## Problems for Chap

4.1 A cyclical engine receives 325 kJ of heat from a reservoir at 1000 K, delivers 200 kJ of work, and rejects 125 kJ of heat to a reservoir at 400 K. Does this engine violate either the First law or the Second law?

4.2 Consider a Carnot cycle of 25% efficiency using water as the working fluid. Heat transfer to the engine takes place at 300oC, during which the water changes from saturated liquid to saturated vapor. Similarly, heat rejection occurs isothermally as the

(a) Sketch this Carnot cycle superimposed on the T-s diagram of water (above).

(b) What is the temperature of the heat sink?

(c) What is the steam quality at the beginning and end of the heat-rejection step?

(d) What is the work done by the engine (per kg of water)?

4.3 Starting from state 1 in the two-phase region, 1 kg of water undergoes the cycle shown below. Steps 1-2 and 3-4 are isobaric and reversible and steps 2-3 and 4-1 are adiabatic and reversible. The system remains in the two-phase region throughout the cycle

(a) Prepare a table with the following column headings: State, Temperature, Pressure, Specific volume, and Quality. Fill in this table for the four states shown in the diagram.

(b) What is the efficiency of this cycle?

0.004 0.0701

specific volume, m^Jkg

0.004 0.0701

specific volume, m^Jkg

4.4 The sketch below shows two identical, interacting Carnot engines. The first operates between a reservoir at 600 K and rejects heat at a temperature T to the second cycle, which rejects its heat to a reservoir at 300 K. If the two cycles have the same efficiency, what is the intermediate temperature T?

4.5 A Carnot engine operating in outer space receives heat from a nuclear power source at Th. The engine rejects heat by thermal radiation at a rate given by QL = AKTL4 , where A is the area of the radiating surface and K is a constant. TL is the temperature of the radiator, which is considered to be the low-temperature reservoir for the engine. Show that if the power is fixed at W, the area of the radiator is a minimum when TL/TH = 3/4. What is the minimum radiator area in terms of , W, K, and TH?

4.6 A 100% efficient, adiabatic turbine in a power cycle of the type shown in Fig. 4.3 is fed with superheated steam at 325oC and 10 MPa and produces an exhaust of 90% quality. Determine:

(a) The exit pressure and temperature.

(b) The volume fraction of steam in the exit gas

(c) The work produced per unit mass of inlet steam.

(d) The condenser in the power cycle is cooled by water at 20oC. What is the cycle efficiency compared to the maximum possible value? Neglect pressure drops in the circuit (except for the pump and turbine) and assume that the pump is reversible and that heat transfer in the boiler is reversible.

4.7 A nuclear rocket is propelled by lithium vapor in a stream of helium. A 50-50 mole percent mixture is contained in the coolant vessel. The gas mixture is heated to 3000 K and 10 MPa in a nuclear reactor and then expanded through an adiabatic nozzle. The exhaust temperature is 500 K.

Coolant vessel

3000 K 10 MPa exhaust nozzle

Ve 500 K

Calculate the nozzle exhaust velocity (Ve) for the following conditions during the expansion:

(a) The lithium remains gaseous throughout the expansion even though it becomes supercooled during the process. Helium and lithium can be treated as ideal monatomic gases.

(b) During the expansion, lithium remains gaseous until the temperature has dropped to 1600 K. At this point, it completely condenses and remains liquid thereafter. The enthalpy change of liquid lithium from 500 to 1600 K is 22.9 kJ/mole and the heat of vaporization at 1600 K is 147.5 kJ/mole.

4.8 Superheated steam enters an insulated pipe attached to a large reservoir at 3 MPa and 350oC. In the pipe, the pressure is 1.6 MPa and the velocity of 550 m/s. The steam mass flow rate is 0.5 kg/s. Calculate:

(a) The exit quality

(b) The exit temperature

(c) The pipe cross sectional area

4.9 A helium gas turbine operates isentropically with an inlet pressure of 3 MPa and an exhaust pressure of 0.1 MPa. The inlet temperature is 800 K.

(a) Although the process is isentropic and the gas is ideal, Eq (3.19) does not apply in this case. Explain why.

(b) How much work can be obtained per mole of gas flowing through the turbine?

(c) Modify Eq (3.19) so that it is similar in form to the shaft work equation derived in Part (b). What is the difference in these two work formulas?

4.10 A helium gas turbine operates adiabatically with the following specifications: inlet temperature and pressure: 1100 K, 3 MPa. Outlet pressure: 0.5 MPa. When built and operated, the outlet temperature is 700 K. What is the efficiency of the turbine?

4.11 A Carnot engine with air as the working fluid operates on the p-v cycle shown in Fig. 4.2. If the high-temperature reservoir is maintained at 200oC and the conditions at point 4 of the cycle are 200 kPa and 0.3 m3/kg, what is the efficiency of the engine?

4.12 A 1000 MW(electric) power plant uses steam as the working fluid and the condenser is cooled by river water (see sketch). The maximum steam temperature is 550oC and the condenser pressure is 10 kPa. Estimate the minimum temperature rise of the river downstream.

4.13 A geothermal supply of hot water at 500 kPa and 150oC is fed to an insulated flash evaporator at the rate of 1.5 kg/s. The flash evaporator contains vapor and liquid water and operates at 200 kPa. Saturated liquid is drained from the bottom of the evaporator and the saturated vapor is fed to a turbine. The turbine has an efficiency of 70% and an exit pressure of 15 kPa.

(a) Draw a flow diagram of the flash evaporator/turbine combination and label all streams entering or leaving the units with their thermodynamic properties(e.g., pressure, temperature, enthalpy, entropy)

(b) Calculate the flow rates of liquid and steam leaving the flash evaporator.

(c) Determine the quality of the steam exiting the turbine. (Hint: first determine the work produced by the turbine if it were isentropic, or 100% efficient)

(d) Determine the rate of entropy production, Sjrr, in the combined flash evaporator/turbine.

4.14 A 100% efficient adiabatic gas turbine normally operates with an inlet gas at 4 MPa pressure and 1000oC. To reduce turbine power, a throttle valve in the line before the turbine inlet is partially closed. This reduces the inlet gas pressure to 3 MPa. The exhaust pressure is 1 MPa in both cases. The gas is ideal and has a heat capacity ratio of 1.4.

At what percentage of full power does the turbine operate in the throttled condition?

### 4.15 in powerpoint slide

4.16 A block of metal at temperature To is cooled by a heat reservoir at Tc, which remains constant during the process. Cooling of the block from To to Tc is accompanied by a quantity QH of heat. If the block and the reservoir had been connected by a reversible heat engine, the quantity of heat absorbed by the reservoir during the block cooling process would have been reduced to QC. What is the ratio QC/QH in terms of To and Tc?

4.17 The mixing equipment shown below preheats 45oC water (input 1) using superheated steam at 250oC (input 2). The unit is adiabatic and operates at 600 kPa. Assuming that the two water streams are saturated, calculate the temperature of the heated water (output 3).

4.18 Consider the Carnot cycles for the open system (slide with Rankine-cycle T-v diagram) and the closed system (Fig. 4.2 of the Reader).

(a) How do the calculations of the net cycle work differ?

(b) Why are the 1 ^ 2 and the 3 ^ 4 processes different?

(c) What do the 2 ^ 3 and the 4 ^ 1 lines in the two plots have in common?

(d) What is different about the fluids that the 2 ^ 3 and the 4 ^ 1 lines in the two plots represent?

(e) Why can't an ideal gas be used in the Carnot-type flow cycle?

Superheated steam, 0.5 (2)

4.19 A reversible, isothermal gas turbine is connected to two closed gas reservoirs. Initially, the upper reservoir contains nT moles of an ideal gas and the lower reservoir is a vacuum. The volumes of the upper and lower reservoirs are VU and VL, respectively. The entire apparatus remains at a constant temperature T during the process. The process consists of flow of gas from the upper reservoir to the lower reservoir through the turbine until the pressures equalize.

(a) How much work is performed? Hint: establish a relation between the differential work SWs and the differential number of moles dnu flowing through the turbine. This relation will contain the ratio of the pressures in the two reservoirs, pU/pL, which must be related to the current number of moles of gas in the upper vessel, nu.

(b) How much heat is absorbed by the turbine in the process? Part (a) need not be solved to answer this part.

4.20 A reversible helium gas turbine operates with inlet and outlet pressures of 4 and 0.5 MPa, respectively. The inlet gas temperature is 1000oC. However, the turbine is not adiabatic, which causes the temperature of the gas moving through the unit to decrease faster than in the absence of heat loss. The pressure-temperature relationship of the gas flowing through the turbine is:

Lower reservoir

Shaft work

### Lower reservoir

Using the 2nd Law only, determine the fraction of the enthalpy loss of the gas between the outlet and inlet of the turbine that escapes as heat to the environment.

4.21 A helium gas turbine operates isentropically with an inlet pressure of 2 MPa and temperature of 300oC. If the unit produces 3 kJ of shaft work per mole of He, what is the outlet pressure?

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### Responses

• eija
Why can't a turbine operate in the two phase region?
2 years ago