# Momentum Theory and Mathematical Analysis for Propeller Forces Explained

## INTRODUCTION

There are many forms of propulsion devices. One such device is a propeller. This theory is one of the oldest theory used for calculation of the thrust, torque and the efficiency of the propeller. We have already learnt how a ship moves forward in our previous article. Now we shall proceed to go into the mathematical details and learn about the momentum theory for propellers.

**WHAT A PROPELLER DOES**

- When a propeller is rotating it imparts momentum to water which causes a force to act on the ship which in-turn moves the ship.

**ASSUMPTION**

***** In this theory, the propeller is considered as an actuator disc.

* An actuator disc is a disc which performs water movement. The disc imparts axial acceleration to the water.

- Let us consider the following:

A - Area of actuator disc,

Po- Ambient pressure ( pressure of water initially and finally ),

dp- Increase in pressure at the tip of the disc,

va- Speed of advance of the screw,

ava- Initial velocity of water,

bva-Final velocity of water,

Now, velocity of water relative to disc is

va + ava = va(1+a)

where a- axial inflow factor

Mass of water acted on per unit time is M=

M= (Density X volume of water)/ time = ρ X area X (length / time) = ρ X area X velocity (since Density =mass/volume)

= ρ A va (1+a)

change of momentum

= mass X velocity

= ρ A va (1+a) X bva

According to Newton’s third law " for every action there is an equal and opposite reaction”, this change of momentum produces thrust and this thrust in-turn moves the ship.

Equating this change of momentum to the thrust we get,

T = ρ A va2 ( 1 + a ) b

We know that the work done by the thrust on water is given by the product of the thrust and the velocity,

= T X ava

= ρ A va2 ( 1 + a )b X ava

= ρ A va3( 1 + a ) X ab

We know that the work done is equal to kinetic energy in water column

ρ A va3( 1 + a ) X ab = work done

kinetic energy in water column

k.e. = (1/2) X m X v2 = (ρ A va (1+a)(bva)2 ) / 2

kinetic energy = work done

ρ A va3( 1 + a ) X ab = ρ A va3( 1 + a ) b2 / 2

a = b/2

- From the above equation it is clear that when water touches the disc half of the velocity is reached ultimately.

The useful work done by the disc is equal to the product of thrust and the forward velocity

Useful work done

= ρ A va2 ( 1 + a )b X va = ρ A va3 ( 1 + a )b

The total work done is the sum of the above two work done:

= ρ A va3 ( 1 + a )b + ρ A va3( 1 + a ) X ab

= ρ A va3 (1+a) [ ab + b ]

- Therefore efficiency = The ratio between the useful work(input) and the total work ( output )

η = (ρ A va3 ( 1 + a )b) /( ρ A va3 (1+a) [ ab + b ] )

η = 1/(1+a)

- The above is called as Ideal efficiency. It means that for a better efficiency the value of ‘a’ should be small so that the efficiency will be large.
- For a given speed and thrust the disc must be large so that the velocity imparted to the water will also be large.

Till now we have considered only the axial velocity imparted to the water, but in case of a real propeller, because of the rotation of the propeller, the water will also have rotational velocity imparted to it by the propeller, and hence it has to be considered. Therefore the overall efficiency becomes,

overall efficiency

η = ( 1 - a’ )/( 1 + a)

where,

a’ - Rotational inflow factor,

This means that, because of the rotational velocity the efficiency is further reduced.

## Limitations of This Theory

- Since the propeller is considered as an actuator disc, the hydraulic leakage losses are neglected. These losses are due to the gap that is present in between the propeller blades.
- When a real propeller rotates, each and every element of the blade contributes thrust to the water but in this theory only the force from the tip of the disc is considered.
- The torque is given by the product of the angular acceleration and the perpendicular distance (ie) the radius of the actuator disc which is not acceptable as in a real propeller there will be gaps in between blades.
- It is considered that the thrust which is produced by the propeller is coming back as it is and acting on the ship, but in reality there will be losses such as wake, etc, due to which the value of thrust will be reduced to an extent.

**Figure Reference:**

Tupper, E.C. (2004) *Introduction to Naval Architecture*, 4th ed. Burlington: Elsevier Butterworth - Heinemann