Problems
Rankine Cycle
A small steam power plant produces 50 MW from a simple Rankine cycle operating between an evaporator pressure of 60 bar and a condenser pressure of 10 kPa. The turbine inlet temperature is 550 ◦C and the quality at its exit is 0.9. Assume that the adiabatic pump is reversible.
1. Draw a sketch of the cycle, and the corresponding p-v- and T-s-diagrams.
2. Make a list with the values of the relevant enthalpies of the cycle.
3. Determine the isentropic efficiency of the turbine.
4. Determine the mass flow of steam in the cycle and the thermal efficiency of the system.
Steam Cycle
Consider a simple steam power plant that develops a power of 100 MW. The condenser pressure is 10 kPa, the pressure in the steam generator is 8 MPa, and the temperature at the turbine inlet is 500 ◦C. The isentropic efficiency of the pump is 75%. At the turbine exit the quality is measured as x = 0.91.
1. Draw a schematic, a T-s-diagram, and a p-v-diagram (both with respect to saturation lines) of the plant.
2. Determine the enthalpies at the corner points, and the thermal efficiency of the cycle.
3. Determine the isentropic efficiency of the turbine.
4. Determine the mass flow of water through the cycle.
5. In the condenser the heat is transferred to a stream of cooling water which changes its temperature by 10 ◦C. Compute the mass flow of cooling water (incompressible liquid with specific heat cw = 4.2 kJ ).
Geothermal Steam Power Plant
In the Larderello (Italy) steam power plant, steam at 5 bar, 220 ◦C is produced by geothermal heating. Assume that the processes in the plant follow the basic Rankine cycle with irreversible turbine (ηT = 0.85) and reversible pump. The condenser temperature is 40 ◦C.
1. Determine the enthalpies at the corner points.
2. Compute the thermal efficiency, and discuss its value as compared to standard steam power plants, which operate at higher pressures and temperatures.
Reheat Rankine Cycle
A steam power plant with a power output of 80 MW operates on a reheat Rankine cycle:
1-2: adiabatic irreversible pump from condenser pressure p1 = 10 kPa to p2 = 10 MPa, state 1 is saturated liquid, isentropic pump efficiency is 95%
2-3: isobaric heating to 500 ◦C
3-4: adiabatic turbine, expansion to p4 = 1 MPa, isentropic turbine efficiency is 80%
4-5: isobaric reheat to 500 ◦C
5-6: adiabatic turbine, expansion to p6 = p1, isentropic turbine effi- ciency is 80%
1. Draw a schematic, and the corresponding p-v- and T-s-diagrams with respect to saturation lines.
2. Determine the thermal efficiency of the system.
3. Determine the mass flow rate.
Another Reheat Cycle
For an ideal (i.e., reversible) Rankine cycle with reheat, the minimum and maximum pressures reached are 10 kPa and 9 MPa, respectively. Moreover, the turbine inlet temperatures of both turbines are 500 ◦C, the quality at the condenser inlet is 90% and the mass flow is 25 kg of steam. Determine:
1. The reheat pressure.
2. The heat input and the power produced.
3. The thermal efficiency.
Refrigerators, Heat Pumps
Draw schematics for vapor refrigeration cycles and heat pumps, as well as the corresponding T-s- and p-v-diagrams (include irreversible processes for compressors). Explain the difference in operating conditions between heat pumps and refrigerators. Also draw T-s-diagrams for Carnot heat pump and refrigerator, and use the diagrams to compute their COP’s.
Refrigerator
A vapor-compression refrigeration system uses R134a as working substance. The pressure in the evaporator is 1.4 bar, and the condenser pressure is 7 bar.
The temperature at the compressor inlet is −10 ◦C, and the working fluid leaves the condenser at a temperature of 24 ◦C. Moreover, the mass flow rate is 0.1 kg , and the isentropic efficiency of the compressor is 67%.
1. Draw a sketch, a p-v-diagram, a T-s-diagram (with respect to saturation lines).
2. Determine the COP of the system.
3. Compute the refrigeration capacity, and the power consumption.
4. Compute COP and power consumption if the isentropic efficiency of the compressor is 85%.
Refrigeration Cycle
A frozen pizza factory requires a refrigeration rate of 200 kW to maintain the storage facility at −15 ◦C. Cooling is performed by a standard vapor-compression cycle, using R134a with the following data: condenser pressure:
700 kPa, evaporator temperature: −20 ◦C, isentropic efficiency of compressor:
75%. The condenser is cooled by liquid water. Use the log p-h diagram for
R134a for the solution of this problem.
Determine the mass flow rate of the refrigerant, the power input to the compressor, the COP, and the mass flow rate of the cooling water when its temperature changes by 10 ◦C.
Vapor Refrigeration Cycle
A refrigerator uses R134a as working fluid which undergoes the following cycle:
1. Draw a schematic, and the process in a T-s-diagram, with respect to saturation lines.
2. Make a table with the values of temperature, pressure, and enthalpy at points 1-4.
3. Compute the coefficient of performance (COP).
4. The refrigerator draws a power of 4 kW. Compute the mass flow and the cooling power.
Heat Pump
A heat pump that operates on the ideal vapor-compression cycle with R134a is used to heat water from 15 to 54 ◦C at a rate of 0.24 kg . The condenser and evaporator pressures are 1.4 and 0.32 MPa, respectively. Determine the COP and the power input to the heat pump.
Heat Pump
A vapor compression heat pump with R134a as cooling fluid is used to keep a house at 20 ◦C. The heat pump has a compressor with isentropic efficiency of 85%, and it draws heat from groundwater which has a temperature of 12 ◦C. The condenser and evaporator pressures are 900 kPa and 320 kPa, respectively, the temperature at the inlet of the throttling valve is 30 ◦C, and the compressor draws saturated vapor.
Compute the COP, the mass flow, and the power consumption if the heating power is 10 kW. Don’t forget to draw schematic and diagrams.
Vapor Heat Pump Cycle
An air conditioning system sucks in a mass flow of 5000 kg of outside air at 10 ◦C, 0.95 bar, and heats it isobarically to 24 ◦C. The air is heated by means of an standard vapor heat pump cycle (R134a), whose compressor has an isentropic efficiency of 0.85. The condenser pressure is 800 kPa. The evaporator is outside the building, and the minimum temperature difference for heat transfer is 10 ◦C. Consider the air as ideal gas with constant specific heats
1. Make a sketch of the system, and draw the corresponding T-s-diagram.
2. Make a table with the values of pressure, temperature and enthalpy at the
corner points.
3. Determine the COP of the cycle.
4. Determine the work required to run the heat pump.
Gas Turbine (Ideal Brayton Cycle)
An ideal Brayton cycle with air as working fluid (variable specific heats) is to be designed such that the minimum and maximum temperatures in the cycle are 300 K and 1500 K, respectively. The pressure ratio is 16.7. Compressor and turbine are both irreversible, with an isentropic efficiency of 0.9 for the turbine and 0.85 for the compressor.
Compute compressor and turbine work per unit mass of air, and the thermal efficiency of the cycle.
Brayton Cycle
This problem compares the calculation with constant and non-constant specific heats, to give an idea of the differences. Air enters the compressor of a simple Brayton cycle gas turbine power plant at 95 kPa and 290 K. The heat transfer rate is 50 MJ and the turbine entry temperature is 1400 K. The pressure ratio is P = Tmax 2(k−1) for maximum power output. Compressor min
and turbine are reversible. Compute the power delivered and the thermal efficiency for:
1. Cold air approximation, that is constant specific heats with values at room temperature.
2. Variable specific heats.
Make tables for the values of pressure and temperature at the relevant process points, and draw diagrams and schematic.
Gas Turbine (Brayton Cycle)
A Brayton cycle delivers a power of 150 MW. The working fluid can be considered as air (ideal gas with variable specific heats), and the following data are known: inlet state p1 = 0.9 bar, T1 = 280 K; state after adiabatic compression p2 = 17.06 bar, T2 = 690 K, maximum temperature in the cycle 1600 K, heating rate 354 MW.
Determine the thermal efficiency of the cycle, compressor and turbine work per unit mass of air, mass flow of air, and the isentropic efficiencies of compressor and turbine.
Brayton Cooling Cycle
A small gas-cooling system operates on the inverse Brayton cycle. The cycle uses argon as cooling fluid. The cycle is used for maintaining a small cold space at −60 ◦C, and rejects heat into the environment at 25 ◦C. Both heat exchangers require a temperature difference of at least 5 ◦C for operation.
The cycle operates between the pressures 1 bar and 4 bar, and isentropic efficiencies of compressor and turbine are 0.75 and 0.85, respectively. Draw schematic and diagrams and determine:
1. The COP.
2. The mass flow required for a cooling power of 0.5 kW, and the required power to run the system.
3. The work losses to irreversibilities in turbine, compressor and both heat exchangers. Discuss the results.
Gas Refrigeration Cycle
A refrigerator for cryogenic applications uses helium as working fluid which undergoes the cycle described below.
1-2: Adiabatic irreversible compression of helium at T1 = −75 ◦C and p1 = 1 bar to p2 = 10 bar, the isentropic efficiency of the compressor is 0.85
2-3: Isobaric cooling until the temperature reaches T3 = 30 ◦C
3-4 : Adiabatic irreversible expansion in turbine to pressure p4 = p1, the isentropic efficiency of the turbine is 0.8
4-1: Isobaric heating to state 1 Helium is an ideal gas with constant specific heats, with cp = 5.196 kJ , R = 2.0785 kJ .
1. Draw a schematic, and the process in a T-s-diagram.
2. Make a table with the values of temperature and pressure at points 1-4.
3. Compute the coefficient of performance (COP).
4. The mass flow is 1.5 kg . Compute the cooling power, and the power needed to run the refrigerator.
Gas Cooling System
A gas refrigeration system operates on the inverse Brayton cycle (1-2: adiabatic compression, 2-3: isobaric heat exchange, 3-4: adiabatic expansion, 4-1: isobaric heat exchange) with air as the working fluid. The compressor pressure ratio is 3. This system is used to maintain a refrigerated space at −23 ◦C and rejects heat to the environment at 27 ◦C. The isentropic efficiency of the compressor is 0.8, but the turbine exhibits no losses. The temperature difference for heat transfer is 10 ◦C. Consider air as an ideal gas with variable specific heats.
1. Draw a schematic, and the corresponding T-s-diagram.
2. Make a list with the enthalpies and temperatures at the corner points of the process.
3. Compute the COP.
4. For the computation above, the knowledge of pressure ratio was enough, so nothing was said about the value of p1. Discuss the choice of this pressure (should it be high or low . . .).