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

Подождите немного. Документ загружается.

APPENDIX L: PROPELLER DESIGN TOOLS 613

r/p

1.o

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

0.8

0.7

0.6 1

0.5 0

1 1.2 1.4 1.6 1.8 2

V~ lVo

Fig. L.3 Ideal propeller performance indicators.

The three ideal propeller performance parameters are displayed together in

Fig. L.3. Comparison reveals that high r/~o can be obtained only at the cost of large

size of

Ap.

Putting this in more precise, quantitative terms, in order for ~7~ to be at

least 0.80, then

Ve/Vo

must be less than 1.5, C~. must be less than 1.25, C~ must

be less than 1.56, and

Ap/Ao

must be greater than 0.8.

These expressions represent the ideal performance--something that can never be

fully achieved in practice. The underlying assumption is that the propeller generates

only the kinetic energy of axial velocity. In reality, several sources of energy loss

occur, including frictional drag on the propeller blades, kinetic energy of rotation

of the wake, and nonuniform axial velocities in the wake. Actual efficiencies are

about 85% of the ideal value, with the most important loss mechanism being

frictional drag. The next section will shed more light on this subject.

Propeller Blade Element or Strip Theory

This amounts to the equivalent of two-dimensional or infinite aspect ratio theory

for propeller blades and is closely related to wing airfoil theory. It deals with the

local conversion of mechanical power into thrust power of the propeller exhaust jet.

To determine the behavior of the entire propeller, it is necessary to integrate from

hub to tip across all blade elements and then to make corrections for end effects.

The purposes of this section are completely served, however, by considering only

one infinitesimal radial increment or blade element.

An important feature of this analysis is that the moving propeller blades do not

"see" the undisturbed, freestream velocity V0. Equation (1.5) has already shown

614 AIRCRAFT ENGINE DESIGN

Tip

Cross- ~Blade

Element~ ~'dr

Total Airfoil Force Tangential Force

~dR ~dQ/r

Lift/__

~\ I ~ Drag

I 4~ ~k I // / / Geometric

s,i Pitch

~ - ~ = Angle of Attack

••

Pitch or

Advance

Relative Wind, V~ Angle

a ')J2r

tan y = D/L = CD/CL

(l+a) V0

tan # =

Vo(l+a)/(1-a')Or

Fig. L.4 Blade element theory nomenclature.

that the axial component of velocity is larger than V0 and is, in the general case,

represented by (1 +

a)Vo,

where a is known as the "axial interference factor."

Similarly, it has been found that the trailing vortex system reduces the relative

tangential velocity to (1 -

a')rfL

where a' is known as the "rotational interference

factor." Whereas a depends primarily upon

Ve/Vo

and can have a wide range of

values (it can even be negative when the thrust is reversed), a' is merely of the

order of 1%.

The thrust on the blade element (Fig. L.4) follows directly from

dF = dL c°s(~ + y) -- P0 {(1 + a)V0} 2 cos(~b -F

Y)CLcdr

(L.10)

cos y 2 sin2~ cos y

so that the total thrust is

frf tip

F = dF (L.11)

APPENDIX L: PROPELLER DESIGN TOOLS 615

The tangential torque on the blade element follows directly from

dQ = rdL

sln(q~ + y) _

cos y 2 sin 2 ~b cos y

p0{(1 + a)V0} 2 sin(~b +

Y)CLC rdr

(L.12)

so that the total torque is

frh 'tip

Q = dQ (L.13)

The most interesting relationship, which allows the evaluation of the local or blade

element efficiency of conversion of mechanical power into thrust power, comes

from the definition of the blade element efficiency Oe.

.__VodF

(V0)( dd@/r )

which, from the relative wind velocity and total airfoil force triangles of Fig. L.4,

becomes

1 -

a' tan 4,

~e •

(L.14)

1 + a tan(4~ + y)

The local conversion efficiency therefore depends primarily upon a, y, and q~.

If a and ~b are held constant, t/e must decrease as y increases, which certainly

makes sense because that decreases dF while increasing

dQ/r.

The behavior of

Eq. (L.14) with constant a and y and changing q~ is more complex, as indicated in

Fig. L.5. It can be shown, for example, that there is a maximum at q~ = 45 ° -

y/2,

at which point

1 - a' tan(45 - y/2)

t]ema x =

(L.15)

1 + a tan(45 + y/2)

The curve also passes through zero at ~b = 0 and 90 ° - y, which correspond to no

thrust power (V0 = 0) and the total force being in the tangential direction, respec-

tively. Finally, it should be noted that, as the propeller aerodynamics approach the

ideal situation where

a' and y are

zero and a is

(Ve - Vo)/2Vo,

then Oe approaches

the ~ of Eq. (L.7), an outcome having great intuitive appeal.

Real propeller blade elements do not, of course, have constant y because 4) at

a given radius varies as a result of flight and propeller speed changes, but have

their lowest y and best performance in the vicinity of 4, = /~ (design angle of

attack). One way to improve the range of operation near minimum y, at the cost

of mechanical complexity, has been to provide "variable pitch" in order to match

/3 to q~ as flight and propeller speeds change. If sufficient turning is provided, this

also makes "reverse thrust" possible.

It should also be noted that the naturally occurring values of q~ are in the range

of 10-15 deg, which has prompted the use of geared propellers in order to reduce

f2 and therefore increase both q~ and Oe.

616 AIRCRAFT ENGINE DESIGN

1.0

0.5

0.0

I I I I I I I I

30 60 90

¢ (o)

Fig. L.5 r/e as a function of ~b(~ = 0.8).

increasing

.CLIC o =oo~

.CL I C ~

=50

.CLIC o

=10

Radial Variations

Because the propeller moves in solid body rotation, ~ must decrease, and Vr

must increase with radius. The first result of this fact is that fl is designed to

decrease with radius in order to maintain the design angle of attack near zero. This

gives the propeller its twisted appearance and makes it impossible to maintain zero

angle of attack at all radii at off-design conditions.

More importantly, the relative Mach number

given by

= (1 +

a)Mo/sin

(L.16)

increases with freestream Mach number M0 and radius and always exceeds M0.

When

approaches unity, blade element shock wave and boundary layer separa-

tion drag cause F to rapidly increase and Oe to decrease below any reasonable value.

Turboprop operation must therefore be restricted to subsonic forward speeds. This

condition is also relieved somewhat by gearing down f2 and thereby increasing ~b.

Real Propeller Behavior

Two questions that often arise while designing propellers cannot be answered

using these fundamental methods, namely, the size and off-design performance.

Sizing is usually done on the basis of static thrust and often uses experimental or

empirical information. Thanks to the brilliant work of Theodorsen 2 on the vortex

theory of propellers, it is possible to understand and compute the observed behavior

APPENDIX L: PROPELLER DESIGN TOOLS 617

of the "standard" correlation parameters for thrust

and horsepower

Cr -

(L.17)

660a (N / 1000)2 (D / 10)

Cp -

(L. 18)

2000a

(N/1000)

(D/10)

where F, N, D, and

lip are

in units of lbf, rpm, fl, and hp, respectively. Note that

these have little to do with C~. and C~o defined earlier, but are used by engineers

to correlate real propeller behavior.

The complete calculations of Theodorsen, done by Amatt et al.,4 show that the

quantity

C3/2

2000

F 3/2

Cp (6605)3/2tr1/2(D/lO)HP

is not only independent of N, but is nearly constant over the usual range of operating

parameters. As a result, it is possible to solve the preceding equation for D and

use the resulting approximate correlation

D = 589aHp2(blades)l/3

(L.19)

as a starting point. This clearly shows that the same thrust can be obtained from

less

lip

by increasing the airflow, i.e., D!

Example

Suppose a four-bladed propeller is to be designed to produce 15,000 lbf of thrust

from 4000 hp at standard sea level static conditions. Then Eq. (L. 19) shows that

{ (15,000)3

}1/2

D = 589(1)(4000)2(4)1/3 = 15.0ft

If the same job were to be done with two rows of propellers, each producing half

the thrust and using half the power, whether on the same or separate engines, then

/ (15'000/2)3 }1/2

D ---- [ 589(~4)1/3 = 10.6ft

Propeller Map

The most accurate method for computing off-design propeller behavior, which

also requires the most initial data, is the "propeller map" containing contours of

constant

Oprop, an

obvious relative of the well-known compressor map. A typical

propeller map is shown in Fig. L.6.

The type of propeller map to be used in any situation depends on which input

parameters are known. In the typical propulsion case of selecting propellers for

618 AIRCRAFT ENGINE DESIGN

0 1.0 2.0 3.0 4.0

J=Vo/ ND

Fig. L.6 Typical propeller map.

turboprop cycles, it is most advantageous to employ

[Eq. (L.18)] and the

"advance ratio" or "slip function"

I10

J ---- -- (L.20)

as the independent variables. The reason for the choice of Cp and J is simply that

HP and Vo are

usually known and the propeller design goal is to determine values

for D and N (which are often fixed for mechanical and control reasons) that result

in acceptable values of the propeller efficiency

rlprop

over the entire mission. Each

set of Cp and J uniquely determines D and N.

An example of this method can be found in Fig. L.6, which is data representative

of a counterrotating, tandem (or dual) propeller operating below the critical relative

Mach number. The benefits of variable pitch have been included by presenting the

highest attainable value of

17pro p

at each point and the effects of gearing appear

directly in both

Ce and J.

Such data can and should be obtained from the propeller

manufacturer.

The general features of this map can be understood in terms of propeller blade

element theory. The highest efficiencies occur in the vicinity of J = Jr, which

APPENDIX L: PROPELLER DESIGN TOOLS 619

Table L.I Example turboprop design

Flight M0/Alt, Power Thrust

rlprop,

Thrust

Label condition kft (avail) (required) % (actual)

J/Ce

A Takeoff 0.088/0 2387hp 66601bf 43.0 66601bf 0.60/1.50

B Sealeveldash 0.529/0 2259 3452 85.0 3576.1 3.88/1.38

C Low cruise 0.454/0 1411 2540 83.7 2570.3 3.30/0.85

D High cruise 0.5/35 619 1152 90.0 4067.8 3.20/1.24

E Loiter 0.2/10 218 916 81.9 1229.3 1.45/0.18

D = 14 ft and N = 663.4 rpm.

corresponds to 4' = 45 deg. Beyond this, the efficiency "islands" are the result of

angle of attack effects on the propeller blades. In the regions where

llpro p

falls off

rapidly, it is clear that stall is occurring because the relative angle of attack is too

positive or too negative.

Five points labeled A through E have been spotted on Fig. L.6 and portray the

"best" solution for the multipurpose transport aircraft of Table L. 1 with many

conflicting requirements, including short takeoff, high-speed sea level dash, and

great range. Also shown in Fig. L.6 are the locations to which each of these five

points would move if D were reduced 5% (longer lines) and if N were reduced

5% (shorter lines). Although

tiptop

could be slightly increased at the four flight

conditions (B through E), particularly by reducing the diameter D, sufficient thrust

for takeoff would no longer be available at condition A. Reversing this reasoning

quickly leads to the conclusion that providing excessive takeoff thrust will unnec-

essarily penalize

Oprop

at the four flight conditions. Other propellers, with different

performance maps, should also be explored for any such application.

Structural Design

The method of designing propeller blades for a specified service life follows

the lines already described for compressor rotor airfoils with one overwhelming

change: Because propeller blades have the protential for so much destruction, they

cannot be allowed to separate. In the jargon of the trade, they must be "prime

reliable." This only increases the need for good analytical tools, exhaustive and

realistic testing, and high-quality manufacturing.

Beyond this, there are significant differences because bending and twisting

moments can be more important than centrifugal forces and because composite

materials with nonisentropic properties (such as wood) are frequently employed.

The net result is that propeller structural design, 4 like many other things, is a field

that requires its own special expertise. Nevertheless, there are certain structural

aspects that warrant some discussion here.

In reviewing the typical turbofan or turbojet engine, a number of components and

design features are readily apparent that optimize inflow angle and flow velocity.

Such is the case with inlet ducts, inlet guide vanes, and variable compressor stator

vanes. As a result of these design features, the engine is nearly impervious to aircraft

altitude changes. The propeller, however, always operates in the freestream and

aircraft pitch, yaw, and roll motions constantly change the inflow angle into the

620 AIRCRAFT ENGINE DESIGN

blades. More often than not, the inflow angle is such that the velocity vector is

not normal to the propeller blade plane of rotation. This condition results in cyclic

aerodynamic loads that peak one time per (1XP) revolution. This cyclic loading

is known as the "IXP" moment and is directly proportional to the product of

inflow angle and dynamic pressure. In aft-mounted (pusher) propeller, the wing

and nacelle flowfield interactions result in other moments, referred to as "2XP,"

etc., that must be considered. These cyclic loads are of paramount importance to

the structural integrity of propellers, especially in view of their "prime reliable"

requirements.

It was just mentioned that the twisting and bending moments on propeller blades

are more significant than those on axial compressor blades. A typical compressor

rotor has upward of 100 blades, giving high solidity (which means greater ability

to absorb power), with a low aspect ratio on the blades (of the order of 1 to 2). In

contrast, a typical propeller has four blades of high aspect ratio (greater than six).

This means that fewer blades must absorb all the power, which translates into very

high loading. This has an impact on the severity of bending moments, which grow

rapidly with radius and are greatest at the root. The high loading also affects the

twisting moments because the pressure distribution on the highly twisted blades

is asymmetric.

These differences between compressor blades and propeller blades highlight the

complexity of structural considerations in propellers. These problems are made

more acute by the need to install anti-ice or de-ice devices on the leading edge of

propeller blades. In compressors this is done by injecting hot air bleed from the

compressor into the inlet, but this is not easily accomplished with a propeller blade

rotating in the freestream. Although there are various methods, such as circulation

of hot oil along the leading edge, the most common is the resistive heater, in which

an electrical heating element is installed along the leading edge of the blade.

Variable Speed vs Constant Speed

One of the advantages of turboprop and turboshaft engines lies in the fact that

their purpose allows for operation at constant speeds. By simply changing the

blade pitch, and hence the load, a pilot can control thrust. Throttle movements

will increase or decrease fuel flow to the engine, with a consequent tendency to

increase or decrease rpm. The propeller governor will sense the change and adjust

propeller pitch, thereby adjusting the load corresponding to engine power output

and hence maintaining constant rpm. Such a mode of operation allows for rugged,

simple engines, constructed with a single compressor tied to the turbine by just

one shaft, which also drives the propeller reduction gearbox. On the other hand,

constant speed does not allow for most efficient propeller operation at all aircraft

airspeeds. Under constant N conditions, given that the diameter is constant, the

advance ratio will vary only with flight airspeed. At the same time, power setting

will change the value of the power coefficient. The end result is that, at low flight

speeds and power settings, the propeller is not operating at optimum efficiency

for the flight condition. To remedy this problem, given a set at low airspeed, the

rotational speed must be decreased in order to increase the value of the advance

ratio. At greater values of J, the operating point on the propeller map moves to the

right and up on efficiency. This can have a significant impact on fuel consumption

APPENDIX L: PROPELLER DESIGN TOOLS 621

and aircraft loiter time. The drawback of the variable speed propeller is that the

engine must now operate in a more conventional way, that is through an extended

range of rotational speeds. This will increase the complexity of controls, structures,

and manufacture, and ultimately cost. It is thus extremely important to take into

consideration the intended duty cycle or mission profile in order to make the most

advantageous choice---constant speed or variable speed.

Turboprop Cycle Performance Characteristics

A novel feature of turboprop engines is the simplicity with which their general

behavior can be understood. Returning to the initial view of the propeller as a

device for converting engine-generated mechanical power into thrust power and

ignoring the usually negligible amount of thrust generated by the core flow, then

from the definition of

T/pro p

Thrust power F Vo

T/prop "z-

Propeller shaft power

Pprop

(L.21)

there follows

F -- T/prop Pprop -- l'lprop eprop

(L.22)

Vo aoMo

Recognizing that

Pprop

is only slightly affected by M0, but does decrease faster

than a0 with altitude, it is possible to conclude immediately that F decreases with

altitude and varies almost inversely with M0 at constant altitude (see Fig. K.7).

The latter has tremendous implications for high-speed turboprop flight. Because

aircraft drag increases in proportion to M E and

T/prop

diminishes with M0, the

Pprop

required to fly faster at fixed altitude varies at least as M 3. Small wonder that the

record speed for a turboprop-powered aircraft, which was set in 1960 by the Soviet

TU-114, is a Mach number of only about 0.77.

Something about static thrust can be learned by differentiating Eq. (L.22) using

UHospital's rule, whence

FIMo=0

Pprop ( dT/prop ~

(L.23)

= a-7- /Mo=O

Even though this thrust is often too great for some other part of the propulsion

system, such as the gears or shafts, it allows the maximum thrust capability of the

engine to be calculated only from the slope of propeller efficiency at M0 = O. An

important conclusion about turboprop thrust behavior, as noted at the outset, is

that it provides high thrust at low speed.

The other important performance parameter is specific fuel consumption, which,

using the definition of overall efficiency

(T/prop × T/TH),

is given by

S = thf =.

aoMo

(L.24)

T/propT/THhpR

622 AIRCRAFT ENGINE DESIGN

where OrH

Pprop/(lhfhpR)

and is largely proportional to Vc~ or a0, so that

S c< -- (L.25)

~prop

which means that S worsens as Mo increases (see Fig. K.8), particularly as Mo = 1

is approached and

Oprop

decreases rapidly. Again, the turboprop is an excellent

device at low Mo--in fact the lower the better.

To find the static specific fuel consumption, Eq. (L.24) is differentiated using

L'Hospital's rule to yield

SIMo=0 =

rlrHhpR(d~prop/dMO)Mo= 0

(L.26)

Example

Suppose that

~prop

0.1 < M0 < 0.8. How do F and S vary with Mo?

From Eq. (L.22)

= lO~lpropmaxPprop

-- ~prop

max

Pprop

aoMo

From Eq. (L.24)

= lOMoOpropma x

for M0 _< 0.1 and

Oprop

= Opropmax for

M0 < 0.1 (constant)

0.1 < Mo < 0.8 (~l/Mo)

S = M0 < 0.1 (constant)

10ITpro

max

hpRrlTH

aoMo

S = 0.1 < Mo < 0.8 (~M0)

Oprop max h pR lTTH

References

IGlauert, H.,

The Elements of Airfoil and Airscrew Theory,

2nd ed., Cambridge Univ.

Press, Cambridge, England, UK, 1947.

2Theodorsen, T.,

Theory of Propellers,

McGraw-Hill, New York, 1948.

3Theodorsen, T., "Theory of Static Propellers and Helicopter Rotors," American

Helicopter Society, Paper 326, May 1969.

4Amatt, W., et al., "Structural Analysis and Blade Design,"

Summary of Propeller Design

Procedures and Data,

Vol. II, USAAMRDL-TR-73-34A, Nov. 1973.