Flight for Range and Endurance

8/10/2019 Flight for Range and Endurance

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Flight for Range and Endurance



Contents

Definition of Power ............................................................................................................................... 4

A Dynamometer ................................................................................................................................ 4

Shaft Horsepower (SHP) .................................................................................................................. 5

Consult the POH for BHP ................................................................................................................ 6

De-rated Engines ............................................................................................................................... 7

Emergency Power ............................................................................................................................. 7

Power Available (Pa) ............................................................................................................................ 7

Propeller Efficiency .......................................................................................................................... 8

The Pa Graph .................................................................................................................................. 11

Power Required (Pr) ........................................................................................................................... 12

The Pr vs. Velocity Graph .............................................................................................................. 13

Flight for Range and Endurance - Jet Airplanes ............................................................................. 15

How a "True" Jet Engine Works .................................................................................................. 16

Thrust Specific Fuel Consumption (TSFC) .................................................................................. 17

Ta = Drag ......................................................................................................................................... 17

Mass Flow Affects The Value of TSFC (the Bypass Engine) ...................................................... 17

Air Temperature Affects the Value of TSFC ............................................................................... 19

Engine RPM Affects the Value of TSFC (effect of air density on thrust produced) ................. 19

Fuel Flow vs. Velocity Graph ......................................................................................................... 21

The Simulation ................................................................................................................................ 22

Maximum Endurance at L/Dmax .................................................................................................... 23

Maximum Range at L1/2/Dmax ......................................................................................................... 23

Effect of Weight on Specific Range (and Endurance) ................................................................. 26

Effect of Altitude on Specific Range (and Endurance) ................................................................ 30

Cruise Control ................................................................................................................................. 31

How Wind Affects Range and Optimum Cruise Speed ............................................................... 33

Flight for Range and Endurance- Propeller Airplane ............................................................................ 35

Two Types of Propeller Airplane ....................................................................................................... 35

Specific Fuel Consumption (SFC) ....................................................................................................... 36

Pa = Pr ............................................................................................................................................... 37

What Affects SFC? ............................................................................................................................. 37

Air Temperature Affects the Value of SFC for Turboprops - but not Piston ..................................... 37

Engine RPM Affects the Value of SFC for Turboprops - Less so for Piston ....................................... 37

Fuel Flow vs. Velocity Graph ............................................................................................................. 38

The Simulation .................................................................................................................................. 39





Aerodynamics Index

Definition of Power

Before we turn to investigating range and endurance for propeller airplanes we must

define power. We must do this because for a propeller driven airplane fuel flow is

proportional to power. We will return to discuss fuel flow and its relationship to power on

the next page. For now we should examine what power is.

Power (P) measures the amount of work (w) the engine does in a unit of time (t.)Mathematically we say that:

P = w/t

Work measures the force that is applied over a given distance (d.) Mathematically we say

that:

w = F x d

F x d may be familiar to you, it is simply torque. So to measure the power output of an engine

all we need to measure is:

Power output of engine = (F x d) / t

A Dynamometer

A dynamometer is a device that measures the power output of an engine. In the past the

method of measurement was to have a "brake like" device clamp onto the output shaft and

measure the torque by equating torque to the amount of braking force required to stop the

engine. Modern dynamometers are much more sophisticated but the term Brake

Horsepower (BHP) has survived.

http://selair.selkirk.bc.ca/training/aerodynamics/index.html





A Dynamometer measures BHP - i.e. the output power of an engine. It works by measuring

torque and rpm (Note that rpm is a unit of time/(2 x pi).) These are the only two items

needed. Mathematically:

BHP = torque x rpm x (unit correction factor)

In the metric system torque is measured in units of Newton x meter. RPM is by definition

revolutions per minute. Power is in units of Watts, which requires the time unit be seconds

rather than minutes so you would need a correction factor of (2 x pi)/60.

In the engineering unit system torque is measured in units of lb x ft. RPM is measured in

revolutions per minute. Power is in units of Horse Power. One horsepower is defined as

33,000 lb x ft / min. The appropriate unit correction factor is (2 x pi)/33,000, or 5255.

Shaft Horsepower (SHP)

Turboprop engines are jet engines that have propellers attached to the front. A jet engine

turns at very high rpm, but produces very little torque. A typical turbine engine runs at

30,000 rpm. This rpm is far too high for a propeller. A propeller turning at such high rpm

would either be just a few inches in diameter or would have tip speeds far in excess of the

speed of sound. The solution is to attach a gearbox to the front of the turbine engine. As a

result the engine turns the gearbox and the gearbox turns the propeller.



The gearbox on a turboprop reduces rpm by a fixed ratio - typically about 15:1.

In a perfect (frictionless) gearbox the power does NOT change as it passes through. At the

input end the rpm is high and the torque is low, at the prop shaft the rpm is low and the

toque is high. (Note that when a gearbox reduces rpm it increases torque. The same is true

of the gearbox on your car. In first gear the rpm of the wheels is low but the torque is high.

In overdrive the wheel rpm is high but the torque is low.)

In the real world there is always some loss of power due to friction with a gearbox, however a

good gearbox is more than 98% efficient. I.E. power output is 98% of power input.

When using a dynamometer it is normal to measure the power at the output shaft of the

gearbox. In such cases the term Shaft Horsepower (SHP) is used to remind us that a gearboxis involved. For the remainder of this course we will use the term BHP and not mention SHP

again. You may take everything said about BHP to be equally applicable to SHP.

Consult the POH for BHP

Every airplane has a pilot operating handbook (POH.) In the POH you will find the rated

BHP of the engine(s.)

The term "normally aspirated" means that the air entering the intake of the engine is at

atmospheric pressure. On a normally aspirated piston engine the full BHP is only available atsea level. As the airplane climbs the power output of the engine decreases, roughly in

proportion to air pressure. Thus at 18,000 feet the engine produces only half as much BHP as

at sea level.

Many piston engines used in airplanes have turbo-chargers. A turbo-charger is a device that

pressurizes the air entering the engine intake. This "fools" the engine into thinking it is still at

sea level. As a result a turbo-charged engine can produce its full rated BHP up to some

altitude at which the turbo charger reaches its limit. Above that altitude the engine starts to

lose power again.

Turboprop engines are normally aspirated. The preceding is a crucial point to realize. All

turboprop engines lose power as altitude is gained. This situation is often unacceptable to the



aircraft designer as it means that full power is not available for takeoff except for those few

airports that are at sea level. The solution is to use a "de-rated engine."

De-rated Engines

If you consult the POH for a turboprop airplane it will tell you the maximum BHP (SHP) of

the engine. Along with this maximum BHP will be maximum torque and rpm values.(

Turboprop engines normally have torque meters.) The POH will usually give a clue that the

engine is de-rated by either saying so directly or stating that full BHP is available up to some

altitude, for example 6000 feet. How is that possible given that the engine is normally

aspirated?

The answer is that the engine can actually produce significantly more power than the rated

BHP at sea level. The POH restrictions on torque and rpm limit the pilot to setting just the

maximum specified BHP. But at sea level you will find that you are not at full throttle when

you reach maximum torque and rpm. The engine actually has a "power reserve." NOTE thatthe same is true of some turbo-charged piston engines.

Should you use the reserve power? Only in an emergency. The reason is that the higher

output stresses the engine. The engine runs hotter and faster and the gearbox is twisted with

more torque than it is designed for. If you run a higher than approved BHP for more than a

few minutes you will destroy the engine.

Emergency Power

Many POHs specify an emergency power option. Even if the POH doesn't, if your life is on

the line push the throttle to full and use all the available power. Just be aware that your

engine that would run quite safely at rated power for hundreds even thousands of hours, will

be destroyed in a matter of minutes if you do this. Your life is worth more, but being a few

minutes behind schedule isn't. So use emergency power ONLY in an emergency.

Power Available (Pa)

So far we have been concentrating on BHP. But the BHP is used to turn a propeller, so it is

really the power that the propeller manages to put into the air that actually moves the



airplane. Imagine the engine roaring away with no propeller attached. The BHP might be

impressive, but we wouldn't be going anywhere. The effective power would be zero.

We call the power that is actually available - i.e. delivered by the propeller Pa. This

sometimes is also called the thrust horsepower (THP.) In mathematical terms:

Pa = x BHP [ is propeller efficiency. is always less than 1.0]

Propeller Efficiency

Propeller efficiency measures the percentage of BHP that is turned into useable power by the

propeller.

Mathematically: = Pa / BHP.

must be evaluated by experimental measurement or complex computer analysis. Howeverthe factors that determine it are quite obvious. They are the same factors that determine the

L/D ratio of an airplane. If you think about it the drag on a propeller is the force that engine

torque must overcome and "Lift" from a propeller is actually thrust. You should recall that we

learned in the drag chapter that L/D ratio depends on three factors; angle of attack, C DP, and

Aspect ratio. The same is true of a propeller. We will look at AOA in a moment. CDp is

measure of how streamline the propeller is and is really not a concern because unless some

propeller manufacturer loses his mind all propellers are very streamlined. We also see that

AR is important. So it is clear that a large diameter propeller with long thin blades is more

efficient than one with short stubby blades. Most propellers meet that criteria, now it is clear

why.

The determining factor for propeller efficiency then seems to be angle of attack (AOA.) We

will now explore that in detail.

The simulation to the left demonstrates how a propeller produces lift. It also shows the

relationship between rpm, TAS and propeller angle of attack.

To use the simulation you change rpm using the throttle lever at the lower left corner. Just

drag the throttle with your mouse. You change airspeed using the Q-key and Z-key.

The green vector represents rpm - or more precisely the velocity of the propeller due torpm.

The red vector represents true airspeed (TAS.)

The blue vector presents the relative wind, which is by definition opposite to the direction

the propeller travels. NOTE that the direction the propeller blade travels is the sum of the

velocity due to rpm and the TAS of the whole airplane.

The side view shows a section of the propeller blade at one specific point along the span of

the propeller. The front view shows a blue circle that you can imagine represents the

distance along the span of the propeller section represented by the side view.



The angle of attack of the propeller blade is the angle between the chord and the relative

wind (blue) vector.

Experiment with the simulation to convince yourself that as rpm increases so too does

angle of attack.

Also convince yourself that as TAS increases angle of attack decreases.

If we took the section at a different distance from the crankshaft the velocity of the

propeller blade due to rpm would be different. Velocity due to rpm is given by the formula

rpm x pi x d (d is the distance from the hub.) Close to the crankshaft velocity would be low,

and further out it would be greater. This affects the angle of attack of the propeller blade.

This explains why a propeller blade is "twisted" as shown in the photograph to the left. The

root of the blade must have a larger blade angle than the tip in order that all points along

the span of the blade strike the relative airflow at the same angle of attack.

To keep a propeller operating at a constant angle of attack the ratio of rpm / TAS must be

constant. This ratio is so important that is given a name. It is called the advance ratio (J.) The

exact definition of advance ratio is:

J = TAS / (rpm x d) [ d is prop diameter]

Propeller efficiency changes with the value of J. A typical plot of vs. J is shown below.



The graph to the left shows that propeller efficiency ( ) is low when J is low. This is

because TAS is low so the angle of attack of the propeller is too high.

As J increases there is some ratio of TAS/J that is optimum. NOTE that each specific ratio

of TAS/J corresponds to a specific propeller AOA.

Previously we learned that the wing must fly at a certain angle of attack to be efficient so it

makes sense that the propeller too must operate at a particular angle of attack to be

efficient. If TAS/J increases too much (probably due to a dive or the pilot suddenly

reducing rpm) the efficiency of the propeller drops off again.

The 10°, 20°, etc. markings represent the average propeller blade angle. Normally this is

the blade angle measured at 75% diameter. A coarser propeller achieves its maximum

efficiency at a higher ratio of TAS/J. This corresponds to say that a coarse propeller is

needed for a faster airplane and a fine pitch propeller is needed for a slow airplane.

The graph shows maximum as 0.85. This is ONLY an example. The actual value depends

on the propeller design. The most important factors being similar to aircraft wings. A large

diameter propeller will be more efficient UNLESS it is turned so fast that it reaches the

speed of sound. If that happens efficiency will suffer.

Most modern propeller airplanes have constant speed propellers. With a constant speed

propeller the blades can "twist" to a certain maximum and minimum blade angle or any

angle in between. In the graph to the left the propeller can vary between 10° and 40° blade

angle.

The graph to the left shows that the efficiency of the different blade angles can be merged.

The blue line shows the efficiency of the constant speed propeller which is now optimum

over a wide range of advance ratios.

The constant speed propeller is vital if an airplane is to have a propeller that is efficient for

both climb and cruise. It is quite easy to install a fixed pitch propeller on an airplane that is

efficient at cruise. The designer simply chooses a propeller that has peak efficiency at the

ratio (cruise TAS)/(d x cruise rpm.) Many single engine trainers have such propellers. But



when the TAS decreases to climb speed and the rpm increases to climb rpm the propeller is

not efficient any more.

Of course in some specialty airplanes, especially float planes or glider tow planes the

designer might choose a fixed pitch propeller that achieves maximum efficiency at the ratio

(climb TAS)/(d x climb rpm.) In this case the airplane will climb wonderfully, but be veryinefficient in cruise.

The constant speed propeller allows the airplane designer, and therefore the pilot, to have

his/her cake and eat it to. The propeller can be at maximum efficiency in both climb and

cruise.

The Pa Graph

Recapping what we know so far; BHP is a constant value that the pilot sets by adjusting the

throttle and propeller controls to get a certain torque and rpm. Propeller efficiency varies withthe ratio TAS/J, except that it is constant over a wide range of this ratio if a constant speed

propeller is installed.

We also know that Pa = x BHP.

The graph to the left shows Pa vs. Velocity.

Notice that the graph is exactly the same shape as the propeller efficiency graph.

Notice that BHP is a constant (red line.) It is really the BHP that you are setting when you

adjust the throttle. Pull the throttle back and the red line would drop down the y-axis.

The blue line represents the actual power (Pa) that the propeller converts to useful power.

Because the graph is plotted against TAS rather than advance the ratio this blue line is

necessarily specific to some particular rpm. When the pilot adjusts the rpm the blue curve

will adjust left or right slightly. The simulation on the upcoming climb page takes that into

account. For simple understanding of aerodynamics this is not really too important

however.

http://selair.selkirk.bc.ca/training/aerodynamics/climb_prop.htm






The graph to the left shows Pa vs. TAS for a fixed pitch propeller.

Note that the fixed pitch propeller can be just as efficient as the constant speed propeller at

some particular speed.

If you have ever flown with a fixed pitch propeller you know that when you throttle back

the rpm decreases. Since depends not on TAS but on the ratio TAS/(d x rpm) the peak of

the blue curve will move to the left (slower speed) as rpm is decreased.

Power Required (Pr) Now that we know a little something about the power produced by the engine the question is

- how much power do we need to fly?

One way to find out would be to go flying at see what power setting we have to set to fly at a

given airspeed. There is nothing wrong with that plan, but there is a simple relationship

between drag and power required that allows us to predict how much power will be required

(Pr) before we go flying.

We will use the abbreviation Pr to represent required power throughout this course. To seehow it is determined review the definition of power given above.

Pr = w/t = F x d/t = F x V [V = TAS]

We know that power equals work/time and we know that work equals force x distance. This

leads to Power = force x distance / time. But distance / time is just velocity. So power = force

x velocity. But what force and what velocity are we talking about? The force is the force

required to move the airplane, i.e. thrust. The velocity is the actual velocity of the airplane,

i.e. the TAS. Thus we get the equation given above.

In the metric system thrust would be measured in Newtons and TAS in meters per second.Multiplying them together will give Pr in Watts.



In the engineering system we measure thrust in pounds and TAS in either Knots or ft/sec.

Power is measured in Horsepower, which as we mentioned above is defined as 33,000 lb x

ft/min. This is equivalent to 550 lb x ft/sec, or 325.7 lb x knots (check these conversions for

yourself.) Using the conversion factor we get:

Pr = T x TAS / 550 [T is thrust in pounds and TAS is in units of ft/sec]

Pr = T x TAS / 325.7 [TAS is in units of KTAS]

The Pr vs. Velocity Graph

We know that in level flight thrust equals drag. So just as we did with the jet airplane on the

previous page we will start with the drag vs. velocity curve when we set out to generate a Pr

vs. Velocity curve. We will assume that:

Pr = D x TAS / 325.7 [D = drag in pounds and TAS is in units of KTAS]

On the graph to the left the blue curve is represents drag. We have seen this curve many

times before.

The magenta curve represents the amount of power that corresponds to the blue drag curve.

To plot the magenta power curve you would make a table of values similar to this:



KTAS

(V)

Drag

(D)D x V / 325.7 Pr

60 1600 60 x 1600 / 325.7 295 HP

80 998 80 x 960 / 325.7 236 HP

100 720 100 x 720 / 325.7 221 HP

etc.

300 1600 300 x 1600 / 325.7 1474 hP

325.7 1900325.7 x 1900 /

325.7

1900

HP

The table shows the process by which you can methodically turn a drag curve into a Pr curve.

Note the special speed of 325.7 KTAS. At that speed we get the result that "X" pounds of

thrust equals "X" horsepower. For this particular airplane there is 1900 pounds of drag at

325.7 KTAS so 1900 HP is required. Obviously these values will change with altitude and

will be different for a given airplane. The things that is true for all airplanes is that at speeds

less than 325.7 KTAS the Pr in units of horsepower will be less than the drag in units of

pounds. Above 325.7 KTAS Pr is more than drag. This fact has lead some people to claim

that 325.7 is a "magic speed." The claim is that airplanes that fly slower than this speed

should be propeller airplanes and airplanes that fly faster should be jets. The reasoning is that

fuel flow is proportional to drag for a jet and power for a propeller airplane. The reasoning

would be valid if both jet and propeller engines had the same specific fuel consumption. But

they do not so this reasoning is invalid.

© Copyright Raymond J. Preston



Aerodynamics Index

Flight for Range and Endurance - JetAirplanes

Flight for range explores the question - how much fuel will the airplane use to get to its

destination? Flight for endurance explores the question - how much fuel does the airplane use

in a unit of time?

We will define the term Specific Range (SR) to be the ratio of groundspeed/fuel flow. The

units of SR are nautical miles per gallon, or nautical miles per pound of fuel. It is better to use

units of nautical miles per pound of fuel because the chemical energy of a unit of fuel

depends on the mass of the fuel. So, as fuel gets colder and therefore more dense you actuallyget more energy from a gallon of fuel, but always get the same amount of energy out of a

pound of fuel.

We will define the term Specific Endurance (SE) to be the ratio 1/Fuel Flow. The units of SE

are hours per gallon, or hours per pound of fuel. The unit hours per pound of fuel is preferred.

The greater the value of SR the further the airplane can fly (for a given amount of fuel

onboard.) It is the equivalent of saying that your car gets 40 miles per gallon while someone

else's only gets 20 miles per gallon. Obviously the one that gets 40 miles per gallon can go

further between fuel stops, or requires less fuel when it does stop to gas up.

The greater the value of SE the longer the airplane can stay in the air. With a car you can

always pull over and shut the engine off. In that case fuel flow (FF) becomes zero and SE

(1/FF) becomes infinite. In other words your fuel will last forever - because you aren't using

any. Unfortunately an airplane must keep moving to stay in the air; therefore you cannot

reduce fuel flow to zero. But, if you fly at the lowest possible fuel flow you will have the

maximum possible SE and will be able to stay in the air as long as possible.

In small propeller airplanes pilots almost never fly at the speed for maximum range. Pilots

tend to fly much faster than ideal, just as we all tend to drive our cars faster than ideal. But,

pilots of jet airplanes, especially on long range transoceanic flights, do try to fly at the speedfor maximum SR. We will spend considerable time discussing the intricacies of how to do

that later.

A pilot would only be concerned about maximizing endurance (SE) if s/he were flying

circles. That could happen - for example the runway may be closed for snow removal, or to

remove FOD etc. In such a case the pilot might circle overhead trying to consume fuel at the

lowest possible rate. In this case we want maximum SE, not SR.






How a "True" Jet Engine Works

A "true" jet engine takes cold air in through an inlet (left side of diagram to left.) The cold

air is compressed as it passes through the compressor section. The compressor consists of a

series of rotating blades that work like a propeller, and stator blades that are fixed.

For a subsonic engine the compressor section is shaped as a convergent chamber - i.e. like a

venturi that causes the air to accelerate as energy is added to it. Just before the combustion

chamber there is an expansion chamber so the air slows down and the pressure rises (as per

Bernoulli's equation.) When the air enters the combustion chamber it is compressed and at

moderate velocity (well below the speed of sound.)

The air then enters the combustion chamber where fuel is injected and ignited. The air and

fuel mixture in the combustion chamber expands and rushes out the back of the combustion

chamber at very high speed.

In the turbine section the rapidly moving air flow is used to turn some turbine blades. Thesework just like a windmill, except the air flowing over them is red hot, so they must be made

of tungsten to prevent them from melting.

The turbine blades are connected to a shaft that turns the compressor blades at the front of the

engine. Thus the engine is self sustaining. You can see the shaft in the middle of the above

diagram.

The turbines remove a small amount of energy from the air, but it is still flowing at highspeed when it leaves the outlet at the rear of the engine (right side of above diagram.) Thrust

results primarily from the difference in velocity of the air entering the inlet and that exiting

the outlet. There is a small amount of additional thrust due to the increase in mass due to fuel

being added, but the mass of fuel is only about 4% of the mass of the air worked on by the

engine, so most of the thrust results from the air being accelerated.

The above explanation is for a true jet engine, which is an engine in which all the air entering

the inlet passes through the combustion section. Modern jet engines are always bypass

engines. A bypass engine is shown below. The bypass engine is more efficient because it

works on a larger mass of air and therefore doesn't need to change the velocity of the air as

much to create thrust.



Thrust Specific Fuel Consumption (TSFC)

Thrust Specific Fuel Consumption (TSFC) is defined as the ratio of fuel flow to thrust

produced by the engine:

We use the abbreviation Ta to represent "thrust available" the letters FF represent fuel flow,

and the units should be pounds of fuel per hour per pound of thrust. (Note: Most small

airplane handbooks specify fuel flow in gallons per hour, which can be converted to pounds

per hour (approximately) by multiplying by 6.)

For JET engines TSFC is very close to being constant over a wide range of airspeeds. In other

words the same amount of fuel flow produces the same amount of thrust at any speed from

zero to close to the speed of sound.

We must emphasize the above point: TSFC is very close to being constant for a JET

engine. This is a critical point. It means that to determine the fuel flow we simply use the

formula: FF = TSFC x Ta. Given that TSFC is a constant this is a very useful and powerful

equation. Let's emphasize it again:

FF = TSFC x Ta [for a jet engine - but NOT for a propeller engine.]

Ta = Drag

Previously we learned that thrust must equal drag in level flight. Therefore we can say that:

FF = TSFC x Drag [for a jet airplane flying level.]

Mass Flow Affects The Value of TSFC (the Bypass

Engine)

A great deal of research goes on every year to improve the TSFC of jet engines. Every

percentage point improvement in TSFC will save hundreds of thousands of pound of fuel

consumed by all the airliners of the world every year. And like they say, a million here anda million there and pretty soon your talking real money.



Even a casual observer will have noticed that jet engines have been getting larger in

diameter as the years go by. The main reason is that the larger the diameter of the engine

the more fuel efficient it is.

The picture to the left shows a modern Boeing 737-700. Compare the engines on thisairplane to the ones on the ProfessionalPilot.ca logo at the top of every page. That airplane

is also a Boeing 737, but a much older one; you can see how much smaller the diameters of

the engines are.

As already noted, a jet engine produces thrust by accelerating air. The thrust equals the mass

of air times the acceleration, this is just Newton's second law:

F=ma - i.e. Thrust (T) = (mass of air) x (acceleration given to the air)

We write:

T = m x dV [dV represents change in velocity. Above we saw how a jet engine changes the

velocity of the air it works on.]

The velocity of the air entering the inlet is zero (more or less.) The air leaving the outlet is

moving much faster. As result the engine leaves the air behind the airplane disturbed (inmotion), which represents a loss of efficiency. The more residual motion the air behind the



engine has the more energy has been wasted. Therefore, an engine is more efficient if a

large mass is given a smaller dV:

The engine to the left is a high bypass jet engine, which means that in addition to the core

of the engine, as described above, there is a large "fan" at the front of the engine that

accelerates a large mass of air. In the picture the fan is the part surrounded by the brownshroud. The air acted on by the fan bypasses the core of the engine, hence the term bypass

engine. This type of engine is also often called a "fan-jet." Fanjets are more efficient than

true jets - i.e. they have lower TSFC. All modern jet airplanes are actually fanjets, not true

jets.

Air Temperature Affects the Value of TSFC

When a true jet engine accelerates air it does so by injecting fuel into the air and igniting the

fuel. The fuel burns which causes the air to accelerate. Two effects are at work; the minor

effect is that each hydrocarbon molecule in the fuel brakes down into many individual carbonand hydrogen atoms that combine with oxygen to produce C02 and H2O. The increased

number of molecules plus the rise in temperature due to combustion causes a substantial rise

in pressure that forces the gas to rush out the back of the engine. Thrust is the reaction force,

as per Newton's third law.

The dominant factor in producing thrust is the temperature rise of the air; therefore, the colder

the air entering the engine the more efficient the engine will be. In the atmosphere

temperature usually decreases with altitude so jet engines are more efficient at higher (colder)

altitudes than at lower (warmer) altitudes. Obviously on a cold winter day it can be cold at

low altitudes so you should remember that it is temperature not altitude that actually matters.

However, see the next point.

Engine RPM Affects the Value of TSFC (effect of air

density on thrust produced)

As we saw above thrust results from accelerating a mass of air. Given that air density

represents the mass per unit volume of air we must expect a jet engine to produce less thrust

at high altitudes where the air is less dense. That is definitely the case.

At 22,000 feet the air is about half as dense as at sea level, so a jet engine will produce abouthalf as much thrust there. Jet airplanes must have engines that produce MUCH MUCH more

thrust at sea level than will be needed in cruise at high altitude. A side benefit is that all jet

airplanes have LOTS and LOTS of thrust available for takeoff and climb. There is however a

disadvantage to having such powerful engines; it makes it impossible to cruise efficiently at

low altitude. The reason is discussed next.

Jet engines operate efficiently only when operated at close to maximum rpm. If you throttle a

jet engine back below a critical rpm it becomes very inefficient. It is important for pilots to

know that.





In the diagram to the left the same airplane has reached 36,000 feet and the pilot is about to

pitch the nose over and accelerate to cruise speed of 500 KTAS.

The engine is still operating at 100%, but in the much less dense air at 36,000 feet the

engine is now producing only 4470 pounds of thrust (see above for reason why.)

Drag is 3670 pounds, but will increase somewhat when the airplane accelerates to 500

knots (what will it be?)

In the left margin of the diagram the drag in cruise at 500 knots has been marked. You can

see that the pilot will not have to pull the throttle back very much. So, the engine will

continue to operate just below 100% and as such will be very efficient. I.E. TSFC will be

"good."

Imagine what would have happened if the pilot had chosen to level off at a lower altitude.

S/he would have been forced to reduce the throttle substantially to match thrust to drag. Theresult would have been a power setting much less than 100% and the TSFC would have been

"poor." It should be clear that the designer of the airframe and the engine must work together

so that both will work optimally together at cruising altitude.

In summary, the amount of thrust produced depends on the density of the air. In above

example 100% rpm produces 4470 lb of thrust at 36,000 feet. At sea level the pilot would

have to throttle back to less than 50% to produce the same 4470 lb of thrust - and the engine

would not be efficient.

NOTE: you can recreate the situation shown above, using the climb performance simulation,

and explore it more completely. We will get to that simulation later in this section.

Fuel Flow vs. Velocity Graph

We are now almost ready to analyze range and endurance performance of a jet airplane. What

we need is a graph relating fuel flow to velocity. Since we know that fuel flow is directly

related to thrust and we also know that thrust equals drag the process is quite simple. We start

with a drag vs. velocity graph and convert it into a fuel flow vs. velocity graph.

In the early days of jet engines the value of TSFC was pretty close to 1.0. In other words

1000 pounds of thrust required 1000 lb/hour of fuel flow, or 20,000 pounds of thrust required20,000 lb/hour of fuel flow, etc. In such a case you can just take a Drag vs. Velocity graph

and re-label it as Fuel Flow vs. Velocity with no further changes.

Thankfully modern bypass engines are more efficient, but still TSFC is usually about 0.8. In

other words 1000 pounds of thrust requires a fuel flow of 800 lb/hour of fuel, or 20,000

pounds of thrust requires a fuel flow of 16,000 lb/hour of fuel.

In the simulation below you can set the value of TSFC and experiment to see how it affectsrange and endurance.



The process of creating a Fuel Flow vs. Velocity graph is really very simple; you simply take

a Drag vs. Velocity graph and adjust the y-axis to represent Fuel Flow by using the definition

of TSFC - i.e. FF - Drag x TSFC.

The Simulation

The simulation creates a Fuel Flow vs. Velocity graph for both a jet and a propeller airplane.

We discuss propeller airplanes later in the course, so you can ignore it for now. To hide the

propeller airplane curve click the "Hide Curve" button in the propeller airplane information

box.

The simulation works by first calculating the drag. You can display the parasite drag and

induced drag curves if you like, although they are hidden by default. The simulation is for

level flight only, and we know that thrust equals drag in level flight. So, the simulationsimply takes all the drag values and multiplies them by TSFC. The result is fuel flow, and

this is presented on the y-axis.

NOTE: a different process is required for the propeller airplane and we will cover that later.

For now ignore the propeller airplane except for comparative interest.

You can set TSFC. (You can also set SFC, which is for the propeller airplane. We discuss

SFC later.)

To operate the simulation use the up and down arrow keys to change AOA. As you change

AOA the airplane speeds up and slows down, but it always remains in level flight.

At the lower left corner you can modify the design of the airplane. To change the aspect ratio

drag the green navlight in or out. To change wing area drag the "S" symbol up or down.

Similarly change weight by dragging the "W" symbol up or down.

Use the E and C key to change density altitude. If you use the shift key you can change

altitude in 10,000 foot increments. Without the shift key altitude changes in 1,000 foot

increments. You can go all the way up to 70,000 feet, but remember that the engine loses

thrust in proportion to air density.

Drag the windsock to change the wind. The simulation properly calculates the best rangespeed taking the wind into account. Our next task is to analyze what speed to fly for

maximum range or endurance. That discussion is below the simulation.

Please NOTE that the simulation does NOT take supersonic effects into account. Every

airplane has some critical speed above which drag rises due to shockwave formation. We

discuss that later. In this simulation there is no allowance for shockwave drag so if you fly at

speeds more than about M=0.85 you are probably fooling yourself about the efficiency of the

airplane.



Maximum Endurance at L/Dmax

In the simulation the y-axis represents FF. Clearly maximum SE occurs at the lowest possible

FF - recall that SE = 1/FF. The lowest possible fuel flow occurs at the bottom of the Fuel

flow vs. Velocity graph. Since fuel flow is proportional to drag it is also true to say that

minimum fuel flow occurs when the airplane flies at L/Dmax. We say that maximum SE

occurs at L/Dmax.

In the drag chapter we learned that L/Dmax occurs at a particular angle of attack. Therefore,

the actual speed the airplane must fly at for maximum SE decreases as fuel is burned and the

airplane gets lighter.

In the drag chapter we also leaned that the total drag curve moves to the right as altitude

increases but L/Dmax does not change with altitude. This leads to a preliminary conclusionthat altitude will not affect endurance. We will explore this conclusion more in a moment.

You should note that maximum endurance speed for a jet airplane is the same as maximum

glide speed. Both occur at the angle of attack for L/Dmax.

How does altitude affect endurance of a jet airplane?

Using the simulation you can see than minimum drag is NOT affected by altitude. Therefore,

aerodynamically altitude has no affect on endurance. However we discussed earlier that the

jet engine must be operated at high rpm to be efficient. And we also noted that colder air

temperature improves TSFC. Therefore, if you have a choice you would fly at altitudes in thestratosphere where the air is coldest and you can keep the engine rpm high to achieve

maximum SE.

How does wind affect endurance of a jet airplane?

Wind affects groundspeed. But when we fly for maximum endurance we are just flying

circles so groundspeed does not matter.

Interesting Point: If you do get caught at low altitude and need to maximize SE (minimize

fuel flow) in an emergency there is a trick you can use if your airplane has four engines rather

than the usual two. Simply shut down two of the engines. Once you do that you can operatethe remaining two at high rpm while flying at L/Dmax angle of attack. This may strike you as a

crazy idea - and I don't seriously recommend it to B747 pilots. But it is actually done in some

military operations.

Maximum Range at L1/2/Dmax

Imagine that you are flying at L/Dmax, then start to reduce AOA below the L/Dmax AOA; the

fuel flow increases, but so does airspeed. Since SR = TAS/FF range actually improvesinitially. Eventually fuel flow starts to rise so rapidly that SR decreases again.



Remember that FF = T x TSFC.

Remember that SR = Speed/FF. For now let use use TAS as the speed value, but it really

should be groundspeed that we use. We will deal with that later.

Therefore, in zero wind SR = TAS/(T x TSFC).

In level flight TAS is tied to the CL1/2 as we saw in the Lift chapter, and Thrust is proportional

to CD. Therefore we conclude that SR will be proportional to the square root of CL divided by

CD or in other words:

The maximum SR will occur when is at its maximum value.

On the Fuel Flow vs. Velocity graph the point for maximum SR can easily be found

graphically. Consider the diagram to the left.

Chose any point on the graph then draw a line from the origin to that point, as shown in the

diagram.



Now consider the angle "R" between the line just drawn and the X-axis. What is the value

of Tangent(R)?

tan(R) = FF/TAS [examine the diagram to convince yourself of this]

Recall that the definition of SR is SR = TAS/FF. In other words:

SR = 1/tan(R)

Summary. The tangent of any point on the FF vs Velocity graph equals the SR for that

point. Now we ask the question - where is SR maximum?

SR must be a maximum where tan(R) is a minimum.

The point for maximum range is easily found by inspection. The point with the minimum

tangent of R is shown in the diagram to the left.

If you got lost in the mathematical development above simply realize that this point is the

one for which the ratio of speed to fuel flow is maximized.

We already know that flying at the point shown to the left corresponds to flying at the angle

of attack for .



In the simulation the computer calculates this angle of attack for you. Notice that if you fly

at this angle of attack you always get maximum range.

Next we will explore how weight and altitude affect SR.

Effect of Weight on Specific Range (and Endurance)

In the simulation the default weight for the airplane is 2400 lb, which is very unrealistically

low for a jet. We will try increasing the weight and see what changes. Before we do we

should record the starting figures. occurs at AOA = 2.1 degrees (if you changed the

default AR change it back to 7.3 and set CDp = 0.024) Change AOA as needed, using the up

and down arrow keys, to 2.1°.

The "X" that represents your current speed and fuel flow should now be at the point where

the tangent line intersects the curve. Make sure the tangent line is visible, but hide the

propeller curve to avoid clutter. Your graph should look like the one to the left.

Record the data in the jet data box:

The information box shows that the fuel flow is 143.5 pph (about 24 gallons per hour)



The specific Range is 0.677 Nm/lb

You can also slow to max endurance AOA and record the maximum endurance, it is

about 1.48 min/lb.

Now we will increase the weight to that of a small personal jet with a wing loading of 60

/b/ft2. To do that increase W to 10,445 lb.

The resulting graph is shown to the left. The CRUCIAL first thing to notice is that

maximum range still occurs at AOA = 2.1°. You may be surprised, but you shouldn't if you

remember our previous analysis that showed that:



We previously said that L/D ratio depends on only three factors, which are expressed in the

above equation:

1.

Angle of Attack (CL)

2. CDp (i.e. how streamlined the airplane is)

3.

Aspect Ratio

You must imagine that we have increased the weight of the airplane without changing it

in any other way. I.E. the airplane is still exactly the same shape, so CDP is the same, as

is AR. So the same optimum angle of attack applies.

But, notice that the airplane must now fly much faster to stay airborne at the higherweight. It now flies 203 KTAS at 2.1° AOA as compared to 97KTAS at 2400 lb.



Because of the way the simulation adjusts the y-axis to keep the graph optimally viewable

it may be difficult for you to fully visualize how and why the fuel flow curve changes withweight. The graph to the left should help.

In this graph the y-axis has been kept constant as the weight changes from 2400 lb to

10,450. Check the values against the graphs above to convince yourself that they are the

same values.

The parasite drag and induced drag curves are visible in this graphic.

As we expect parasite drag does NOT change when weight changes.

As we also expect, induced drag increases when weight increases. The graph showsinduced drag at 2400 lb, 6200 lb and 10,450 lb.

The drag curve, and therefore the FF vs Velocity curve must always shift up and to the

right as weight increases. It is actually better to think of it as rolling up from the lower left

as induced drag increases but parasite drag does not change.

We can easily see that the L/Dmax point is higher up at higher weight. So just as common

sense would predict the heavier airplane will consume more fuel and have a lower SE.



The tangent lines drawn from the origin to the point ("X" point) have been left out to

avoid clutter, but we can easily see that if we drew them in SR would get worse as W

increases.

In summary, we know that an airplane always achieves maximum endurance at the L/Dmax

AOA, which does NOT change with weight. But the actual value of SE does decrease at

higher weight. We also know that maximum range always occurs at the AOA, which

does NOT change with weight. But the actual value of SR decreases as weight increases.

Both of these effects are pretty much what common sense would have predicted.

Effect of Altitude on Specific Range (and Endurance)

To understand how altitude affects range and endurance of a jet airplane you must recall what

we previously learned about how drag changes with altitude. We learned that parasite drag

decreases with altitude. This was fairly intuitive since we expect the less dense air to cause

less drag. We also learned that induced drag INCREASES with altitude. That is because the

airplane must fly at a higher angle of attack for a given TAS at the higher altitude.

It is often easier to simply remember that both induced and parasite drag are always the same

for a given EAS. So, at higher altitude where TAS > EAS the drag curve shifts to the right. If

we ignore the effect of temperature on TSFC for a moment then the FF vs Velocity curve will

mirror the changes in the drag curve. The graph below shows FF vs Velocity at sea level and

at 45,000 feet for the personal jet (W=10.450) mentioned above.

The graph shows the induced and parasite drags at both sea level and 45,000'.



The parasite drag decreases and the induced drag increases with altitude. The result is that

the drag curve moves to the right.

Since FF = TSFC x Drag the FF curve also moves to the right.

The horizontal green line shows that minimum fuel flow does not change with altitude - IFTSFC does not change. Remember that we said earlier that TSFC will decrease when

temperature decreases. To simulate that when you use the simulation above you should

manually reduce TSFC as altitude increases. Remember that temperature only decreases up

to the tropopause (36,000') then becomes constant.

Based on the above we would expect a jet to achieve maximum endurance at the

tropopause or above (i.e. in the stratosphere.)

The above graph makes it pretty clear that altitude is NOT much of a factor in maximum

endurance for a jet. However, it is equally clear that altitude is a MAJOR FACTOR in range.

As we learned previously maximum range always occurs at the AOA for . The

corresponding points are marked with "X" on the graph.

We also learned that SR = 1/tan(R) where R is the angle as labeled in the graph above. It is

quite clear that the angle R gets less and less as altitude increases. In other words SR becomes

better and better as altitude increases. NOTE that this improvement is in addition to any

improvement due to temperature drop. Therefore, SR continues to increase as the jet climbs

into the stratosphere (unlike SE which becomes constant.)

Please NOTE that while the above analysis indicates that SR will become infinite if youclimb to infinity there is a practical limit - actually two practical limits. The first limit is

engine thrust. We already learned that jet engines produce less thrust at altitude. So, the

engine will run out of thrust at some altitude and the pilot won't be able to climb any higher.

The second practical limit is the speed of sound. The graph shows that the true airspeed the

airplane cruises at gets greater and greater at higher and higher altitudes. At some altitude the

airplane will be going so fast that shockwaves will begin to form. This will increase parasite

drag and invalidate the assumptions that underlay our analysis. We now turn to a more

detailed analysis of this point.

Cruise Control

On long range flights a jet pilot practices a technique known as cruise control. Its a bit more

complicated than using a cruise control button on a modern car - but not much.

To understand cruise control you only need to know two things. One we have reviewed over

and over, the other is a point we have only implied:

1.

Maximum Range occurs at the AOA for

2.

Every jet airplane has a maximum Mach number that it MUST never exceed



The objective of cruise control is to fly BOTH at and maximum Mach number.

To understand cruise control follow through the following example using the simulation. We

will assume we jet transport pilots - as such setup the simulation with the following data:

Altitude = sea level W = 800,000 lb

S = 5334

AR = 10.1

Once you have set the above values you will notice that you need to fly at AOA = 2.9° to

achieve maximum SR - use the up and down arrow keys to go to that AOA.

Notice that you are only flying at TAS = 292 and you Mach number is only 0.44 (values are

shown on the airspeed indicator.)

Now let us assume that this airplane has a maximum Mach number of 0.85. What can we do

to get the TAS up to Mach = 0.85 while remaining at our present angle of attack (2.9°)?

The only thing we can do is climb. Start a climb - you will see that your EAS remains at 293,

but your TAS and mach number increase with altitude (Mach number = TAS / speed of

sound) When you reach 32,000 feet you will be cruising at Mach = 0.85. STOP your climb

here. If you go higher you will have to increase AOA to stay below the Mach limit, but if you

had leveled off lower you would either have been flying slower than Mach 0.85, or would

have used an AOA less than .

You should be able to see that there is just ONE ALTITUDE at which we achieve ourobjective of flying at both and maximum Mach number.

What happens as the flight progresses and we burn fuel?

To answer this question reduce W to 700,000 - but don't change any other airplane parameter.

The reduction in weight reduced the wing loading from 150 lb/ft2 to 131 lb/ft2. The

simulation is programmed to stay at the same AOA, so AOA is still 2.9° as desired. But, the

EAS has dropped to 273 KEAS and Mach number is down to 0.79.

We have just learned that as fuel is burned the pilot MUST slow down to stay at the optimum

angle of attack. But, is there any way we can keep our Mach number up to the desired Mach

0.85?

Yes - try climbing. You will find that if you climb to 35,000 feet you can regain Mach 0.85.

You have just made your first STEP CLIMB - welcome to the world of jet pilot aviation.

In cruise control the pilot makes a series of step climbs as fuel is burned. That way s/he can

keep the airplane at and maximum Mach number. Apparently sometimes you can have

your cake and eat it too :)

The above system is called cruise control.



How Wind Affects Range and Optimum Cruise Speed

Previously I mentioned that when we calculate SR we should use:

SR = groundspeed / FF.

Groundspeed is just TAS +/- headwind or tailwind. It should be obvious that SR is better with

a tailwind and poorer with a headwind. But should we still fly at ? The answer is NO!

In the simulation you can change the headwind/tailwind by dragging the windsock icon at the

lower right corner of the graph.

To graphically allow for the headwind simply realize that we need to subtract the headwind

from the TAS to get groundspeed. On the graph to the left you can see how that is done.

In this example there is a 100 knot headwind. To plot this graphically we start the tangent

line at 100 knots that way the speed value is 100 knots less than if we started from the

origin.

The tangent line touches the FF vs. Velocity curve at the revised speed for best range.

You can see that the angle "R" is greater than it would be if the line was drawn from the

origin. Thus SR is worse with a headwind, which is what common sense should have told

us.



The computer calculates the best angle of attack for SR

with the wind and without. This information is displayed

in the Jet A/C data box, as shown to the left.

The airplane in this example is the same jet used in the

cruise control example above. We saw previously that it

must fly at 2.9° AOA for maximum SR in zero wind. But

with the 100 knot headwind it must be flown at 2.1° AOA.

This corresponds to changing airspeed from 273 KEAS to

300 KEAS. (Note that using the same cruise control

concept discussed above the pilot must descend to 31,000

feet, due to the higher required EAS.)

It is important to realize that we must always fly faster

when flying into a headwind. Ideally we should slow

down slightly with a tailwind. The same graphical method

can be used to find the optimum speed with a tailwind. Try

it in the simulation above to see.




Aerodynamics Index

Flight for Range and Endurance- Propeller

AirplaneBefore reading the material on this page it is assumed that you have studied the material on

the page "Flight for Range and Endurance- Jet Airplane."

It is also necessary that you understand the meaning of Power

Just as with the jet, we will define the term Specific Range (SR) to be the ratio of

groundspeed/fuel flow. The units of SR are nautical miles per pound of fuel.

We will define the term Specific Endurance (SE) to be the ratio 1/Fuel Flow. The units of SE

are hours per pound of fuel.

Two Types of Propeller Airplane

There are two types of propeller airplanes:

1.

Piston - also known as reciprocating - engine

2. Turbine - also known as turboprop - engine

A reciprocating engine has pistons that move

up and down as shown in the stylized picture

to the left

A piston engine converts fuel into power. That

is important to know because as you recall a

jet engine turned fuel directly into thrust. This

contrast explains all the differences in the way propeller and jet airplanes are designed and

flown.

Piston engines can be further subdivided as:

1. Normally aspirated

2. Turbo charged

Review the power page for the definition of

these terms. Don't confuse turbo charging and

turboprop. The two are completely different.



http://selair.selkirk.bc.ca/training/aerodynamics/range_jet.htm



http://selair.selkirk.bc.ca/training/aerodynamics/power.htm












A turboprop engine is really a jet engine but with more turbines installed so that almost all

the energy is removed from the outflow. Review the earlier discussion of jet engines. At thattime we said that the turbines only removed a small amount of energy from the hot airflow

exiting through the exhaust pipe of the jet engine; just enough to turn the compressor. In a

turboprop we try to remove all the energy so that the central shaft can turn a propeller in

addition to the compressor blades. Of course not quite all the energy is removed so there is

always a small but generally insignificant amount of thrust produced directly by a turboprop

engine.

On the Power page you learned that a gearbox is required to drive the propeller.

Specific Fuel Consumption (SFC)

For the jet engine we defined TSFC as FF/thrust.

For propeller engines we define Specific Fuel Consumption (SFC) as the ratio of fuel flow to

BHP produced by the engine. Sometimes this is also known as BSFC for Brake specific fuel

consumption, but in this text we will use the shorter form SFC:

Recall that previously we learned that Pa = x BHP. Solving for BHP we get BHP = Pa / .

This value was substituted in to get the final form of the above equation.

On the power page we defined the term Pa to represent power available taking propeller

efficiency into account. The letters FF represent fuel flow, and the units of SFC should be

pounds of fuel per horsepower.

For both piston and turboprop engines SFC is very close to being constant over a wide range

of BHP. In other words a certain amount of fuel flow produces a certain amount of power -this relationship is independent of airspeed.

In summary: SFC is very close to being constant for a PROP engine. Therefore, to

determine the fuel flow we simply use the formula: FF= SFC x Pa / . Given that SFC is a

constant this is a very useful and powerful equation. Let's emphasize it again:

FF = SFC x Pa / [for a prop engine - note how this is different than jet engine.]







Pa = Pr

As defined on the power page the required power is referred to as Pr. When the airplane flies

level Pa = Pr. In a climb of course Pa must be greater than Pr and in a descent Pa must be less

than Pr.

On the Power page we learned how to calculate Pr. Therefore, FF = SFC x Pr / [for a prop

airplane flying level.]

What Affects SFC?

When we examined jet engines we saw that the diameter of the jet engine was a determinant

of TSFC. For a propeller airplane the same is true; larger diameter propellers are more

efficient. This give an advantage to turboprops over piston engines because the gearbox of

the turboprop provides an opportunity to turn the propeller quite slowly. Redline rpm on a

typical turboprop is close to 2000 while most piston engines redline at closer to 3000 rpm.

Redline is the maximum rpm the engine is to be operated at. Redline rpm is important

because the velocity of the propeller blade is proportional to both diameter and rpm. If the

diameter is too large the propeller velocity will approach the speed of sound. If the propeller

velocity gets too close to the speed of sound shockwaves will form and the propeller

efficiency will begin to suffer. Generally speaking we want a propeller that turns as slow as

possible and has a diameter as large as possible, consistent with the tip speed being less thanapproximately 80% of the speed of sound. (Note that while not common, gearboxes are

sometimes attached to piston engines also. This is entirely to facilitate slow turning and

therefore larger diameter propellers to improve efficiency.)

Air Temperature Affects the Value of SFC for

Turboprops - but not Piston

Previously we learned that jet engines are more efficient when air temperature is cold.

Because a turboprop is really just a jet engine with a propeller the same logic applies to it.

For piston engines temperature has a much smaller effect. You will find the effect of

temperature is so small that you can ignore it for piston engines.

Engine RPM Affects the Value of SFC for Turboprops -

Less so for Piston

As with jet engines turboprop engines must be operated at close to 100% rpm to be efficient.

Just like the jet most turboprops have abundant reserve power at sea level in order to facilitate

flight at higher altitudes. If the pilot chooses to fly at very low altitudes the engines will have

to be throttled back to well below optimum rpm. As much as possible we should try to avoidoperating turboprops this way.



Piston engine efficiency is not directly affected by the rpm but indirectly it is because piston

engines lose efficiency if the throttle is not fully open. To set a particular power setting with a

piston engine the pilot sets a certain rpm and manifold pressure combination. The same

amount of power can be produced with a high manifold pressure and low rpm or a high rpm

and low manifold pressure. The SFC will not be the same however. SFC will be lower (good)

with a high manifold pressure and low rpm. The reason is that reducing the manifold pressureis done by partially blocking the engine intake (that is what a throttle is) which creates a

vacuum in the intake manifold. It takes energy for the engine to draw air into the cylinders on

the intake stroke if there is a vacuum in the manifold, consequently the engine is less

efficient.

With turbo charged piston engines the throttle is always fully open any time manifold

pressure is more than atmospheric pressure so the engine efficiency is optimum. With

normally aspirated piston engines efficiency is less than optimum anytime the throttle levers

are anywhere other than full forward. Anyone who has flown such an airplane knows that by

the time you climb to about 6000 feet or higher you can cruise with full throttle. If you chose

to fly at a lower altitude you will have to reduce throttle and thus will suffer a slight rise inSFC.

In summary:

Turboprops gain efficiency with altitude up to the tropopause because the temperature keeps

dropping. Turboprops will also lose efficiency if operated at low rpm, which tends to happen

if the pilot chooses to fly at very low altitude.

Piston engines are not significantly affected by either altitude or temperature. Extremely low

altitude, (less than 6000) should be avoided if possible in order to facilitate full throttle

operation. But even if this advice is ignored the effect on SFC is typically less than 2%.

NOTE: piston engines are usually manually leaned while jet and turboprop engine fuel flows

are electronically controlled. The pilot is the most likely culprit causing high SFC for piston

engines. If the pilot does not properly lean the engine then everything said below is

completely invalid.

Fuel Flow vs. Velocity Graph

The process of creating a Fuel Flow vs. Velocity graph for a propeller airplane is more

complex than for a jet because it is a two step process. First we convert a Drag vs. Velocity

graph into a Pr vs. Velocity graph. Then we use the relationship FF = SFC x Pa / . Now we

simply say that Pa = Pr and we have FF = SFC x Pr / [ - note how this is different than jet

airplane.]

Using the above equation we adjust the y-axis of the Pr vs Velocity graph thereby creating a

FF vs Velocity graph.

Review the process for generating the Pr vs. Velocity graph on the power page.







The Simulation

The simulation creates a Fuel Flow vs. Velocity graph for both a jet and a propeller airplane.

Previously we looked at the jet curve. Now we will look at the propeller curve.

The simulation works by first calculating the drag (D.) You can display the parasite drag and

induced drag curves if you like, although they are hidden by default. For the propeller

airplane the computer then calculates Pr using the formula Pr = (D x TAS)/325.7. Using the

Pr values fuel flow is then calculated using the equation FF = SFC x Pr / . These values are

plotted as the magenta curve.

NOTE: That a fixed value of p (propeller efficiency) is assumed. The value appears just left

of the propeller data box. If you click on it you can change the value.

You can set SFC. (You can also set TSFC, which is for the jet airplane.) The default value of

0.62 lb/hr/HP is typical of piston engines.

To operate the simulation use the up and down arrow keys to change AOA. As you change

AOA the airplane speeds up and slows down, but it always remains in level flight.

At the lower left corner you can modify the design of the airplane. To change the aspect ratio

drag the green navlight in or out. To change wing area drag the "S" symbol up or down.

Similarly change weight by dragging the "W" symbol up or down. You can also click on the

Oswald efficiency number (e) to change it.

Use the E and C key to change density altitude. If you use the shift key you can change

altitude in 10,000 foot increments. Without the shift key altitude changes in 1,000 foot

increments. You can go all the way up to 70,000 feet, but remember that a real airplane as a

service ceiling.

Drag the windsock to change the wind. The simulation properly calculates the best range

speed taking the wind into account. Our next task is to analyze what speed to fly for

maximum range or endurance. That discussion is below the simulation.

Please NOTE that the simulation does NOT take supersonic effects into account. Every

airplane has some critical speed above which drag rises due to shockwave formation. We

discuss that later. In this simulation there is no allowance for shockwave drag so if you fly at

speeds more than about M=0.85 you are probably fooling yourself about the efficiency of the

airplane.



Maximum Endurance at L3/2/Dmax

In the simulation the y-axis represents FF. Clearly maximum SE occurs at the lowest possible

FF (recall that SE = 1/FF.) Previously we saw that for a jet airplane max SE occurred at

L/Dmax (see the red curve.) For the propeller airplane you can see that it must be flown

considerably slower. The lowest possible fuel flow for a propeller airplane occurs at the angle

of attack that corresponds to L3/2/Dmax. I recommend that you not worry about why it is this

specific ratio and instead remember that max SE always occurs at a greater angle of attack

than L/Dmax.

As with any specific L/D it occurs at a particular angle of attack. Therefore, the TAS the

airplane must fly at for maximum SE decreases as fuel is burned and the airplane gets lighter.

NOTE that the simulation is based on the assumption that the jet airplane and propellerairplane are identical except for the engines. Yet the propeller airplane must fly slower - i.e.

at a higher angle of attack for maximum endurance. It seems almost intuitive that the jet

airplane achieves maximum endurance when drag is minimum. But for the propeller airplane

it is actually on the backside of the drag curve at maximum endurance. This is because power

is proportional drag x velocity. So when drag is minimum slowing a little more, even though

it increases drag, actually decreases power due to the reduced velocity.

How does wind affect endurance of a propeller airplane?

Wind affects groundspeed. But when we fly for maximum endurance we are just flying

circles so groundspeed does not matter.

Maximum Range at L/Dmax

Previously we saw that a jet airplane achieves maximum endurance at t L/Dmax. and

maximum range at L1/2/Dmax. Now we see that a propeller airplane achieve maximum range at

L/Dmax. The maximum endurance speed of a jet is the maximum range speed of an identical

propeller airplane. Why would minimum drag be the proper condition for maximum range

for a propeller airplane?

The reason is quite straight forward. Remember that SR = GroundSpeed/FF. Initially let us

assume zero wind so we can say that SR = TAS/FF. Now recall that Pr and therefore FF is

proportional to Drag x TAS. Substituting that into the SR equation we get:

SR = TAS / FF TAS / (Drag x TAS) [TAS cancels] therefore:

SR 1/Drag

In other words maximum SR occurs when Drag is a minimum, which is the defining

condition for L/Dmax.



The maximum SR will occur when L/D is at its maximum value.

The method of finding maximum range graphically is the same as it was for the jet airplane

we examined earlier. The graph to the left is for a jet but the process is the same for a

propeller FF vs. Velocity graph (see below.)

The tangent line drawn from the origin to the FF vs Velocity graph touches the graph at the

speed for maximum range, just as in the example to the left.

Proof:

tan(R) = FF/TAS [examine the diagram to convince yourself of this]

Recall that the definition of SR is SR = TAS/FF. In other words:

SR = 1/tan(R)

SR max occurs when tan(R) is minimum.



Effect of Weight on Specific Range (and Endurance)

Review the analysis of how weight affects range and endurance for a jet airplane. The

analysis is the same for a propeller airplane. Both range and endurance decline at higher

weights due to the increase in induced drag.

Just as with the jet, the propeller airplane must fly at a specific angle of attack for range and

endurance (L/Dmax and L3/2/Dmax respectively.) We have said before but it cannot be

emphasized too much that this means the airplane must slow down when weight decreases.



Effect of Altitude on Specific Range and Endurance

Previously we saw that altitude had a huge effect on SR for jet airplanes. But it actually has

almost no effect on propeller airplanes.

As the drag curve shifts to the right with altitude the power curve shifts up because power

drag x velocity (and shifting to the right represents an increase in velocity even though

drag does not change.)

How does altitude affect endurance of a propeller airplane?

In the graph to the left you can see the bottom of the FF curve moves UP with altitude.

Therefore maximum endurance occurs at sea level for a propeller airplane.

Of course this does not take changes in SFC that occur with altitude into account.

Previously we saw that altitude has only a minor effect on piston engines, so we expect a

piston airplane to achieve maximum endurance at or very close to sea level.

Altitude does affect the SFC of a turboprop because of temperature, and more so because of

the ability to turn higher rpm at altitude. Most turboprops will achieve maximum endurance



at less than 10,000 feet however, which contrasts with jets, which achieve maximum

endurance at 36,000 feet or above (see jet page.)

How does altitude affect the Range of a propeller airplane?

The graph to the left shows that SR does not change with altitude. Recall from above thatSR is maximum when the angle "R" is minimum. You can see that R does not change with

altitude, therefore SR does not change.

As with the discussion of endurance, this analysis assumes that altitude does not affect SFC.

We know that for a piston airplane altitude has no substantial effect on SFC although we

should fly high enough to operate at full throttle. This leads to the conclusion that piston

airplanes achieve maximum - or the same - range at any altitude from about 6000 or higher.

Turboprop engines become more efficient with altitude due to declining temperature.

Therefore we expect a turboprop to have slightly improving SR up to 36,000 feet (thetropopause.) However, SR will not change much between 10,000 and 36,000.

Should propeller airplanes fly high or low?

This is a very important question for pilots to deal with. The jet pilot has a relatively

straightforward decision compared to the propeller pilot.

For a propeller airplane it seems that we can fly at any altitude we desire. This is true in zerowind.

The most important point, and one that cannot be emphasized too much is that the mainfactor determining cruise altitude for a propeller airplane is wind. On most days wind gets

stronger with altitude so it would be foolish to climb into a stronger headwind given the

graph for range shown above. It would be equally foolish NOT to climb if a stronger tailwind

is available. Pilots who do not heed this advice are being wasteful with fuel.

On days when the wind is light any altitude will do, but it is IMPORTANT to note that while

the airplane achieves the same SR at all altitudes, TAS increases with altitude. In other words

you will get to your destination faster at higher altitude. Since the old saying that time is

money is pretty close to being true where airplanes are concerned it "pays" to fly higher. It

does NOT pay in fuel, but it does pay in other ways.

For a turboprop the dropping SFC with altitude and the monetary value of time usually means

that you will climb into a slight headwind. But should not climb into a strong headwind. The

exact optimum altitude on a given real world day requires considerable analysis, see below.



Cruise Control for the Propeller Airplane

Review cruise control for the jet airplane.

There is no equivalent of jet cruise control for propeller airplanes because propeller airplanes

DO NOT have maximum Mach numbers. For a propeller airplane cruise control is a complex

task of analyzing all the costs of which fuel is only one. For example it costs money to do

maintenance on the airplane every 50 or 100 hours, so the more revenue it can generate in

100 hours the more economical it is. It is often wise to consume more fuel to fly faster and

complete more revenue flying in a given number of hours. But fuel is not cheap either, so if

you fly too fast the fuel costs will offset the increased revenue. When you add wind into the

equation there are so many variables that a computer program is required to determine the

optimum cruising altitude and angle of attack.

How Wind Affects Range and Optimum Cruise Speed

SR = groundspeed / FF.

Groundspeed is just TAS +/- headwind or tailwind. It should be obvious that SR is better with

a tailwind and poorer with a headwind. But should we still fly at L/Dmax? The answer is NO!

In the simulation you can change the headwind/tailwind by dragging the windsock icon at thelower right corner of the graph.



The graph to the left is for a jet but the same analysis applies to the propeller airplane. We

must fly faster (lower AOA) with a headwind and slow down (greater AOA) with a

tailwind.

Many pilots use the rule of thumb "increase speed by half the amount of the headwind."

For example speed up by 25 knots for a 50 knot headwind. The simulation determines theactual speed needed. If you try comparing that to the rule of thumb you will see that the

rule is reasonably good.

Most propeller airplane pilots don't actually slow down with a tailwind despite the theory

that shows that they should. As fuel gets more expensive in the coming years we will see if

this changes.


Date post:	02-Jun-2018
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Flight for Range and Endurance

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