SOME CONSIDERATIONS RELATING TO GAS TURBINE PERFORMANCE

I

In order to understand, at least qualitatively, why a particular configura­tion of a gas turbine engine performs as it does, let us consider a few basic principles. The ideal thermodynamic cycle for the gas turbine engine is shown in Figure 6.29a, where it is compared to the cycle for the piston engine. The Otto cycle, which approximates the piston engine thermodynamics, consists of an, isentropic compression of the gas followed by a rapid combustion at

Figure 6.29a A comparison between the Otto and Brayton cycles.

nearly constant volume. The gas then expands isentropically, forcing the piston ahead of it. Unlike the piston engine, the gas turbine engine involves a continuous flow of the working gas. The Brayton or constant pressure cycle, which approximates the actual gas turbine cycle, begins with an isentropic compression of the air from ambient conditions. Part of this compression occurs prior to the compressor stages as the air enters the engine inlet. Following the compression, burning occurs at constant pressure, resulting in increased volume and total temperature. The air then expands isentropically through the turbines and jet nozzle to the ambient static pressure. In a turboprop or turboshaft engine, nearly all of the expansion occurs within the turbines in order to drive the compressor and produce shaft power. In a turbojet engine, an appreciable amount of expansion occurs after the turbines in order to produce the high-momentum jet.

The heat that is added to the flow per unit weight of gas is given by

<?i„=Cp(T3-T2) (6.66)

while the heat rejected is

<?ои,= Ср(Г4-Г1) (6.67)

Cp is the specific heat at constant pressure, as used previously in Chapter Five. The work output per unit weight of gas equals the added heat minus that which is rejected. The thermal efficiency equals the work output divided by the added heat. Thus,

Г4-Т,

T3-T2

Let r denote the compression ratio, p2lpі (or p3/p4). Since compression and expansion are both assumed to be isentropic,

Tl = Tl= ,(•>■—і )h

T T4

Thus, Equation 6.68, in terms of the compression ratio, can be written as

As stated previously, the compression ratio r is achieved partly in the inlet (ram pressure), and the remainder is achieved through the compressor. The pressure increase across the compressor, at a constant rpm, as a first approximation, is proportional to the mass density, pc, just ahead of the compressor.

Ap oc p<

If r0 denotes the value of r for static sea level operation,

P =Po(r0- 1) —

p0 is, of course, the standard sea level value of mass density.

For isentropic compression in the inlet up to the compressor, the ambient mass density and pc are related by

Pc. Г1 + (y — 1 l2)MjyKy~n

p® L 1 + (y – 1/2)M2 J

where М» is the free-stream Mach number and M is the local Mach number just ahead of the compressor. The pressure ratio, r, thus becomes

r^pc + Ap

P cc

or

pc/p® is given by

Thus r finally becomes

r = [/(M, М„)]’йт-,) + Г-~г^- [/(M, Moo)]I/(l,_,)

V

where

1 + (y — H2)Mj 1 + (y – 1/2)M2

Equation 6.75 is substituted into Equation 6.70 an expression for the thermal efficiency results that is a function of 5, r0, and M.

The effect of pressure ratio, altitude, and free-stream Mach number on the ideal thermal efficiency is shown in Figure 6.29b. This figure assumes the ratio of M to M„ just before the compressor to equal approximately zero. This is a fairly reasonable assumption, since values of this ratio up to at least 0.4 affect t) by less than 1%. Figure 6.29b shows the effect of varying one parameter at a time while keeping the other two parameters at their normal values. Increasing from zero to 0.8 is seen to result in a 7% improvement in 7j. The efficiency also improves with altitude, increasing by approximately 6% in going from sea level to 40,000 ft (12,200 m). Doubling the pressure ratio, r0, from 10 to 20 results in a 16% improvement in tj.

With regard to the production of thrust, r/ does not tell the whole story, tj is simply a measure of how efficiently the air passing through the engine is

being used. The heat added to the flow, per unit weight, is given by

Qm = vCp(T3 – T2)

Thus, for the same efficiency, if T3 is increased or the mass flow increased, the thrust will be increased.