1American Institute of Aeronautics and Astronautics

Effect of Fan Pressure Ratio and Bypass Ratio onBusiness Jet Range Performance

Roy Y. Myose,*

Wichita State University, Wichita, KS 67260-0044

Trevor Young†

Mechanical & Aeronautical Engineering Department, University of Limerick, Limerick, Ireland

and Ismael Heron‡

Eclipse Aviation Corporation, Albuquerque, NM 87106

A model for the aircraft performance including a Brayton turbofan engine cycle analysis was developed inorder to study the effect of varying the fan pressure ratio and bypass ratio on the range performance of amedium-size business jet. In the classic flight mechanics treatment of aircraft performance, maximizing range

L Drequires maximizing C /C . This classic approach, however, assumes that thrust specific fuel consumption is1/2

a constant. Previous studies, which include the effect of variation in thrust specific fuel consumption, typicallyseek to determine the optimum flight Mach number under a constant altitude - constant speed flight condition.This project was motivated by a desire to study what effect variations in flight conditions such as speed, lift-to-drag ratio, or thrust settings have on aircraft performance. An increase in range of approximately 2% to 8%

L Dwas obtained, depending upon the bypass ratio, by maximizing U(L/D)/TSFC compared to the case where C /C1/2

was maximized.

Nomenclature

L/D = lift-to-drag ratioM = Mach number

rfP = fan pressure ratioT = static temperature (subscript 2 indicates compressor inlet location)

0T = total temperature (subscript 04 indicates engine burner exit location)

= thrust

TSFC = thrust specific fuel consumptionU = flight speed (i.e., true air speed)W = aircraft weight" = coefficient in approximate TSFC relationship, eq. (10)$ = turbofan bypass ratioD = air density

Introduction

In the classic flight mechanics treatment of jet aircraft performance, the range is found to be a function of the1

flight speed (i.e., true air speed) U, lift-to-drag ratio L/D, thrust specific fuel consumption TSFC, and aircraft weightW according to the relationship:

(1)

In eq. (1), the range is given in nautical miles if flight speed is given in knots and thrust specific fuel consumption isgiven in lb/hr/lb (i.e., fuel consumption rate in lb/hr for each pound of thrust produced). In order to obtain some generaltrends, most textbooks assume that the flight speed, lift-to-drag ratio, and thrust specific fuel consumption are constants.Under these simplifying assumptions, eq. (1) becomes the well-known Breguet range equation for jets, which is givenby:

Professor, Department of Aerospace Engineering, Associate Fellow AIAA.*

Senior Lecturer, Senior Member AIAA.†

Flight Test Engineer, Member AIAA.‡

6th AIAA Aviation Technology, Integration and Operations Conference (ATIO)25 - 27 September 2006, Wichita, Kansas

AIAA 2006-7706

Copyright © 2006 by Myose, Young, and Heron. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2American Institute of Aeronautics and Astronautics

(2)

Maintaining a constant lift-to-drag ratio implies that the airplane flies a cruise-climb, in which a gradual increase inheight occurs as the weight is reduced (other solutions to eq. (1) are also possible). For given initial and final aircraftweights, maximizing range means that U(L/D)/TSFC must be maximized. Most textbooks focus on the aircraftperformance aspects rather than on propulsion analyses, by simply assuming that TSFC does not vary. In this case,maximizing range requires that U(L/D) be maximized. In level flight, where the lift L is equal to the aircraft weight W,the lift coefficient is given by:

(3)

Dwhere D is the air density and S is the wing reference area. The drag coefficient C may be adequately represented, atspeeds below which significant compressibility drag rise occurs, as a parabolic function of the lift coefficient (seeAnderson , for example):1

(4)

D,0where C is the drag coefficient at zero lift and the induced drag factor K is given by:

(5)

In eq. (5), e is the Oswald span efficiency factor and AR is the wing aspect ratio, which is given by wing span squareddivided by the wing area (b /S). The lift-to-drag ratio can be obtained by taking eq. (3) and dividing it by eq. (4), which2

results in the following:

(6)

It can be shown (cf. Anderson ) that the maximum lift-to-drag ratio value is not a function of weight nor of speed and1

is given by:

(7)

Although eq. (7) specifies the maximum lift-to-drag ratio value, this does not correspond to the situation where theproduct U(L/D) is maximized. The relationship for velocity times lift-to-drag ratio can be obtained by noting that lift

Lis equal to weight in level flight and that the drag can be written in terms of the drag coefficient. Based on eq. (3), C 1/2

is equal to the square root of weight divided by dynamic pressure and wing area. This means that the relationship forvelocity times lift-to-drag ratio can be written as:

(8)

For a given weight and flight altitude, the bracketed term in eq. (8) is a fixed value. Thus, maximizing U(L/D) means

L D L Dmaximizing C /C . It can be shown (cf. Anderson ) that maximum C /C occurs at a speed corresponding to the1/2 1 1/2

requiredtangent point of the thrust required curve. For steady, level flight, the thrust required is equal to the total dragD, which can be expressed in terms of drag coefficient given in eq. (4). This means that thrust required can be writtenas:

(9)

D,0where C and K are essentially fixed constants for a given aircraft, for speeds up to the drag rise speed, while flightat a particular altitude defines the air density D. Thus, for a given weight W, the thrust required is a function of velocity

3American Institute of Aeronautics and Astronautics

squared.Up until this point, the thrust specific fuel consumption was assumed to be constant. This is not the case, however,

in actual engines. Attempts to quantify the effect of variations in thrust specific fuel consumption have been studiedby a number of researchers in the past. The focus of these studies was to determine the optimum flight Mach number2-4

under a constant altitude - constant speed flight condition. A convenient model (see Martinez-Val and Perez , for3

example) that may be used to represent TSFC at constant altitude is:

(10)

where " varies from 0.4 to 0.7 on high bypass ratio turbofan engines, 0.2 to 0.4 on low bypass ratio turbofan engines,and 0 on turbojet engines. However, " = 0 means that TSFC is a fixed constant, independent of Mach number.Although TSFC does not vary much with speed when a turbojet engine is at full throttle, it is nevertheless not a fixed

0constant. In a previous study, Myose et al. showed that eq. (10) with TSFC = 0.92 lb/hr/lb and " = 0.08 was a5

reasonably good fit for the turbojet at full throttle. Interestingly, however, the coefficient values had to be changed whenthe engine was throttled back. All aircraft under cruise flight conditions are flown with the engine thrust reduced sothat it matches the thrust required given in eq. (9). For a medium-size business jet under cruise flight condition, the

0reduced thrust requirement resulted in a change in the coefficient values to TSFC = 0.84 lb/hr/lb and " = 0.5. Thus,an analytic approach such as that given in eq. (10) results in some error in the aircraft performance calculations if a fixedpair of coefficients are used.

The present investigation is an extension of Myose et al. , which considered several different flight strategies5

including constant speed, constant lift-to-drag ratio, constant thrust, always maximizing U(L/D), and always maximizingU(L/D)/TSFC for a turbojet engine powered medium-size business jet. Although a number of different flight strategiesprovided good range performance, the best range was obtained by always maximizing U(L/D)/TSFC. Earlier in theintroduction section, a comment was made that flight mechanics textbooks tend to focus on the aircraft performanceaspects rather than the details of the engine’s thermodynamic performance. Conversely, the large number of excellentjet propulsion textbooks tend to focus on the engine’s thermodynamic performance rather than its performance afterthe engine is mated to an aircraft. Although aircraft designers in industry have access to engine "decks" (i.e., specificengine performance data) from manufacturers, this information is typically not available to the general public.Consequently, the goal of the present paper is to consider what effect fan pressure ratio and bypass ratio have on therange performance of a medium-size business jet. This requires a model for not only the aircraft performance portion,but also the propulsion analysis, which is described in the following section.

Aircraft and Engine Model for a Medium-Size Business Jet

The aircraft considered in this study is a medium-size business jet (see specifications in Table 1). The parasite dragcoefficient value is somewhat on the high side so these specifications correspond to an older legacy-type business jet.A rough estimate for the amount of fuel consumed prior to the start of cruise was necessary. Raymer suggests that the6

typical aircraft weight at the start of cruise is 95.5% of the gross take-off weight. (The use of this value provided arealistic starting point for the study, but does not, in fact, influence the results presented herein.) This corresponds toan aircraft weight of 19,100 lb at the start of cruise for the medium-size business jet specified in Table 1. Obviously,the cruise start weight would vary depending on the cruise altitude chosen. In the present study, the cruise altitude wasfixed to 35,000 ft, and, for the sake of simplicity, the cruise start weight rounded-off to 19,000 lb. The end of cruiseweight was taken at the aircraft empty (zero fuel) weight. Two other assumptions were made regarding the aircraftresponse during cruise. The lift required to match the aircraft weight at all times during the cruise was assumed to beautomatically obtained by angle of attack adjustments made by the auto-pilot. This is, in essence, what the auto-pilotperforms in order to hold the altitude constant during cruise. Finally, high speed (transonic) drag was not included inthe calculations of this work. This is not a problem on older legacy-type business jets since they typically cruise atmoderate Mach numbers when range is being maximized.

The engines being considered in this study provide a total thrust of 8,000 lb (from two engines) under sea-leveltake-off conditions. Engine component parameters such as the efficiency and ratio of specific heats given in Table 2are typical of those found in most jet engines. When engine manufacturers provide performance data, the information7

is always given in terms of 100% ram air recovery. Once the engine is mated to an aircraft and brought to flightcondition, however, there is a loss in pressure recovery by the engine inlet (i.e., in the diffuser section). Figure 1, takenfrom Hill and Peterson , provides the typical diffuser efficiency as a function of Mach number. An assumption was7

2made that the diffuser with the efficiency given in Figure 1 provides compressor inlet Mach number of M = 0.3. In

a, take-offorder to attain the sea-level take-off thrust specified in Table 1, a certain amount of air mass flow rate m0 mustbe provided. By specifying the compressor inlet Mach number and knowing the required air mass flow rate, the

2compressor cross-sectional area A is then defined. This physically sizes the engine so that the air mass flow rate atcruise speed and altitude can be determined from:

4American Institute of Aeronautics and Astronautics

(11)

CHere, the subscript 2 indicates conditions corresponding to the compressor inlet, g is the ratio of specific heats for thecompressor (given in Table 2), and R is the gas constant for air [R =1716 ft-lb/(slug- R)]. The standard Braytono

turbofan cycle analysis model provides thrust specific fuel consumption and thrust per air mass flow rate for a given7

flight speed and altitude. The cruise air mass flow rate value determined from eq. (11) is then multiplied with the thrustper air mass flow rate result from the Brayton cycle analysis to determine the raw amount of thrust produced by the

04turbofan engine at flight conditions. At maximum throttle, the burner exit temperature T is set to the maximum

04allowable value of 2800 °R. If a thrust less than the maximum value is desired, then the burner exit temperature T isreduced below the maximum allowable until the desired thrust is obtained. One additional assumption was maderegarding turbojet engine response. Hill and Peterson notes that there is a one-to-one relationship between shaft speed7

04and fuel flow rate, which in turn would depend upon the burner exit temperature T . Thus, a change in engine throttlesetting is assumed to be analogous to a change in the burner exit temperature in the results that follow.

Compared to the turbojet engine considered in Myose et al. , there were two additional variable parameters with5

the turbofan engine. The first variable parameter was the bypass ratio, which compares the amount of air mass flowrate going through the bypass fan against the air mass flow rate going through the gas turbine core. In this study, fourdifferent engine types were considered: (1) a turbojet with $=0 (the results of which were reported earlier in Myose etal. ), (2) low bypass ratio turbofan with $=1, (3) moderate bypass ratio turbofan with $=3, and (4) high bypass ratio5

turbofan with $=5. The turbofan engine’s bypass ratio is not necessarily a fixed quantity, but can change between sea-level take-off condition and cruise speed at altitude. However, this change in bypass ratio tends to occur at lowaltitudes. Consequently, it will be assumed that the bypass ratio at sea-level take-off is the same as the bypass ratio at8

cruise speed and altitude for a given turbofan engine type.

rfThe second parameter that was varied in this study was the fan pressure ratio P . Once the engine type is selected(i.e., bypass ratio is fixed), the fan pressure ratio effectively determines the distribution of the available kinetic energyoutput between the gas turbine core’s exhaust and the bypass fan’s exhaust. As illustrated schematically in Figure 2,

e,fanrfincreasing the fan pressure ratio P increases the bypass fan exhaust velocity u while decreasing the gas turbine core

e,coreexhaust velocity u . Fan pressure ratios well in excess of 2 are found on multi-stage fans (i.e., the low pressurecompressor section) such as those commonly used in low bypass ratio turbofan engines. On moderate to high bypassratio turbofans, fan pressure ratios tend to be limited to about 1.8 using a single stage fan blade. In this case, the fan9

f Cefficiency 0 should be better than that of a multi-stage compressor, which has an efficiency 0 of about 85%. Eventhough there should be an efficiency improvement with increased bypass ratio using a single stage fan blade, no attemptwas made to vary the fan efficiency. Finally, a larger bypass ratio increases the engine cross sectional area, which, inturn, would increase the aircraft drag. There was no attempt to model this increase in drag on the high bypass ratioturbofan. A number of past studies have considered the problem of fan pressure ratio optimization for turbofans;9,10

however, these studies tend to consider high or ultra-high bypass ratio turbofans, which are typically used on largertransport aircraft.

Results

The effect of fan pressure ratio and bypass ratio variations on turbofan performance was considered under fullthrottle and partial throttle conditions at the start and end of cruise. Figure 3 shows the TSFC and thrust at full throttleas a function of speed and fan pressure ratio for (a) low, (b) moderate, and (c) high bypass ratio turbofan engines atcruise altitude. For comparison sake, the thrust required at the start and end of cruise are also shown. Except for lowand high speed operations, the engine provides excess thrust compared to thrust required for cruise. In general, the full

rfthrottle fuel efficiency improves (i.e., TSFC is reduced) as the fan pressure ratio P is increased. As discussed earlier(for Figure 2), increasing the fan pressure ratio increases the exhaust velocity through the bypass fan, which is moreefficient than the gas turbine core. However, continually increasing the fan pressure ratio does not provide the optimumperformance. For reasons to be discussed later (i.e., for the partial throttle end of cruise condition), the moderate fanpressure ratio cases indicated by the red solid-line curves were selected for detailed analysis. Figure 4 overlays theTSFC and thrust results as a function of speed for the various bypass ratio turbofans at full throttle. Also shown are theturbojet results ($=0) and the thrust required at the start and end of cruise. In general, the higher bypass ratios improvethe fuel efficiency (i.e., TSFC is reduced), but the thrust output is reduced. It should be noted that the variation in TSFCwith speed is shallow at the lower bypass ratios whereas the variation is steeper at the higher bypass ratios. This isconsistent with the TSFC relationship given in eq. (10). In the present study, the coefficients were " = 0.15 for lowbypass ratio ($=1), " = 0.25 for moderate bypass ratio ($=3), and " = 0.34 for high bypass ratio. This compares with0.2 # " # 0.4 for low bypass ratio and 0.4 # " # 0.7 for high bypass ratio suggested by Martinzez-Val and Perez.3

Figure 5 presents the TSFC results as a function of bypass ratio and speed for the partial throttle case at the start

04of cruise (i.e., for aircraft weight of 19,000 lb). Here, the burner exit temperature T was reduced until the engine thrust

5American Institute of Aeronautics and Astronautics

matched the thrust required at the start of cruise. In general, the partial throttle fuel efficiency is significantly improvedover (i.e., lower TSFC compared to) the full throttle case. It should be noted that the tangent point for the thrust required

L Dcurve (i.e., where C /C is a maximum) does not correspond to the point where TSFC is a minimum. It therefore1/2

L Dfollows that, although maximizing C /C maximizes U(L/D), it will not maximize U(L/D)/TSFC. Figure 6 shows the1/2

partial throttle TSFC as a function of speed and fan pressure ratio for (a) low, (b) moderate, and (c) high bypass ratioturbofan engines at the end of cruise (i.e., for aircraft weight of 14,000 lb). For comparison sake, the thrust required

L Dat the end of cruise and the velocity for maximum C /C are also shown. At this cruise condition, increasing the fan1/2

pressure ratio to too high a level results in a deterioration in the fuel efficiency (i.e., increases the TSFC) for themoderate and high bypass ratio turbofans. In fact, this deterioration in fuel efficiency occurs right around the speedswhere maximum range would be achieved. As the turbofan is throttled back to match the low thrust requirement, theraw amount of thermal energy available from the burner is significantly reduced. With a relatively high fan pressureratio, most of the available thermal energy is used to drive the bypass fan and very little remains for the gas turbine coreexhaust. This significantly erodes the efficiency of the gas turbine core to the point of reducing the overall efficiencyof the turbofan for the end of cruise condition. Figure 7 overlays the TSFC results for the various bypass ratios at the

L Dstart and end of cruise. It is clear that the velocity for maximum C /C does not correspond to the point where TSFC1/2

L Dis a minimum. Thus, maximizing range requires a speed that is less than the velocity for maximum C /C .1/2

Figure 8 presents the range achieved by the high, moderate, and low bypass ratio turbofans as well as the turbojetdriven medium-size business jet. Here, the business jet was flown in a manner that continually maximizedU(L/D)/TSFC during the entire cruise cycle. Also shown are the range achieved by turbofans with a less than optimumfan pressure ratio. Figure 9 presents the range achieved by the same turbofan (i.e., optimum fan pressure ratio case)when the business jet is flown to continually maximize U(L/D) compared to continually maximizing U(L/D)/TSFC.Improvements in the range performance by about 2% (for turbojet) up to about 8% (for high bypass ratio turbofan) wasachieved by flying in a manner that continually maximizes U(L/D)/TSFC. The aircraft characteristics that are requiredto continually maximize U(L/D)/TSFC are presented in Figure 10. An earlier investigation showed that optimizing the5

range performance on a turbojet powered medium-size business jet involved a lift-to-drag ratio that was effectively aconstant throughout the cruise. This condition is illustrated by the solid-line curve in Figure 10a. Also shown in thefigure are the lift-to-drag ratio variations for turbofans with bypass ratios that are low (dash-dot curve), medium (dash-dot-dot curve), and high (dotted curve). As the bypass ratio is increased, a larger reduction in the lift-to-drag ratio isrequired by the end of cruise. However, the variations in the lift-to-drag ratio are still relatively modest (less than about3%) irrespective of engine type. Figure 10b shows that the variation in cruise speed required for range optimizationdecreases as the bypass ratio is increased. However, the cruise speed variations are again relatively modest irrespectiveof whether the engine is a high bypass turbofan or a turbojet.

At first glance, the thrust required curves for the start (W=19,000 lb) and end of cruise (W=14,000 lb) shown inFigure 3 do not seem to show much of a difference. However, the thrust axis scale has been compressed in this figurein order to incorporate the overlaying of TSFC information. Figure 10c shows that the thrust must be reduced by about30% between the start and end of cruise in order to optimize the range. Figure 3 shows that there are large variationsin TSFC as the speed is varied. However, the cruise speed variations shown in Figure 10b are relatively modest. Thus,the variation in TSFC illustrated in Figure 10d is also relatively small. Finally, Figure 10e shows the product of flightspeed, lift-to-drag ratio, and reciprocal of TSFC, which would be incrementally added up during the cruise to providethe maximum range performance.

Summary

A model for the aircraft performance including a Brayton turbofan engine cycle analysis was developed in orderto study the effect of varying the fan pressure ratio and bypass ratio on the range performance of a medium-size business

L Djet. It was noted that the speed at which C /C is a maximum does not correspond to the point where TSFC is a1/2

L Dminimum. It follows that by maximizing C /C the parameter U(L/D) will be maximized, but this will not necessarily1/2

maximize the full range of the parameter U(L/D)/TSFC. An increase in range of approximately 2% to 8% was obtained,

L Ddepending upon the bypass ratio, by maximizing U(L/D)/TSFC compared to the case where C /C was maximized.1/2

References

Anderson, J.D., Aircraft Performance and Design, McGraw-Hill, New York, 1999.1

Bert, C.W., "Prediction of Range and Endurance of Jet Aircraft at Constant Altitude," Journal of Aircraft, Vol. 18, No.2

10, 1981, pp. 890-892.

Martinzez-Val, R. and Perez, E., "Optimum Cruise Lift Coefficient in Initial Design of Jet Aircraft," Journal of Aircraft,3

Vol. 29, No. 4, 1991, pp. 712-714.

Cavcar, M. and Cavcar, A., "Comparison of Generalized Approximate Cruise Range Solutions for Turbojet/Fan4

6American Institute of Aeronautics and Astronautics

Aircraft," Journal of Aircraft, Vol. 40, No. 5, 2003, pp. 891-895.

Myose, R., Young, T., and Sim, G., September 2005, "Comparison of Business Jet Performance Using Different5

Strategies for Flight at Constant Altitude," AIAA paper 2005-7326, 5th AIAA ATIO Conference, Crystal City, VA.

Raymer, D.P., Aircraft Design: A Conceptual Approach, AIAA Education Series, Reston, 1999.6

Hill, P. and Peterson, C., Mechanics and Thermodynamics of Propulsion, Addison-Wesley, New York, 1992.7

Mattingly, J.D., Elements of Gas Turbine Propulsion, McGraw-Hill, 1996.8

Guha, A., "Optimum Fan Pressure Ratio for Bypass Engines with Separate or Mixed Exhaust Streams," Journal of9

Propulsion and Power, Vol. 17, No. 5, 2001, pp. 1117-1122.

de Luis, J. and Mavris, D.N., September 2004 "Prediction Methodology of an Optimum Turbofan Engine Cycle,"10

AIAA paper 2004-4363, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY.

American Institute of Aeronautics and Astronautics

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0.85

0.86

0.87

0.88

0.89

0.90

0.91

0.92

0.93

0.94

0.95

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Mach Number

Diff

user

Eff

icie

ncy

Table 1 – Specifications for the medium-size business jet.

Gross take-off weight 20,000 lb Empty (zero-fuel) weight 14,000 lb

Parasite drag coefficient CD,0 = 0.02

Oswald efficiency factor e = 0.8

Wing area S = 300 ft2

Span b = 50 ft

Table 2 – Specifications for the turbojet and turbofan engines.

Pressure ratio η γ

Sea-level take-off thrust (total from two engines) [lb] 8,000

Diffuser efficiency ηf & ratio of specific heats γd varied (Fig. 1) 1.4

Bypass fan pressure ratio Prf, efficiency ηf, & ratio of specific heats γf varied (Fig. 2) 0.85 1.4

Compressor pressure ratio Prc, efficiency ηc, & ratio of specific heats γc 30 0.85 1.37

Burner pressure ratio Prb & ratio of specific heats γb 1 1.35

Jet fuel heating value Qr [ft-lb/(slug-oR)] 4.83x108

Maximum cycle temperature T04 [oR] T04 < 2800

Turbine efficiency ηt & ratio of specific heats γt 0.9 1.33

Nozzle efficiency ηn & ratio of specific heats γn 0.98 1.36

Bypass fan nozzle efficiency ηnf & ratio of specific heats γnf 0.98 1.4

Figure 2 – Turbofan exhaust velocity schematic. Increasing fan pressure ratio Prf increases the fan exhaust Figure 1 – Turbojet diffuser efficiency as a function of Mach number from velocity ue,fan, but this causes the core Hill and Peterson.7 exhaust velocity ue,core to decrease.

American Institute of Aeronautics and Astronautics

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TSFC at max thrustPrf = 1.8

2.02.2

Max engine thrust

W = 19,000 lbThrust required

W = 14,000 lb

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ust (

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(a) Low bypass ratio (β = 1) turbofan.

Prf = 1.4

TSFC at max thrust 1.61.8

Max engine thrust

W = 19,000 lb

W = 14,000 lb

Thrust required

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(b) Moderate bypass ratio (β = 3) turbofan.

Figure 3 – Full throttle TSFC (red upper-most curves) and maximum engine thrust (blue curves) as a function of fan pressure ratio (Prf) and speed for turbofan at cruise altitude (35,000 ft). Also shown is the thrust required (green center-line curves) as a function of aircraft weight and speed at cruise altitude (35,000 ft).

American Institute of Aeronautics and Astronautics

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Prf = 1.2

TSFC at max thrust 1.4

1.6

Max engine thrust

W = 19,000 lb

W = 14,000 lb

Thrust required

0.0

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(c) High bypass ratio (β = 5) turbofan.

Figure 3 (continued).

TSFC at max thrust β0

135

Max engine thrust 0β

1

35

W = 19,000 lbThrust required

W = 14,000 lb

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ust (

lb)

Figure 4 – Full throttle TSFC (red solid-line curves) and maximum engine thrust (blue dashed-line curves) as a function of bypass ratio (β) and speed at cruise altitude (35,000 ft). Also shown is the thrust required (green center-line curve) as a function of aircraft weight and speed at cruise altitude (35,000 ft).

American Institute of Aeronautics and Astronautics

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0

TSFC forW = 19,000 lb

β

Velocity for max CL

1/2/CD

13

5Thrust for

W = 19,000 lb

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Figure 5 – Partial throttle TSFC (red upper-most curves) as a function of bypass ratio (β) and speed at the start of cruise (W = 19,000 lb). Turbofan maximum cycle temperature was adjusted until the engine thrust was equal to the thrust required (blue lower-most curve) at cruise altitude (35,000 ft) for this aircraft weight.

TSFC forW = 14,000 lb

Velocity for max CL

1/2/CD

1.8

Prf = 1.82.0

Prf = 2.2

2.2

Thrust forW = 14,000 lb

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ust (

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(a) Low bypass ratio (β = 1) turbofan.

Figure 6 – Partial throttle TSFC (red upper-most curves) as a function of fan pressure ratio (Prf) and speed at the end of cruise (W = 14,000 lb). Turbofan maximum cycle temperature was adjusted until the engine thrust was equal to the thrust required (blue lower-most curve) at cruise altitude (35,000 ft) for this aircraft weight.

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TSFC forW = 14,000 lb

Velocity for max CL

1/2/CDPrf = 1.4

1.6

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(b) Moderate bypass ratio (β = 3) turbofan.

TSFC forW = 14,000 lb

Velocity for max CL

1/2/CDPrf = 1.2

1.41.6

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0.9

1.0

0 100 200 300 400 500 600

Speed (knots)

TSF

C (l

b/hr

/lb)

0

1000

2000

3000

4000

5000

6000

Thr

ust (

lb)

(c) High bypass ratio (β = 5) turbofan.

Figure 6 (continued).

American Institute of Aeronautics and Astronautics

12

β

W = 19,000 lb

0

Velocity for max CL

1/2/CD

W = 19,000 lb

TSFC

Thrust

βW = 14,000 lb

0

W = 14,000 lb

11335

5W = 19,000 lb

W = 14,000 lb

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 100 200 300 400 500 600

Speed (knots)

TSF

C (l

b/hr

/lb)

0

1000

2000

3000

4000

5000

6000

Thr

ust (

lb)

Figure 7 – Partial throttle TSFC as a function of bypass ratio (β) and speed at cruise start (red solid line curves) and cruise end (red dashed line curves). Turbofan maximum cycle temperature was adjusted until the engine thrust was equal to the thrust required (blue curves) at cruise altitude (35,000 ft) for the relevant aircraft weight.

β

β

β

β

β

β

β

β

β

β

2400 2500 2600 2700 2800 2900 3000 3100 3200 3300

= 5 Prf = 1.6

= 5 Prf = 1.4

= 5 Prf = 1.2

= 3 Prf = 1.8

= 3 Prf = 1.6

= 3 Prf = 1.4

= 1 Prf = 2.2

= 1 Prf = 2.0

= 1 Prf = 1.8

= 0 (turbojet)

Range (nautical miles)

Figure 8 – Maximum range as a function of bypass ratio (β) and fan pressure ratio (Prf) at cruise altitude (35,000 ft). Maximum range was obtained by adjusting the speed and lift-to-drag ratio until U(L/D)/TSFC was maximized.

American Institute of Aeronautics and Astronautics

13

β

β

β

β

β

β

β

β

2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300

= 5 Max U*(L/D)/TSFC

= 5 Max U*(L/D)

= 3 Max U*(L/D)/TSFC

= 3 Max U*(L/D)

= 1 Max U*(L/D)/TSFC

= 1 Max U*(L/D)

= 0 Max U*(L/D)/TSFC

= 0 Max U*(L/D)

Range (nautical miles)

Figure 9 – Range as a function of bypass ratio (β) and flight strategy at cruise altitude (35,000 ft). The aircraft was flown by adjusting the speed and lift-to-drag ratio to maximize either U(L/D) or U(L/D)/TSFC.

β

0

5

31

Max L/D=16.18

15.0

15.2

15.4

15.6

15.8

16.0

16.2

14,00015,00016,00017,00018,00019,000Weight (lb)

L/D

(a) Lift-to-drag ratio.

Figure 10 – Performance characteristics as a function of bypass ratio (β) when the aircraft is flown to maintain maximum U(L/D)/TSFC throughout cruise at altitude (35,000 ft).

American Institute of Aeronautics and Astronautics

14

β0

531

0.5

0.6

0.7Mach

250

300

350

400

450

14,00015,00016,00017,00018,00019,000Weight (lb)

Spee

d (k

nots

)

(b) Flight speed.

β0

5

31

900

1000

1100

1200

1300

14,00015,00016,00017,00018,00019,000Weight (lb)

Thru

st (l

b)

(c) Engine thrust.

Figure 10 (continued).

American Institute of Aeronautics and Astronautics

15

0β

5

3

1

0.40

0.50

0.60

0.70

14,00015,00016,00017,00018,00019,000Weight (lb)

TSFC

(lb/

hr/lb

)

(d) Thrust Specific Fuel Consumption.

0

β5

3

1

7000

8000

9000

10000

11000

14,00015,00016,00017,00018,00019,000Weight (lb)

U(L

/D)/T

SFC

(nau

tical

mile

s)

(e) Flight speed times lift-to-drag ratio divided by TSFC.

Figure 10 (continued).