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Thread: [Acro] Fwd: Prop thrust equation?
Message: [Acro] Fwd: Prop thrust equation?
Follow-Up To: ACRO Email list (for List Members only)
From: "Doug Carter" <douglcarter at hotmail.com>
Date: Tue, 18 Dec 2001 18:43:48 UTC
"----Original Message Follows----
From: Paul Stambaugh <pstambau at yahoo.com>
To: "Dr. Guenther Eichhorn" <acro at gf24.de>
Subject: [Acro] Prop thrust equation?
Date: Tue, 18 Dec 2001 07:03:14 -0800 (PST)
Does anyone have the equation to calculate prop thrust?"
From a NASA website:
Most general aviation or private airplanes are powered by propellers. 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.
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. Propellers usually have between 2 and 6
blades. The blades are usually long and thin, and a cut through the blade
perpendicular to the long dimension will give an airfoil shape. Because the
blades rotate, the tip moves faster than the hub. So to make the propeller
efficient, the blades are usually twisted. 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, we know that 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
There is no pressure-area term because the pressure at the exit is equal to
the free stream pressure. 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 = ps0 + .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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Guided Tours
Propellers:
Turboprops:
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Go to...
Beginner's Guide Home Page
Learning Technologies Home Page
http://www.grc.nasa.gov/WWW/K-12
NASA Glenn Home Page
http://www.grc.nasa.gov
NASA Home Page
http://www.nasa.gov
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by Tom Benson
Please send suggestions/corrections to: benson at grc.nasa.gov
Last Updated Mon, Oct 22 04:43:50 PM EDT 2001 by Ruth Petersen 9/6/2000
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