Calculating necessary Horsepower

[Breadboard]

Calculating necessary Power to generate required Thrust

Hello,
This is my first post on this site and I hope someone can help. I am building an experimental air vehicle for my final dissertation in my Transport Engineering degree course. I am trying to track down a formula I need but can't figure it out, and am tearing my hair out !

I need to calculate the combustion engine size (hp) necessary to:

Lift a given weight vertically off the ground, when driving a propeller of fixed given pitch and given diameter, running at a given rpm, for a given drag co-efficient. Assuming air density and altitude to be at sea level.

Can anyone please help,
Thankyou very much in advance. :)

 

[Breadboard]

No ideas anyone?

 

[Astronuc]

Is this like a helicopter?

One then needs to determine the necessary thrust and work back throught the torque and the power is just the rotational velocity * torque

http://hyperphysics.phy-astr.gsu.edu/hbase/rke.html#rke

http://www.sparknotes.com/physics/rotationalmotion/rotationaldynamics/section2.rhtml

http://www.machinedesign.com/ASP/viewSelectedArticle.asp?strArticleId=57854&strSite=MDSite&catId=0

 

[Breadboard]

Thanks for the reply Astronuc. Yes its kind of like a helicopter. However it has a fixed prop, not a variable pitch prop as in a heli. Therefore the calculations should be simpler.

To phrase it differently, I want to calculate the thrust necessary using my set of fixed values which I know, and then as you say I can figure out the hp:

Weight
Prop rpm
Prop Pitch
Prop Diameter
Air Density
Gravitational Force
Estimated Vehicles Drag Co-efficient

Therefore:

Thrust Required = The above factors sorted somehow!

But I am not sure how the above factors relate to each other in such a formula and hence can't figure out the formula. I know there is one because many moons ago I knew it but can't track it down. From memory it contained all the above 'except' the Prop Pitch value. :(

So basically I need to know the formula, or maybe someone can help me figure it out?

 

[Breadboard]

I think I may be mixing up my constants and my formula, but my memory is a bit hazy about this! Anyway, I know the force required to push up, because I can convert my objects weight into Newtons pushing down. Therefore the force pushing up must be larger by any amount that that pushing down. But I think the formula used the constants above to calculate the thrust in pounds generated by a specified prop, which I could then convert into horsepower to calculate my required engine size. Does this help anyone give me the necessary formula?

 

[RainmanAero]

Hi Breadboard:

Weight
Prop rpm
Prop Pitch
Prop Diameter
Air Density
Gravitational Force
Estimated Vehicles Drag Co-efficient

Therefore:

Thrust Required = The above factors sorted somehow!

But I am not sure how the above factors relate to each other in such a formula and hence can't figure out the formula. I know there is one because many moons ago I knew it but can't track it down. From memory it contained all the above 'except' the Prop Pitch value. :(

So basically I need to know the formula, or maybe someone can help me figure it out?

You've actually got at least two different design problems mixed in here. So let's see if I can help you derive what you need, because any single forumula that contained all of these things would only be valid for certain aircraft and engine design assumptions.

First, I assume your design point is steady-state cruise flight (i.e. you will need more thrust for takeoff and climb than the solution for steady-state cruise). At constant cruise speed and level flight, we know that Lift=Weight and Thrust=Drag. You don't mention that you know the aircraft's lift coefficient, so in this case weight does us no good. To size the engine for optimum cruise we focus on the Thrust=Drag condition. This gives us the minimum thrust required to maintain level flight at whatever the design cruise airspeed is (this is a performance issue, and aircraft performance curves can be developed to define the airspeed at which the absolute minimum thrust is required).

So if you know your cruise airspeed, the thrust required becomes:

Treq = Drag = CD*0.5*rho*Airspeed^2*(Aircraft Wing Reference Area)

(rho is your air density at your cruise altitude).

Now as far as the the specifics of the engine and propeller design, you would need to know more about the propeller aerodynamics to be able to calculate the thrust it could produce at any given aircraft airspeed and any given RPM. For another thing, even with fixed-pitch propellers, while their incidence is fixed with respect to the airstream, the pitch usually varies from the spinner (hub) out to the tip to achieve maximum efficiency (since each radial distance from the spinner is operating at a different absolute speed, the optimum pitch angle decreases as you move out towards the propeller's tip).

One of the principle design parameters for a propeller engine is called the Advance Ratio. The equation for this is:

J = Advance Ratio = Aircraft Airspeed/(Prop Speed in Rev/Sec)/(Prop. Diameter)

Before we go any further, let me make sure you understand this so far and give you a chance to ask questions or provide feedback.

Rainman

 

[Breadboard]

Rainman thankyou very much for all your help so far. Ironically another site has just given me a formula also. Literally within the last 5min after trying to locate this for over a month I have two! lol

Anyway, I need to calculate the values for the object to rise vertically from a stationary position at ground level. Not in flight with lift generated by lift surfaces.

1) In your formula above Treq = Drag = CD*0.5*rho*Airspeed^2*(Aircraft Wing Reference Area) what does the value 'CD' represent please?

2) The prop pitch varies from hub to tip, but when purchasing they have a given pitch value, can this value not be used to give a relatively accurate approximation for the prop?

The other formula I have come across is:

Ct = T / p . n^2 . D^4
Where:
Ct = Thrust Co-Efficient
T = Thrust
p = rho
n = rev per sec
D = Diameter

3) I think this formula is more appropriate however I don't know what 'Ct' or 'T' are measured in. T would undoubtedly be Newtons but 'Ct' ?

4) And what's the relationship between them so far as the difference between T and Ct? ie: How do I calculate Ct?

I also found another formula that I think is not what I need, but it could be relevant. Its:

F = .5 x r x A x [Ve^2 - Vo^2]
Where:
F = Thrust
r = Air Density
A = Prop Disc Area
Ve = Exit Velocity
Vo = Aircraft Velocity

5) My major problem with this formula was that I couldn't figure out the value for Ve. Any ideas?

6) Last but not least, I am going to be using either a 4 or a 5 bladed propeller. How does this change my formulas if at all? I am thinking that for example with a four bladed prop I would just double the resultant thrust. Or would I have to increse my powers in the formulae by a factor of two? (ie: ^4)

Thanks for any help. Its really great to have someone to ask this of.

Esentially what I am trying to establish in the end is what hopsepower engine I need to lift my object when I know its weight, my prop diameter, pitch, and rpm.

 

[RainmanAero]

Hi again, Breadboard:

Anyway, I need to calculate the values for the object to rise vertically from a stationary position at ground level. Not in flight with lift generated by lift surfaces.

Ahh, OK I see. Then the Treq=Drag equation I provided is not applicable. In fact, if you are lifting it vertically then only drag upon the body underneath the propeller is relevant, and it would be acting in the same direction as weight. You can use the same drag equation I gave you, but just drop the "=Treq" part. This drag force would be added to the weight that you would have to overcome.

1) In your formula above Treq = Drag = CD*0.5*rho*Airspeed^2*(Aircraft Wing Reference Area) what does the value 'CD' represent please?

That would be the drag coefficient which you said was a given. To use this equation given the vertical lifter you describe above, you would have to replace Airspeed with the exit velocity of the propeller (which you refer to in a later equation). And instead of the wing reference area, you would need to know the frontal projection area of the body underneath the propeller, as this would be the area used to arrive at the non-dimensional drag coefficient.

2) The prop pitch varies from hub to tip, but when purchasing they have a given pitch value, can this value not be used to give a relatively accurate approximation for the prop?

In that case, the manufacturer of the prop is typically quoting you the pitch of the propeller blade at a distance 75% of the way from the hub to the tip. You MAY be able to use this along with the thrust coefficient, but I am going to have to think about how...

The other formula I have come across is:

Ct = T / p . n^2 . D^4
Where:
Ct = Thrust Co-Efficient
T = Thrust
p = rho
n = rev per sec
D = Diameter

3) I think this formula is more appropriate however I don't know what 'Ct' or 'T' are measured in. T would undoubtedly be Newtons but 'Ct' ?

'Ct' is a dimensionless coefficient, just like the drag coefficient. If you check the dimensions you will see that Ct turns out with reduced units of "1/rev^2", which is essentially non-dimensionalize to the inverse of the square of the rotational rate of the prop.

4) And what's the relationship between them so far as the difference between T and Ct? ie: How do I calculate Ct?

Just like an airplane manufacturer specifies the drag of a specific airplane design by the drag coefficient (CD), a propeller manufacturer will quantify the thrust performance of a specific propeller design via the Ct. Therefore, Ct should be a number specified by the propeller manufacturer for any given design. You then scale it to the thrust you need by making it larger (D) or making it rotate faster (n). This is why I believe you should be able to use the pitch factor along with the Ct. For a given blade pitch, the manufacturer should be able to tell you what the Ct is for that specific propeller design (with the noted amount of pitch at 75% of the blade radius).

I also found another formula that I think is not what I need, but it could be relevant. Its:

F = .5 x r x A x [Ve^2 - Vo^2]
Where:
F = Thrust
r = Air Density
A = Prop Disc Area
Ve = Exit Velocity
Vo = Aircraft Velocity

This is the standard propeller equation which relates the force produced by the propeller to the difference in airstream energy (velocity^2) between in front of the propeller (Vo) and behind the propeller (Ve). This difference in velocity is tied directly to a specific propeller design (Ct and the pitch).

5) My major problem with this formula was that I couldn't figure out the value for Ve. Any ideas?

This is directly related to the aerodynamics of the propeller (pitch and Ct) and what rotation rate it is turning. If you were given a Ct by the manufacturer (for the specific pitch), then you could solve the Ct equation for "T" and set that equal to the "F" of this equation. If you then assume Vo=0 (you are hovering with no axial airspeed approaching the propeller), you could then solve this equation for Ve.

6) Last but not least, I am going to be using either a 4 or a 5 bladed propeller. How does this change my formulas if at all? I am thinking that for example with a four bladed prop I would just double the resultant thrust. Or would I have to increse my powers in the formulae by a factor of two? (ie: ^4)

It is not that easy. Propeller thrust does not scale linearly with number of blades... mostly because of aerodynamic interference effects between the blades. The aerodynamic force created by one blade creates the dreaded "downwash drag" on the propellers behind it. Depending on the aerodynamic design of the propeller blade itself, adding blades could actually create more drag and thus be less thrust. This is where "simple" propeller design assumptions break down and computational fluid dynamics and differential equations are needed to accurately model and analyze the propeller flowfield.

I would not worry about trying to model the effects of multiple blades. That is too detailed for what you are doing. All you really want to "tailor" is the velocity difference from front to back of the propeller shown in your "F" equation above. It should be as large as you need to lift your weight (and oppose the drag the propeller creates) but no larger. The reason is that when you analyze propeller efficiency, it (prop efficiency) actually goes DOWN (becomes less efficient) as the difference between Vo and Ve goes up. This is generally why propellers are not used for aircraft that have to fly at Mach numbers above about 0.6 because their efficiency is poor compared to turbofan jet engines.

Rainman

 

[Breadboard]

Thanks Rainman. That has cleared up a huge amount. So this is the equation I need:

Ct = T / p . n^2 . D^4
Where:
Ct = Thrust Co-Efficient
T = Thrust
p = rho
n = rev per sec
D = Diameter

I will e-mail the prop manufacturer today and get a value for Ct.

Once I have Ct and solve for T. Can I then simply compare my T Newtons thrust to the force pushing my object down in Newtons, then change the prop diameter and rpm until it overcomes the force holding it down, and convert the T Newtons into pounds thrust and henceforth into Horsepower to get my necessary engine size for my object?

Then of course I'd have to check that said engine can turn my prop size at the necessary rpm.

Thanks