Problems in chapter 9 CB Thermodynamics

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1 Problems in chapter 9 CB Thermodynamics 9-82 Air is used as the working fluid in a simple ideal Brayton cycle that has a pressure ratio of 12, a compressor inlet temperature of 300 K, and a turbine inlet temperature of 1000 K. Determine the required mass flow rate of air for a net power output of 70MW, assuming both the compressor and the turbine have an isentropic efficiency of (a) 100% and (b) 85%. Assume constant specific heats at room temperature. Solution: Answers: (a) 352 kg/s, (b) 1037 kg/s 9-85 A gas-turbine power plant operates on the simple Brayton cycle between the pressure limits of 100 and 1600kPa. The working fluid is air, which enters the compressor at 40 C at a rate of 850m 3 /min and leaves the turbine at 650 C. Using variable specific heats for air and assuming a compressor isentropic efficiency of 85% and a turbine isentropic efficiency of 88%, determine (a) the net power output, (b) the back work ratio, and (c) the thermal efficiency. Solution: Answers: (a) 6083kW with T 3 = C, (b) 0.535, (c) 37.4% 9-86 A gas-turbine power plant operates on a simple Brayton cycle with air as the working fluid. The air enters the turbine at 800kPa and 1100K and leaves at 100kPa and 670K. Heat is rejected to the surroundings at a rate of 6700kW, and air flows through the cycle at a rate of 18kg/s. Assuming the turbine to be isentropic and the compresssor to have an isentropic efficiency of 80%, determine the net power output of the plant. Account for the variation of specific heats with temperature. Solution: Answer: 2979 kw 9-87 For what compressor efficiency will the gas-turbine power plant in Problem 9-86 produce zero net work? Solution: Answer: 9-90 How does regeneration affect the efficiency of a Brayton cycle, and how does it accomplish it? Solution: Answer: 9-94 A gas turbine for an automobile is designed with a regenerator. Air enters the compressor of this engine at 100kPa and 20 C. The compressor pressure ratio is 8; the maximum cycle temperature is 800 C; and the cold air stream leaves the regenerator 10 C cooler than the hot air stream at the inlet of the regenerator. Assuming both the compressor and the turbine to be isentropic, determine the rates of heat addition and rejection for this cycle when it produces 150 kw. Use constant specific heats at room temperature. Solution: Answer: 303kW and 153kW Problems in chapter 10 CB Thermodynamics Consider a 210MW steam power plant that operates on a simple ideal Rankine cycle. Steam enters the turbine at 10MPa and 500 C and is cooled in the condenser at a pressure of 10kPa. Show the cycle on a T-s diagram with respect to saturation lines, and determine (a) the quality of the steam at the turbine exit, (b) the thermal efficiency of the cycle, and (c) the mass flow rate of the steam. 1

2 Solution: Answers: (a) 79.3 %, (b) 40.2 %, (c) 165 kg/s A simple Rankine cycle uses water as the working fluid. The boiler operates at 6000kPa and the condenser at 50kPa. At the entrance to the turbine, the temperature is 450 C. The isentropic efficiency of the turbine is 94%, pressure and pump losses are negligible, and the water leaving the condenser is subcooled by 6.3 C. The boiler is sized for a mass flow rate of 20kg/s. Determine the rate at which heat is added in the boiler, the power required to operate the pumps, the net power produced by the cycle, and the thermal efficiency. Solution: Answers: kw, 122 kw, kw, 30.3 % Consider a coal-fired steam power plant that produces 175 MW of electric power. The power plant operates on a simple ideal Rankine cycle with turbine inlet conditions of 7 MPa and 550 C and a condenser pressure of 15 kpa. The coal has a heating value (energy released when the fuel is burned) of 29.3 MJ/kg. Assuming that 85 % of this energy is transferred to the steam in the boiler and that the electric generator has an efficiency of 96 %, determine (a) the overall plant efficiency (the ratio of net electric power output to the energy input as fuel) and (b) the required rate of coal supply. Solution: Answers: (a) 31.5 %, (b) 68.3 t/h Consider a steam power plant that operates on the ideal reheat Rankine cycle. The plant maintains the boiler at 5000kPa, the reheat section at 1200kPa, and the condenser at 20kPa. The mixture quality at the exit of both turbines is 96.0%. Determine the temperature at the inlet of each turbine and the cycle s thermal efficiency. L w-p urbin Rehe Pu1np Q) Figure 1: Assignment Solution: Answers: 327 C, 481 C, 35.0 % A steam power plant operates on the reheat Rankine cycle. Steam enters the high-pressure turbine at 12.5MPa and 550 C at a rate of 7.7kg/s and leaves at 2MPa. Steam is then reheated at constant pressure to 450 C before it expands in the low-pressure turbine. The isentropic 2

3 efficiencies of the turbine and the pump are 85% and 90%, respectively. Steam leaves the condenser as a saturated liquid. If the moisture content of the steam at the exit of the turbine is not to exceed 5%, determine (a) the condenser pressure, (b) the net power output, and (c) the thermal efficiency. Pump Q) Figure 2: Assignment Solution: Answers: (a) 9.73 kpa, (b) 10.2 MW, (c) 36.9 % The closed feedwater heater of a regenerative Rankine cycle is to heat 7000 kpa feedwater from 260 C to a saturated liquid. The turbine supplies bleed steam at 6000kPa and 325 C to this unit. This steam is condensed to a saturated liquid before entering the pump. Calculate the amount of bleed steam required to heat 1kg of feedwater in this unit. Bleed te m from turbine Figure 3: Assignment

4 Solution: Answer: kg/s A steam power plant operates on an ideal regenerative Rankine cycle. Steam enters the turbine at 6MPa and 450 C and is condensed in the condenser at 20 kpa. Steam is extracted from the turbine at 0.4 MPa to heat the feedwater in an open feedwater heater. Water leaves the feedwater heater as a saturated liquid. Show the cycle on a T-s diagram, and determine (a) the mass flow rate of steam through the boiler and (b) the thermal efficiency of the cycle Solution: Answers (a) 1017 kj/kg (b) 37.8 % Repeat previous problem by replacing the open feedwater heater with a closed feedwater heater. Assume that the feedwater leaves the heater at the condensation temperature of the extracted steam and that the extracted steam leaves the heater as a saturated liquid and is pumped to the line carrying the feedwater. Solution: Answers (a) kj/kg (b) 37.8% Consider a steam power plant that operates on the ideal regenerative Rankine cycle with a closed feedwater heater as shown in the figure. The plant maintains the turbine inlet at 3000kPa and 350 C; and operates the condenser at 20kPa. Steam is extracted at 1000kPa to serve the closed feedwater heater, which discharges into the condenser after being throttled to condenser pressure. Calculate the work produced by the turbine, the work consumed by the pump, and the heat supply in the boiler for this cycle per unit of boiler flow rate. boiler Figure 4: Assignment Solution: Answers: 741 kj/kg, 3.0 kj/kg, 2353 kj/kg A steam power plant operates on an ideal reheat- regenerative Rankine cycle and has a net power output of 80MW. Steam enters the high-pressure turbine at 10MPa and 550 C and leaves at 0.8MPa. Some steam is extracted at this pressure to heat the feedwater in an open 4

5 feedwater heater. The rest of the steam is reheated to 500 C and is expanded in the low-pressure turbine to the condenser pressure of 10kPa. Show the cycle on a T-s diagram with respect to saturation lines, and determine (a) the mass flow rate of steam through the boiler and (b) the thermal efficiency of the cycle Solution: Answers (a) 54.5kg/s (b) 44.4% Heat exchanger problems HEX 001 A feedwater train with five closed feed water heaters and no pumps. The feedwater heaters heat the feedwater to a temperature of 250 C. The condensor temperature is 80 C. Determine the optimum temperature between the heaters. The optimum division of the temperatures in the feed water train was given in the lecture as T k T 1 = T 1 T 2 =... = T n 1 T n = k (1) which can be expressed as where T k T n k n k = n T k T n (2) Temperature after the condenser [K] Temperature after nth feed water heater[k] Constant Number of feed water heaters Solution: Answer: T 2 = C, T 3 = C, T 4 = C, T 5 = C HEX 002 A power plant has three feedwater heaters. The steam temperature is 5 C above the temperature of the feedwater that leaves the feedwater heater. There is no sub-cooling in the feedwater heater and there are no pressure losses. The final feedwater temperature is not known, T n Steam flow before turbine 100 kg/s Inlet pressure 100 bar Inlet temperature 500 C Turbine isentropic efficiency 87 % Grädigkeit 5 C Condenser pressure 0.58 bar Temperature after the condenser 85 C Calculate for the following cases T n T n 1 =1.08, 1.1 and 1.12 Solution: (Answer for first case: T k = 85 C, T 1 = C, T 2 = C och T 3 = 178 C) 5

6 HEX 003 In a factory that produce olive oil the oil must be cooled after it has been pressed. This is performed in two heat exchangers (see figure) with fresh water. The olive oil is cooled from 45 C to 23 C. Calculate the cooling water flow if the oil flow is V oil = 450dm 3 /h. The density for the oil is ρ oil = 915kg/m 3 and the specific heat for oil is c p,oil = 1.65kJ/(kgK). Water has the density ρ w = 1000kg/m 3 and a specific heat of c p,w = 4.18kJ/(kgK). T oil = 23 C T oil = 45 C T water = 13 C T water = 19 C Figure 5: Assignment HEX 003 Solution: (Answer: No answer at the moment.) HEX 004 Oil with a temperature of 100 C is to be cooled in a counter-current heat exchanger. Cooling water is available at a temperature of 10 C. The heat transfer area in the heat exchanger is 50m 2 and the flow of oil is 1.8 kg/s and the cooling water flow is 1.7kg/s. The following data is known. The oil coolers heat transfer coefficient, U = 116W/(m 2 K). The specific heat of the oil, c v = 1.98kJ/(kgK) and the specific heat of the water is c k = 4.19kJ/(kgK). Calculate the oil and water temperature at the exit of the heat exchanger and calculate the amount of heat transfered by the heat exchanger. Solution: (Answer: T 1 = C, T 2 = C and Q = 229.4kW) Problems in chapter 11 CB Thermodynamics Refrigerant-134a enters the compressor of a refrigerator as superheated vapor at 0.20 MPa and 5 C at a rate of 0.07 kg/s, and it leaves at 1.2 MPa and 70 C. The refrigerant is cooled in the condenser to 44 C and 1.15 MPa, and it is throttled to 0.21 MPa. Disregarding any heat transfer and pressure drops in the connecting lines between the components, show the cycle on a T-s diagram with respect to saturation lines, and determine (a) the rate of heat removal from the refrigerated space and the power input to the compressor, (b) the isentropic efficiency of the compressor, and (c) the COP of the refrigerator. Solution: Answers: 9.42 kw, 3.63 kw, 74 %, 30.3 % An actual refrigerator operates on the vapor compression refrigeration cycle with refrigerant- 22 as the working fluid. The refrigerant evaporates at 15 C and condenses at 40 C. The isentropic efficiency of the compressor is 83 %. The refrigerant is superheated by 5 C at the compressor inlet and subcooled by 5 C at the exit of the condenser. Determine (a) the heat removed from the cooled space and the work input, in kiloj/kg and the COP of the cycle. Determine (b) the same parameters if the cycle operated on the ideal vapor-compression refrigeration cycle between the same evaporating and condensing temperatures. The properties of R-22 in the case of actual operation are: h 1 = kJ/kg, h 2 = kJ/kg, h 3 = kJ/kg. The properties of R-22 in the case of ideal operation are: h 1 = kJ/kg, h 2 = kJ/kg, h 3 = kJ/kg, Note: 6

7 state 1: compressor inlet, state 2: compressor exit, state 3: condenser exit, state 4: evaporator inlet. Solution: Answers: and A heat pump with refrigerant-134a as the working fluid is used to keep a space at 25 C by absorbing heat from geothermal water that enters the evaporator at 50 C at a rate of kg s and leaves at 40 C. The refrigerant enters the evaporator at 20 C with a quality of 23 % and leaves at the inlet pressure as saturated vapor. The refrigerant loses 300 W of heat to the surroundings as it flows through the compressor and the refrigerant leaves the compressor at 1.4 MPa at the same entropy as the inlet. Determine (a) the degrees of subcooling of the refrigerant in the condenser, (b) the mass flow rate of the refrigerant, (c) the heating load and the COP of the heat pump, and (d) the theoretical minimum power input to the compressor for the same heating load. Expan ion valve 20 c x=0.23 t Water 50 C Evaporator 40 C t Compressor CD at. vapor Figure 6: Assignment 1142 Solution: Answers:3.8 C, kg/s, 3.07 kw and 4.68, kw A two-stage compression refrigeration system operates with refrigerant-134a between the pressure limits of 1.4 MPa and 0.10 MPa. The refrigerant leaves the condenser as a saturated liquid and is throttled to a flash chamber operating at 0.6 MPa. The refrigerant leaving the low-pressure compressor at 0.6 MPa is also routed to the flash chamber. The vapor in the flash chamber is then compressed to the condenser pressure by the high-pressure compressor, and the liquid is throttled to the evaporator pressure. Assuming the refrigerant leaves the evaporator as saturated vapor and both compressors are isentropic, determine (a) the fraction of the refrigerant that evaporates as it is throttled to the flash chamber, (b) the rate of heat removed from the 7

8 refrigerated space for a mass flow rate of 0.25 kg/s through the condenser, and (c) the coefficient of performance. Solution: Answers:0.2528, Q L = 28.57kW, COP R = 2.50 Problems in chapter 15 CB Thermodynamics In a combustion chamber, ethane C 2 H 6 is burned at a rate of 8kg/h with air that enters the combustion chamber at a rate of 176 kg/h. Determine the percentage of excess air used during this process. Solution: Answer: 37% A fuel mixture of 60% by mass methane CH 4 and 40% by mass ethanol C 2 H 6 O, is burned completely with theoretical air. If the total flow rate of the fuel is 10kg/s, determine the required flow rate of air. Solution: Answer: 139 kg/s Methane CH 4 is burned with dry air. The volumetric analysis of the products on a dry basis is 5.20% CO 2, 0.33% CO, 11.24% O 2, and 83.23% N 2. Determine (a) the air-fuel ratio and (b) the percentage of theoretical air used. Solution: Answers: (a) 34.5 kg air /kg fuel, (b) 200% Determine the enthalpy of combustion of methane CH 4 at 25 C and 1atm, using the enthalpy of formation data from Table A-26. Assume that the water in the products is in the liquid form. Compare your result to the value listed in Table A-27. Solution: Answer: kj/kmol Calculate the higher and lower heating values of a coal from Illinois which has an ultimate analysis (by mass) as 67.40% C, 5.31% H 2, 15.11% O 2, 1.44% N 2, 2.36% S, and 8.38% ash (non-combustibles). The enthalpy of formation of SO 2 is kJ/kmol. Solution: Answer: kj/kg, kj/kg Diesel fuel C 12 H 26 at 25 C is burned in a steadyflow combustion chamber with 20% excess air that also enters at 25 C. The products leave the combustion chamber at 500K. Assuming combustion is complete, determine the required mass flow rate of the diesel fuel to supply heat at a rate of 2000kJ/s. Solution: Answer: 49.5 g/s A gaseous fuel mixture that is 40 percent propane C 3 H 8 and 60 percent methane CH 4 by volume is mixed with the theoretical amount of dry air and burned in a steadyflow, constant pressure process at 100kPa. Both the fuel and air enter the combustion chamber at 298K and undergo a complete combustion process. The products leave the combustion chamber at 423 K. Determine (a) the balanced combustion equation, (b) the amount of water vapor condensed from the products, and (c) the required air flow rate, in kg/h, when the combustion process produces a heat transfer output of kJ/h. 8

9 h 0 f, kj/kmol M, kg/kmol c p kj/(kgk) C 3 H 8-103, CH 4-74, CO 2-393, CO -110, H 2 O (g) -241, H 2 O (l) -285, O N Solution: Answer: (c) 50.1 kg/h Estimate the adiabatic flame temperature of an acetylene C 2 H 2 cutting torch, in C, which uses a stoichiometric amount of pure oxygen. Solution: Answer: 8832 C with c p,h2 O = kJ/(kgK) and c p,co2 = 1.234kJ/(kgK) Acetylene gas C 2 H 2 at 25 C is burned during a steady-flow combustion process with 30% excess air at 27 C. It is observed that 75000kJ of heat is being lost from the combustion chamber to the surroundings per kmol of acetylene. Assuming combustion is complete, determine the exit temperature of the product gases. Solution: Answer: 2301 K Liquid propane C 3 H 8 enters a steady-flow combustion chamber at 25 C and 1atm at a rate of 0.4kg/min where it is mixed and burned with 150% excess air that enters the combustion chamber at 12 C. If the combustion products leave at 1200K and 1atm, determine (a) the mass flow rate of air, (b) the rate of heat transfer from the combustion chamber, and (c) the rate of entropy generation during this process. Assume T 0 = 25 C. Solution: Answer: (a) 15.7kg/min, (b) 1730kJ/min, (c) 38.07kJ/min/K, with s C3 H 8 (l) = kj/(kmolk) and Ṡgen = kJ/K/kmol C3 H A liquid-gas fuel mixture consists of 90% octane C 8 H 18, and 10% alcohol C 2 H 5 OH, by moles. This fuel is burned with 200% theoretical dry air. Write the balanced reaction equation for complete combustion of this fuel mixture. Determine (a) the theoretical air-fuel ratio for this reaction, (b) the product-fuel ratio for this reaction, (c) the air-flow rate for a fuel mixture flow rate of 5kg/s, and (d) the lower heating value of the fuel mixture with 200% theoretical air at 25 C. Solution: Answer: (a) 14.83kg air /kg fuel, (b) 30.54kg product /kg fuel, (c) 148.3kg/s, (d) 43760kJ/kg fuel Extra gas turbine problems In these assignments please use constant specific heats according to For air: c pa = 1.005kJ/(kgK) k =

10 For combustion gas: c pg = 1.148kJ/(kgK) k = The gas constant for air and combustion gases, R = kj/kg. In order to calculate the fuel-air-ratio you will need the following set of equations x 1 = (700 T 02 ) x 2 = (700 ) T T 03 x 3 = ( T ) x 1 x1 2 + x 2 x 3 f = η b The lower heating value of the fuel is kj/kg which can be used to calculate the efficiency of a cycle according to the following equation η = Ẇ net ṁ fuel LHV 1. A compressor has an isentropic efficiency of 0.85 at a pressure ratio of 4.0. Calculate the corresponding polytropic efficiency, and thence plot the variation of isentropic efficiency over a range of pressure ratio from A peak-load generator is to be powered by a simple gas turbine with free power turbine delivering 20 MW of shaft power. The following data are applicable: Compressor pressure ratio 11.0 Compressor isentropic efficiency 0.82 Combustion pressure loss 0.4 bar Combustion efficiency K Gas-generator turbine isentropic efficiency 0.87 Power-turbine isentropic efficiency 0.89 Mechanical efficiency (each shaft) 0.98 Ambient conditions, p a, T a 1 bar, 288 K Calculate the air mass flow required and the SFC. 3. A gas turbine for use on a large airliner uses a single-shaft configuration with air bled from the compressor discharge for aircraft services. The unit must provide 1.5 kg/s bleed air and a shaft power of 200 kw. Calculate (a) the total compressor air mass flow and (b) the power available with no bleed flow, assuming the following Compressor pressure ratio 3.80 Compressor isentropic efficiency

11 Combustion pressure loss 0.12 bar 1050 K Turbine isentropic efficiency 0.88 Mechanical efficiency (compressor rotor) 0.99 Mechanical efficiency (driven load) 0.98 Ambient conditions, p a, T a 1 bar, 288 K [ṁ a = 4.676kg/s, Ẇ = kW] 4. A single-shaft gas turbine for electric power generation has been steadily developed over time. Cycle data for three versions are given below. A being the initial version A B C Polytropic efficiency (compressor) Polytropic efficiency (turbine) Compressor pressure ratio Compressor pressure loss (%) (K) Rotor cooling bleed (%) Airflow (kg/pers) Assume combustion efficiency and mechanical efficiency are both 99 % and ignore inlet and exhaust pressure losses. (a) Calculate the power and SFC for each version. (b) Calculate the percentage improvement from version A. (c) Calculated the exhaust gas temperature for each version and comment on their effect for a cogeneration plant. [SFC = kg/(kw h), kg/(kw h) and kg/(kw h) ] EGT = K, K and K T 03 T 04 = K, K and K] 5. Using the data from Example 3 and assuming a reheat pressure of 13 bar, evaluate the power and thermal efficiency for reheat temperatures of 1525 K, 1425 K and 1325 K. Would this be a good strategy for part-load operation? [240, 214.6, MW, efficiency almost constant] 6. The following data refer to an intercooled gas turbine, with a twinspool gas generator and a free power turbine: LP compressor pressure ratio 5.5 HP compressor pressure ratio 7.5 Air temperature after intercooler 300 K 1550 K Rotor cooling bleed 5 % LP compressor isentropic efficiency

12 HP compressor isentropic efficiency LP turbine isentropic efficiency 0.89 HP turbine isentropic efficiency 0.88 Power-turbine isentropic efficiency 0.89 Neglect inlet, intercooler and exhaust pressure losses. Calculate (a) Airflow required for an output of 100 MW. (b) Thermal efficiency. (c) Exhaust gas temperature. [192.97kg/s, 43.7%, 421 C] 7. A gas turbine is to be designed for continuous duty at a rating of 4MW. Compare the following two cycles and comment on their suitability for this role. Also comment on their potential for use in a small cogeneration plant. Regenerative cycle Pressure ratio K Simple cycle: Pressure ratio K 8. The following data apply to a regenerative cycle industrial gas turbine: Compressor pressure ratio 8.5 LP compressor isentropic efficiency K Gas generator turbine isentropic efficiency 0.87 Power-turbine isentropic efficiency 0.88 Mechanical efficiency (both shafts) 0.99 Combustion efficiency 0.99 Combustion pressure loss (of comp. delivery press.) 4.2 % Regenerator effectiveness 0.90 Regenerator pressure loss cold side (of compressor delivery pressure) 1.5 % hot side (of atmospheric pressure) 2.0 % Mass flow kg/s Ambient conditions, 1.013bar and 15 C 12

13 (a) Calculate the power output and thermal efficiency. [ kw, 41.8%] (b) If this engine were to be used in a combined cycle application, what changes would you suggest? [Pressure between turbines = kpa] 9. A closed-cycle gas turbine is to be used in conjunction with a gas-cooled nuclear reactor. The working fluid is helium (c p = 5.19kJ/(kgK) and γ = 1.66). The layout of the plant consists of two-stage compression with intercooling followed by a heat-exchanger; after leaving the cold side of the heat-exchanger the helium passes through the reactor channels and on to the turbine; from the turbine it passes through the hot side of the heat-exchanger and then a precooler before returning to the compressor inlet. The following data are applicable: Compressor and turbine polytropic efficiencies 0.88 Temperature at LP compressor inlet 300 K Pressure at LP compressor inlet 14 bar Compressor pressure ratios (LP and HP) 2.0 Temperature at HP compressor inlet 300 K Mass flow of helium kg/s Reactor thermal output (heat input to gas turbine) MW 1285 K Pressure loss in precooler and intercooler (each) 0.34 bar Pressure loss in heat-exchanger (each side) 0.27 bar Pressure loss in reactor channels 1.03 bar Helium temperature at entry to reactor channels 700 K Calculate the power output and thermal efficiency, and the heat-exchanger effectiveness implied by the data. [214.5 MW, 0.429, 0.782] 13