = 640 kJ/kg; H_{V} = 2749 kJ/kg). Determine the rate of addition of superheated steam if mixing is done adiabatically.

**Problem 5.3: **An ideal gas with heat capacity C_{P} = 29.7 kJ/kmol K flows steadily through a long capillary tube at 5 bar and 350 K and leaves at 2 bar. What is its exit temperature?

**Problem 5.4: **3600 kg/h of superheated steam at 200 kPa and 673 K enters a turbine with a velocity of 100 m/s. The inlet to the turbine is at an elevation of 10 m and the exit is at an elevation of 3 m. The steam leaves the turbine at a velocity of 150 m/s and is 98 % dry at a pressure of 10 kPa. What is the power output of the turbine if the energy loss from it is estimated to be 40000 kJ/h?

**Problem 5.5: **A tank has a volume of 3 m^{3} and contains 1400 kg liquid water in equilibrium with its vapor, which fills the remainder of the tank. The temperature and pressure are 505 K and 2902 kPa respectively (enthalpy of saturated vapor = 2802 kJ/kg; enthalpy of saturated liquid = 1000 kJ/kg; the specific volume of saturated liquid = 1.213×10–3 m3/kg; specific volume of saturated vapor = 68.89×10^{–3} m^{3}/kg). A quantity of 1000 kg water at 340 K (H = 280 kJ/kg) is to be pumped into the tank without removing any steam. How much heat must be added during this process, if the pressure and temperature in the tank remain at their initial values?

**Problem 5.6: **A turbine is fed with steam at 78 bar and 698 K (enthalpy = 3213 kJ/kg) at a rate of 1000 kg/h. Saturated steam at 5 bar (H = 2749 kJ/kg) is withdrawn from one point in the turbine at a rate of 250 kg/h. The remaining steam leaves the turbine saturated at 1 bar (H = 2676 kJ/kg). Determine the power output from the turbine, if it operates adiabatically.

**Problem 5.7: **A tank contains 1 kg steam at a pressure of 2100 kPa and a temperature of 648 K (enthalpy and specific volume are 3192 kJ/kg and 137.76×10^{–3} m^{3}/kg). It is connected through a valve to a vertical cylinder containing a frictionless piston. The piston is loaded with a weight such that a pressure of 700 kPa is necessary to support it. Initially, the piston is at the bottom of the cylinder. The valve is opened slightly so that steam flows into the cylinder until the pressure is uniform throughout the system. The final temperature of steam in the tank is found to be 513 K (enthalpy and specific volume of the superheated steam at 700 kPa and 513 K are 2933 kJ/kg and 329.23×10^{–3} m^{3}/kg). Calculate the temperature of steam in the cylinder, if no heat is transferred from the steam to the surroundings.

**Problem 5.8: **A well-insulated closed tank has a volume of 70 m^{3}. Initially, it contains 23000 kg water distributed between liquid and vapor phases at 300 K. Saturated steam at 1100 kPa is admitted to the tank until the pressure reaches 700 kPa. Determine the amount of steam added.

**Problem 5.9: **An evacuated tank is connected to a pipe carrying steam at 1400 kPa and 598 K (enthalpy = 3097 kJ/kg) through a valve. The valve is opened and the tank is filled with steam until the pressure is 1400 kPa, and then the valve is closed. Assume that the process is adiabatic and the kinetic and potential energies are negligible. Determine the final temperature of the steam.

**Problem 5.10: **A tank of volume 0.3 m^{3} initially contains saturated steam at 345 kPa. It is connected to a pipeline carrying steam at 1400 kPa and 598 K through a valve. The valve is then opened and steam from the line flows into the tank till the pressure is equal to 1400 kPa. Calculate the mass of steam that flows into the tank.

**Problem 5.11: **A rigid and insulated tank of volume 1 m^{3} initially contains air at 300 K and 10 bar. A valve is opened and the gas is discharged until the mass of air in the tank is reduced by half. Determine the temperature and pressure of the gas left in the tank.

**Problem 5.12: **An insulated rigid tank is initially evacuated and kept in a room. The atmospheric air in the room is at 101.3 kPa and 300 K. A valve is opened and air is allowed to enter the tank. The valve is closed when the pressure in the tank reaches 101.3 kPa. The air can be assumed to be an ideal gas with constant specific heat. What is the final temperature of the air in the tank?

**Problem 5.13: **A rigid insulated tank of volume 1 m^{3} contains an ideal gas (molecular weight = 29, g = 1.4) at a pressure of 200 kPa and temperature 400 K. The tank is connected to a pipeline carrying the same gas at 5 MPa and 650 K through a valve. The valve is opened and the gas is allowed to enter the tank till the pressure reaches 5 MPa. Determine (a) the temperature attained by the gas in the tank and (b) the amount of gas admitted into the tank.

**Problem 5.14: **A rigid tank 0.1 m^{3} in volume initially contains saturated water vapor at 425 K. The tank is connected by a valve to a supply line that carries steam at 1400 kPa and 523 K. Now the valve is opened and steam is allowed to enter the tank. Heat transfer takes place with the surroundings such that the temperature in the tank remains constant at 425 K at all times. The valve is closed when it is observed that one-half of the volume of the tank is occupied by liquid water. The specific volume of saturated liquid and saturated vapor at 425 K are 1.093×10–3 and 0.3749 m3/kg, respectively. The internal energy of saturated liquid and saturated vapor at 425 K are 639.7 and 2561.2 kJ/kg, respectively. The enthalpy of superheated steam at 1400 kPa and 523 K is 2927.2 kJ/kg. Determine (a) the amount of steam that has entered the tank, and (b) the heat transferred.

**Problem 5.15: **A rigid tank of volume 1 m^{3} initially contains equal volumes of water and water vapor at 523 K. Water is discharged as a saturated liquid from the bottom of the tank at a constant flow rate by opening a valve. The contents in the tank are kept at a constant temperature of 523 K by transfer of heat. The valve is closed when one-half of the initial mass has been discharged. Determine the quality (mass fraction of vapor) of the vapor-liquid mixture left in the tank and the heat transferred.

**Problem 5.16: **An ideal gas confined in a piston-cylinder assembly at a pressure of 101.3 kPa and 300 K occupies a volume of 0.015 m^{3} initially. The cylinder is connected through a valve to a pipeline through which the same gas is flowing at a steady state at a pressure of 700 kPa and 400 K. The valve is opened and the gas is admitted into the cylinder keeping the pressure inside constant at 101.3 kPa. The valve is closed when the volume of gas in the cylinder has become double the original volume. Assume the specific heat at constant pressure and constant volume are, respectively, 29.4 and 21 kJ/kmol K. Determine the final temperature of the gas in the cylinder.

**Problem 5.17: **Air, assumed to be an ideal gas with molar heat capacity C_{P} = 30 kJ/kmol K, is flowing through a pipe of diameter 0.15 m at a rate of 0.3 m^{3}/s at 100 kPa and 300 K before entering a compressor. A cooler removes heat from the compressed gas at a rate of 75.0 kJ/s. The gas at 315 K and 550 kPa is then carried away through a pipe of diameter 0.03 m. What is the power input to the compressor?

**Problem 5.18: **A pump is used to transfer a solution of density 1200 kg/m^{3} from a mixing vessel to a storage tank through a pipe of diameter 0.08 m at a velocity 1 m/s. The level difference between the liquid in the mixing vessel and the storage tank is 20 m. Both tanks are open to the atmosphere. Frictional loss is estimated to be 300 W. Determine the pressure increase over the pump. What is the power input to the pump?

**Problem 5.19: **A pump is used to transfer a solution of density 1250 kg/m^{3} at a rate of 12 m^{3}/h from an open storage tank to the top of an absorption tower which is operated at a pressure of 500 kPa. The pump discharges into the tower through openings equivalent in area to a 25×10^{–3} m pipe. The point of discharge is 30 m above the level of solution in the tank. The pump intake is through a pipe of diameter 50×10^{–3} m which extends to a depth of 2 m below the level of solution in the tank. The friction head in the suction line is estimated to be 1.5 m of water and that in the discharge line is 10 m of water. If the efficiency of the pump is 70 %, what is the power input to the pump? What pressures will be indicated by the pressure gauges at the inlet and exit of the pump?

**Problem 5.20: **Steam at 700 kPa and 553 K enters a nozzle with negligible velocity and discharges at a pressure of 475 kPa. Determine (a) The exit velocity (b) The cross-sectional area at the nozzle exit for a flow rate of 0.5 kg/s.

**Problem 5.21: **Steam at 1400 kPa and 598 K enters a convergent-divergent nozzle with negligible velocity. The nozzle may be assumed to act isentropically. The cross-sectional area at the throat is 6.5×10^{–4} m^{2}. Determine the state of the steam at the discharge end of the nozzle if the pressure there is 350 kPa. What is the mass flow rate of the steam?

**Problem 5.22: **Calculate the maximum Mach number at the discharge of the divergent section of the convergent-divergent nozzle under the conditions given in Example 5.4, assuming that supersonic velocity is realized in the nozzle.

**Problem 5.23: **Air expands through a nozzle from a negligible initial velocity to a final velocity of 350 m/s. What is the temperature drop of air, if air is assumed an ideal gas with C_{P} = (7/2)R?

**Problem 5.24: **Steam at 6000 kPa and 773 K enters a converging-diverging nozzle and discharges to a constant pressure region at 2000 kPa. If the expansion in the nozzle is isentropic, determine the velocity and temperature of the discharge steam.

**Problem 5.25: **Steam at 700 kPa and 573 K enters a nozzle with a velocity 30 m/s. The nozzle operates isentropically. Determine the area of cross-section at a point in the nozzle where the pressure is 400 kPa, as a fraction of the inlet area. Take the necessary data from the steam tables.

**Problem 5.26: **Steam at 700 kPa and 573 K enters a nozzle with a velocity 30 m/s. The nozzle operates isentropically and the steam may be assumed to behave as an ideal gas with g = 1.3. Determine: (a) The critical pressure ratio and the velocity at the throat. (b) The discharge pressure, if the Mach number at the discharge is 2.

**Problem 5.27: **A wind tunnel is fed with air through a nozzle at Mach number 1.5 and temperature 300 K. The diameter of the discharge end of the diverging section of the nozzle is 0.15 m. Assume air to be an ideal gas with C_{P} = 30 kJ/mol K. Calculate the temperature and pressure of the air fed to the nozzle.

**Problem 5.28: **Discuss the effect of clearance on the work required and on the volumetric efficiency of a multi-stage adiabatic compressor.

**Problem 5.29: **Saturated steam at 175 kPa is compressed adiabatically to 650 kPa in a centrifugal compressor at a rate of 1.5 kg/s. The compressor efficiency is 75 %. What is the power requirement of the compressor and what are the enthalpy and entropy of steam leaving the compressor?

**Problem 5.30: **A single-stage compressor is used to compress 1500 m^{3}/h of ammonia gas at 255 K and 100 kPa to 550 kPa. The isentropic compression efficiency is 75 % and the volumetric efficiency is 85 %. Calculate: (a) The power required for compression (b) Piston displacement in m3/s. (Refer to Perry and Chilton, Chemical Engineer’s Handbook for the T-S diagram of ammonia.)

**Problem 5.31: **A single-stage compressor is used to compress 800 m^{3}/h of carbon dioxide measured at 288K and 1 bar from its initial state of 0.5 bar and 300 K to a final pressure of 1.5 bar. A volumetric efficiency of 75 % and a compression efficiency of 85 % may be assumed. Assuming adiabatic compression, calculate the power required for driving the compressor, the piston displacement in m^{3}/s, and the discharge temperature.

**Problem 5.32: **A two-stage compressor is used to compress 800 m^{3}/h of carbon dioxide measured at 288 K and 1 bar from its initial state of 0.5 bar and 300 K to a pressure of 1.5 bar with intercooling to 300 K. A compression efficiency of 85 % may be assumed in each stage. Calculate the power required to run the compressor and the discharge temperature.

**Problem 5.33: **Determine the coefficient of performance of an ideal Carnot engine operating between a low temperature 280 K and a surrounding temperature 300 K.

**Problem 5.34: **The work output from a Carnot engine operating between two thermal reservoirs at 500 K and 300 K respectively, is utilized by a Carnot refrigeration machine for absorbing heat at the rate of 4 kJ/s from a cold room at 270 K and discarding heat to the surroundings at 300 K. Determine the quantity of heat absorbed by the engine at 500 K. If the COP of the refrigerator and the efficiency of the engine are two-third of the ideal values, what is the quantity of heat absorbed by the engine at 500 K?

**Problem 5.35: **The work output of an ideal Carnot engine operating between two thermal reservoirs, one at 1000 K and the other at 300 K is utilized to drive the compressor of a vapour-compression refrigeration unit working on Freon-12. The heat rejected by the engine is 30 kW. The refrigerator operates between 240 K and 300 K. The enthalpy of saturated Freon-12 liquid at 300 K = 61.9 kJ/kg, the enthalpy of saturated Freon-12 vapors at 240 K = 172.8 kJ/kg. Determine the COP, the refrigerator capacity, and the circulation rate of the refrigerant.

**Problem 5.36: **An ordinary vapour-compression cycle uses steam as the working fluid. The steam leaves the condenser at 303 K and is evaporating at 278 K. The enthalpy of saturated vapor at 278 K is 2510.6 kJ/kg and the enthalpy of saturated liquid at 303 K is 125.78 kJ/kg. Calculate the circulation rate for a refrigeration load of 1000 MJ/h.

**Problem 5.37: **A vapor-compression refrigerator employing Freon-12 works between pressure limits of 182.5 kPa and 960.6 kPa. The heat transfer from the condenser is found to be 72 kJ per minute and the heat absorbed in the evaporator is 3200 kJ/h. The refrigerant vapor leaves the evaporator in a saturated state. Calculate: (a) The refrigerant flow rate through the system in kg per minute (b) The energy input to the compressor and (c) The COP of the system. The enthalpy of saturated vapor at 182.5 kPa = 181.2 kJ/kg and the enthalpy of saturated liquid at 960.6 kPa = 76.2 kJ/kg.

**Problem 5.38: **Ammonia is being used in an ordinary vapor-compression machine rated at 5 ton. The evaporator is at 273 K and the condenser is at 303 K. The saturation pressures of ammonia corresponding to these temperatures are 4.29 bar and 11.67 bar respectively. The allowable temperature rise for cooling water in the condenser is 10 K. The enthalpy of saturated liquid and vapor at 273 K are 168 kJ/kg and 1300 kJ/kg respectively. The enthalpy of saturated liquid and vapor at 303 K are 300 and 1327 kJ/kg respectively. The enthalpy of superheated vapor leaving the compressor at 11.67 bar is 1445 kJ/kg. Determine the following: (a) The theoretical minimum horsepower to drive the unit (b) The refrigerant circulation rate (c) The cooling water circulation rate (d) The coefficient of performance.

**Problem 5.39: **A standard vapor-compression refrigeration unit using ammonia produces a refrigeration equivalent to 210 kJ/minute. The unit operates between a condenser temperature of 308 K and a refrigerator temperature of 258 K. Assuming that the compression process is reversible adiabatic and the vapor leaves the refrigerator saturated, calculate the COP and the ammonia circulation rate. The following data are available:

T (K) | P^{S} (bar) | H_{L} (kJ/kg) | H_{V} (kJ/kg) |

308 | 12.05 | 324 | 1474 |

258 | 2.44 | 113 | 1430 |

The enthalpy of vapor leaving the compressor is 1650 kJ/kg.

**Problem 5.40: **A refrigerator with Freon-12 as refrigerant operates with an evaporator temperature of 248 K (P = 1.2 bar, S = 0.7130 kJ/kg K, H = 176.22 kJ/kg) and a condensation temperature of 298 K (P = 6.4 bar, S = 0.224 kJ/kg K, H = 59.17 kJ/kg). The saturated liquid leaving the condenser is passed through an expansion valve and an evaporator. The vapor leaving the evaporator is saturated. (a) If the refrigerator is rated at 1.5 ton, what is the circulation rate of Freon-12? (b) By how much the circulation rate would be reduced, if the throttle valve were replaced by a turbine? (c) The liquid leaving the condenser is passed through a counter-current heat exchanger where it gives off its heat to the vapor leaving the evaporator. The liquid leaving the condenser is at 298 K and the vapor leaving the evaporator is at 248 K. In the exchanger, the vapor gets heated to 292 K (P = 1.2 bar, H = 203.53 kJ/kg, S = 0.8164 kJ/kg K). What would be the circulation rate of Freon-12? (d) What is the COP in each of the above cases?

Enthalpy of superheated vapor at 6.4 bar and having an entropy 0.7130 kJ/kg K = 204.69 kJ/kg. Enthalpy of superheated vapor at 6.4 bar and having an entropy 0.8164 kJ/kg K = 241.90 kJ/kg. Enthalpy and entropy of superheated vapor at P = 1.2 bar and T = 248 K are 203.53 kJ/kg K and 0.8164 kJ/kg K respectively.

**Problem 5.41: **A refrigerating machine using ammonia as the refrigerant is employed for producing 500 kg/h of ice from water. Ammonia boils at 266 K and condenses at 293 K. The water in the condenser gets heated from 283 K to 288 K. Calculate the theoretical minimum power of the compressor and the rate of circulation of cooling water. The latent heat of fusion of water is 339.1 kJ/kg.

**Problem 5.42: **A refrigeration system requires 1 kW of power for a refrigeration rate of 3 kJ/s. Determine: (a) The coefficient of performance(b) The heat rejected by the system (c) The lowest temperature that the system can maintain if the heat is rejected at 308 K.

**Problem 5.43: **An ideal vapor-compression unit with Freon-12 as refrigerant operates between an evaporator temperature of 243 K and a condenser temperature of 308 K. If the power input to the compressor is 50 kW, what is the refrigeration capacity (in tons) of refrigeration? The enthalpy of saturated liquid Freon-12 at 308 K is 69.55 kJ/kg. The enthalpy of saturated vapor at 243 K is 174.2 kJ/kg. The enthalpy of superheated vapor leaving the compressor is 200 kJ/kg.

**Problem 5.44: **A cold room is to be maintained at 261 K using an air-refrigeration system which should absorb 1000 kJ/minute. Cooling water is available at 293 K. Air leaves the compressor at 506.5 kPa and later expanded to 101.3 kPa. Assume air to behave as an ideal gas and calculate COP and power requirements. Take C_{P} = 1.008 kJ/kg K and g = 1.4.

**Problem 5.45: **A heat pump is used for heating the inside of a building in the winter and for air-conditioning in the summer. The average winter temperatures are 278 K outside and 293 K inside. The average summer temperatures are 303 K outside and 299 K inside. A 5 K temperature approach is allowed in all cases. Determine the work required in both cases as a fraction of heat input assuming the ideal cycle.

**Problem 5.46: **Nitrogen at 200 K and 200 bar is expanded reversibly through an adiabatic turbine to saturation and is then allowed to pass through a throttle valve to a pressure of 1 bar. (a) What percentage of the gas is liquefied? (b) The gas leaving the separator is passed through a heat exchanger kept between the turbine and the valve for cooling the high-pressure stream to the valve. A 5 K approach is desired at the hot end of this exchanger. Determine the percent of nitrogen liquefied. (Refer to Perry and Chilton: Chemical Engineer’s Handbook for enthalpy data of nitrogen.)

**Problem 5.47: **Air is to be liquefied in a Linde-liquefaction system. The air enters the throttle valve at 300 K and 100 bar and is expanded there to 1 bar. The flow rate of air is 85 m3/h at a temperature of 289 K and pressure of 1 bar. Assume no heat losses, zero temperature difference at the warm end of the exchanger, and adiabatic compression. Determine: (a) The rate of production of liquid in kg/h and the fraction of air liquefied (b) The rate of production of liquid and the fraction of air liquefied if a heat loss of 2.5 kJ/kg and a temperature approach of 15 K are to be accounted.

**Problem 5.48: **The low-pressure side of the throttle valve in the Linde process for the liquefaction of methane is maintained at 1 bar (the enthalpy of saturated liquid and vapor at 1 bar are 285 kJ/kg and 797 kJ/kg respectively). The gas leaves the compressor at 60 bar and 300 K (H = 1140 kJ/kg). The uncondensed gases are passed through the heat exchanger where it gets heated to 295 K (H = 1189 kJ/kg). Determine: (a) The fraction of the gas liquefied (b) The temperature of the gas at the high-pressure side of the valve.

**Problem 5.49: **In the Linde process for liquefaction, air is compressed from a pressure of 1 bar to 200 bar. Air at a rate of 200 kg/h is treated. The air entering the compressor is at 298 K (H = 510 kJ/kg) and that leaving it is cooled to 298 K (H = 474 kJ/kg). The air is throttled to a pressure of 1 bar. The enthalpy of saturated liquid at 1 bar is 92 kJ/kg. Heat loss from the unit is estimated to be 8.3 kJ/kg of air. Determine: (a) The rate of liquefaction (b) The power requirement.

**Problem 5.50: **An ideal regenerative cycle operates with steam supplied at 2800 kPa and 773 K and a condenser pressure of 5 kPa. Extraction points provided are at 350 kPa and 75 kPa, one closed, and the other open. Neglecting pump work, calculate the thermal efficiency of the plant.

**Problem 5.51: **In a steam power plant operating on the Rankine cycle, the turbine is supplied with superheated steam at 2600 kPa and 573 K. The steam leaving the turbine containing 93 % vapor is sent to a condenser operated at 13.6 kPa. The feed water pump takes in water at 13.0 kPa and 319 K and delivers water to the boiler at 2900 kPa. Superheated steam as it leaves the boiler is at 2800 kPa and 598 K. Determine the following for kg steam flowing through the plant: (a) Pump work (b) Turbine work (c) Heat transfer in the line between boiler and turbine (d) Heat transfer in boiler (e) Heat transfer in the condenser.

**Problem 5.52: **In a steam power plant, steam is supplied to the high-pressure turbine at 2800 kPa and 648 K. It is expanded to 558 kPa and sent to the boiler where it is heated to 558 kPa and 648 K. It is then expanded to a final pressure of 2.5 kPa in the low-pressure turbine. Determine: (a) The ideal reheat cycle efficiency (b) The ideal Rankine cycle efficiency in the absence of reheating.

**Problem 5.53: **A steam turbine power plant operates on a regenerative cycle. Steam enters the turbine at 3500 kPa and 710 K. A fraction of the steam is extracted at 211 kPa and sent to an open heater and the remainder is condensed at 7 kPa. Neglecting pump work, determine the thermal efficiency.

**Problem 5.54: **In a steam power plant of the regenerative type, steam is supplied to the turbine at 2800 kPa and 648 K. The condenser is operated at 7 kPa. The steam is extracted from the turbine at 684 kPa and 81.5 kPa and sent separately to two closed heaters. Determine the thermal efficiency of the cycle.

**Problem 5.55: **The turbine of a 1-MW steam power plant is supplied with superheated steam at 3000 kPa and 573 K, where it is expanded to the condenser pressure of 5 kPa. The isentropic efficiency of the turbine is 85 %. The saturated liquid leaving the condenser is pumped to the boiler pressure by means of the feed water pump, the thermodynamic efficiency of which is 80 %. Determine: (a) The efficiency of the ideal Rankine cycle (b) Thermal efficiency of the cycle (c) The rate of production of steam.

**Problem 5.56: **The high-pressure turbine of a steam power plant is supplied with superheated steam at 3000 kPa and 773 K. After expansion to 600 kPa, the exhaust steam is returned to the boiler to heat it to 773 K and 600 kPa. It then enters the low-pressure turbine and expanded to 5 kPa. The exhaust steam leaving the turbine is taken to a condenser operated at 5 kPa. The saturated liquid leaving the condenser is pumped to the boiler pressure by means of the feed water pump. The isentropic efficiency of the turbine is 0.85 and that of the pump is 0.7. What is the thermal efficiency of the plant?

**Problem 5.57: **Superheated steam at 3000 kPa and 573 K is supplied to the turbine of a steam power plant operating on the regenerative cycle. A fraction of the steam at 400 kPa is extracted from the turbine and is sent to the feed water-heater and the remainder is expanded to the condenser pressure of 10 kPa and admitted to the condenser. The saturated liquid leaving the condenser is pumped to a pressure of 400 kPa and enters the heater where it exchanges heat with the extracted steam. The exit streams leaving the heater are combined together and pumped to the boiler. Determine the thermal efficiency of the cycle.

**Problem 5.58: **A thermal power plant operating on Rankine cycle is rated at 1 MW. Superheated steam at 5000 kPa and 673 K enters the turbine where it is expanded to the condenser pressure of 7 kPa. The saturated liquid leaving the condenser is pumped to the boiler pressure, the isentropic efficiency of the pump being 75 %. The isentropic efficiency of the turbine is 80 %. Determine: (a) Thermal efficiency of the plant (b) The rate of production of steam.

**Problem 5.59: **An 80-MW steam power plant is operated on Rankine cycle. The turbine is fed with superheated steam at 8600 kPa and 773 K where it is expanded to the condenser pressure of 10 kPa. The saturated liquid leaving the condenser is pumped to the boiler. The isentropic efficiencies of both the pump and the turbine are 75 %. Determine: (a) The thermal efficiency of an ideal Rankine cycle for the stated conditions (b) Thermal efficiency of the cycle (c) Rate of production of steam (d) Rate of heat input in the boiler and the condenser.

**Problem 5.60: **Superheated steam enters the turbine of a steam power plant operating on a regenerative cycle at 4100 kPa and 700 K. After expansion to 404 kPa, some of the steam is extracted from the turbine for the purpose of heating the feed water in an open feed water heater. The pressure in the feed water heater is 404 kPa and the water leaving it is saturated liquid at 404 kPa. The steam not extracted is expanded to a pressure of 7 kPa. What is the thermal efficiency of the cycle?

**Problem 5.61: **It is desired to determine the effect of turbine inlet pressure on the performance of a Rankine cycle. Steam enters the turbine at 650 K and exhausts at 14 kPa. Calculate the cycle thermal efficiency and moisture content of steam leaving the turbine for the turbine inlet pressure of 700 kPa, 3500 kPa, 7000 kPa and 14000 kPa.

**Problem 5.62: **It is desired to study the effect of turbine inlet temperature on the performance of an ideal Rankine cycle. Steam enters the turbine at 3500 kPa and exhausts at 14 kPa. Calculate the cycle thermal efficiency and moisture content of the steam leaving the turbine for the inlet temperatures of 573 K, 623 K, 873 K and 923 K.

**Problem 5.63: **An air-standard Otto cycle operates with a compression ratio of 8 and a maximum temperature per cycle of 1400 K. If the temperature and pressure at the beginning of the compression stroke are 300 K and 100 kPa respectively, determine the following assuming air to be an ideal gas with constant specific heats C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K: (a) The heat supplied (b) The work produced (c) The thermal efficiency.

**Problem 5.64: **Determine the thermal efficiency and the work output per kg of air of an air-standard Otto cycle with a compression ratio of 8. The temperature and pressure at the beginning of the compression stroke are 300 K and 100 kPa. Heat supplied to the engine is 2000 kJ/kg. Assume air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K.

**Problem 5.65: **For an air-standard Otto cycle the temperature and pressure at the beginning of compression stroke are 310 K and 100 kPa respectively. The maximum temperature and pressure permitted are 3000 K and 7000 kPa respectively. Assuming air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K, determine (a) The compression ratio (b) The thermal efficiency (c) The net work output from the engine.

**Problem 5.66: **In an air-standard Otto cycle, the work output per kg air is 1000 kJ and the temperature at the end of heat addition is 3500 K and that at the end of compression process is 800 K. Determine the compression ratio of the engine assuming air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K.

**Problem 5.67: **In an air-standard Diesel cycle, the temperature, pressure, and volume at the beginning of the compression stroke are respectively 300 K, 100 kPa, and 0.05 m^{3}. The pressure at the end of the compression stroke is 4000 kPa and the heat supplied is 500 kJ/kg. Assuming air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K, determine: (a) The compression ratio (b) The cut-off ratio (c) The work done per cycle (d) The thermal efficiency.

**Problem 5.68: **An air-standard Otto cycle operates on 5 kg of air with a compression ratio of 10. If the temperature and pressure at the beginning of compression stroke are 310 K and 80 kPa and the heat supplied is 500 kJ, determine the following, assuming air to be an ideal gas with constant specific heats of C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K. (a) The pressure and temperature for each step in the process (b) The net work output (c) The thermal efficiency.

**Problem 5.69: **Compare the thermal efficiency of an Otto cycle of compression ratio 8 and operating with air (g =1.4) with that of an air-standard Diesel cycle of the same compression ratio and a cut-off ratio of 2. How does the comparison change if the cut-off ratio is 3?

**Problem 5.70: **A 6-cylinder reciprocating engine operates on an air-standard Diesel cycle, each cylinder having 115 mm bore and 125 mm stroke and a speed of 2000 rpm. The temperature and pressure at the beginning of compression stroke are 300 K and 100 kPa respectively and the maximum temperature permitted is 1650 K. The clearance volume is one-eighth of the stroke volume. Assume air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} =0.718 kJ/kg K. Calculate: (a) The compression ratio (b) The temperature and pressure after compression (c) The thermal efficiency (d) The power output from the engine.

**Problem 5.71: **5.71 For an air-standard Diesel cycle, the temperature and pressure at the beginning of the compression stroke are 325 K and 100 kPa. The pressure after compression is 4000 kPa. The heat supplied is 600 kJ/kg of air. Assume air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K. Determine (a) The compression ratio (b) The cut-off ratio (c) The work output (d) The thermal efficiency.

**Problem 5.72: **In an air-standard Diesel cycle, the pressure and temperature at the beginning of the compression stroke are 100 kPa and 295 K respectively, and the heat supplied is 1500 J/mol, what are the compression ratio and the expansion ratio of the cycle if the pressure at the end of combustion step is 400 kPa. Assume air to be an ideal gas with C_{P} = (7/2)R and C_{V} = (5/2)R.

**Problem 5.73: **An air-standard Diesel cycle operates with a compression ratio of 16. If the temperature and pressure at the beginning of compression stroke are 310 K and 100 kPa and the heat supplied is accompanied by an increase in entropy of 1.2 kJ/kg K determine the following, assuming air to be an ideal gas with constant specific heats of C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K. (a) The maximum temperature (b) The cut-off ratio (c) The heat supplied (d) The thermal efficiency.

**Problem 5.74: **In an air-standard Diesel cycle operating with 5 kg air, the temperature and pressure at the beginning of the compression stroke are 310 K and 80 kPa. The heat rejected per cycle is 4000 kJ and the maximum temperature per cycle is 3000 K. If the efficiency of the cycle is 60 percent determine the compression ratio assuming air to be an ideal gas with constant specific heats of C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K.

**Problem 5.75: **An air-standard Diesel cycle has an initial pressure and temperature of 100 kPa and 300 K. The compression ratio is 14. Temperature at the end of heat supply is 1980 K. The specific heat ratios are 1.37, 1.34, and 1.31 respectively for the compression, heat supply, and expansion processes. Determine: (a) The thermal efficiency (b) The work output from the cycle per kg air (c) The entropy change for the heat supply process.

**Problem 5.76: **In an air-standard Diesel cycle, the compression ratio is 16 and the cut-off ratio is 3. The conditions at the beginning of the compression stroke are 100 kPa and 300 K. Assuming air as an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K, determine the following: (a) The heat supplied (b) The net work output (c) The thermal efficiency.

**Problem 5.77: **In air-standard dual cycle, the temperature and pressure at the start of the compression stroke are 300 K and 100 kPa. The compression ratio is 15 and the maximum temperature is 3000 K. The maximum pressure is 7000 kPa. Determine: (a) The work done per kg air (b) The heat supplied. Assume air to be an ideal gas with C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K.

**Problem 5.78: **In a gas-turbine cycle, the air enters the compressor at 100 kPa and 300 K and leaves at 500 kPa. The maximum temperature is 1200 K. Assume a compressor efficiency of 80 percent, a turbine efficiency of 85 percent and a pressure drop between the compressor and turbine of 15 kPa. Take C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K. Determine: (a) The compressor work (b) The turbine work (c) The cycle efficiency.

**Problem 5.79: **In an air-standard gas-turbine cycle with a pressure ratio of 8, the temperature and pressure of the air entering the compressor are 300 K and 100 kPa. The permissible maximum temperature is 1300 K. Determine: (a) The temperature and pressure at each point in the cycle (b) The compressor work (c) The turbine work (d) The thermal efficiency.

**Problem 5.80: **A gas-turbine power plant operates with a pressure ratio of 6. The temperature of the air entering the compressor is 300 K and the maximum permissible temperature in the turbine is 1100 K. Determine: (a) The efficiency of an ideal gas-turbine cycle if g = 1.4, (b) The thermal efficiency of the power plant if the compressor and the turbine operate adiabatically but reversibly with efficiencies 83 percent and 86 percent respectively.

**Problem 5.81: **The pressure ratio across the compressor of an air-standard Brayton cycle is 4. The conditions of air at the beginning of the compression stroke are 290 K and 100 kPa. The maximum temperature in the cycle is 1100 K. Assume constant specific heats C_{P} = 1.005 kJ/kg K and C_{V} = 0.718 kJ/kg K. The rate of airflow is 10 kg/s. Determine: (a) The compressor work (b) The turbine work (c) The thermal efficiency of the cycle (d) The mean effective pressure, if this cycle were utilized for a reciprocating machine.

**Problem 5.82: **5.82 A stationary gas-turbine power plant operates on the ideal Brayton cycle and delivers 20,000 hp to an electric generator. The maximum temperature and pressure are 1100 K and 420 kPa respectively and the minimum temperature and pressure are 290 K and 100 kPa. (a) What is the power output of the turbine? (b) What fraction of the output of the turbine is used to drive the compressor? (c) What is the rate of circulation of air in kg/s?

**Problem 5.83: **The compressor of a gas-turbine power plant operating on Brayton cycle is supplied with air at 300 K and 100 kPa. The temperature of air at the inlet to the turbine is 900 K. The pressure ratio is 4. The pressure drop in the combustion chamber is 10 kPa and that in the exit line from the turbine is 5 kPa. Determine: (a) The temperature and pressure at each point (b) The thermal efficiency of the cycle.