Mattingly J.D., Heiser W.H., Pratt D.T. Aircraft Engine Design

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520 AIRCRAFT ENGINE DESIGN

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APPENDIX C: GAS TURBINE ENGINE DATA 521

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Appendix D

Engine Performance:

Theta Break and Throttle Ratio

Aircraft engine designers have recently begun to capitalize on a fundamental

cycle design choice most frequently referred to as either the theta break or throttle

ratio in order to precisely tailor their machines for the expected range of flight

conditions. In the development that follows, you will see that the theta break and

throttle ratio are properties of the engine control system, so that this exploration

takes us beyond the traditional boundaries of aerothermodynamics. This excursion

has the added advantage of demonstrating how aircraft engines behave away from

their reference or design conditions.

Although this development is based largely on a simplified model of the basic

aircraft engine, namely, the single-shaft, uncooled turbojet, experience shows that

the results apply equally well to all families of turbine engines. A special benefit

of this development is the derivation of straightforward, transparent, algebraic

equations that allow the basic workings of turbine engines to be understood

and regulated. This presentation is based on material and nomenclature found in

Chapters 4 and 5 of this textbook and draws heavily upon the foundational material

found in Refs. 1 and 2.

Dimensionless Freestream Total Temperature

Because the freestream total temperature Tt0 will be seen to exert a strong

influence on the internal and overall behavior of the turbojet engine, it is useful to

define a practical, dimensionless form. The dimensionless ratio of the freestream

total temperature to the sea level static temperature of the standard atmosphere is

called theta 0 and given the symbol 00. It is equal to the corrected freestream total

temperature [cf. Eq. (5.22)] and is written as [cf. Eq. (1.1)or (4.5a-CPG)]

Tto

T0(l+ Yc - 1M°2)2

0 0 ~" -- = 0"C r

(D.1)

Tsta Tstd

Figure D. 1 displays contours of constant 0o for the standard atmosphere of Ref. 3

and a wide range of flight conditions. The most important message of Fig. D. 1 is

that 00 can be used as a surrogate to condense various combinations of altitude and

Mach number into a single parameter. That is, every point in the flight envelope

possesses a specific value of 00, although the converse is not true. We will therefore

refer hereinafter only to 0o rather than the associated flight conditions.

To increase your fluency with theta 0, it may be helpful to consider that for the

standard atmosphere 1) 00 = 1 at sea level static conditions; 2) 00 can be greater

or less than 1; 3) 00 depends only on Mach number above 37 kft, the start of the

523

524 AIRCRAFT ENGINE DESIGN

Alt

(lift)

00 = 0.8 0.9 1.0 1.1 1.2 1.3 1.4

40 1

/ /

30 / 1 t

/ //

/ /

o /

/, .... , ....

0.0 0.5 1.0 1.5 2.0 2.5

Fig. D.1 Theta 0 as a function of flight altitude and Mach number for the standard

atmosphere of Ref. 3 with -y = 1.4.

constant static temperature tropopause; and 4) the entire range of 00 for modem

aircraft operations is only about 0.8 < 00 < 1.4. Particular attention should be

paid to the fact that the definition of theta 0 applies equally well to nonstandard

atmospheres (e.g., Refs. 4 and 5), even though the standard atmosphere is generally

found in textbooks because it is convenient, familiar, and relevant.

Single-Shaft Uncooled Turbojet Compressor Behavior

The behavior of a given single-shaft, uncooled turbojet compressor, some-

times referred to as its off-design behavior, can be evaluated as follows. To begin,

Eq. (4.21 a) for the power balance between the compressor and turbine is simplified

and solved for the compressor total temperature ratio to yield

1 cptTt4

(D.2)

"r c -- 1 = r/m(l -- fl)(l + f)(1 --

rt)Tstd

CpcO0

Equation (D.2) reveals that compressor total temperature ratio depends only

upon turbine total temperature ratio rt, throttle setting

Tt4,

and flight conditions 00.

The other quantities in Eq. (D.2) are constant. The turbine total temperature ratio

is now examined more closely.

APPENDIX D: ENGINE PERFORMANCE 525

Employing the logic and assumptions of Sec. 5.2.4, except that the choking

downstream of the single shaft turbine occurs in the fixed area exhaust nozzle

throat rather than the entry of the low-pressure turbine, the same two conclusions

are reached, namely, the turbine total temperature ratio rt and turbine total pressure

ratio zrt are constant. This result is valid for all important throttle settings, although

it may fail when choking no longer occurs at very low throttle settings.

Consequently, Eq. (D.2) now reveals that re varies only with the ratio of Tta to

00. Moreover, the compressor total pressure ratio zr~, derived from the efficiency

relationships of Eq. (4.9c-CPG), also varies only with the same ratio, to wit

yc I I ~

Tt4 ~ ~

r~ = [1 + r/c(v~ - 1)1 ~T-~ = ~ 1 + C'l-z--I (D.3)

Uo/

where

1 Cpt

(D.4)

C1 = OcOm(1 -fl)(1

+ f)(1 -

rt) Tstd Cp c

It is imperative that the reader pauses to absorb the significance of these

results. First, the compressor responds primarily to the throttle setting and the

flight condition. The causal chain of events is evidently remarkably short. Second,

r~ and Zrc increase as the ratio

Tt4/O0

increases. For example, Zrc will increase when

flight Mach number and Tt4 are fixed and the aircraft climbs to higher altitudes,

and decrease when flight altitude and

Tt4

are fixed and the aircraft is accelerated

to higher Math numbers.

Turbomachinery Limits: The Role of the Control System

Control systems must be designed to prevent aircraft engines from destroying

themselves (see Appendix O). Two of the most stringent turbomachinery limita-

tions are the maximum allowable values ofrrc and Tt4. The control system situation

can be easily grasped with the aid of Fig. D.2, which graphically depicts Eq. (D.3)

for a typical, example compressor.

Figure D.2 makes several things clear. First, if00 is decreased from a large initial

value while

Tt4

is fixed at

Tt4max,

then

7rcmax

is reached when 00 = 1.1. As 00 is

further decreased,

Tt4

must then be reduced in order to protect the compressor from

stalling. In fact, according to Eq. (D.3), the ratio of

Tt4/Oo

must remain constant

at 3300/1.1 = 3000 in order to maintain zr¢ constant at ~cmax. Second, if 00 is

increased from a small initial value while holding

Tt4/Oo

constant at 3000, then

Zrc will remain fixed at Zrcmax

as Tt4

increases until, of course,

Ztnma x

is reached

at 00 = 1.1. As 0o further increases, Zrcmax cannot any longer be maintained, and

rrc gradually decreases. Third, and of

paramount

importance, is the fact that Jrc

and Tt4

simultaneously reach their upper limits

only

at 00 = 1.1. Finally, Fig. D.2

shows how 00 and

Tt4

may be chosen to reach lower values of rrc.

The Theta Break

The unique point, so visually striking in Fig. D.2, at which the control logic

must switch from limiting Zrc to limiting

Tt4,

is known as the theta break, or 00

brea~.

Returning briefly to Fig. D. 1, you will find it very convenient to visualize that at any

526 AIRCRAFT ENGINE DESIGN

,,l, I I .... I''''1''

~c max

Theta

Break

rt4 max

,,~1

I I

10 i i i i i i i i i , i i i i i i

0.8 0.9 1.0 1.1 1.2 1.3

Fig. D.2 Compressor pressure ratio as a function 00 of and

Tt4

for a compressor with

a reference point of 7% = 7% max = 20, Tt4

= Tt4 max ~"

3300 o R (1833 K), and 00 = 1.1 [i.e.,

C1 = 0.0004512 1/°R (0.0008122 l/K) in Eq. (D.3)].

point in the flight envelope to the left of the theta break :rc = ~rc

max

and

Zt4 < Tt4

max,

while at any point to the right of the theta break ~rc < ~r~ max and

Tt4 = Tt4 max.

The

relationship of the instantaneous value of 00 to 00 break has important consequences

to engine cycle performance. On the one hand, when 00 <

Oobreak and

Tt4 < Ttamax,

the specific thrust of the engine is less than its inherent material capabilities would

make possible. On the other hand, when 0o

OObreak

and nc < ~cmax, the specific

fuel consumption is more than its inherent thermal efficiency would make possible.

The designer would therefore strongly prefer to have the engine always operate

at or very near 00 =

Oobreak,

but this is impossible because every aircraft has a

flight envelope with a range of 00 (see Fig. D. 1). The best available compromise

is to chose

OObreak

that provides the best balance of engine performance over the

expected range of flight conditions.

It is interesting to note that because early commercial and military aircraft

primarily flew at or near 00 = 1 they were successfully designed with

OObreak =

Consequently, several generations of propulsion engineers took it for granted that

aircraft engines always operated at Jr~ max and

Tt4

max

under standard sea level static

conditions. However, the special requirements of more recent aircraft such as the

AAF of this textbook (00 = 1.151 at supercruise) have forced designers to select

theta breaks different from one. These engines may operate either at Jr~max or

Tt4max

at standard sea level static conditions, but never both.

APPENDIX D: ENGINE PERFORMANCE 527

The Throttle Ratio

We now determine how the designer can set the theta break for the engine.

Assuming for the moment that

OObreak

> 1, which is the usual case, and returning

to our earlier conclusion regarding the constancy of the ratio

Tt4/Oo,

it follows

immediately that, for the standard atmosphere

Ttamax Tt4

-- --

Tt4SLS

(D.5)

OObreak O0

because

OOSLS

= 1, and that the throttle ratio, or

TR,

is therefore given by

Tt4 max

TR -- -- -- OObreak

(D.6)

Tt4 SLS

The result is surprisingly simple. The engine must merely be designed to have

the

Tt4s~

given by Eq. (D.5) at standard sea level static conditions and must have

a control system that limits zrc to 7rc max and

Zt4

to Ttnma x.

Everything else follows

directly. The terms throttle ratio and theta break are used interchangeably in the

propulsion industry. This should present no problems because, as Eq. (D.6) shows,

they are identical.

It is interesting and useful to determine the sea level flight Mach number at which

the theta break is reached. Returning to Eq. (D.1) and recognizing that 0 = 1 at

standard sea level conditions, it follows immediately that

MObreak ]------"~(OObreak-

1) (D.7)

=VYc- 1

Equation (D.7) offers the option of selecting a design or reference point at

M0break and standard sea level static conditions with zr~ = ~cmax and

rt4

= rt4raax.

For example, if it is desired to have

00break ~-

1.1, then Eq. (D.7) shows that

MObreak

= 0.707.

It is satisfying to find that the results of AEDsys performance calculations always

precisely and simultaneously obey both Eqs. (D.6) and (D.7).

In the rare case that 00 < 1, a different concept is needed because Tt4 sLs =

Tt4

max

and ~ < rrcmax. One approach would be to use Eq. (D.3) to determine the value

of ~ that exists at standard sea level static conditions. Figure D.1 shows that there

is no relevant

MObre~k

for this case. Another approach would be to specify the

standard day altitude (h) for M0 = 0, as obtained from Eq. (D.1), as the altitude at

which

0 =OObreak

(D.8)

We could, of course, refer to this as the

hbreak.

The Compressor or Engine Operating Line

Another pleasant discovery is that it is now possible to define and construct the

required compressor or engine operating line with little further ado (cf. Secs. 5.3.2

and 7.4.3). It is often surprising, but nevertheless true, that this critical component

characteristic can be determined from first principles. Applying conservation of

528 AIRCRAFT ENGINE DESIGN

mass to the compressor and turbine, and using the mass flow parameter of Sec. 1.9.3

and the corrected flow quantity definitions of Sec. 5.3.1, we find that

7rc zrbPstdA4MFP(M4) O0

= C2Y( c

(D.9)

rhc2 = (1 -/5)(1 + f)

where M4 and MFP(M4) are constant because the turbine entry is presumed to be

choked, and

~b PstdA4MFP(M4)

C2 = (D. 10)

(1 - fl)(1 + f)

Figure D.3 corresponds to the example compressor of Fig. D.2 with a corrected

mass flow rate of 100 lbm/s at the theta break. The straight lines portray Eq. (D.9)

for several constant values of the ratio Tt4/Oo. It should be noted that, because :re

cannot be less than one (and M4 drops as the turbine entry unchokes at very low

power), the straight lines do not continue to the origin.

ffc

'''I'''I'''I' 'I' '

Operating Line

T,4 / 0o

3000 °R

2500 °R

2000 °R

1500 °R

0 20 40 60 80 100 120

mc2 (Ibm/s)

Fig. D.3 The required operating line for a compressor with a reference point of

71" c = 71"cmax = 20, rt4 -- Tt4ma x =

3300°R (1833 K),

OObreak --

1.1, and the2 = 100 lbm/s

(45.36 kg/s) [i.e., Ct = 0.0004512 1/°R (0.0008122 l/K) and C2 = 273.9 Ibm x/T--R/s

(92.60 kg v/-K/s) in Eq. (D.11)].

APPENDIX D: ENGINE PERFORMANCE 529

Equation (D.9) may now be combined with Eq. (D.3) to yield the required

compressor operating line, which also appears on Fig. D.3:

rhc2 = Cazrc

(D.11)

zr (J c- l~-/-r" - 1

Because the operating line connects the dots on a series of rays radiating from

the origin, it always has the characteristic shape found in Fig. D.3, which is quite

similar to that of Figs. 5.5 and 7.El 1.

The nature of the predicted and observed compressor operating line is obvi-

ously dictated by the choking of the fixed A4 turbine inlet guide vane. We are

therefore frequently asked at this point whether superior turbine engines could be

designed if the choking assumption were revised. Indeed, a great deal of attention

(Refs. 1 and 6) has been given to experimental turbines having a variable A4 for cy-

cle purposes. This work is stimulating and worth investigating. However, because

the vast preponderance of aircraft engines are deliberately designed to operate with

choked, fixed A4 turbine guide vanes, it is the appropriate model for this textbook.

Uncooled Nonafterburning Single-Shaft Turbojet Performance:

break

This line of attack is now carried to a higher level, namely, the evaluation of the

overall performance of the entire engine. You will see that this has many benefits,

including added insight into the behavior of engines and strong analytical support

for the performance correlations of Secs. 2.3.2 and 3.3.2. The inspiration for this

work is Sec. 7.3 of Ref. 1, and several intermediate steps are given next to guide

the reader. Once again, even though the analysis strictly pertains only to a narrow

class of engines, experience shows that it is broadly applicable. To the best of our

knowledge, this is the first time that the complete algebraic analysis of overall

engine performance

with control limits imposed

has been presented.

In this first case, the engine is operating at maximum thrust with 00

OObreak,

so that Tt4/00, rc, rCc, rt, and zrt are constant. Six quite reasonable assumptions

are made that retain the underlying physics and provide adequate accuracy while

greatly simplifying the analysis. They are that f and/3 are negligible compared

with one, that Zrd is constant, that t/m = 1, that the engine is always perfectly

expanded (i.e., P9 = P0), and that in Eq. (4.18)

cptTt4

is negligible compared with

hp R (i.e., the energy density of the fuel greatly exceeds that of the combustion

gases).

Following Ref. 1 closely, the uninstalled specific thrust of the engine is given

by the expression

F--a°( Mo V9 )

tho gc Voo - Mo

(D.12)

where

MOVoo = 1- r rd c rb , r.

(D.13)