Chapter 9

Compare the thermal efficiency of a two-stage gas turbine with regeneration, reheating and intercooling to that of a three-stage gas turbine with the same equipment when \((a)\) all components operate ideally, \((b)\) air enters the first compressor at \(100 \mathrm{kPa}\) and \(20^{\circ} \mathrm{C},(c)\) the total pressure ratio across all stages of compression is \(16,\) and \((d)\) the maximum cycle temperature is \(800^{\circ} \mathrm{C}\)

Electricity and process heat requirements of a manufacturing facility are to be met by a cogeneration plant consisting of a gas turbine and a heat exchanger for steam production. The plant operates on the simple Brayton cycle between the pressure limits of 100 and 1000 kPa with air as the working fluid. Air enters the compressor at \(20^{\circ} \mathrm{C}\). Combustion gases leave the turbine and enter the heat exchanger at \(450^{\circ} \mathrm{C},\) and leave the heat exchanger of \(325^{\circ} \mathrm{C},\) while the liquid water enters the heat exchanger at \(15^{\circ} \mathrm{C}\) and leaves at \(200^{\circ} \mathrm{C}\) as a saturated vapor. The net power produced by the gas-turbine cycle is \(1500 \mathrm{kW}\). Assuming a compressor isentropic efficiency of 86 percent and a turbine isentropic efficiency of 88 percent and using variable specific heats, determine \((a)\) the mass flow rate of air, \((b)\) the back work ratio and the thermal efficiency, and \((c)\) the rate at which steam is produced in the heat exchanger. Also determine \((d)\) the utilization efficiency of the cogeneration plant, defined as the ratio of the total energy utilized to the energy supplied to the plant.

A turbojet aircraft flies with a velocity of \(1100 \mathrm{km} / \mathrm{h}\) at an altitude where the air temperature and pressure are \(-35^{\circ} \mathrm{C}\) and \(40 \mathrm{kPa} .\) Air leaves the diffuser at \(50 \mathrm{kPa}\) with a velocity of \(15 \mathrm{m} / \mathrm{s}\), and combustion gases enter the turbine at \(450 \mathrm{kPa}\) and \(950^{\circ} \mathrm{C}\). The turbine produces \(800 \mathrm{kW}\) of power all of which is used to drive the compressor. Assuming an isentropic efficiency of 83 percent for the compressor, turbine, and nozzle, and using variable specific heats, determine ( \(a\) ) the pressure of combustion gases at the turbine exit, ( \(b\) ) the massflow rate of air through the compressor, \((c)\) the velocity of the gases at the nozzle exit, and \((d)\) the propulsive power and the propulsive efficiency for this engine.

An air standard cycle with constant specific heats is executed in a closed piston-cylinder system and is composed of the following three processes: \(1-2 \quad\) Constant volume heat addition \(2-3 \quad\) Isentropic expansion with an expansion ratio \(r=V_{3} / V_{2}\) \(3-1 \quad\) Constant pressure heat rejection (a) Sketch the \(P\) -v and \(T\) -s diagrams for this cycle (b) Obtain an expression for the back work ratio as a function of \(k\) and \(r\) (c) Obtain an expression for the cycle thermal efficiency as a function of \(k\) and \(r\) (d) Determine the value of the back work ratio and efficiency as \(r\) goes to unity What do your results imply about the net work done by the cycle?

Consider the ideal regenerative Brayton cycle. Determine the pressure ratio that maximizes the thermal efficiency of the cycle and compare this value with the pressure ratio that maximizes the cycle net work. For the same maximumto- minimum temperature ratios, explain why the pressure ratio for maximum efficiency is less than the pressure ratio for maximum work.

Using EES (or other) software, study the effect of variable specific heats on the thermal efficiency of the ideal Otto cycle using air as the working fluid. At the beginning of the compression process, air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\). Determine the percentage of error involved in using constant specific heat values at room temperature for the following combinations of compression ratios and maximum cycle temperatures: \(r=6,8,10,12\) and \(T_{\max }=1000,1500,2000,2500 \mathrm{K}\)

Using EES (or other) software, determine the effects of pressure ratio, maximum cycle temperature, and compressor and turbine efficiencies on the net work output per unit mass and the thermal efficiency of a simple Brayton cycle with air as the working fluid. Air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\) at the compressor inlet. Also, assume constant specific heats for air at room temperature. Determine the net work output and the thermal efficiency for all combinations of the following parameters, and draw conclusions from the results.

Using EES (or other) software, determine the effects of pressure ratio, maximum cycle temperature, regenerator effectiveness, and compressor and turbine efficiencies on the net work output per unit mass and on the thermal efficiency of a regenerative Brayton cycle with air as the working fluid. Air is at \(100 \mathrm{kPa}\) and \(300 \mathrm{K}\) at the compressor inlet. Also, assume constant specific heats for air at room temperature. Determine the net work output and the thermal efficiency for all combinations of the following parameters..

Using EES (or other) software, determine the effect of the number of compression and expansion stages on the thermal efficiency of an ideal regenerative Brayton cycle with multistage compression and expansion. Assume that the overall pressure ratio of the cycle is \(18,\) and the air enters each stage of the compressor at \(300 \mathrm{K}\) and each stage of the turbine at \(1200 \mathrm{K}\). Using constant specific heats for air at room temperature, determine the thermal efficiency of the cycle by varying the number of stages from 1 to 22 in increments of 3. Plot the thermal efficiency versus the number of stages. Compare your results to the efficiency of an Ericsson cycle operating between the same temperature limits.

An Otto cycle with air as the working fluid has a compression ratio of \(10.4 .\) Under cold-air-standard conditions, the thermal efficiency of this cycle is \((a) 10\) percent (b) 39 percent \((c) 61\) percent \((d) 79\) percent \((e) 82\) percent