Practice Problems: Laws Of Thermodynamics

Question 1:
A mechanical engineer is evaluating a coal-fired power plant that operates on a simple Rankine cycle. The steam enters the turbine at 8 MPa and 500°C and exits at 10 kPa. The turbine has an isentropic efficiency of 85%. Given the following property data:
- At turbine inlet: h₁ = 3399 kJ/kg, s₁ = 6.726 kJ/kg·K
- At 10 kPa (saturated): h_f = 191.8 kJ/kg, h_fg = 2392.8 kJ/kg, s_f = 0.6493 kJ/kg·K, s_fg = 7.5009 kJ/kg·K
- Isentropic exit enthalpy: h₂s = 2130 kJ/kg
What is the actual work output per unit mass of steam flowing through the turbine?
(a) 950 kJ/kg
(b) 1079 kJ/kg
(c) 1270 kJ/kg
(d) 1350 kJ/kg

Question 2:
A HVAC engineer is designing a vapor-compression refrigeration system for a commercial building. The refrigerant R-134a enters the compressor as saturated vapor at -10°C and exits at 1.0 MPa and 60°C. The mass flow rate is 0.15 kg/s. Given the following properties:
- At compressor inlet: h₁ = 241.4 kJ/kg
- At compressor exit: h₂ = 290.8 kJ/kg
- At condenser exit (saturated liquid at 1.0 MPa): h₃ = 105.3 kJ/kg
- At evaporator inlet: h₄ = 105.3 kJ/kg
What is the compressor power requirement?
(a) 5.2 kW
(b) 7.4 kW
(c) 9.8 kW
(d) 12.3 kW

Question 3:
A process engineer is analyzing a gas turbine power plant operating on the Brayton cycle. Air enters the compressor at 100 kPa and 25°C and is compressed to 800 kPa. The compressor has an isentropic efficiency of 82%. Assume air behaves as an ideal gas with c_p = 1.005 kJ/kg·K and k = 1.4. Given:
- Inlet temperature: T₁ = 298 K
- Pressure ratio: r_p = 8
- Isentropic efficiency: η_c = 0.82
What is the actual exit temperature of the compressor?
(a) 445 K
(b) 482 K
(c) 524 K
(d) 567 K

Question 4:
A thermal systems engineer is evaluating an Otto cycle engine for automotive application. The engine has a compression ratio of 9.5 and operates with air as the working fluid. At the beginning of compression, the air is at 100 kPa and 27°C. The maximum temperature in the cycle is 1800 K. Assume air properties: c_v = 0.718 kJ/kg·K and k = 1.4. What is the thermal efficiency of this Otto cycle?
(a) 52.4%
(b) 58.5%
(c) 61.2%
(d) 64.8%

Question 5:
A refrigeration engineer is designing a cascade refrigeration system. The upper cycle uses R-134a and the lower cycle uses R-410A. The heat exchanger between the two cycles operates at -10°C on the R-134a side and -15°C on the R-410A side. If the heat transfer rate in the intermediate heat exchanger is 45 kW and the mass flow rate of R-134a is 0.25 kg/s, what is the change in enthalpy of R-134a across the heat exchanger?
(a) 120 kJ/kg
(b) 150 kJ/kg
(c) 180 kJ/kg
(d) 210 kJ/kg

Question 6:
A power plant engineer is analyzing a combined heat and power (CHP) system. The system produces 50 MW of electrical power and 30 MW of thermal energy for district heating. The total fuel energy input to the system is 95 MW (based on lower heating value). What is the overall efficiency of this CHP system?
(a) 68.4%
(b) 74.2%
(c) 84.2%
(d) 92.6%

Question 7:
An energy systems engineer is evaluating a steady-flow air compressor that takes in air at 14.7 psia and 70°F and compresses it to 150 psia. The compressor has a polytropic efficiency of 80% with n = 1.3 for the polytropic process. The mass flow rate is 5 lbm/s. For air, R = 53.35 ft·lbf/lbm·R and c_p = 0.24 Btu/lbm·°F. What is the power required by the compressor?
(a) 178 hp
(b) 212 hp
(c) 245 hp
(d) 289 hp

Question 8:
A mechanical engineer is designing a closed feedwater heater for a steam power plant. Steam is extracted from the turbine at 1.2 MPa and 250°C (h = 2935 kJ/kg) and mixed with feedwater at 1.2 MPa and 60°C (h = 251.5 kJ/kg). The exit state is saturated liquid at 1.2 MPa (h = 798.6 kJ/kg). If the feedwater flow rate is 100 kg/s, what is the required steam extraction rate?
(a) 21.5 kg/s
(b) 25.6 kg/s
(c) 29.8 kg/s
(d) 33.2 kg/s

Question 9:
A consulting engineer is analyzing a Diesel cycle engine with a compression ratio of 18 and a cutoff ratio of 2.2. The air at the beginning of compression is at 95 kPa and 20°C. Assume air-standard analysis with k = 1.4 and c_v = 0.718 kJ/kg·K. What is the thermal efficiency of this Diesel cycle?
(a) 58.7%
(b) 62.3%
(c) 65.8%
(d) 69.2%

Question 10:
A thermal engineer is evaluating a steam turbine that receives superheated steam at 6 MPa and 450°C (h = 3302.9 kJ/kg, s = 6.7219 kJ/kg·K) and exhausts at 50 kPa. Assume isentropic expansion. At 50 kPa: h_f = 340.5 kJ/kg, h_fg = 2305.4 kJ/kg, s_f = 1.0910 kJ/kg·K, s_fg = 6.5029 kJ/kg·K. What is the quality of the steam at the turbine exit?
(a) 0.852
(b) 0.866
(c) 0.891
(d) 0.912

Question 11:
A process engineer is designing a heat exchanger where hot exhaust gases at 450°C cool to 200°C while heating water from 30°C to 85°C. The exhaust gas flow rate is 2.5 kg/s with c_p = 1.08 kJ/kg·K. Assuming no heat loss to surroundings, what is the required water flow rate?
(a) 3.2 kg/s
(b) 4.9 kg/s
(c) 6.1 kg/s
(d) 7.8 kg/s

Question 12:
An automotive engineer is analyzing a turbocharger system where the turbine drives the compressor. The turbine receives exhaust gases at 650°C and 250 kPa with a mass flow rate of 0.08 kg/s. The gases exit at 450°C and 105 kPa. Assume c_p = 1.15 kJ/kg·K for exhaust gases. If the mechanical efficiency is 92%, what power is available to drive the compressor?
(a) 14.2 kW
(b) 16.9 kW
(c) 18.4 kW
(d) 21.3 kW

Question 13:
A refrigeration engineer is designing an absorption chiller system. The system has a cooling capacity of 500 kW and requires 600 kW of heat input to the generator. The cooling water removes 1100 kW of heat from the absorber and condenser. What is the coefficient of performance (COP) of this absorption refrigeration system?
(a) 0.45
(b) 0.67
(c) 0.83
(d) 1.12

Question 14:
A power systems engineer is evaluating a regenerative Rankine cycle. Steam leaves the boiler at 8 MPa and 500°C (h = 3399 kJ/kg) and expands to 0.8 MPa where some steam is extracted for feedwater heating. The remaining steam expands to 10 kPa. For isentropic expansion from 8 MPa to 0.8 MPa, the enthalpy at 0.8 MPa is 2950 kJ/kg. If the extraction fraction is 0.15, what is the turbine work per kg of steam entering?
(a) 1025 kJ/kg
(b) 1158 kJ/kg
(c) 1242 kJ/kg
(d) 1389 kJ/kg

Question 15:
A mechanical engineer is analyzing an ideal Stirling cycle operating between temperature limits of 1200 K and 350 K. The compression ratio is 8. The working fluid is helium with R = 2.077 kJ/kg·K. At the beginning of isothermal compression, the volume is 0.002 m³ and pressure is 100 kPa. What is the thermal efficiency of this Stirling cycle?
(a) 58.3%
(b) 65.7%
(c) 70.8%
(d) 78.2%

Question 16:
A cryogenic systems engineer is designing a Joule-Thomson expansion valve for a natural gas liquefaction plant. Natural gas enters the valve at 10 MPa and -50°C with enthalpy h₁ = 425 kJ/kg and exits at 2 MPa with enthalpy h₂ = 380 kJ/kg. The mass flow rate is 5 kg/s. Assuming adiabatic process with negligible kinetic energy change, what is the temperature change if c_p = 2.2 kJ/kg·K?
(a) -18.2°C
(b) -20.5°C
(c) -25.3°C
(d) -30.1°C

Question 17:
A thermal engineer is evaluating a gas turbine cogeneration system. The turbine produces 25 MW of electrical power. The exhaust gases leave the turbine at 480°C with a mass flow rate of 80 kg/s and c_p = 1.09 kJ/kg·K. These gases are used in a heat recovery steam generator (HRSG) where they are cooled to 150°C. What is the heat recovery rate in the HRSG?
(a) 22.8 MW
(b) 28.8 MW
(c) 34.2 MW
(d) 41.6 MW

Question 18:
A process engineer is analyzing an adiabatic mixing chamber where two streams of water mix at steady state. Stream 1: 3 kg/s at 80°C (h₁ = 335 kJ/kg). Stream 2: 2 kg/s at 30°C (h₂ = 126 kJ/kg). The mixed stream exits at 100 kPa. Assuming negligible kinetic and potential energy changes, what is the exit temperature? (Use c_p = 4.18 kJ/kg·K)
(a) 52°C
(b) 58°C
(c) 61°C
(d) 67°C

Question 19:
An HVAC engineer is designing a ground-source heat pump system for a residential building. The heat pump operates between ground temperature of 10°C and indoor temperature of 22°C. The heat pump delivers 15 kW of heating to the building and has a COP of 3.8. What is the power input required to operate the compressor?
(a) 2.85 kW
(b) 3.26 kW
(c) 3.95 kW
(d) 4.52 kW

Question 20:
A plant engineer is evaluating an air compressor system that operates continuously. The compressor takes in 0.5 m³/s of air at 101 kPa and 20°C (ρ = 1.20 kg/m³) and compresses it to 700 kPa. During compression, the air is cooled such that 50 kW of heat is removed. The exit air temperature is measured at 180°C. For air, c_p = 1.005 kJ/kg·K. What is the compressor power input?
(a) 92.4 kW
(b) 96.5 kW
(c) 102.3 kW
(d) 146.5 kW