Jet Propulsion/Thermodynamic Cycles

Gas turbines are based on the Brayton cycle.

All jet engines and gas turbines are heat engines that convert thermal energy into useful work. The useful work may be in the form of mechanical power, as from a shaft which may be used to drive a propeller, a vehicle, a pump, an electric generator, or any other mechanical device. In Jet engine applications the work is in producing compressed air and combustion products which are then accelerated to provide reaction propulsion.

Efficiency

Table 2.1: Efficiency definitions
Name	Definition	Formula	Symbol	Typical Range
Thermal efficiency	input thermal energy vs. output work.		ηTh	5%-50%
Propulsive efficiency	work transmitted to vehicle vs. total engine output work; propulsive work delivered to the total mechanical energy produced by the engine.		ηPr	5-40%
Combustion efficiency	input chemical energy vs. output thermal energy		ηCo	90-99%

Thermodynamic cycles

Brayton Cycle

The Brayton cycle is the fundamental constant pressure gas heating cycle used by gas turbines. It consists of

0-2: isentropic compression

2-3: constant pressure heating

3-4: isentropic expansion

4-0: constant pressure cooling (absent in open cycle gas turbines)

Ramjet cycle

A ramjet uses the open Brayton cycle. In the diagram below a 2D supersonic intake is shown, downstream of which is a divergent subsonic diffuser. Fuel is then injected into the compressed air and evaporates producing a mixture that is ignited when it reaches the flame front. The flameholders provide the turbulent circulation necessary to stabilize the flame, since deflagration velocities are usually much smaller (<10m/s) than the average velocity of air in the combustor. The combustion products are then exhausted through the nozzle.

To understand how thrust is produced if we assume that the flow of fuel is negligible compared to the air mass then the exhaust flow will be at approximately the same Mach number as the input flow. However the total temperature of the exhaust is much higher and the exit velocity will be correspondingly higher than the input velocity. This difference in velocity (and momentum) produces thrust.

The temperature rise in the intake-diffuser is related to the freestream Mach number $M_{0}$:

${\frac {T_{2}}{T_{0}}}=1+{\frac {\gamma -1}{2}}M_{0}^{2}$

Maximum efficiency is reached if temperature rise in combustor is small.

$\eta ={\frac {[(\gamma -1)/2]M_{0}^{2}}{1+[(\gamma -1)/2]M_{0}^{2}}}$

where $\gamma =c_{p}/c_{v}\approx 1.4$ is the ratio of specific heats of air.

Ramjets are inefficient at subsonic speeds and their efficiency improves at supersonic speeds.

At hypersonic speeds the compression and dissociation processes make full diffusion unattractive and supersonic combustion is being researched. A Scramjet slows the air down to low supersonic speeds and then burns high flame velocity fuels such as hydrogen or methane.

Turbojet cycle

Adding a compressor to a ramjet powered by a turbine in the exhaust allows increased combustor inlet temperature, and a consequent increase in possible thermal efficiency. The turbine however is limited in the temperature it can handle, so maximum power is also limited.

In the T-S diagram below the presence of the compressor allows us to raise the combustor inlet temperature (3). The raising of the combustor segment increases the cycle area and the thermal efficiency.

Addition of an afterburner (5-6) allows thrust augmentation as can be seen from the increased area of the diagram shown below. The afterburner operates in the higher entropy range and has lower efficiency than the base turbojet.

Turbofan cycle

A turbofan diverts some of the pressure energy of the core flow to power a fan which moves a larger mass flow, providing an increase in thrust and propulsive efficiency.

Turbofans normally have two or three shafts. Since the diameter of the fan is larger the same tip speed can be achieved at a lower rpm than the smaller diameter compressor and two shafts become necessary. The alternate method is to employ a gearbox to step down the shaft speed which is used in some smaller turbofans. In most turbofans however a multistage LP turbine is used to extract the same energy with smaller stage loadings and lower tangential velocity. The smaller diameter HP compressor is run with one or two turbine stages with higher tangential velocity than the LP turbine.

Cycle Losses

The compression incurs several loss mechanisms:

Shock:

Separation

Tip clearance

Seal clearance

Blockage

Whirling