Gas Power Cycles

Abstract

An important application of thermodynamics is the analysis of power cycles through which the energy absorbed as heat can be continuously converted into mechanical work. A thermodynamic analysis of the heat engine cycles provides valuable information regarding the design of new cycles or for improving the existing cycles. In this chapter, various gas power cycles are analyzed under some simplifying assumptions.

Problems

Problem 14.1

In a power plant operating on a Rankin cycle, steam at 100 bar enters a turbine and expands to 0.1 bar. Calculate the minimum temperature of the steam, which enters the turbine to ensure a quality of 0.9 at the exit of the turbine.

Problem 14.2

In a thermal power plant, steam enters a turbine at 500 °C, and the condenser is maintained at 0.1 bar. It is required that the quality of steam at the turbine exit should be at least 0.9. Determine the pressure at which steam should be supplied to the turbine.

Problem 14.3

In a thermal power plant operating on a Rankine cycle, superheated steam is produced at 3 MPa and 300 °C and feeds to a turbine where it expands to the condenser pressure of 5 kPa. The saturated liquid coming out of the condenser is fed to a pump, which the isentropic efficiency of 0.80 has been achieved. Calculate the thermal efficiency of the power plant if the isentropic efficiency of the turbine is 0.85. Determine the rate of steam production if the power output of the plant is 1 MW. Also, calculate the efficiency of the corresponding ideal Rankine cycle.

Problem 14.4

In a power plant, employing a Rankin cycle with reheat modification, the steam enters the turbine at 3 MPa and 500 °C. After expansion to 0.6 MPa, the steam is reheated to 500 °C and expanded in a second turbine to a condenser pressure of 5 kPa. The steam leaves the condenser as saturated liquid. Calculate the thermal efficiency of the plant if the isentropic efficiency of the pump is 0.6 and the isentropic efficiency of the turbine is 0.8. Use Fig. 14.1 below (Fig. 14.31).

Fig. 14.31

Sketch of Problem 14.2

Problem 14.5

Determine the efficiency of a steam power plant operating on a Carnot cycle with isothermal energy addition as heat at 3 MPa and an isothermal energy rejection as heat at 5 kPa. At the beginning of the isothermal energy addition, the fluid is a saturated liquid and at the end of the isothermal energy addition, it is in a saturated vapor state. Use Fig. 14.2 below (Fig. 14.32).

Fig. 14.32

Sketch of Problem 14.5. Carnot cycle on a T-s diagram

Problem 14.6

In the test of a turbine in a thermal power plant, the following data has been recorded. Superheated steam at 3 MPa and 300 °C enters the turbine. The steam leaves the turbine at 5 kPa with a moisture content of 0.15. Determine the isentropic efficiency of the turbine. Use Fig. 14.3 below (Fig. 14.33).

Fig. 14.33

Sketch of Problem 14.6. Rankin cycle on a T-s diagram

Problem 14.7

A gas turbine unit has a pressure ratio of 10/1 and a maximum cycle temperature of 700 °C. The isentropic efficiencies of the compressor and turbine are 0.82 and 0.85, respectively. Calculate the power output of an electric generator geared to the turbine when the air enters the compressor at 15 °C at the rate of 15 kg/s. Take C_P = 1.005 kJ/kg K and γ = 1.4 for the compression process, and take C_P = 1.11 kJ/kg Kand γ = 1.333 for the expansion process (Fig. 14.34).

Fig. 14.34

Sketch of Problem 14.7

Problem 14.8

Calculate cycle efficiency and the work ratio of the plant in Problem 14.7, assuming that C_P for the combustion process is 1.11 kJ/kg K.

Problem 14.9

A gas turbine has an overall pressure ratio of 5 and a maximum cycle temperature of 550 °C. The turbine drives the compressor and an electric generator, the mechanical efficiency of the drive being 97%. The ambient temperature is 20 °C, and air enters the compressor at a rate of 15 kg/s; the isentropic efficiencies of the compressor and turbine are 80% and 83%. Neglecting changes in kinetic energy, the mass flow rate of fuel, and all pressure losses, using Fig. 14.5 below, calculate:

• (i) The power output

• (ii) The cycle efficiency

• (iii) The work ratio (Fig. 14.35)

Fig. 14.35

Sketch of Problem 14.9

Problem 14.10

In a marine gas turbine unit, a high-pressure (HP) stage turbine drives the compressor, and a low-pressure (LP) stage turbine drives the propeller through suitable gearing. The overall pressure ratio is 4/1, the mass flow rate is 60 kg/s, the maximum temperature is 650 °C, and the air intake, turbine, and LP turbine are 0.8, 0.83, and 0.85, respectively, and the mechanical efficiency of both shafts is 98%. Neglecting kinetic energy changes, and the pressure loss in combustion, using Fig. 14.6, calculate:

• (i) The pressure between turbine stages

• (ii) The cycle efficiency

• (iii) The shaft power (Fig. 14.36)

Fig. 14.36

Sketch of Problem 14.10

Problem 14.11

In a gas turbine generating set, two stages of compression are used with an intercooler between stages. The high-pressure (HP) turbine drives the HP compressor, and the low-pressure (LP) turbine drives the LP compressor and the generator. The exhaust from the LP turbine passes through a heat exchanger, which transfers heat to the air leaving the HP compressor. There is a reheat combustion chamber between turbine stages which raises the gas temperature to 600 °C, which is also the gas temperature at entry to the HP turbine. The overall pressure ratio is 10/1, each compressor having the same pressure ratio, and the air temperature at entry to the unit is 20 °C. The heat exchanger thermal ratio may be taken as 0.7, and intercooling is complete between compressor stages. Assume isentropic efficiencies of 0.8 for both compressor stages and 0.85 for both turbine stages and that 2% of the work of each turbine is used in overcoming friction. Neglecting all losses in pressure, and assuming that velocity changes are negligibly small, using Fig. 14.7, calculate:

• (i) The power output in kilowatts for a mass flow of 115 kg/s

• (ii) The overall cycle efficiency of the plant (Fig. 14.37)

Fig. 14.37

Sketch of Problem 14.11

Problem 14.12

For the unit of Problem 14.11, calculate the cycle efficiency obtainable when a heat exchange is fitted. Assume a thermal ratio of 0.75.

Problem 14.13

A motorcar gas turbine unit has two centrifugal compressors in series giving an overall pressure ratio of 6/1. The air leaving the high-pressure (HP) compressor passes through a heat exchanger before entering the combustion chamber. The expansion is in two turbine stages, the first stage driving the compressors and the second stage driving the car through gearing. The gases leaving the low-pressure (LP) turbine pass through the heat exchanger before exhausting to atmosphere. The HP turbine inlet temperature is 800 °C, and the air inlet temperature to the unit is 15 °C. The isentropic efficiency of the compression is 0.8, and that of each turbine is 0.85; the mechanical efficiency of each shaft is 98%. The heat exchanger thermal ratio may be assumed to be 0.65. Neglecting pressure losses and changes in kinetic energy, using Fig. 14.8, calculate:

• (i) The overall cycle efficiency

• (ii) The power developed when the air mass flow is 0.7 kg/s

• (iii) The specific fuel consumption when the calorific value of the fuel used is 42,600 kJ/kg and the combustion efficiency is 97% (Fig. 14.38)

Fig. 14.38

Sketch of Problem 14.12

Problem 14.14

In a gas turbine generating station, the overall compression ratio is 12/1, performed in three stags with pressure ratios of 2.5/1, 24/1, and 2/1, respectively. The air inlet temperature to the plant is 25 °C, and intercooling between stages reduces the temperature to 40 °C. The high-pressure (HP) drives the HP and intermediate-pressure compressor stages; the LP turbine drives the LP compressor and the generator. The gases leaving the LP turbine are passed through a heat exchanger, which heats the air leaving the HP compressor. The temperature at inlet to the HP turbine is 650 °C, and reheating between turbine stages raises the temperature to 650 °C. The gases leave the heat exchanger at a temperature of 200 °C. The isentropic efficiency of each compressor stage is 0.83, and the isentropic efficiencies of the HP and LP turbines are 0.85 and 0.88, respectively. Take the mechanical efficiency of each shaft as 98%. The air mass flow is 140 kg/s. Neglecting pressure losses and changes in kinetic energy, and taking the specific heat of water as 4.19 kJ/kg K, using Fig. 14.9 below, calculate:

• (i) The power output in kilowatts

• (ii) The cycle efficiency

• (iii) The flow of cooling water required for the intercoolers when the rise in water temperature must not exceed 30 K

• (iv) The heat exchanger thermal ratio (Fig. 14.39)

Fig. 14.39

Sketch of Problem 14.14

Problem 14.15

A simple ideal Brayton cycle with air as the working fluids has a pressure ratio of 11. The air enters the compressor at 300 °K and the turbine at 1200 °K. Accounting for the variation of the specific heats with temperature, determine (a) the air temperature at the compressor and turbine exits, (b) the back-work ratio, and (c) the thermal efficiency. Assume that:

• (i) Steady operating conditions exit.

• (ii) The air-standard assumptions are applicable.

• (iii) Kinetic and potential energy changes are negligible.

• (iv) The variation of specific heat with temperature is to be considered (Fig. 14.40).

Fig. 14.40

Sketch of Problem 14.15

Problem 14.16

Consider an ideal gas turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each stage of the compressor and turbine is 3. The air enters each stage of the compressor at 300 °K and each stage of the turbine at 1200 °K. Determine the back-work ratio and the thermal efficiency of the cycle, assuming (a) no regenerator is used and (b) a regenerator with 75% effectiveness is used. Use constant specific heats at room temperature. Assume that:

• (i) Steady operating conditions exit.

• (ii) The air-standard assumptions are applicable.

• (iii) Kinetic and potential energy changes are negligible.

• (iv) The variation of specific heat with temperature is to be considered (Fig. 14.41).

Fig. 14.41

Sketch of Problem 14.16

Problem 14.17

A simple ideal Brayton cycle with air as the working fluid has a pressure ratio of 11. The air enters the compressor at 300 K and the turbine at 1200 K. Accounting for the variation of the specific heats with temperature, determine:

• (a) The air temperature at the compressor and turbine exits

• (b) The back-work ratio

• (c) The thermal efficiency

Use the two following drawings (Fig. 14.42) for a simple ideal Brayton cycle.

Fig. 14.42

(a) A gas turbine engine with recuperator, (b) T-s diagram of a Brayton cycle with regeneration

Problem 14.18

Consider an ideal gas turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each stage of the compressor and turbine is 3. The air enters each stage of the compressor at 300 K and each stage of the turbine at 1200 K. Use Fig. 14.43 to determine the back-work ratio and the thermal efficiency of the cycle, assuming:

• (a) No regenerator is used.

• (b) A regenerator with 75% effectiveness is used. Use constant specific heats at room temperature.

  Fig. 14.43

  

  (a) A gas turbine engine with two-stage compression with intercooling, two-stage expansion with reheating and regeneration. (b) T-s diagram of an ideal gas turbine with intercooling reheating and regeneration

For this problem, use the same data that is given in Problem 14.17 as follows:

$$ \left|\begin{array}{l}\begin{array}{l}{T}_1=300\;\mathrm{K}\\ {}{h}_1=300.19\;\mathrm{kJ}/\mathrm{g}\\ {}{\Pr}_1=1.386\end{array}\\ {}\\ {}\begin{array}{l}{T}_6=1200\;\mathrm{K}\\ {}{h}_6=1277.79\;\mathrm{kJ}/\mathrm{g}\\ {}{\Pr}_6=238\end{array}\end{array}\right. $$