The Wright brothers used twin-pusher
propellers
on their
powered aircraft
between 1903 and 1905.
The details of how a
propeller generates thrust is very
complex, but we can still learn a few of the fundamentals using the
simplified momentum theory presented here.
This modern theory is slightly different from the theory developed
by the Wright's to design their propellers.
Propeller Propulsion System
On the slide, we show a schematic of a propeller propulsion system
at the top and some of the equations that define how a propeller
produces thrust at the bottom. The details of propeller propulsion
are very complex because the propeller is like a rotating wing.
The Wright brothers are generally credited with being the first ones
to look at the problem this way.
The Wright propellers had 2 blades. The blades were designed to be
long and thin, and a cut through the blade perpendicular to the long
dimension gives an airfoil shape. Because
the blades rotate, the tip moves faster than the hub. So to make the
propeller efficient, the blades are twisted from hub to tip.
The angle
of attack of the airfoils at the tip is lower than at the hub because
it is moving at a higher velocity than the hub. Of
course, these variations make analyzing the airflow through the propeller a very
difficult task. Leaving the details to the aerodynamicists, let us
assume that the spinning propeller acts like a disk through which the
surrounding air passes (the yellow ellipse in the schematic).
The engine, shown in white, turns the propeller and does work on
the airflow. So there is an abrupt change in pressure across the
propeller disk. (Mathematicians denote a change by the Greek symbol
"delta" ( ).
Across the propeller plane, the pressure changes by "delta p"
( p). The
propeller acts like a rotating wing. From airfoil
theory, we know that the pressure over the top of a lifting wing
is lower than the pressure below the wing. A spinning propeller sets
up a pressure lower than free stream in front of the propeller and
higher than free stream behind the propeller. Downstream of the disk
the pressure eventually returns to free stream conditions. But at
the exit, the velocity is greater than free stream because the
propeller does work on the airflow. We can apply Bernoulli's
equation to the air in front of the propeller and to the air
behind the propeller. But we cannot apply Bernoulli's equation across
the propeller disk because the work performed by the engine violates
an assumption used to derive the equation.
Simple Momentum Theory
Turning to the math, from the basic thrust
equation, the amount of thrust
depends on the mass flow rate through the
propeller and the velocity change through the propulsion system. Let us
denote the free stream conditions by the subscript "0", the conditions at
the propeller by the subscript "p", and the exit conditions
by the subscript "e". The thrust (F) is equal to the mass flow rate (m dot)
times the difference in velocity (V).
F = [m dot * V]e - [m dot * V]0
The mass flow through the
propulsion system is a constant, and we can determine the value at
the plane of the propeller. Since the propeller rotates, we can
define an area (A) that is swept out by the propeller of blade length
(L). Through this area, the mass flow rate is density (r) times velocity (Vp),
times area.
m dot = r * Vp * A
Substitute this value for the mass flow rate into the thrust equation to get the
thrust in terms of the exit velocity, entrance velocity, and velocity
through the propeller.
F = r * Vp * A * [Ve - V0]
We can use Bernoulli's equation to relate the
pressure and velocity ahead of and behind the propeller disk, but not through
the disk. Ahead of the disk the total pressure (pt0) equals the static pressure
(p0) plus the dynamic pressure (.5 * r * V0 ^2).
pt0 = p0 + .5 * r * V0 ^2
Downstream of the disk,
pte = p0 + .5 * r * Ve ^2
At the disk itself the pressure jumps
delta p = pte - pt0
Therefore, at the disk,
delta p = .5 * r * [Ve ^2 - V0 ^2]
The force on the propeller disk is equal to the
change in pressure times the area (force/area x area = force)
F = delta p * A
If we
substitute the values given by Bernoulli's equation, we obtain
F = .5 * r * A * [Ve ^2 - V0 ^2]
Combining the two expressions for the the thrust (F) and solving for Vp;
Vp = .5 [Ve + V0]
The airspeed through the propeller disk
is simply the average of the free stream and exit velocities.
[Note that this thrust is an ideal number that does not
account for many losses that occur in practical, high speed
propellers (like tip losses). The losses must be determined by a more
detailed propeller theory, which is beyond the scope of these pages.
The complex theory also provides the magnitude of the pressure jump
for a given geometry. The simple momentum theory, however, provides a
good first cut at the answer and could be used for a preliminary
design.]
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