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Flight for Range and Endurance- Propeller Airplane

Before 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.

 

   

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 that time 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 than approximately 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 avoid operating 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 pressure is 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 in SFC.

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 propeller airplane 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 that SR 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 (the tropopause.) 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 zero wind.

The most important point, and one that cannot be emphasized too much is that the main factor 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 the lower 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 the actual 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.

 

© Copyright Raymond J. Preston