Post on 06-Mar-2015
Axial Momentum Theory:
Axial Momentum theory tells how the propeller works. In this theory propeller is
considered as an actuator disc or circular disc, instead of consisting of blades
which is rotating in air. The action of the actuator disc is to increase the pressure
field in the fluid across the propeller disc. Therefore the disc generates the thrust.
In this theory how the propeller changes the pressure field is not explained. There
is an assumption that the pressure field changes across the disc. The fluid which
passes through the still disc with velocity called free stream velocity. Free stream
velocity is equal and opposite to the forward propeller.
Figure 1-Actuator Disc
P0 – Initial pressure value at upstream and downstream of the propeller
Pressure drops down to the negative value P1 and then increases to due to the
action of propeller to P2. After passes through the propeller the pressure drops to
initial value.
When there is change in pressure there will be change in velocity. Velocity is
gradually increasing in the flow due to continuity there will not be abrupt change
across the disc.
V1= Increament velocity at the disc; V2 = Increament velocity at downstream.
Assume that the velocity is constant across the disc (i.e. V+V1 is constant). Here
only the velocity change in axial direction is considered.
Mass of fluid passing through the disc = ( ) ( )
Then
Thrust produced by the propeller = [( ) ] ( )
Rate of change of momentum in the entire fluid gives the thrust force generated in
the propeller.
Substitute the Value of mass in equation (2), we get
( )
Power delivered by the propeller = Work done by the thrust
= ( ) ( )
Work done by the thrust = Rate of change of Kinetic Energy
[( )
] ( )
( )[
]
( ) [ ]
( )
( ) ( ) ( )
( ) ( )
[ ]
( )
From the final equation, increament in velocity at the disc is half of the
downstream velocity.
In non-dimensional form,
( )
Propeller efficiency:
Output power = Power used by the aircraft.
( )
( )
Where
√
( )
Design Engineers always try to increase the efficiency of the propeller. From the
above conditions ‘a’ is always positive but less than one. always less than one.
In order to increase the efficiency from the equation (9), change in will cause
increase or decrease in .
In the above equation and T cannot be changed. Only the variable
can be changes to change . This means that bigger the propeller efficiency is
more.
From the above equations, the final results are
1.
2. For higher , the propeller diameter should be more.
Momentum Theory including rotation or Impulse Theory
This is similar to momentum theory but it includes rotational velocity. In this case
there will be rotational velocity impart by the propeller to the fluid. The slip-stream
contracts at the disc and goes far away.
Actuator Disc rotating with angular velocity,
In this case we are considering only the elemental area but not the whole disc area
because the rate of change of angular momentum changes with radius from the axis
of the disc.
Elemental area, ( ) ( )
Across the disc, velocity is constant
[( ) ]
( ) ( )
[ ]
( ) ( )
Work done in elemental thrust = change in translational kinetic energy
( )
[( )
]
( )(
) ( )
Work done by the torque = rate of change of rotational kinetic energy
[ ]
Increamental rotational velocity at the downstream is twice the increamental
rotational velocity of fluid at the disc.
( )
By the additional of rotational velocity,
Efficiency also changes due to the addition of rotational velocity. depends on
increamental translational velocity and increamental rotational velocity.
Translational velocity increament depends on diameter of the propeller and
rotational velocity increament depends on r.p.m of the disc.
So depends on two factors
1. Diameter
2. Rpm
; increase if rpm decreases for the same thrust.
Blade Element Theory
The simple momentum theory provides an initial idea regarding the performance of
a propeller but not sufficient information for the detailed design. Detailed
information can be obtained through analysis of the forces acting on a blade
element like it is a wing section. The forces acting on a small section of the blade
are determined and then integrated over the propeller radius in order to predict the
thrust, torque and power characteristics of the propeller. BET explains how
actually the propeller produces thrust by observing certain amount of power or
torque, and it depends on the shape of the propeller. Let us consider that the
propeller blade consist number of blade elements rather than considering the whole
blade. Integration of the thrust which produces by an each element of blade will
gives the total thrust which produced by the blade. If there is ‘Z’ number of blades,
then the thrust generated by one blade multiplied with Z gives the total thrust of
propeller. When the blade element moves in fluid, it is subjected to axial velocity
and tangential velocity of the fluid. The axial velocity and tangential velocity
together generate force on the blade element. The resultant force is composed of
two vectors in axial and tangential direction. Thrust is the force which acts in an
axial direction and tangential direction force generate moment around the axis of
rotation called torque. The sum of all the axial forces gives total axial force. For
tangential calculate the moment around rotation axis and sum all the moment to get
total torque.
A differential blade element of chord c and width dr, located at a radius r from the
propeller axis, is shown in Figure 1. The element is shown acting under the
influence of the rotational velocity, ωr, forward velocity of the airplane, V, and the
induced velocity, w. Vector sum of these velocities produce
Figure 1: Propeller blade element with velocity and force diagram
The section has a geometric pitch angle of its zero lift line of β. If it is assumed that
V and ωr are known, then calculation of the induced velocity w is desired to find αi,
and consequently the section angle of attack α. knowing α and the section type, Cl
and Cd can be calculated, then the differential lift and drag of the section will
follow. However, w depends on dL which in turn depends on w. Thus the problem
is closely related to the finite wing problem but is more complicated because of the
helicoidal geometry of the propeller.
When the propeller rotates in clockwise direction, fluid rotates in opposite
direction. The blade element at an angle subject to flow generates lift force and
drag force due to action of flow i.e. induced drag without considering viscosity.
Tangential velocity = ; n = rotation per second
Propeller is rotating in one direction and the fluid is impinging on the each section
of the blade. VR is the resultant velocity at an angle that falls on the blade. Axial
velocity is perpendicular to because air passes perpendicular to the blade.
;
is the geometric helix angle of the element measured between the zero-lift line of
the element and the rotor disc.
is the angle between the relative velocity and the chord.
is the angle between the resultant velocity and the plane of rotation.
Geometric angle of attack can be calculated from by knowing V and .
Lift and drag depends on the angle of attack. The elemental lift expressed by the
blade element is
The elemental drag is found to be
Where and are 2-D aerodynamic characteristics of the blade section. From
the force diagram,
[
]
[
]
Then
[ ]
From the force diagram
( )
If drag is equal to zero then efficiency will be equal to one. In this case we have
considered that there is no change in velocity.
Both axial and tangential velocities undergo changes due to propeller action. Due
to the propeller action the flow is pulled and reduced in rotational direction i.e.
velocity induced (increament or decrement) due to propeller action.
decreases slightly
− increases slightly
By considering the induced velocity,
( )
( )
( )
( )