Problems
Regenerative Boiler
The boiler for a power plant is fitted with a regenerator of effectiveness 81.2% to preheat the incoming air before it is heated by burning of coal. Specifically, the system draws environmental air at T0 = 7 ◦C, the flame temperature is 827 ◦C, and the boiler exhaust is at 667 ◦C.
1. Determine the preheat temperature, the final exhaust temperature, the work potential of the boiler exhaust, and the work potential of the final exhaust.
2. Determine the entropy generation in the regenerator, and the corresponding work loss.
Mixing Chamber
Steam at 100 bar, 600 ◦C is throttled to 10 bar, and fed into an adiabatic mixing chamber where it is mixed with compressed liquid water at 10 bar, 50 ◦C. The exiting mass flow is 100 kg of saturated liquid at 10 bar. For steady state operation, determine
1. The mass flows of steam and liquid that enter.
2. The rate of entropy generation due to throttling, and the rate of entropy generation due to mixing.
3. The associated work loss.
As always: draw a sketch and a T-s-diagram.
Closed Feedwater Heater
In a closed feedwater heater, a mass flow of 200t/h of compressed liquid water at 80 bar, 40 ◦C is heated by heat exchange with a stream of saturated liquid-vapor mix at x = 0.95 , p = 10 bar.
For the case that both streams leave the heat exchanger in saturated liquid state determine the mass flow of liquid-vapor mix, the entropy generation rate, and the work loss.
Steam Power Plant with Regeneration and Reheat
A steam power plant operates on a reheat-regenerative Rankine cycle with an open feedwater heater. Steam enters the high-pressure turbine at 100 bar, 550 ◦C, and leaves at a pressure of 8 bar as saturated vapor. Some steam is extracted at this pressure to heat the feedwater in an open feedwater heater which provides saturated liquid. The rest of the steam is reheated to 500 ◦C and then expanded in the low pressure turbine to the condenser pressure of 10 kPa.
The isentropic efficiency of the low pressure turbine is 0.95; all pumps can be considered to operate reversibly.
1. Draw a schematic and a T-s-diagram of the process, numerate corresponding points in schematic and diagram, and name the different devices (pump, turbine, etc.)
2. Make a list with the values of enthalpy at all relevant points of the process.
3. Compute the ratio of mass flow diverted to the feedwater heater after the first turbine.
4. Compute the thermal efficiency of the cycle.
5. The net power output of the plant is 100 MW. Determine the mass flow through the high pressure turbine.
6. Determine total entropy generation rate and work loss of the cycle.
Vapor Power Plant with Regeneration
A power plant operates on a regenerative vapor power cycle with one closed feed water heater according to the following process:
1-2: Isentropic compression of saturated water from condenser pressure
0.04 bar to 60 bar.
2-3: Isobaric heating in the closed feedwater heater to 141.3 ◦C.
3-4: Isobaric heating in the steam generator to 60 bar, 550 ◦C.
4-5: Isentropic expansion into the condenser.
Some steam is extracted from the turbine at 4 bar to heat feedwater in a closed feedwater heater. This part of the steam undergoes the following two processes:
6-7: Isobaric cooling and condensation at 4 bar of diverted steam to saturated liquid state.
7-8: Throttling of condensate exiting the feedwater heater into the condenser.
1. Draw a schematic and a T-s-diagram of the process.
2. Make a list of the values of enthalpies at the points 1 to 8.
3. Compute the percentage of mass flow diverted into the feedwater heater at point 6.
4. Determine the thermal efficiency of the cycle.
5. Determine the mass flow rate into the turbine, if the net power developed is 320 MW.
6. Determine total entropy generation rate and work loss of the cycle.
Steam Power Plant with Two Feedwater Heaters, One Open, One Closed
Consider an ideal steam regenerative Rankine cycle with one open and one closed feedwater heater. Steam enters the turbine at 12.5 MPa, 550 ◦C, the condenser pressure is 10 kPa. Steam for the closed feedwater heater is ex- tracted from the turbine at 0.8 MPa and for the open feedwater heater at 0.3 MPa. The feedwater is heated to the condensation temperature of the stream for the closed feedwater heater. The extracted steam leaves the closed feedwater heater at saturated state and is throttled into the open feedwater heater.
1. Draw a schematic of the process, and the corresponding T-s-diagram.
2. For a power output of 250 MW determine the mass flow rate through the
steam generator, and the mass flows into the feedwater heaters.
3. Determine the thermal efficiency of the cycle.
4. Determine the entropy generation in the throttle, and estimate the corresponding work loss.
Steam Power Plant with Reheat and Two Feedwater Heaters, One Closed, One Open The boiler pressure in a reheat steam power plant is 150 bar, the reheat pressure is 14 bar, and the condenser pressure is 10 kPa. For both turbines, the inlet temperature is 500 ◦C. After the high pressure turbine, some steam is bled-off and routed to the closed feedwater heater where it is fully condensed, and then pumped into the boiler feedwater. The remaining steam is reheated, and then runs through the low pressure turbine. Part of the flow is bled-off from the turbine at a pressure of 4 bar, while the main flow expands into the condenser. The diverted flow is mixed in the open feedwater heater with the flow that is pumped in from the condenser. The resulting mixture, which is in the saturated liquid state, is then pumped to boiler pressure before it enters the closed feedwater heater.
1. Draw a schematic of the process. Use the following numbering of processes: 1-2: Low pressure feedwater pump (from condenser). 3-4: Second feedwater pump (after open feedwater heater). 6-7: High pressure steam generator. 7- 8: High pressure turbine. 8-9: Reheat. 9-11: Low pressure turbine. 10: Bled- off for open feedwater heater. 11-1: Condenser. 12-13: Third feedwater pump.
2. Draw the corresponding T-s-diagram. Use different colors (or different line styles) to show the process curves for the main flow and the two bled-off flows.
3. Determine the enthalpies at all relevant states, based on the following assumptions: Reversible pumps and turbines, exit of open feedwater heater is saturated liquid (state 3), perfect heat exchange in closed feedwater heater, so that T5 = T12.
4. Determine the thermal efficiency of the plant.
5. Determine the three mass flows when the plant delivers a power of 500 MW.
6. Determine the overall entropy generation of the system, and the work loss to irreversibilities.
7. Determine the thermal efficiency of a standard reheat plant with the same pressures and turbines. Explain why the feedwater heaters improve efficiency.
Steam Power Plant with Reheat and Two Feedwater Heaters, One Closed, One Open
Repeat the previous problem, now considering irreversible pumps (isentropic efficiency ηP = 0.85) and turbines (isentropic efficiency ηT = 0.92).
Steam Power Plant with Reheat and Two Feedwater Heaters, One Open, One Closed A reheat steam power plant has one closed feedwater heater (c.f.w.h.) and one open feedwater heater (o.f.w.h.). The boiler pressure is 150 bar, the reheat pressure is 15 bar, and the condenser pressure is 10 kPa. For both turbines, the inlet temperature is 500 ◦C. After the high pressure turbine, some steam is bled-off and routed to the o.f.w.h. The remaining steam is reheated, and then runs through the low pressure turbine. Part of the flow is bled-off from the turbine at a pressure of 5 bar. This flow is further routed through the c.f.w.h., where it fully condenses, and is then pumped into the o.f.w.h. The main flow expands into the condenser, which it leaves as saturated liquid. This flow is pumped to the o.f.w.h. pressure, heated in the c.f.w.h., and then mixed with the other flows in the o.f.w.h. The resulting mixture in the o.f.w.h., which is in the saturated liquid state, is then pumped to boiler pressure.
1. Draw a schematic of the process. Use the following numbering of processes: 1-2: Low pressure feedwater pump (from condenser). 2-3 and 9-11: Closed f.w.h. 4-5: Second feedwater pump (after o.f.w.h.). 5-6: High pressure steam generator. 6-7: High pressure turbine. 7-8: Reheat. 8-10: Low pressure turbine. 9: Bled-off for closed f.w.h. 10-1: Condenser. 11-12: Third feedwater pump.
2. Draw the corresponding T-s-diagram. Use different colors (or different line styles) to show the process curves for the main flow and the two bled-off flows.
3. Determine the enthalpies at all relevant states, based on the following assumptions: Reversible pumps and turbines, exit of o.f.w.h. is saturated liquid (state 4), perfect heat exchange in closed feedwater heater, so that T3 = T11.
4. Determine the relative amounts of the relevant mass flows.
5. Determine the thermal efficiency of the plant.
6. Determine the three mass flows when the plant delivers a power of 500 MW.
7. Determine the overall entropy generation of the system, and the work loss to irreversibilities.
8. Determine the thermal efficiency of a standard reheat plant with the same pressures and turbines. Explain why the feedwater heaters improve efficiency.
Steam Power Plant with Reheat and Two Feedwater Heaters, One Open, One Closed
Repeat the previous problem, now considering irreversible pumps (isentropic efficiency ηP = 0.85) and turbines (isentropic efficiency ηT = 0.92).
Cogeneration Power Plant
A cogeneration power plant with reheat produces 3 MW of power and supplies 7 MW of process heat. Steam enters the isentropic high-pressure turbine at 8 MPa and 500 ◦C and expands to a pressure of 1 MPa. At this pressure, part of the steam is extracted from the turbine and routed to the process heater; this stream leaves the process heater as compressed liquid at 120 ◦C. The remaining steam is reheated to 500 ◦C and then expanded in the isentropic low-pressure turbine to the condenser pressure of 15 kPa. The condensate is pumped to 1 MPa and then mixed with the stream of compressed liquid that comes from the process heater. The mixture is then pumped to the boiler pressure.
1. Make a schematic of the cycle, and draw the corresponding T-s-diagram.
2. Determine the heat input, the relative amount of steam running through the process heater, and the utilization factor.
Cogeneration Steam Power Plant with Regeneration
A small power plant that produces 30 MW of power operates on a regenerative vapor power cycle with one closed feedwater heater according to the following process:
Steam of 125 bar, 550 ◦C (state 1) enters the high pressure turbine where it is expanded isentropically to 10 bar (state 2). 50% of this steam are reheated to 500 ◦C (state 3) and then expanded in the low pressure turbine to the condenser pressure 0.075 bar (state 4). After condensation to saturated liquid state (state 5) this stream is pumped isentropically to 10 bar (state 6) and routed into the open feedwater heater. Part of the steam extracted after the high pressure turbine (state 2) is used for process heating. For this, the steam passes through a heat exchanger which it leaves as compressed liquid at 60 ◦C (state 7) that is fed into the open feedwater heater. The rest of the extracted steam of state 2 is directly routed into the feedwater heater. The water leaving the feedwater heater is in the saturated liquid state (state 8); an isentropic pump increases its pressure to the boiler pressure (state 9).
1. Draw a schematic and a T-s-diagram of the process.
2. Make a table with enthalpies at the relevant states of the process.
3. Determine the mass flows through the boiler and the process heater.
4. Determine the utilization factor of the plant.
District Heating
A 40 MW power plant is to be build to supply electrical power to a small town in the North where, due to the low average temperature, a large amount of space heating is required. One proposal suggest to set the condenser pressure a bit higher, so that the condenser is at temperature TC1, and to use the removed heat for district heating. An alternative proposal suggests to set the condenser to the lower temperature TC2 so that the turbine delivers more work, which can then be used to run heat pumps between TC1 and TC2. Discuss these proposals and make a recommendation to town council. Use thermodynamic arguments (of course!), it might be helpful to draw pictures with energy flows and temperatures.
Standard Vapor Cooling Cycle with Ammonia
A standard vapor refrigeration cycle operates with ammonia as cooling fluid. The maximum and minimum pressures reached are 1.5 atm and 10 atm, re- spectively. The adiabatic compressor draws saturated vapor, and has an isen- tropic efficiency of 0.9. The ammonia vapor leaving the compressor is cooled, condensed and further cooled to 20 ◦C before it enters the throttling device.
Draw the process into the log p-h diagram for ammonia, and find the enthalpies and temperatures at all principal points. For a cooling power of 2 kW, determine the power consumption and the COP.
Two-Stage Refrigeration Cycle
1. Draw a schematic, and the corresponding T-s-diagram for a two-stage refrigeration cycle with an open heat exchanger.
2. Indicate all principal points in both diagrams by numbers, and indicate the different elements by name (compressor, throttle, etc.).
3. Compute the mass flow ratio between upper and lower cycle in terms of enthalpies.
4. Give the expression for the COP of the system in terms of enthalpies and mass flow ratio.
Note: The next three problems compare cooling cycles running between the same upper and lower pressures. There is some data overlap, and to simplify proceedings an irreversible compressor is considered only in the first cycle. For all three, start with drawing a sketch, and the T-s-diagram
Standard Vapor Cooling Cycle with R134a
A standard vapor refrigeration cycles operates with R134a as cooling fluid between the pressures 1.2 MPa and 0.1 MPa, respectively. The adiabatic com- pressor draws saturated vapor, and has an isentropic efficiency of 0.9. The vapor leaving the compressor is cooled and fully condensed before it en- ters the throttling device. Draw a schematic and the T-s-diagram, and then determine:
1. The COP for the cycle with irreversible and with reversible compressor.
2. The mass flow rate and the power consumption for a cooling power of
200 kW.
3. Determine entropy generation rates for each process, the overall entropy generation rate, and work loss of the cycle. Assume TH = 30 ◦C and TL = −20 ◦C.
Two-Stage Refrigeration Cycle with Flash Chamber
A two-stage compression refrigeration system operates with R134a between the pressures 1.2 MPa and 0.1 MPa. The refrigerant leaves the condenser as saturated liquid and is throttled to a flash chamber operating at 0.4 MPa. The refrigerant leaving the low-pressure compressor at 0.4 MPa is also routed to the flash chamber.
The saturated vapor leaving the flash chamber is compressed to the con- denser pressure by the high-pressure compressor, while the saturated liquid leaving the flash chamber is throttled to the evaporator pressure. The re- frigerant leaves the evaporator as saturated vapor and both compressors are isentropic. Draw a schematic and the T-s-diagram, and then determine:
1. The fraction of mass flows running through the two compressors.
2. The COP, and compare to that of the previous problem.
3. The two mass flow rates and the power consumption for a cooling power of 200 kW.
4. Determine entropy generation rates for each process, the overall entropy generation rate, and work loss of the cycle. Assume TH = 30 ◦C and TL = −20 ◦C.
Two-Stage Refrigeration Cycle with Heat Exchanger
A two-stage cascade refrigeration system operates with R134a between the pressures 1.2 MPa and 0.1 MPa. Heat exchange between the two cycles takes place in an adiabatic counter-flow heat exchanger where the pressures are 0.32 and 0.4 MPa, respectively. In both cycles, the refrigerant is in saturated liquid state at the condenser exit and at saturated vapor state at the compressor inlet. Draw a schematic and the T-s-diagram, and then determine:
1. The fraction of mass flows running through the two compressors.
2. The coefficient of performance.
3. The two mass flow rates, and the power consumption for a cooling power of 200 kW.
4. The total entropy generation rate and work loss of the cycle. Assume TH = 30 ◦C and TL = −20 ◦C.
Two-Stage Refrigeration Cycles
Repeat the previous two problems for the case where the compressors have an isentropic efficiency of 0.85.
Refrigeration Cycle with Intercooling
A vapor-compression refrigeration cycle operates at steady state with ammonia as working fluid according to the following cycle:
1-2: Adiabatic irreversible compression of saturated vapor at p1 =
1.75 bar to p2 = 5 bar, isentropic compressor efficieny is ηC = 0.8.
2-3: Isobaric cooling to 20 ◦C.
3-4: Adiabatic irreversible compression to p4 = 12 bar, isentropic com- pressor efficieny is ηC = 0.8.
4-5: Isobaric heat rejection in condenser; state 5 is saturated liquid. 5-6: Throttling into the evaporator, p6 = p1.
6-1: Isobaric evaporation to state 1.
1. Draw a schematic and plot the process in a T-s-diagram.
2. Find the enthalpies at points 1-6.
3. Determine the coefficient of performance.
Advanced Cooling Cycle
In a hot climate, a two-stage cascade refrigeration system operates with refrigerant R134a. The evaporator temperature of the low pressure stage is −20 ◦C, and the condenser temperature of the high pressure stage is 50 ◦C. Heat ex- change between the cycles takes place in a counter-flow heat exchanger where the pressures are 0.4 and 0.5 MPa, respectively. In both cycles, the refrigerant is in saturated liquid state at the condenser exit, and in saturated vapor state at the compressor inlet. The isentropic efficiency of both compressors is 0.8.
1. Draw a T-s-diagram of the cycle with respect to saturation lines.
2. Make a list of the enthalpies and entropies at states 1 through 8.
3. Determine the ratio of mass flows entering the upper and the lower compressor.
4. Determine the COP of the cycle, and the power requirement for a cooling power of 120 kW.
5. Determine entropy generation rates for all processes, the overall entropy generation rate, and the work loss of the cycle. Assume TH = 40 ◦C and TL = −10 ◦C.
Regenerative Gas Cooling System
A regenerative gas refrigeration cycle uses helium as working fluid. The helium enters the compressor at 100 kPa and −10 ◦C and is compressed to 300 kPa. Then, it is cooled to 20 ◦C by heat exchange with a cooling water flow. Next, the helium enters the regenerator where it is cooled further before it enters the turbine. Helium leaves the refrigerated space at −25 ◦C and enters the regenerator. Assume isentropic operation of turbine and regenerator, and determine
1. The temperature at the turbine inlet.
2. The COP of the cycle.
3. he net power input required for a mass flow rate of 0.45 kg .
Helium behaves as an ideal gas; it is monatomic, and thus has