Propeller Coefficient of Thrust

Summary

This video delves into the coefficient of thrust for a propeller, a topic previously touched upon in terms of advance ratio. The script explains the relationship between aircraft velocity, propeller rotational velocity, and the angle of advance, using the formula for propeller efficiency involving the coefficients of thrust and torque. Drawing from MIT's reference, the video employs dimensional analysis to derive the thrust equation, considering factors like air density, viscosity, and flight velocity. It also explores the Reynolds and Mach numbers' impact on propeller performance, ultimately defining the coefficient of thrust and torque, and their role in calculating engine power and propeller efficiency.

Takeaways

• 🛫 The video discusses the coefficient of thrust for a propeller, a topic related to the efficiency and performance of aircraft propellers.
• 📉 The script introduces the concept of advance ratio, which is crucial for maintaining high propeller efficiency at different speeds.
• 🔄 The angle of attack is explained as related to aircraft velocity and propeller rotational velocity, which is critical for understanding propeller dynamics.
• 📚 The video references MIT materials to delve deeper into the coefficient of thrust and torque, indicating the use of authoritative sources.
• 🔧 Dimensional analysis is used to break down the thrust equation into its fundamental components, showing a scientific approach to understanding propeller physics.
• 🔢 The thrust equation is derived from fundamental principles, with variables representing propeller diameter, rotational velocity, air density, and more.
• 📉 The Reynolds number is identified as a key factor in the thrust equation, relating to the flow of air over the propeller blades.
• 🌀 The Mach number is also discussed, highlighting its importance in understanding the velocity of air over the propeller in relation to the speed of sound.
• 🔄 The advance ratio is revisited, showing its connection to the efficiency of the propeller and how it relates to the angle of attack.
• ⚙️ The torque equation is explained, showing how it is related to the lift and drag forces acting on the propeller blades.
• 🚀 The efficiency of the propeller is tied back to the relationship between thrust, torque, and the advance ratio, completing the loop on the main topic of the video.

Q & A

What is the main topic of the video?

The main topic of the video is the coefficient of thrust for a propeller, including its relationship with advanced ratio and efficiency.

What is the advanced ratio and why is it important for propeller efficiency?

The advanced ratio is the ratio of the aircraft's velocity to the product of the rotational velocity of the propeller and its diameter. It is important because it helps maintain the propeller's efficiency at different speeds by changing the blade angle.

How is the angle of advance related to the aircraft velocity and propeller velocity?

The angle of advance is related through the formula where the tangent of the angle of advance is equal to the aircraft velocity times the propeller velocity (rotational velocity).

What is the formula for the rotational velocity in radians per second?

The formula for the rotational velocity in radians per second is Omega = (RPM / 60) * 2 * pi, where RPM stands for revolutions per minute.

What is the efficiency equation of a propeller mentioned in the video?

The efficiency equation of a propeller mentioned in the video is Efficiency = (1 / (2 * pi)) * (Coefficient of Thrust / Coefficient of Torque) * J.

What does the MIT reference discuss regarding the coefficient of thrust and coefficient of torque?

The MIT reference discusses the dimensional analysis of the coefficient of thrust and coefficient of torque, expressing them as functions of various factors including propeller diameter, rotational velocity, air density, air viscosity, bulk modulus, and flight velocity.

What is dimensional analysis and how is it used in the context of the video?

Dimensional analysis is a method to convert one set of units into another or to check the dimensional consistency of an equation. In the video, it is used to derive the relationships between thrust, torque, and various factors affecting them.

What are the indices a, b, c, d, e, and f in the thrust equation, and what do they represent?

The indices a, b, c, d, e, and f in the thrust equation represent the powers to which the variables (Diameter, Rotational velocity, Density, Viscosity, Bulk modulus, and Flight velocity) are raised, respectively.

How are the Reynolds number and Mach number related to the thrust and torque equations?

The Reynolds number and Mach number are related to the thrust and torque equations as they represent the ratio of inertial forces to viscous forces and the ratio of the flow velocity to the local speed of sound, respectively. They are used to account for the effects of fluid dynamics on the propeller's performance.

What is the final expression for the efficiency of a propeller in terms of the coefficient of thrust, coefficient of torque, and the advanced ratio?

The final expression for the efficiency of a propeller is Efficiency = (1 / (2 * pi)) * (CT / CQ) * J, where CT is the coefficient of thrust, CQ is the coefficient of torque, and J is the advanced ratio.

How does the video script explain the relationship between power input and power output for a propeller?

The video script explains that the power input to a propeller is the torque times the rotational velocity (Omega), and the power output is the thrust times the flight velocity. The efficiency of the propeller is then the ratio of power output to power input.

Outlines

00:00

🚀 Introduction to Propeller Thrust Coefficient

The speaker begins by expressing their intent to discuss the coefficient of thrust for a propeller, a topic they discovered online. They connect this to previous videos on the benefits of advanced ratio for propellers, explaining how to maintain efficiency at different speeds by adjusting blade angles. The speaker introduces the angle of advance and its relation to aircraft and propeller velocities, using the rotational velocity (Omega) formula to derive the advanced ratio. They express interest in the coefficient of thrust and torque, referencing MIT materials to explore these concepts further, promising to explain them based on the notes they've taken.

05:04

📚 Dimensional Analysis of Propeller Thrust

In this paragraph, the speaker delves into the MIT reference material to understand the thrust of a propeller as a function of various factors including diameter, rotational velocity, air density, viscosity, bulk modulus, and flight velocity. They apply dimensional analysis to these factors, assigning indices to each and balancing them according to the units of force (mass times acceleration). The speaker explains the units for each factor and how they relate to the overall thrust equation, aiming to simplify the complex formula into a more comprehensible form.

10:06

🔍 Calculating the Coefficients for Propeller Thrust

The speaker continues by rearranging the expressions obtained from the dimensional analysis to solve for the coefficients a, b, and c, which are related to the propeller diameter, rotational velocity, and other factors. They substitute these values back into the thrust equation, simplifying it further. The speaker also explains how certain terms in the equation relate to the Reynolds number and the Mach number, providing a deeper understanding of how these factors influence propeller performance.

15:08

🌀 Understanding the Advanced Ratio and Its Impact on Thrust

This paragraph focuses on the advanced ratio and its role in calculating thrust. The speaker explains how the advanced ratio is derived and its significance in the overall thrust equation. They also discuss the relationship between the advanced ratio and the Reynolds number, as well as its connection to the Mach number. The speaker integrates these concepts into the thrust equation, illustrating how they contribute to the overall propeller performance.

20:11

🔧 Propeller Efficiency and the Relationship Between Thrust, Torque, and Power

The speaker concludes by discussing the efficiency of the propeller, which is derived from the relationship between thrust, torque, and power. They explain how the power produced by the engine is delivered to the propeller and how this power output is related to the thrust and velocity. The speaker uses the formula for efficiency to show the relationship between the coefficient of thrust, the coefficient of torque, and a factor J, which encapsulates the Reynolds number, the tip number, and the advanced ratio. The summary ties back to the initial formula, reinforcing the understanding of propeller efficiency.

Keywords

💡Coefficient of Thrust

The coefficient of thrust is a dimensionless number that relates the thrust generated by a propeller to the fluid density, the velocity of the fluid, and the area of the propeller disc. In the video, it is central to understanding the efficiency of a propeller, as it is part of the formula that calculates the propeller's efficiency. The script discusses how this coefficient is derived from various factors such as air density, rotational velocity, and flight velocity.

💡Advanced Ratio

The advanced ratio is a term used in aeronautics to describe the ratio of the forward speed of an aircraft to the speed of the tip of the propeller. It is crucial for maintaining the efficiency of the propeller at different speeds. The video script explains that to keep the propeller's efficiency reasonably high, the blade angle must be changed at different speeds, which is directly related to the advanced ratio.

💡Angle of Advance

The angle of advance is the angle between the direction of the relative wind and the pitch of the propeller blade. It is used to calculate the thrust generated by the propeller. In the script, the angle of advance is discussed in the context of its tangent being equal to the product of the aircraft velocity and the rotational velocity of the propeller.

💡Rotational Velocity

Rotational velocity, often denoted by the Greek letter Omega (Ω), is the rate at which an object rotates around a fixed axis, measured in radians per second. In the context of the video, it is used to calculate the angle of advance and is essential for understanding the dynamics of the propeller and its interaction with the air.

💡Efficiency of Propeller

The efficiency of a propeller is a measure of how well it converts the power supplied by the engine into thrust. The script explains that the efficiency is given by the ratio of the coefficient of thrust to the coefficient of torque, modified by the advance ratio, which is a critical concept for understanding propeller performance.

💡Dimensional Analysis

Dimensional analysis is a method used in physics and engineering to derive relationships between physical quantities by identifying their units. In the script, dimensional analysis is used to express thrust as a function of various factors like propeller diameter, rotational velocity, air density, and flight velocity, which helps in understanding the underlying physics of propeller thrust.

💡Reynolds Number

The Reynolds number is a dimensionless quantity used in fluid dynamics to predict flow patterns in different fluid flow situations. In the video, it is related to the ratio of the propeller's rotational velocity to the air's viscosity and is used to understand the behavior of air flowing over the propeller blades.

💡Mach Number

The Mach number is the ratio of the velocity of an object moving through a fluid to the speed of sound in that fluid. It is used to describe the speed of the aircraft relative to the speed of sound. The script discusses how the Mach number is related to the velocity of the propeller and the speed of sound, which is important for understanding the aerodynamic effects at different flight speeds.

💡Bulk Modulus

The bulk modulus is a measure of a fluid's resistance to compression. It is used in the script to describe one of the factors that influence the thrust generated by a propeller. The bulk modulus is related to the air's ability to withstand changes in pressure, which is important when considering the forces acting on a propeller.

💡Torque

Torque is the rotational force that tends to cause rotation about a specified axis. In the context of the video, torque is related to the force generated by the propeller and the distance from the axis of rotation. The script explains that torque is a critical factor in calculating the power required by the engine to rotate the propeller.

💡Power

Power is the rate at which work is done or energy is transferred. In the video, power is discussed in terms of the engine's output (torque times rotational velocity) and the power delivered by the propeller (thrust times velocity). The script uses power to explain the efficiency of the propeller and the relationship between input power and output thrust.

Highlights

Exploration of the coefficient of thrust for a propeller based on online findings.

Connection between advanced ratio and propeller efficiency at varying speeds.

Explanation of the angle of advance and its relation to aircraft and propeller velocities.

Derivation of rotational velocity in radians per second using RPM and 2π.

Introduction to the efficiency equation involving the coefficient of thrust and torque.

Interest in the coefficient of trust and coefficient of torque, leading to MIT reference exploration.

Assumption that thrust is a function of various factors including diameter, rotational velocity, air density, etc.

Use of dimensional analysis to break down the thrust equation into fundamental units.

Conversion of mass, length, and time units to MLT system for dimensional consistency.

Calculation of thrust indices for mass, length, and time to achieve dimensional balance.

Substitution of derived indices back into the thrust equation to simplify the expression.

Identification of thrust components related to Reynolds number, a function of velocity and characteristic length.

Relation of Mach number to velocity and speed of sound, and its role in thrust calculation.

Introduction of advanced ratio as a factor in the coefficient of thrust.

Expression of thrust as a function of air density, rotational velocity, diameter, and advanced ratio.

Calculation of torque as a product of lift force, drag force, and distance, resulting in D to the power of 5.

Engine power is defined as torque times rotational velocity, leading to power in and out of the propeller.

Efficiency of the propeller is derived from the ratio of power out to power in.

Final expression of propeller efficiency involving coefficients of thrust and torque, and the advanced ratio.

Transcripts

hello and in this video I want to go

through

um something I found online about the

coefficient of thrust for a propeller

so

in previous videos that I did I spoke

about the benefit of the advanced ratio

for propeller so if this is the advanced

ratio here on the x-axis

and we can see

um

to keep the efficiency of the propeller

reasonably High we need to change the

blade angle at different speeds and

um that makes sense when you're using a

constant speed propeller what changes

the the blade angle

and

on that previous video I spoke about

this angle of Advance here and we said

that the angle of events are the tan of

the angle of Advance was equal to the

aircraft velocity times the propeller

velocity the rotational velocity of the

propeller

and uh

if we look at this Omega R well Omega is

the rotational velocity in radians per

second

and uh thus RPM divided by 60 multiplied

by 2 pi so that just becomes radians per

second by 2 pi so if I go Omega r

it's n 2 pi by R and 2R is D so I get n

d Pi so I can put in here ND pi

and you can see that that's very similar

here to this Advanced ratio so the

advanced ratio is is provided as the

velocity of over ND

and there is an equation then that says

that the efficiency of the propeller is

1 over 2 pi the coefficient of thrust

over the coefficient of torque times J

so there's the J and the one over Pi so

J and one over Pi here

um

and I was interested on this

part of the expression this coefficient

of trust and coefficient of torque

and I was looking online and

um

I came across this reference from MIT

and they actually go through the the

coefficient of the thrust and the

coefficient of torque

and what I thought it might be

beneficial is if I made a video on on

that so I'm I'm basically just taking

their notes and um and explaining it

okay so what they said was let's assume

that the thrust is a function of the

propeller diameter

the rotational velocity

okay so the revs per second

the density of the air the viscosity of

the air the bulk modulus of the air and

the flight velocity V

and then they just said okay well that's

it thrust is some constant times D to a

power of a

n to the power of B rho to the power of

C Nu to the power of d

K to the power of V and V to the power

of f

so on I've explained up what all of

these are over here on on the right hand

side

so

um

what they were going to do then was use

dimensional analysis

so

the thrust the thrust is just a force

and forces mass by acceleration so

basically saying mass by acceleration is

into this constant times

on all these factors here

now acceleration is defined as the rate

of change or velocity or change of uh

distance or time squared

meters per second squared okay that's

the units of it meters per second

squared so my

expression here mass by acceleration is

really mass times distance over time

squared meters

per second squared

and if we use these letters to symbolize

Mass distance or length and time MLT we

can we can change this expression so we

can say

that m is mass okay so there's there's

the mass the distance s well that's a

length so that's l

and the time T we're going to use the

capital letter T so that's t squared but

it's the T to the minus two okay because

it's per per second squared

so MLT to power -2 and that's equal to

this constant whatever that is times all

of these factors to these different

indices

all right so

um then

the MIT

um

slide went on to uh to to to turn this

into uh some dimensional analysis as

well

so D to the power of a well D is a

distance so it's just the length so

that's length to the power of a

uh the revs per second well it's per

second so that's

time to the power of

a minus B okay so because it's one over

one over t

density is kilograms per meter cubed so

kilograms is mass

meter cubed is meters is distance so

parameter cube is distance to power -3

and that's all to the power of C

and then the viscosity knew

um viscosity is

um what is it measured in is meter

squared per per second so meter squared

so that's length squared per second so D

to the minus 1 t to the minus one sorry

all to the power d

and the bulk modulus was Newtons per

meter squared so I've done that down

here Newtons per meter squared when

Newtons is mass by acceleration

um meter squared is just length squared

acceleration is distance per second

squared so that's distance L length

per second squared t squared the L will

cancel so I get m l to the minus 1 and T

to the minus 2. so

the bolt modulus is an alpha minus 1 t

to the minus two e as we're to the power

of E here

and finally

the flight velocity V well that's just

distance

per second so it's l for length per

second so t to the pi and so on okay and

that's all to the power of f

Okay so

there's our our Force our thrust and

it's equal to K times L to the power of

a

T minus B

ml minus 3C

L squared t to the minus 1 Al to the

power d

ml to the minus 1 t to the minus 2 all

to the power of e and l t to the minus

one R to the power of f

okay so that's my thrust so my thrust is

equal to uh to this equation when I'm

using dimensional analysis

so let's compare uh the indices so over

here I have m to the power of one

there's m

I have m to the power C there is MC m to

the power of e m e and just from the

power of indices m

to power C by m to the power of E is

equal to m to the power of C plus e

so this is really m to the power of one

divided across by m i get one is equal

to C plus e

and I do the same then for l so if I

compare the L indices here we got L to

the power of one so there's l

we got L to the power of a

L to the power of minus 3C

L to the power of 2D

L to the power of minus E

and L to the power of f

okay again I'm going to

um

add the indices so if I multiply these

it's the same as adding the indices so I

have L to the power of a minus 3C plus

two D minus E plus F and if we divide

across by L so L goes this uh

into there once

um

sorry if I just that's wrong to say if I

just compare the indices so L to the

power of one so one is equal to a minus

3C plus 2D minus E plus f

and finally we'll do it for T so over

here we have t to the minus 2.

and that's equal to T to the minus B

minus

uh

sorry t to the power minus d

t to the power minus 2 e

and T to the power minus one f

so t to the power minus two is the same

as t to the power minus B minus D minus

two e f

just comparing the indices then so minus

two is equal to minus B minus D minus

two e minus F and If I multiply by minus

one

I get 2 is equal to B plus d plus two e

plus plus f

okay

so that was my expression from the M

coefficients and I've just rearranged

that to get a value of for C

and this was my expression for the L

coefficients

and I've just rearranged that to get

um

a value for a

okay and this was my expression for the

T coefficients and I've just rearranged

that to get a value for B and I'm going

to substitute these back in here for a

be

and C

okay so uh when we do that

we get L to the

um sorry

we get D to the power of a now becomes D

to the power of well a was four minus

two e minus D minus f

n to the power of B is now n to the

power of

uh two minus D minus two e minus f

uh roll with c

and uh

C is equal to 1 minus E

so that becomes rho to the power of 1

minus E

and we still have Nu to the power of d k

to the power of V and V to the f

okay so uh we have D to the power 4 so

I'm going to take that out

there's D to the power of four we then

squared we're going to take that outside

and we have row to the one so we're

going to take that out

so if we look at the coefficients then

uh with d

okay so

um we had Nu to the power d

and then we add the diameter here to

minus 2D so that's

-2

D and we'd n here

to the power of minus D so that's just n

to the power of d

so grouping you know the powers that he

um that's what I get

for k then we've K to the power of E and

we have D to the power of minus 2E so d

to the minus 2 e

n to the minus 2E so that's N squared e

and rho

to the minus E that's row

uh

to the minus E here

and finally uh to the power of f we add

V to the F and then we had D to the

minus F so there's D to the minus F and

there's n to the minus f

okay so now our so don't forget this is

our thrust is equal to

these values here and then some function

of of of

these factors

okay so if I look at

this part of it first

so Nu over D Squared n

so that relates to the Reynolds number

so if I look at renin's number

um Reynolds number is the velocity V

times the characteristic length L over

um

over the the viscosity

and so normally

um you would say you know if this was in

if this this is an air file

and also this is the air going over it

so that's the velocity and this is the

the characteristic length L which is

normally the cordland okay

so that's what the remnants number is so

if I

um

if I invert that then 1 over re becomes

Nu over VL

and if I have a propeller blade

the

air coming in

is that way and then this is the air due

to rotational

and this is the airflow over the

propeller so it is proportional to this

Omega r

okay so the velocity is proportional to

this Omega R and Omega is just radians

per second so you would say

um

Omega

is equal to 2 pi n

and then our

so

2R is D so we got Pi n d so I could say

the velocity here

is proportional to

ND

and the propeller

okay so we said this length here is the

characteristic length but you know the

calc the propeller sorry tables here

so the length is

you know some taper ratio the overall

Dimension d

so this becomes the so what I'm saying

is this is really

proportional to new all over

in D times D which is Nu Over N D

Squared okay

which is what we have here

so this is really

the

one over to Redlands number

that's proportional or it's a function

of one over the randless number

if we look at the second part of it then

um if I look at the Mach number

well Mach number is the velocity over

the speed of sound so again if this is

my propeller blade you know it's the

velocity over over the blade divided by

the speed of sound so I've just

explained that this this velocity is

proportional to

Omega R which is effectively

proportional to ND

I'll do a capital D that's the way it

should be

and

um we'll keep that in mind okay the

speed of sound here is gamma RT where

gamma's uh the ratio specific heat of

air

and it's a constant it's 1.4 and this

the r is the universal gas constant

so this is uh 1.4

and this is 287.

that's joules per uh

the Calvin

dark per kilogram joules per kilogram

and this is the temperature

okay now if I take the um

if I take the universal gas law

um which is you know the density is

equal to P over RT and if I transpose

that I get t is equal to the pressure

over the density times r

so I'm going to substitute that back in

here

so now my Mach number is proportional to

well Omega R is basically

sorry I'll start again

I want to get rid of the square root so

I'm going to square both sides

okay so the Mach number squared is equal

to

um

Omega r squared which is really

proportionate that N squared D Squared

saw that from here

and then we have if we Square this we

get gamma RT and instead of T then I'm

going to put in p p over or

and the RS will cancel

so we just get one over gamma which is

the same as multiplying with one over

row sorry which is the same as

multiplying by row

so we get rho D Squared N squared on

over gamma and this is a constant so we

could just say it is proportional to rho

D Squared N squared which is what we

have here

so I could say this is proportional to 1

over the Mach number

uh squared

or 1 over Mach number squared

so it is proportional to the Mach number

okay and eventually uh finally this

value here where we we opened up this

um this screencast looking at this and

this is called the advanced raycoj

all right so I'm going to put all of

that together

and we're going to say that the thrust

is equal to some value rho N squared D

Squared and it's a function of the

random number the tip number and the

advanced ratio and we can

lump this

and that into some coefficient and we

call that the coefficient of thrust

okay so now we're saying when we this

was always the trust here this part of

it here

so the thrust is some coefficient of

trust rho N squared D to the power 4.

and if I look at the the torque

well if I have you know a lift force

from the air file so that's the trust I

will have a drag force and that's acting

at a distance D to give us torque

so we'll call this

coefficient of torque row N and then

we're just multiplying this by this

value D so we get D to the 5. so the

torque is

CQ row N squared D to the power of 5.

all right so we're nearly there now this

is a long video uh so the power

produced by the engine power is torque

times Omega and we've seen that Omega is

just 2 pi n

so that's the power delivered by the

engine

to power in the power out

so the propeller is delivering some

power now and that's equal to the thrust

times the velocity

so

um so

that's a force

by distance over time

Force by distances work work per second

is uh Power so

um

the power out then is thrust times

velocity so if I want to look at the

efficiency I want to get what do we get

out

from what do we put in

so what we get out is trust times

velocity so instead of thrust now I'm

going to say it's CT rho N squared D to

the power of 4.

times the velocity

all over the torque which is CQ

row N squared D to the power of 5.

2 pi n

for

Omega

and

there's my formula again and then things

just start canceling out so that goes

that goes that goes that goes that goes

that comes just d

so we get C T times V all over CQ d 2 pi

n

just separate those components out so

the one over two Pi that's that

c t over CQ is there and then I get V

left at V over to uh sorry V over ND

and we said this uh was equal to J

so the efficiency is equal to 1 over 2

pi CT CQ over J

and that's where we started from

so that's where that all comes from

um and I'm very thankful for that

website there on the on MIT and I hope

that's of some benefit to you all