ppl_05_e2
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Customer: Oleg Ostapenko E-mail: ostapenko2002@yahoo.com
Customer: Oleg Ostapenko E-mail: ostapenko2002@yahoo.com
CHAPTER 8: PROPELLER THRUST
DEFINITION OF TERMS.
Before we begin our discussion of the Principles of Flight aspects of propeller theory, here are some basic illustrations and defnitions of propeller components, and technical terms describing the function of propellers, without which any discussion of propeller theory would be extremely diffcult.
Blade Shank (Root).
The Blade Shank or Root is the section of the blade nearest the hub to which the blade is attached. The hub forms the end of the propeller shaft which is turned by the engine.
Blade Tip.
The Blade Tip is the outer end of the |
Figure 8.5 Propeller nomenclature. |
blade farthest from the hub. |
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Plane of Rotation.
The Plane of Rotation is an imaginary plane perpendicular to the propeller shaft. It is the plane which is described when the blades rotate (see Figure 8.6).
Spinner.
The spinner is the fairing fitted over the hub of the propeller, in order to reduce drag.
Blade Chord Line.
If the propeller blade is viewed end-on, from tip to root, and a cross-section is taken across the blade, it can be seen that the blade is of aerofoil shape. This means that the blade’s section has a chord line, just as a wing cross-section does. The Blade Chord Line is an imaginary straight line joining the centre of curvature of the leading edge of the propeller blade to the blade’s trailing edge.
Figure 8.6 Plane of rotation.
Figure 8.7 Blade cross-section and chord line.
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CHAPTER 8: PROPELLER THRUST
Blade Angle or Blade Pitch.
The Blade Angle or Blade Pitch is the angle between the blade chord line and the plane of rotation. The blade angle changes along the length of the propeller blade, decreasing from root to tip. This twist in the propeller blade can be seen in Figure 8.5, and will be dealt with later in the chapter. The “mean blade angle” of a propeller is the blade angle at the three-quarters blade length position, measured from blade root to tip. Fine-pitch propellers have a small mean blade angle. Propellers with larger mean blade angles are called coarse-pitch propellers.
Blade Angle of Attack.
The Blade Angle of Attack is the angle between the chord line of any given blade element and the relative airfow which meets the propeller blade when the propeller is rotating. The propeller operates at its most effcient at an angle of attack of around 2 to 4 degrees.
Geometric Pitch.
The Geometric Pitch is the distance the propeller would travel forward in one complete revolution, if it were to advance through the air at the blade angle, just
as a wood screw penetrates a wooden block with one turn of the screwdriver.
Figure 8.10 Geometric pitch.
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CHAPTER 8: PROPELLER THRUST
Effective Pitch.
In fight, the propeller will hardly ever advance through the air at the Geometric Pitch. Air is a fuid, not a solid medium like wood. Propeller Slip will almost always be present. The distance that the propeller actually moves forward with one revolution is called the Effective Pitch.
Figure 8.11 Effective pitch.
Propeller Slip.
The difference between Geometric Pitch and Effective Pitch is called Propeller Slip.
Figure 8.12 Propeller slip.
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CHAPTER 8: PROPELLER THRUST
Helix Angle.
As the propeller rotates and advances through the air (following the line of Effective Pitch), the actual path that the blades follow describes a helix. The Helix Angle is the angle between the Plane of Rotation of the propeller and the path of the Effective Pitch.
Figure 8.13 (Top) and 8.14 (Bottom) showing the helix angle.
PROPELLER THEORY.
Two Theories of Propeller Thrust?
So far, you have learnt (and, if you have stood behind a propeller-driven aircraft with its engine running, have doubtless experienced) that a propeller accelerates a large mass of air rearwards, thus generating thrust in accordance with Newton’s 2nd and 3rd Laws. You have also read in the defnitions above that propeller blades are aerofoils and are set at a given angle to the plane of rotation and so meet the air at an angle of attack. Furthermore, you will recall from an earlier chapter, in this Principles of Flight book, that aerofoils which meet the relative airfow at certain angles of attack generate an aerodynamic force called lift by virtue of the pressure distribution above and below the aerofoil. Rotating propeller blades, then, would seem to be able to generate thrust in the form of a “horizontal lift force” in accordance with the theories of the Swiss scientist Bernoulli, and as illustrated in Figure 8.16.
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Customer: Oleg Ostapenko E-mail: ostapenko2002@yahoo.com
CHAPTER 8: PROPELLER THRUST
As we have already suggested, there appear to be two theories here which apply to the generation of thrust by a propeller. We must investigate this situation further.
If we examine a propeller from close quarters (remembering to observe appropriate precautions, if the propeller is attached
to an aeroplane), perhaps the most obvious feature we observe is that the propeller blades are twisted along their length, as shown in Figure 8.15. The blade angle, in fact, decreases from hub to tip.
We will learn more about blade twist later on, but we can perhaps feel intuitively that the propeller twist may help to explain how air is accelerated rearwards when the propeller rotates, reinforcing the Newtonian theory of thrust that we have already read about.
On the other hand, because of the wing-like, aerofoil structure of the blades, we are maybe ready to admit, too, that the airfow over the rotating blades produces similar aerodynamic effects to those produced by the airfow over a wing.
Figure 8.15 A propeller blade is twisted along its length with the blade angle decreasing from hub to tip.
Figure 8.16 illustrates the arrangement of forces which likens propeller thrust to the lift force produced by a wing. We will return to this diagram later.
Figure 8.16 The rotating-wing analogy of propeller thrust.
Some books on elementary propeller theory, while they may mention that thrust is produced by the reaction to a rearwards acceleration of air, then go on to examine, in depth, only the “wing-theory” explanation of thrust. However, because it is diffcult for anyone who has spent time around aeroplanes to dismiss the rearwards acceleration of air explanation of propeller thrust, no treatment of the subject can be complete unless the Newtonian theory is paid serious attention.
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CHAPTER 8: PROPELLER THRUST
There is both
a “Newtonian” and a
“Bernoulli” explanation of thrust, which some aerodynamicists regard as being, ultimately, the same explanation from two slightly different perspectives.
Consequently, both the “rearwards acceleration of air” theory and the “wing theory” of thrust will be covered in this chapter. If you fnd this state of affairs bewildering, you should draw comfort from the fact that some aerodynamicists defend the view that the two theories ultimately both give rise to a single, identical explanation of thrust, and that the two theories really give the same explanation from two slightly different perspectives.
In this chapter, however, for the sake of simplicity, we will consider the concept of propeller thrust from the points of view of the two separate theories, frst, we will look more closely at the simplifed momentum theory of propeller thrust, and, then, we will consider the propeller as a type of rotating wing (the Bernoulli theory).
PROPELLER THRUST AND SIMPLIFIED MOMENTUM THEORY.
Figure 8.17 Thrust can be explained as a forwards-acting reaction to a rearwards acceleration of air..
Simplifed momentum theory, as applied to propellers, teaches that a mass of air is accelerated rearwards by the propeller (Figure 8.17), and that the reaction to this rearwards acceleration of air at the propeller blades, gives rise to the thrust force which drives the aircraft forwards. This explanation of thrust is sometimes known as the Newtonian explanation because it explains propeller thrust using Newton’s 2nd and 3rd Laws. That is, that “an applied force is proportional to the rate of change of momentum of a body caused by that force” (Newton’s 2nd Law), and “every action has an equal and opposite reaction” (Newton’s 3rd law).
m(Ve |
– Vo) |
(3) |
The equation: Thrust = |
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t |
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which we met earlier in this chapter shows us mathematically, that thrust is produced by the reaction to the rearwards velocity increase (i.e. acceleration) imparted to a given mass of air fowing through the propulsion system.
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CHAPTER 8: PROPELLER THRUST
As we have already established, if we assume that the mass of air fowing into the propeller disc is equal to the mass of air fowing out of the propeller disc, then we may also assume that the rate of mass fow m is constant.
t
Therefore, thrust may be considered as being proportional to the increase in velocity
(Ve – Vo), imparted to the air. |
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In other words, if: m = constant |
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t |
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Thrust h (Ve - Vo) |
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For a fxed-pitch propeller (that is a propeller whose blade angle cannot be varied by |
For a fixed |
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the pilot), experiments show that the increase in velocity imparted to the air passing |
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pitch propeller, |
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through the propeller disc, (Ve – Vo), is greatest when the aircraft is stationary, with |
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thrust is at |
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the propeller blades rotating at maximum revolutions per minute (RPM). Now, it can |
a maximum |
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readily be appreciated that when the aircraft is stationary, Vo, the velocity of the air |
when the aircraft is stationary |
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fowing into the propeller disc, will be very small. However, air is indeed induced, or |
under full power (maximum |
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drawn, into the propeller disc, even though the aircraft is not moving forward. Were |
RPM). |
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it not to be so, there could be no acceleration of air rearwards and the aircraft could |
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not begin to move. In contrast to Vo, however, when the aircraft is stationary and the |
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engine under full power; the velocity of the air leaving the propeller disc, Ve, is very |
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high. Therefore, velocity difference of the accelerated air, Ve – Vo, is at a maximum |
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when the aircraft is stationary under full power, at maximum RPM. (Provided, that is, |
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that the propeller tips do not approach the speed of sound, a proviso that we shall |
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assume for our explanation of propeller theory.) |
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Consequently, from the equation: |
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Thrust = |
m(Ve – Vo) |
(3) |
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t |
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if (Ve – Vo) is a maximum, we can deduce that thrust is also at a maximum.
Thus, we see that the thrust of a fxed-pitch propeller is greatest when a pilot applies full power to begin the take-off run.
Torque.
You should note, too, at this point, that, as well as accelerating air rearwards, the propeller blades also give rise to a drag force (whose magnitude varies with blade angle of attack and blade velocity), acting in the opposite direction to propeller rotation, which balances the turning force of the engine as depicted in Figure 8.16. The propeller’s drag force is often called “propeller torque” while the engine’s turning force, or, more accurately expressed, turning moment, is called “engine torque”. The magnitude of the torque produced by the propeller enables the propeller to absorb the power of the engine. Torque is a by-product of thrust. Without torque, there could be no thrust, and the propeller would overspeed to the destruction of both it and the engine.
The propeller’s
drag force is called propeller
torque.
Propeller torque balances engine torque. Without torque there could be no thrust.
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CHAPTER 8: PROPELLER THRUST
The thrust from a fixed-
pitch propeller decreases with increasing aircraft flight speed.
The RPM of a fixed-pitch
propeller will, at a constant
power setting, increase with increasing aircraft flight speed.
The Variation of Thrust with Speed for a Fixed-Pitch Propeller.
Having established that propeller thrust is greatest when the aircraft is stationary under full power, let us see how the thrust from a fxed-pitch propeller varies with the aircraft’s forward velocity. As aircraft speed increases, Vo increases, too, being equal to the aircraft’s forward speed, plus a small value of induced velocity caused by the propeller’s rotation. Ve, on the other hand, increases by a much smaller amount with increasing airspeed, because, as we shall see, increasing airspeed causes the blade’s angle of attack to decrease.
Therefore, with Vo increasing more rapidly than Ve, as the aircraft gathers speed, the value Ve – Vo must decrease, causing the propeller to impart a progressively diminishing acceleration, or velocity increase, to the air passing through its disc.
Consequently, again from Equation (3), we see that the thrust from a fxed-pitch propeller decreases with increasing aircraft fight speed. With decreasing thrust, caused by a decreasing angle of attack, the propeller torque which resists the engine’s turning moment also decreases, and so propeller RPM will increase with increasing aircraft speed. If fight speed could continue to increase indefnitely, the aircraft would eventually reach a speed at which no further increase in velocity could be imparted to the air passing through the propeller disc. At this fight speed, Ve – Vo would equal zero, and propeller thrust would also be zero.
In level fight, of course, this zero-thrust speed cannot be reached by the aircraft, because the aircraft’s maximum achievable forward speed in level fight is limited to the highest speed at which the propeller can still generate suffcient thrust to balance aircraft drag. Also, with zero thrust, propeller torque would also be zero. There would, thus, be no force to oppose engine torque and the engine RPM would increase to destruction.
It is possible, of course, to increase the forward speed of an aircraft beyond its maximum level-fight speed by entering a dive. In a steep dive, under power, the aircraft does, indeed, approach nearer to the theoretical speed where the propeller produces zero-thrust. As a result, propeller torque decreases rapidly, too , and, if the pilot does not throttle back, engine RPM will continue to increase. This is why, with an aircraft powered by a fxed-pitch propeller, it is easy is to exceed maximum permissible engine rotational speed in a dive. In a dive, therefore, pilots must take care not to “red-line” the engine.
PROPELLER POWER AND PROPELLER EFFICIENCY.
You may remember from your Physics lessons at school that Power is defned as the
Rate of Doing Work, and that Power may be expressed using the formulae:
Work Done Power = __________
time taken
or, because Work Done = Force × Distance through which the Force moves,
Power = |
Force × Distance |
(5) |
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time taken |
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Customer: Oleg Ostapenko E-mail: ostapenko2002@yahoo.com
CHAPTER 8: PROPELLER THRUST
Let us apply Equation (5) to the case of an aircraft in fight. You know that the Force which drives an aircraft forward is called Thrust. You may also have learned that the term Distance is an expression of “velocity”.
Time
Therefore, the power required to drive an aircraft forward at a given velocity is expressed by:
Power = Thrust × Aircraft Velocity ............(6) |
For practical |
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purposes, |
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In the specifc case of the propeller, |
propeller |
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power is equal |
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Propeller Power = Thrust × Vo ............(7) |
to thrust times aircraft velocity. |
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where Vo is the velocity of air fowing into the propeller disc.
For our Effciency considerations, later on, you must remember that, even when the aircraft is stationary, Vo has a small positive value when the propeller is turning. Consequently, as we have already discovered, under normal operating conditions, Vo will always be a little greater than the aircraft’s forward velocity.
Let us now examine propeller power and propeller effciency. Propeller power, or propulsive power, obviously comes ultimately from the engine, and, in order to help it fy effciently, an aircraft should develop maximum possible propulsive power at the expense of the smallest possible power output from the engine. Propeller effciency, then, is an expression of what proportion of engine power output is converted into propulsive power.
Propeller Effciency = |
Propeller Power |
............(8) |
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Engine Power |
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And, from Equation (7) |
Thrust × Vo |
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Propeller Effciency = |
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Engine Power ............(9) |
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Now, you have already learnt that a propeller develops its maximum thrust when it is stationary under full power, at the beginning of its take-off run. You can see, however, from Equation (9) that, when the aircraft is stationary, even though the thrust is at a maximum, propeller effciency is low, because Vo is very small.
Of course, Vo increases as aircraft forward speed increases. You have learnt, too, however, that for a fxed-pitch propeller, thrust diminishes as aircraft forward speed increases, and that, at a certain value of forward speed, thrust will reduce to zero.
What this means for propeller effciency, in practical terms, is that the effciency of a fxed-pitch propeller will increase up to a given aircraft speed, but will, thereafter, as aircraft speed increases still further, diminish, eventually approaching zero as the aircraft nears its theoretical zero-thrust speed, and thrust approaches zero. In fact, a fxed-pitch propeller will operate at its optimal effciency at one value of airspeed only.
The optimal
efficiency of a fixed-pitch
propeller is
achieved at one value of aircraft airspeed, only.
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CHAPTER 8: PROPELLER THRUST
PROPELLER DIAMETER.
Though it is beyond the scope of this book, a further relationship between the power and effciency of propellers, which is interesting to consider, is that, for a given power, both propeller effciency and thrust increase as the diameter of the propeller increases. This is the reason why man-powered and solar-powered aircraft, which at the end of the 20th Century achieved impressive performances for distance fown, used large diameter propellers which turned slowly, imparting a small acceleration to a relatively large mass of air.
Increasing the propeller’s diameter will also lead to an increase in propeller torque, and so large-diameter propellers would theoretically be effective in absorbing the power produced by high-performance piston engines. Unfortunately, though, there are practical and physical limitations to a propeller’s diameter.
Firstly, a large diameter propeller would make it impossible to achieve ground clearance, unless the aircraft had an impractically long undercarriage.
Secondly, because the rotational velocity of any element of a propeller blade increases with increasing distance from the axis of rotation (the propeller hub), the tips of a propeller blade are moving through the air at a much greater velocity than those parts of the blade nearer to the hub. The tips of a large-diameter propeller could, therefore, approach the speed of sound.
Near, or at, the speed of sound, airfow characteristics associated with compression and shock waves would cause propulsive effciency losses as well as greatly
increasing propeller noise.
Let’s look a little further into this latter statement.
The rotational speed of a point moving in a circular path is defned as the linear velocity of that point around the path’s circumference.
We can see from Figure 8.18 that the length of the circumference traced out by a point on a propeller blade increases as the distance of that point from the centre of rotation of the propeller increases. When a propeller is rotating at constant RPM, all points along the length of the blade take the same amount of time to make one revolution. Obviously, though, the elements of the propeller blade furthest from the hub
have to travel a greater distance in that time. Therefore, the greater the distance of a blade element from the hub, the greater its rotational speed, for any given value of propeller RPM. It can be proven that, at constant angular velocity, (N), measured in revolutions per minute, RPM, the rotational speed of a propeller element, at distance r from the axis of rotation of the propeller, is equal to 2 × π × r × N .
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