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,

( )

( )

( )

( )