EVALUATION OF PERFORMANCE REQUIREMENTS

Chapter 5. Design for performance

5.4. EVALUATION OF PERFORMANCE REQUIREMENTS

Note that this equation does not represent an actual aircraft polar; it refers to the initial climb-out after takeoff. To include drag due to engine failure at low thrust/

weight ratios, E may be reduced by approx-imately 4% for wing-mounted engines and 2%

for engines mounted on either side of the fuselage tail.

Although the accuracy of eq. 5-29 for the approach and landing configurations is probably not so good, due to the higher ra-tio of flight speed to stalling speed, it still remains. a useful first-order approx-imation.

where T is the installed thrust, as derived from the engine manufacturers' brochure for a given engine, or assumed equal to a cer-tain percentage of the uninstalled thrust when the engine has not yet been sized

( c f. Section 5. 3. 4. ) .

The drag coefficient, as given by eq. 5-9, is dependent on the lift coefficient,

C = W/S

L hpM2 (5-31)

and eq. 5-30 can be rewritten as follows:

T

w

^(5-32)

The zero-lift drag coefficient may be elab-orated in terms of the wing area and the installed thrust. Any drag prediction meth-od available to the designer may be used for this purpose. In this text the semi-statistical method of Section 5.3. is used to derive the following result:

2 2 +

Tto .7kw 6M nAe

- = ---~---2----(5-33)

wto T/Tto- .76M d 3

where

kw wto

----w

d1 r (CDS)w

+ 1\CD rRe uc rt

--s-comp

d2 (CDS) f Po

rRe r uc rt wto (5-34)

These terms are explained in Sections 5. 3. 2.

and 5.3.3.

In equation 5-33 the three terms in the nu-merator are associated with wing profile drag, induced drag, fuselage drag and em-pennage drag. The denominator may be inter-preted as an effective lapse ratio of the engines installed in nacelles, including internal and external nacelle or intake scoop drag. The value for T/Tto can be

ob-tained from non-dimensional thrust curves or generalized data in Appendix H.

The minimum thrust at a given airspeed and altitude is found when the wing loading is equal to:

(5-35)

This condition is identical to the condi-tion for minimum drag (or L/D-max) of the wing plus that part of the tailplane con-tribution which is proportional to the wing drag. The factor d 1 is thus proportional to the wing profile drag coefficient; the order of magnitude is .008 - .010, for air-craft with retractable undercarriage.

b. Propeller aircraft.

The equivalent horsepower required to fly at a given speed and altitude is given by:

(5-36)

The power/weight ratio in the sea level static (SLS) condition is derived in the same way as for jet aircraft:

(5-37)

For wing-mounted engines, the factors d 1 and d 2 are defined analogous to eq. 5-34, while

(5-38)

This effect of compressibility on the drag can generally be ignored. For a tractoren-gine in the fuselage nose, d 3

is equal to:

0 and d 1

(5-39)

The power lapse ratio P/Pto must be derived from the engine manufacturer's data. For the installed efficiency np typical values may be assumed as presented in Section 5. 3. 4.

•o

t

I

WEIGHT 750,000 LB.

0 ~ 35

"'

I 0 ::l30

,_ 5

"

₂₅

20

15 15

~ 14

~ ¹³

12

11

I

ALTITUDE 25,000 FT.

.7 .8

MACH NUMBER

a. With compressibility effects on drag;

100% (ML/D)max = 14.8

t

0 0

•o

S! 35

20

15

,.

0 13

~ :::e 12

.6 .7 .8 .9

MACH NUMBER

b. No compressibility effects on drag;

100% (ML/D)max = 14.8

Fig. 5-10. Range performance of a high-subsonic long-range jet transport

5.4.2. Range performance

A specified range or radius of action must be achieved with a given payload or maximum fuel capacity, taking into account fuel reserves for holding and diversion. An in-dication of the cruise procedure will gen-erally be mentioned: (initial) cruise al-titude, long-range or high-speed condition.

a. Jet aircraft.

Although range performance depends upon the cruise procedure, the Breguet equation is useful as a basis for an initial prediction of cruise fuel. On the conditions of con-stant angle of attack, airspeed and specif-ic fuel consumption (climb cruise) , the range is:

R (5-40)

or:

(5-41)

where wi is the initial weight and wf the cruise fuel weight. For a given fuel frac-tion and engine type, the primary paramete1 in this equation is M L/D, the range param-eter. The operational variables (cruise al-titude, Mach number), the wing loading and the drag polar are the primary variables.

For medium- and long-range aircraft, the amount of fuel consumed is large and it is necessary to aim at optimum flight condi-tions.

For a given drag polar and wing loading, lines of constant M L/D can be plotted on a speed-altitude diagram: Fig. 5-10 ; the lower part of this figure is an intersection for one altitude. The inclination of the tangent from the origin to the c 0/CL -curve represents the condition for maximum M L/D.

Considering flight at a specified sub-critical Mach

number and variable altitude first, the maximum range is obtained at an altitude where L/D is maxi-mum, hence

(5-42)

corresponding to the minimum drag speed MMD.

The locus of this condition for each altitude is indicated in the upper part of Fig. 5-10.

For a specified altitude and variable Mach number, ignoring compressibility effects on the drag, the condition for maximum M L/D is:

(5-43)

resulting in:

(5-44)

'

corresponding to a Mach number equal to /; MMD"

The conditions for CL according to (5-42) and (5-44) are incompatible and no absolute optimum combination of M and altitude can be obtained. This is confirmed in Fig. 5-10, indicating that in the absence of

com-tude is significant (point B) • The maximum cruise rating of the engines determines the flight speed, and a typical extra drag of some 20 co\lllts is ac-ceptable. An intermediate condition is the cost-economical cruise, resulting in a favorable combi-nation of fuel costs and block time effects on op-erating costs (Section 11-S).

The designer's problem with respect to range performance is to choose a favorable combination of speed, altitude and airplane geometry, to obtain the best - or at least a satisfactory - range performance and to estimate the amount of fuel.

Flight speed variation has a major effect on the fuel required, but also on the de-sign of the wing (sweep angle, section shape), the structural weight, engine s.f.c., and problems of stability and con-trol. Optimization of the design - Mach number is a very complex study and this pa-rameter is usually specified in a rather arbitrary manner in the design requirements.

pressibility M L/D continues to increase with alti- Cruising altitude has a direct effect on

tude. fuel weight. When the installed engine

For high-subsonic speeds, a rapid rise in drag and thrust is based on the cruise condition, the subsequent deterioration of the range is observed engine size required increases with alti-beyond the drag-critical Mach number. Although the tude, as the density decreases, and for a complex character of the flow does not allow an an- given specific thrust the inlet diameter alytical treatment of compressibility effects, Fig. must increase as well. The weight of engine 5-10 indicates that a definite condition for maximum plus fuel is a minimum for some altitude

M L/D is now present. The locus of CL for this con- below the altitude for maximum L/D. This dition is correspondingly modified and intersects case is elaborated analytically by Kiichemann the locus of (L/D)max at the optimum combination of in Ref. 5-4, resulting in an optimum condi-M and altitude. tion for CL depicted in Fig. 5-11. For

long-range aircraft the minimum fuel requirement The condition for maximum specific range (i.e. the dominates while on short hauls the amount distance travelled per pound of fuel consumed) is of fuel consumed is less and the engine slightly different from (M L/D)max due to the ef- weight is the main factor, leading to a rel-fects of altitude and speed on engine s.f.c. In atively low optimum for CL in cruising operational practice the flight speed will always flight.

be some 10-20% above MMD in horizontal flight in Obviously the matter is more complicated in order to obtain positive speed stability and to real life:

avoid buffet during maneuvers. A typical long-range _ Cruise fuel is only part of the total fuel cruise condition results in 98% of the maximum spe- load.

cific range (point A in Fig. 5-lOal. In operations -Fuel weight and engine weight do not have where fuel consumption is not a dominant factor, the the same significance from the point of high-speed cruise at a somewhat lower cruise alti- view of achieving minimum operating costs.

1.0 given by eq. 5-35 can be used to find a

~~~~

^{. .10}

...:::::; .16 first-order approximation for the

L/D-.8 .6

.2

OL---~---L----~----L---~

0 ^.1 .4 .5

Fig. 5-11. Optimum cruise conditions accor-ding to the criterion of Klichemann and Weber (Ref. 5-4)

- For short-haul aircraft the engine thrust is frequently determined by the takeoff field length or an engine-out climb re-quirement.

- Oxygen system requirements are dependent on cruising altitude (e.g. FAR 121.327-333), which may be a deciding factor.

-Air Traffic Control considerations affect the choice of cruising altitude.

For an initial estimation of ~he amount of trip fuel required, the fuel consumed for cruising is derived from eq. 5-41 :

wfcr = 1-exp(- R cT;re co) (5-45)

wto ao M CL

where the initial weight is assumed to be approximately equal to Wto' Additional fuel is also used during takeoff, climb to and descent from cruise altitude, approach and landing. Fig. 5-2 is based on the following assumptions:

a. The cruise fuel according to eq. 5-45 is the dominant factor in the fuel contribu-tion.

b. For transport aircraft the wing loading

ra.tio of the wing.

r.. The wetted area of the fuselage is the primary parameter for the fuselage drag area; other drag contributions are assumed proportional to wing plus fuselage drag.

Fig. 5-2 can be used, provided the fuselage dimensions are known, for example from a fuselage layout drawing or the data from Figures 3-11 and 3-12. Engine s.f.c. is deduced from the engine manufacturer's brochure or from the data listed in Chapter 4.

Reserve fuel consists of various contribu-tions (cf. Table 11-2). One important item is holding fuel, which is proportional to the airplane minimum drag, hence inversely proportional to

lA.

The following empiri-cal correlation has been found to give a good approximation for transport aircraft:

CT/18

.18 (transport a/c) (5-46)

lA

where CT/18 is the same quantity as used in Fig. 5-2. A fuel reserve for 3/4 hour ex-tra flying time should be allowed for bus~

ness jets and executive aircraft. This can simply be translated into an equivalent range increment:

t;R (business and

executive jets) (5-47) Fig. 5-2 is thus valid, provided an equiv-alent range is used equal to R + t;R.

If the effects of varying parameters like wing loading and aspect ratio are to be assessed, the aircraft weight distribution must be computed in terms of these param-eters. The details of these calculations will not be dealt with here;. a simplified example in Section 5.5.3 shows the effects of wing area variation on the weight dis-tribution of a long-range aircraft.

b. Propeller aircraft.

The Br~guet range equation for propeller

159

aircraft is:

R ~ L wi

cp i5 in w - w _{i f} ^(5-47)

A similar picture to Fig. 5-10 can be drawn up. If s.f.c. and propeller efficie~

cy variations are ignored, the specific range can be shown to be a maximum - for minimum airplane drag, if the alti-tude is fixed and flight speed variable, - for minimum power, if the flight speed is fixed and altitude variable.

As in the case of jet aircraft, these con-ditions are incompatible and no absolute optimum exists on the basis of flight me-chanics. In general, range performance continues to improve with altitude until the available engine power becomes the lim-iting factor.

Fig. 5-1 can be used for an initial esti-mation of fuel weight. The wing aspect ra-tio is the primary factor in obtaining a high L/D ratio and good range performance.

It is the only parameter used in this fig-ure to characterize the aerodynamic per-formance, as no considerable improvement was found when fuselage dimensions were introduced.

5.4.3. Climb performance

Climb performance may be specified in the form of:

a. Operational requirements, derived from desired performance capabilities in normal operating conditions, e.g.

- Rate of climb at sea level, clean con-figuration, all engines operating.

- Service ceiling altitude for max. rate of climb = 100 ft/min, .5 m/sec), clean configuration, all engines operating orone engine inoperative.

b. Airworthiness requirements, to ensure adequate performance for safety in normal and critical conditions, e.g.

- Minimum climb gradient in various config-urations (takeoff, en route, landing), one engine inoperative or all engines operating:

flaps deflected or retracted, flying at or above a specified flight speed. This item of performance is of particular interest for jet-propelled transports and will be dealt with in more detail in Chapter 11.6.

- Rate of climb at a specified altitude, one engine inoperative. Frequently the rate of climb is related to the stalling speed

(flaps down). This item of performance is of particular interest to piston-powered transports and all light aircraft.

Climb requirements may be categorized as follows.

a. Rate of climb at a fixed flight speed b. Rate of climb at optimum flight speed c. Climb gradient at a fixed flight speed d. Climb gradient at optimum flight speed In actual practice cases, b. and c. are by far the most important ones; case a. can be found in airworthiness requirements for transport aircraft with reciprocating en-gines. Case d. may incidentally occur where a takeoff climb gradient cannot be fulfilled at v 2min for transport category aircraft ( "overspeed") .

A special case of operational climb per-formance - the time to climb to a given altitude - will not be discussed here as no analytical procedure is available to convert such a requirement into combina-tions of design variables.

The various cases will first be dealt with as a general performance problem. Some ex-amples and applications will then be presented in order to illustrate the pro-cedure.

A useful general term for specifying climb performance is Specific Excess Power, SEP:

(5-48)

Excess ·power is available for climbing, accelerating and making turns. In the case of a steady climb at n = L/W ~ 1, SEP is identical to the rate of climb C, provided the angle of climb y is not too large (approximation: cosy = 1).

For horizontal flight at n 1, the SEP is

equivalent to the rate of increase of ki-netic energy and is therefore a measure of the time required to accelerate from one speed to another. In a horizontal turn at a specified rate of turn or load factor, the SEP represents the maneuvering and ac-celeration capability. Note that the di-mension of SEP is length opr unit time and not power.

a. Jet aircraft.

The thrust required is derived from eq.

5-48:

T SEP + CD

w --v

ⁿ CL (5-49)

1. In the ent for n

case 1,

of a specified climb gradi-steady flight:

dh/dt SEP (5-50)

y = --v-

--v

and eq. 5-49 yields:

T y + CD

(5-51)

w

_CL

The lift and drag coefficients are given by eqs. 5-31 and 5-9. Hence:

2 T l.,;yM CD

WL(J2S)

y + ⁰ (5-52)

w

^{W/(pS) +} _~yM2 11Ae

The minimum value for T/W corresponds to the minimum CD/CL ratio with

~ CL =\CD 11Ae at M =I WL!J2S) _o h\'CD 11Ae

I

0

=

M(L/D)max and consequently:

(5-53)

(5-54)

2. For a specified rate of climb for dV/dh=

0, n=1 and arbitrary V, eq. 5-49 is mod-ified to:

rn

^{2 2}

~= SE~{a

(:•) +

nV~ ~(~)

+ (:•)

I

^(5-55)

where

!.;

M' = ( n W/ (J2S) _{h 'c:;;Ae'}

I

⁼

^rn

^M _{(L/D) max}

\ Do

(5-56)

If the Mach number is specified, this e-quation can be used directly to obtain T/W.

However, in most cases a condition for M is sought for which the T/W ratio is a min-imum. A plot of T/W vs. M, together with a typical thrust lapse curve (Fig. 5-12)

in-.5

.4 .3

.2

.1

-<

^{/ /}

/

/'TYPICAL / ' THRUST CURVE

M Fig. 5-12. Required thrust in a climb with specified SEP

dicates that an acceptable approximation for T/W is found at the Mach number for which eq. 5-55 has a minimum value. This condition is:

(5-57)

The solution is presented in graphical form in Fig. 5-13. In general, it is ac-ceptable to calculate CD as follows:

w 0

c

D0 = 1.1 (d1 + d2 ~) S p 0 (5-58)

where the drag due to powerplant installa-tion is assumed at 10% of c₀ and d 1 and d 2

0

1.0

DRAG IGNORED

.6

2 6 8

are defined by eq. 5-34.

The general result for a specified rate of climb may be simplified for two special cases.

Case A: steady flight at low altitude, all engines operating, in order to achieve a specified rate of climb C at n 1. The contribution of the induced drag (second term in eq. 5-57) to the optimum T/W can be neglected. From Fig. 5-13 it is obvious that for sufficiently large values of C this approximation is acceptable. The con-dition for M is:

.5

.3

IGNORED

400

0

0 PURE JET TRAINERS,FIGHTERS lOW BPR JET TRANSPORTS

e HIGH BPR JET TRANSPORTS

2

800 FTo/MINM 1200

_c_ .rw;Ts

Fig. 5~14. Correlation of thrust/weight ratio and maximum rate of climb

162

10 12 14

~~

2n M

c

00 C/a W yC₀ pS

0

Fig. 5-13. Condition for the minimum thrust Mach number in a climb with specified SEP (jet aircraft)

(5-59)

Substitution into eq. 5-55 yields:

T 3 1/31

~Foo

^12/3

w

^{2 y} _yw/(ps) ^{- - - - +}

CD

1 r

¹³

+ ^{2 0} Vw/(ps>' _(5-60)

yl/3 ^11Ae

fyc;

_a D _{0 •}

In most cases this expression is quite accurate, provided C corresponds to low altitude performance with all engines operating. In the derivation of T/W, it was assumed that dV/dt = 0. In operational practice, however, a rate of climb is usually es-tablished in a flight with constant EAS or CAS. It can be shown that the acceleration necessary for flying at constant EAS at low altitude is given by:

(5-61)

This may be translated into an additional engine thrust required for a given C:

C X W )1/3

/;T _w = .567

£(;

_{a yc} ^pS

0 0

(5-62)

Instead of a detailed calculation; for which many data must be available, a statistical correlation of T/W and C//W7S at sea level may -be used, as given in Fig. 5-14.

Case B: flight at high altitude with low rate of climb, in order to achieve a spec-ified service ceiling. The contribution of

SEP in eq. 5-57 can be neglected and the Mach number for minimum T/W is equal toM'.

At the service ceiling, C = 100 ft/min (.5 m/sec), hence

.00147 (5-63)

and the thrust required at the ceiling is:

rc;:'

yc0 ^1t Ae'

!=2nY ;,;.g+.00123 o (5-64)

w nAe 18 ynw/ (ps)'

where the relative temperature 8 and the ambient pressure p refer to the ceiling.

The ratio of thrust at the ceiling tostat-ic thrust must be used to convert this T/W value into Tt₀/Wto•

b. Propeller aircraft.

For n = 1 the available power is equal to the power required to climb plus the power necessary to balance the drag. Assuming steady flight, SEP = dh/dt = C, and eq.

5-48 is modified into:

(5-65)

If C is defined at a given flight speed, the lift and drag coefficients are known and eq. 5-65 can be used directly.

To find the flight speed for which the power required to climb reaches a minimum, assuming a parabolic drag polar, the P/W ratio for given rate of climb is modified to:

l

^{c0 C 1/2 ,}

P _ 1 C + o + L 2W

w-;;-

_p

(c

_L ^3/2

11Ae)~

^(5-66)

For a given altitude and engine rating, P/W is affected by np and CL. For a parabolic polar, the value of the term:

c 1/2 + _L __

nAe

is minimum for cL =y3 c 0 nAe'. However, propeller efficiency gene~ally improveswith increasing airspeed and in practice themost favorable speed is roughly 20% higher than the speed for the minimum power required to

balance the drag. The result is:

at M

c 1/4

Do

\/W'

(nAe) 3/4 . pS

1 . 0 9 4Yw/(!)S) yc0 nAe'

0

(5-67)

The engine power* is generally the maximum continous (equivalent) power for turboprop engines or the rated (METO) power for

re-~iprocating engines, at the ambient ^condi~

tions for which C is specified.

In the case of a service ceiling, eq. 5-67 can be used by taking C = 100 ft/min

(0.5 m/s), hence

.00147 (5. 68)

while in (5-67) the speed of sound and the static pressure refer to the service ceil-ing.

c. Applications.

Airworthiness climb requirements are to be found in:

FAR 23.65 and 67

SFAR 23. Amendment 1 ch. 6 FAR 25.65 and 67, 25. 117-119-121 BCAR Chapter 02-4.

A survey of the most pertinent data for transport aircraft will be given in Chapter 11 Section 11.5. It is not intended to deal with all possible requirements in this sec-tion. Instead, some examples will be pres-ented to illustrate applications of the formulas already derived and the importance of climb requirements, with particular ref-erence to civil aircraft engine sizing.

Example 1

The rate of climb specified at sea level for a sub-sonic jet trainer is 4500 ft/min (23 m/s), corre-sponding to c/a = .0672. The weight is 7000 lb

(3170 kg), wing areaS= 210 sq.ft (19.5 m2) and

*divide (5-67) by 550 i f P is in hp, W in lb and C in ft/sec

163

A = 5.5, hence W/(p₀S) = .0156 and ~Ae

e = .8. Furthermore c 0 = .019.

To use Fig. 5-13, we c~lculate

13.8 for

1

^{W/ (p S)}

^I~

M' =

~y Jc:

^{Me . 209 and C/ao 2M'}

^v

^{(!fAe C} D ^{= 4. 33} 0 0

The figure indicates M' /M = • 59; hence M = • 35 for minimum T/W. Using eqs. 5-55 and 5-62, we calculate:

T w ^.309 for steady flight

f1T W = 013 for acceleration (constant EAS)

total T/W • .322

For a thrust lapse rate of .85 at M = .35, the takeoff SLS thrust/weight ratio must be at least . 38. The approximate equation 5-60 for steady flight yields T/W = .314, as compared with .309 for the "exact" sol uti on.

Example 2

For a twin-engined subsonic jet passenger trans-port, the ser ... ·ice ceiling with one engine failed is specified at 15,000 ft (4570 m). The following data are pertinent to the aircraft.

w 75,000 lb (34,000 kg); S = 850 sq.ft (79 m2 );

c 0 .018; A= 8.5 and e = .85, hence TrAe = 22. 7.

Fo~ engine failure, an 8% decrease of e is assumed:

TrAe = 20.9. At an altitude of 15,000 ft, W/(pS) .072 and 8 ~ .897. According to eq. 5-56,

M'

1

^.072

^I~ -

⁴¹

• 7 -/.018x20.9

We now calculate from eq. 5-55:

SEP/a ~ ~ 0645 2M' V CD •

0

From Fig. 5-13 we find that M' /M

=

.975, and there-fore the speed for minimum thrust at this altitude is M = .42. Equation 5-56 yields: T/W ~ .0624. The engine thrust lapse at an altitude of 15,000 ft and with M = .42 is .47. The thrust/weight ratio required at SLS is: Tt₀/Wto = .265. The effect of flying at constant EAS is neglected in view of the low rate of climb at the service ceil~ng.

Example 3

An important airworthiness requirement for civil aircraft is the so-called second-segment climb

gra-164

dient, laid down for example in FAR 25.121 (b). It states that with one engine inoperative, flaps in takeoff position, landing gear retracted, engines in the takeoff rating and out of ground effect, a specified minimum climb gradient is to be obtained at the takeoff safety speed v2 • This requirement must be met for all operational ambient conditions and may, especially on hot and high airfields, limit the authorized takeoff weight.

In the case of a subsonic passenger transport e-quipped with 3 engines, for example, the required climb gradient is 2. 7%. The ambient condit.ions are:

sea level, temperature 95 F (i.e. 35 C, ISA + 20 C).

Other data:

wto= 2~0,000 lb (95,000 kg), s ~ 2,060 sq.ft (192 m ) ; V2 ^~ 1.2 Vs with CL-max = 2.40 for the takeoff. Aspect rat:io: A= 7.5; Section 5.3.5. is used to estimate the lift/drag ratio in symmetrical flight with flaps deflected at CL = CL -max/ ( 1. 2) 2 1.67.

We find: C0 /CL = .094 for a slatted wing, hence C0/CL = .10, assuming an increment of 5% for drag due to engine failure. The Mach number at v_{2 is:}

From eq. 5-51 we now find: T/W = .027 + .10 .127 for two engines operating at M = .2 and 95 F (35 C).

For a thrust lapse rate of .75, including the effect of non-standard temperature, the total takeoff thrust {SLS, ISA) must be at least:

N Tto/Wto = N ~1

e

T/Tto = .254 T/W

For Wto ~ 210,000 lb (95,000 kg), the thrust per engine must be at least 17,780 lb {8,054 kg).

Example 4

Light aircraft with W < 6,000 lb (2720 kg) must comply with the requirement FAR 23.67. With takeoff power at sea level, undercarriage down an? flaps in the takeoff position:

C ~ 300 ft/min, or c/a • .0045 and

c • 11.5 v8 ft/min;

where v5 = ~qui valent stalling speed (knots).

1

The first requirement can be substituted directly into eq. 5-67. The second can be evaluated as follows: