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

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266 AIRCRAFT ENGINE DESIGN

A general form of the enthalpy radial distribution equation is obtained by com-

bining Eqs. (iii) and (iv), giving

dht 1 ( du dv ~)

-- u + v + (8.22)

dr gc -~r -d-;r

This equation prescribes the relationship between the radial variation of the three

variables:

ht, u,

and v. The designer may specify the radial variation of any two,

and Eq. (8.22) allows the radial variation of the third variable to be determined.

For constant work,

htis

constant with respect to radius, and Eq. (8.22) becomes

du dv v 2

u ~rr + v ~rr + --r = 0 (8.23)

In this case, if the radial variation of either u or v is prescribed, Eq. (8.23) allows

the other to be determined. The traditional approach is to specify the swift velocity

v, as follows.

2) Swirl distributions:

We assume the following general swift distribution at entry and exit to the rotor

(F~) n (F-~) n

vl= a -b rm

and v2= a +b

(8.24)

r r

where a and b are constants. From the Euler pump equation [Eq. (8.10)], the work

per unit mass flow is

ogr(v2 --

Vl)

2bogrm

Aht -- -- (8.25)

gc gc

which is independent of radius. The constant b in Eq. (8.24) is determined from

the enthalpy rise across the rotor using Eq. (8.25) and, as will be shown next, the

constant a in Eq. (8.24) is related to the degree of reaction at the mean radius

rm.

Three cases of the swirl distribution as considered next correspond to n = -1,

n =0, andn = 1.

Free vortex swirl distribution

(n = -1). Equation (8.24) becomes

vl = (a - b)

--rm

and

/9 2

(a + b)

--.rm

(8.26)

r r

Because v varies inversely with radius, this is known as "flee-vortex" flow. Thus,

if the stator airfoils preceding station 1 produce the flow

vlr = Vlmrm

and the

rotor airfoils modify the flow to

v2r = V2mrm

at station 2, then the Euler equation

confirms that this is a constant work. Furthermore, substitution of the free vortex

swirl distribution into Eq. (8.23) gives

--~0

which requires that the axial velocity u not vary with radius. Equation (8.26) also

shows that, as long as r does not vary substantially from

(say ± 10%), the airfoil

DESIGN: ROTATING TURBOMACHINERY 267

and flow properties will not vary much from the original mean-line design. Using

Eq. (8.8) for a repeating stage (v3 = Vl), the degree of reaction is given by

°Rc 1 a (~)2

= - (8.27)

o)r m

where the constant a in Eq. (8.26) is obtained by evaluating the preceding expres-

sion at the mean radius. Thus

= wrm

°Rcm )

(8.28)

where

°Rcm

is the degree of reaction at the mean radius.

Free vortex aerodynamics played a prominent role in the history of turbomachin-

ery. Before high-speed computation became commonplace, this approach enabled

designers to understand and cope with the most prominent features of radial vari-

ation (see Refs. 3, 7, and 10).

Exponential swirl distribution (n = 0). Equation (8.24) becomes

Vl = a - b rm and

1) 2 =

a + b rm (8.29)

r r

Substitution of the exponential swirl distribution into Eq. (8.23) and integration

gives (Ref. 11)

uZ = uZm _ 2 (a21n r ab )

+ - ab (8.30a)

u~ = uZm - 2 (a21n r ab )

r/r~ + ab (8.30b)

For the case where Ulm = Uzm, the degree of reaction for a repeating stage

(V3 = V1) is given by (see Ref. 11)

°Re = 1 + 1 - 2 (8.31)

o)r m

First-power swirl distribution (n = 1). Equation (8.24) becomes

r a r

vl=a---b rm and v2= +b rm (8.32)

rm r

Substitution of the exponential swirl distribution into Eq. (8.23) and integration

gives (Ref. 11)

}

U 2

= U2m

2 a 2 -k- ab In r _ a2 (8.33a)

}

U 2 = U2m

2 a 2 - ab In r _ a2 (8.33b)

268 AIRCRAFT ENGINE DESIGN

0.5

First power _

Exponential ~

-1

r/r m

1.5

Fig. 8.6 Radial variation of the degree of reaction, u

For the case where

Ulm = bl2m ,

the degree of reaction for a repeating stage

(V3 = V1) is given by (see Ref. 11)

a { (rf~) }

°Rc

= 1 + 2 In - 1 (8.34)

O)F m

Equations (8.27), (8.31), and (8.34) give the radial variation of the degree

of reaction for the free-vortex, exponential, and first power swirl distributions,

respectively, for repeating stages (1/3 = 1/1) with

Ulm = Uzm.

The value of the

constant a is evaluated at the mean radius and is given by Eq. (8.28) for all three

cases.

Consider the case where the degree of reaction at the mean radius is 0.5 (a =

COrm~2).

Results for this case from Eq. (8.27), (8.31), and (8.34) are plotted in

Fig. 8.6 for the range 0.5 <

r/rm

< 1.5. These results show that it is more

difficult to design rotor airfoils at r <

and stator airfoils at r >

rm.

In fact,

because

°Rc

----- 0 at

r/rm

= 0.707 for the free-vortex and at

r/rm

= 0.6065 for

the first-power swirl distributions, the rotor will actually experience accelerating

flow for smaller radii, whereas this is never the case for the stator. For these

reasons, modem compressor design has looked to non-free-vortex (nonconstant

work) machines, but these are absolutely dependent upon large computers for their

definition.

DESIGN: ROTATING TURBOMACHINERY 269

Table 8.1 Range of axial flow compressor design parameters

Parameter Design range

Fan or low-pressure compressor

A T, per stage

Pressure ratio for one stage

Pressure ratio for two stages

Pressure ratio for three stages

Inlet corrected mass flow rate

Maximum tip speed

Diffusion factor

High-pressure compressor

AT~ per stage

Inlet corrected mass flow rate

Hub/tip ratio at exit

Maximum rim speed at exit

Diffusion factor

Maximum exit temperature

60-100°F(35-55 K)

1.5-2.0

2.0-3.5

3.5-4.5

40-42 lbm/(s-ft 2)

[195-205 kg/(s-m2)]

1400-1500 ft/s

[427-457 m/s]

0.50-0.55

60-90°F (35-50 K)

36-38 lbm/(s-ft 2)

[175-185 kg/(s-m2)]

0.90-0.92

1300-1500 ft/s

[396-457 m/s]

0.50-0.55

1700-1800 ° R (945-1000 K)

The three swirl distributions have been incorporated into COMPR to allow you to

explore this method for improving the degree of reaction of your designs at small

radii. However, we recommend that you begin with the free-vortex distribution

because that is most frequently referred to in the open literature.

8.2.1.6 Range of compressor design parameters.

Table 8.1 contains

ranges of design parameters that can be used as guides in the preliminary design of

axial flow compressors. Additional information about the shape and construction

of axial flow compressors can be found by examining the AEDsys Engine Pictures

files.

8.2.2 Turbine Aerodynamics

8.2.2.1 Constant axial veloci~ adiabatic, selected Mach number,

mean-line stage design.

The primary goal of this section is to develop a realis-

tic

turbine stage performance model that will reveal the behavior of the important

aerodynamic and thermodynamic quantities and serve as an initial input to the com-

plete numerical calculations of TURBN. This is in the same spirit as Sec. 8.2.1,

but the design of turbines is different from that of compressors for a number of

reasons, including the following:

1) The engine cycle performance models of this textbook require that the turbine

stage entrance stator (a.k.a. inlet guide vane or nozzle) be choked and all other

stator and rotor airfoil rows be unchoked.

270 AIRCRAFT ENGINE DESIGN

2) The density of the working fluid changes dramatically, so that compressibility

or Mach number effects must be included.

3) The turbine generates rather than absorbs power.

4) High inlet temperatures require that heat transfer and cooling be considered.

5) There are no wide-ranging rules for choosing turbine flow and airfoil geome-

tries, such as the compressor diffusion factor.

It is not possible, therefore, to provide quite such comprehensive and general

methods for turbine stages as has been done for compressor stages in Sec. 8.2.1.

Nevertheless, it is possible to analyze, explore, and understand the behavior of

a truly representative class of turbine stages, subject only to the assumption of

constant axial velocity and enforcement of the stator and rotor airfoil relative exit

Mach number constraints stated above. Because both of these conditions faithfully

reflect the engine cycle assumptions of Chapters 4 and 5 and current design practice,

the resulting solutions will closely resemble real designs.

A typical turbine stage and its velocity diagrams are shown in Fig. 8.7. The

Euler turbine equation [Eq. (8.10)] gives the energy per unit mass flow exchanged

between the rotor and the fluid for constant radius as

(.or

ht2 - hi3 : Cpt(Zt2 - Zt3) :

--(1)2 --I- 1)3)

(8.35)

One can see from the velocity triangles of Fig. 8.6 that because of the large angle

a2 at the stator (nozzle) exit and the large turning possible in the rotor, the value

of v3 is often positive (positive or3). As a result, the two swirl velocity terms on the

right side of Eq. (8.35) add, giving larger power output.

Because of compressibility effects, the rotor degree of reaction

Rt has a defini-

tion that is more suitable to turbines, namely, the rotor static enthalpy drop divided

by the stage total enthalpy drop. For calorically perfect gases this becomes, in the

Stator Rotor

Station: 1 2 2R 3R 3

V 1~~ /V3R/~°Jr

u2 u 2 V3R~

V2~ ~'~2~V2R v 2R V3R

v 2

u V3

Fig. 8.7 Typical turbine stage and velocity diagrams.

DESIGN: ROTATING TURBOMACHINERY 271

turbine stage nomenclature of Fig. 8.6,

h2 - h3 T2 - T3

° Rt -- htl -

ht-------~3 -- Ttl

-- Tt3 (8.36)

This definition retains the same physical insight as the equivalent compressor defi-

nition, while allowing the compressible flow turbine analysis to proceed smoothly

despite large density changes. Please note that the static pressure drop is roughly

proportional to the static temperature drop, and that the stator degree of reaction

is approximately the difference between the rotor degree of reaction and one.

Assumptions

1) M2 and MaR are given.

2) Two-dimensional flow (that is, no property variation or velocity component

normal to the flow).

3) Constant axial velocity (ul = u2 = u3).

4) Constant mean radius.

5) Adiabatic flow in the stator and rotor.

6) Calorically perfect gas with known Yt and

Rt.

The derivations that follow reduce the full turbine stage analysis of Sec. 9.5 in

Ref. 11 by incorporating the preceding assumptions. Please note that making all

velocities dimensionless by dividing them by the velocity gv/~cptTtl denoted as

V' (a known quantity at the entrance of the stage that corresponds to the kinetic

energy the fluid would reach if expanded to a vacuum, also referred to as the

maximum or vacuum velocity) makes the equations more compact and the results

more universal. We use the symbol f2 for the dimensionless rotor speed

wr.

Thus,

V' = v/~cptTtl

(8.37)

o)?"

f2 - (8.38)

This approach cannot be used to design a series of stages as it was for compres-

sors because the effect of steadily decreasing static temperatures, when combined

with constant velocities, would eventually lead to supersonic velocities every-

where, contradicting one of the central design constraints. Thus, this method is

primarily used to reveal the capabilities of individual turbine stages.

Analysis

1) Total velocity at station 2

V2 _ / (gt -

1) M2

~7 -- Vl + (Yt - 1)/2/2 (8.39)

2) Stage axial velocity

u V2

-- COS Ot 2

W V'

3) Tangential velocity at station 2

(8.40)

= ~ sin

Ct 2

(8.41)

V' W

272 AIRCRAFT ENGINE DESIGN

4) Rotor relative tangential velocity at station 2

V2R V2

W W

5) Rotor relative flow angle at station 2

I)2R / V I

tan

f12 -- --

u/V'

6) Rotor relative total temperature

Tt2RTtl

_ l.nt_~2 (~ v2~g')

7) Rotor relative flow angle at station 3

/Tt2R/Ttl (Vt- 1)M2R

tan f13 =V

u2/V'2 1 +--~t --1-~3n/2 - 1

8) Rotor flow turning angle

= f12 + f13

9) Tangential velocity at station 3

l) 3

V I

10) Stage exit flow angle

tan ot 3 --

11) Static temperature at station 3

-- - -- tan f13 -

v3 / V'

u/V'

T3 Tt2 R / Tt l

Ttl 1 .Jr_ ()It -

1)M2R/2

12) Stage temperature ratio

Tt3

rts--

Ttl

13) Rotor degree of reaction

T 3 b/2

1 + tan 2 c~3

Ttl V '2

°gt -- V2R -- V2R

2(1 - rt,)V 'e

14) Rotor solidity based on axial chord

2COS 2/33(tan

/~2 +

tan

/~3)

(8.42)

(8.43)

(8.44)

(8.45)

(8.46)

(8.47)

(8.48)

(8.49)

(8.50)

(8.51)

Crxr -- (8.52)

Cx Z

DESIGN: ROTATING TURBOMACHINERY 273

General solution.

We may now use this analysis to explore the behavior of

turbine stages. The results will be more easily understood if, before proceeding,

we circumscribe the results by means of three useful generalizations.

First, we must consider two different types of stages, namely those having

either choked or unchoked stators. The former are required as the entrance stages

for every turbine and will be represented in our computations by an M2 of 1.1.

The latter are required for all other turbine stages and will be represented in our

computations by an M2 of 0.9. Other supersonic or subsonic stator Mach numbers

should be selected and evaluated by the reader.

Second, higher values of

M3R

always improve stage performance, provided that

some margin to avoid rotor choking is provided. The reader should also inde-

pendently confirm this assertion. Consequently, M3R is set equal to the highest

practical value of 0.9 in our computations.

Third, the analysis lends itself well to the intuitively appealing examination

of the variation of turbine stage properties for the expected ranges of ~2 and ~.

The open literature strongly suggests that the best performance is obtained when

60 deg < oe2 < 75 deg (for example, Ref. 11, Sec. 9.5). The open literature also

concludes that larger values of f2 are better, the upper limit being presently in the

range 0.2 < g2 < 0.3. The ensuing computations will justify these observations. The

computations use ~,'t = 1.30 and

gcCpt =

7378 ft2/(s2-°R), although the equations

are formulated to allow you to choose any desired values.

General results.

The results of the computations are presented in Figs. 8.8-

8.13. Their contents, and the corresponding consequences, will now be described

in turn. To avoid unnecessary repetition and give the primary conclusion the em-

phasis it deserves, we begin by noting that

every

measure of aerodynamic and

0.90

0.88

0.86

0.84

(~,~ / ~,,

)

0.82

0.80

0.78

' ' ' ' I ' ' ' ' I ' ' ' ' I ....

M3R = 0.9

.~ 7, = 1.3

_ f2 = 0.2

M 2 =1.1 --

, , , , I , , , , I , , , , I , , i

65 70 75

1~2 (o)

Fig. 8.8 Stage total temperature ratio.

274 AIRCRAFT ENGINE DESIGN

160

150

140

130

120

110

100

M3R'

' = o9'

i , , ,

, I ' ' ' ' I ' ' ' ' t

i.~ "~ M: = 0.9 ....

M 2 =1.1 --

I I I I I I I I I I I I I I I I I I I

65 70 75 80

a: (%

Fig. 8.9 Rotor flow turning angle.

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

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

M3R

=0.9,

=1.3 f~=0.3

~=0.2

~=0.3

~2=0.2

M 2 = 0.9 ....

M~ =l.l --

, , , , I , , , , I , , , , I , , ,

65 70 75

a2 C)

Fig. 8.10 Rotor degree of reaction.

DESIGN: ROTATING TURBOMACHINERY 275

a3 40

M~=o9 ' .... ' .... '

.... t

= 1

M 2

=1.1 --

, , , ,

, , , , I , , , , I i J i

65 70 75 80

a 2 (°)

Fig. 8.11 Stage exit flow angle.

a~r

1.80

1.60

1.40

1.20

1.00

0.80

0.60

0.40

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

M3R = 0.9

~ y, =1.3

M 2

=0.9 .... "" -.. ~ ~-"

M 2 = 1.1 -- "~.

65 70 75 80

a2 (o)

Fig. 8.12 Rotor solidity based on Z = 1.