5: The Brayton Cycle

The operation of ramjet and gas turbine ("jet") engines can be expressed, in its most basic form, as a "Brayton Cycle" or "Gas Turbine Cycle" . There are four steps involved:

Step	Thermodynamic Process
1. Compress the working fluid (air)	Adiabatic Compression
2. Add heat to the fluid	Constant Pressure Heat Addition
3. Extract work from the fluid by allowing it to expand	Adiabatic Expansion
4. Cool the fluid	Constant Pressure Heat Extraction

Note: Step 4 occurs in the atmosphere after the engine has passed. We don't worry about it because we don't need to re-use the same air, or pay to cool it.

Examples

Brayton Cycle Analysis

Net work output = work done by the system in Step (3) minus the work put into the system in Step (1).

Net heat input = heat put in during Sep (2), assuming that we don't pay to cool the air, and cannot recover the heat that is lost in the exhaust gases.

Ideal Cycle Efficiency = (Net work output) / (Net heat input)

Net work output per unit mass flow rate =

Net heat input =

Efficiency

Dividing throughout by specific heat ,

Now, the process from (A) to (B) is isentropic, as is (C) to (D). Also, pressure is constant from B to C, and from D to A. Thus,

and

Thus,

These are extremely useful results. Note:

1. As the engine "overall pressure ratio" increases, efficiency increases towards 1. This means that to get high efficiency, we must use a high pressure ratio.

2. efficiency = (heat added - heat wasted) / (heat added)

Thus, to increase efficiency,

a). Increase Tc, the highest temperature in the engine

b). Bring T_D down as close to T_A as possible: expand through the nozzle, and extract the maximum work possible.

c). Reduce T_B: take out heat from the compressor using heat transfer to the fuel (this is considered in supersonic-combustion engines which use cryogenic fuel).

3. If T_D <T_A, you get efficiency > 1: a perpetual motion machine. Thus, we see that in reality, the exhaust temperature cannot be lower than the ambient temperature.

EFFICIENCY

Efficiency is an expression of the degree of perfection.

Efficiency is also the ratio of the Useful Output to the Input.

Thermal Efficiency is the ratio of the rate of addition of kinetic energy to the working fluid to the rate of addition of heat in the form of heat of reaction of the fuel.

For a single-flow jet engine (only one exhaust),

where qR is the heat released per unit mass of fuel consumed. For a shaft-powered engine,

where is the shaft power.

Propulsive Efficiency is the ratio of thrust power to the rate of addition of kinetic energy.

If we ignore the pressure thrust term, and assume that f is very much smaller than 1,

Overall efficiency is the product of these two efficiencies.

Note:

1. Thermal efficiency depends on

a) thermodynamic cycle

b) combustion efficiency (how much of the heat is really released?)

2. Propulsive efficiency depends on how close the exit velocity is to the flight velocity. We can easily show that this efficiency reaches 1.0 when the two velocities are equal, but then the thrust is zero. To get a given thrust with high propulsive efficiency, it is thus better to accelerate a large mass flow rate through a small velocity increment, than to accelerate a small mass flow rate through a large velocity increment.