Propeller Static & Dynamic Thrust Calculation

by panther3001 | September 18, 2013 | (8) Posted in Tips

Originally Written: 16 July 2013

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In order to see a more detailed version of this article, and the most up-to-date version of this article as I make changes in the future, see its original posting location here: http://electricrcaircraftguy.blogspot.com/2013/09/propeller-static-dynamic-thrust-equation.html


Please rate this article after you read.


Propeller Thrust Equation, & Downloadable Excel Spreadsheet Thrust Calculator:

DOWNLOAD MY EXCEL SPREADSHEET THRUST CALCULATOR HERE (click link, then go to File --> Download)


Figures above: a preview of what is to come - Static Thrust (left) & Dynamic Thrust (right).

 

I have been interested in propellers for a very long time.  I've also been interested in how they produce thrust, and how forward velocity affects that thrust.  Therefore, I've done a lot of thinking about it, and put a lot of time into understanding them better.  Here is an equation that I came up with to quantify the thrust produced by propellers.  I wanted it to be a simple approximation, with a minimal number of inputs.  Therefore, it uses only the propeller's pitch and diameter (from the numbers on the front of the prop), and the RPMs at which the prop is spinning (this can be measured from a basic optical tachometer such as the one shown in the picture below).  That's it!

Here is the equation.  

The expanded form is shown to help you see where some of the numbers come from.  The simplified form is shown you help you put the equation into a calculator or Excel spreadsheet easier.

is static or dynamic thrust (it is called static thrust if V0 = 0), in units of newtons (N); RPM is propeller rotations per minute; pitch is propeller pitch, in inches; d is propeller diameter, in inches; and V0 is the forward airspeed, freestream velocity, or inflow velocity (depending on what you want to call it), in m/s.

If you want thrust in other units: to convert newtons to grams, multiply newtons by 1000/9.81.  To then convert grams to ounces, multiply grams by 0.035274.  To convert ounces to pounds, divide ounces by 16.  

Note: the equation has a hard-coded atmospheric density of 1.225kg/m^3, which is the "standard day" (avg. annual) density at sea level.  Therefore, it will provide a thrust estimate assuming you are at sea level.

 

Example:

DOWNLOAD MY EXCEL SPREADSHEET THRUST CALCULATOR HERE (click link, then go to File --> Download)

Here is a thrust example, to demonstrate the use of the equation above:  An airplane has a 10x6 propeller (10 in. diameter, 6 in. pitch), spinning at 10,600 RPMs when at full throttle on the bench.  How much static thrust is it producing?  Answer: using the equation above, the propeller is producing 1619g, 1.619kg, or 3.57lbs of static thrust.  Download the spreadsheet above to change the values for your application.

At what airspeed will it produce zero thrust (ie: what is it's max thrust-producing airspeed)?  Answer:  ~60mph.  Note: the 60mph is also the pitch speed of the propeller, which is an underestimate of the actual max thrust-producing airspeed, since I have not yet corrected the dynamic-thrust portion of the equation for the effects of things such as camber of the propeller and the unloading of the prop with increasing airspeed.

 

How Accurate Is This Equation?

Short answer:

Static Thrust:

It's a pretty good to decent ball-park estimate for all props, and a really good estimator for some props.  For static thrust, consider it accurate to within +/- 26% for 95% of all cases.  Slow Fly (SF) props are the least accurate, and usually produce much more static thrust than the equation estimates.  

The plot below shows how well the equation approximates the thrust, when compared to actual, measured static thrust values:

Dynamic Thrust:

For dynamic thrust, consider the equation to be an underestimate of what the propeller is actually doing, by 15~30% when you extrapolate it out using the equation with the RPM value from a static test run.  For extrapolating out dynamic thrust from a static test run, a good guess is that the actual zero-thrust airspeed will be around 15~30% higher than what the equation says.   In other words, if the equation says you get zero thrust at 60mph, you might actually get zero thrust somewhere in the range of 69mph~78mph (60mph x 1.15 = 69mph, and 60 x 1.30 = 78mph).  As I get more dynamic thrust data, I'll work on correcting this.

One dynamic thrust plot is shown below, comparing actual, measured values to calculated results from my equation above:

 

For more details, more in-depth explanations, and additional figures, see the article at its original posting location here: http://electricrcaircraftguy.blogspot.com/2013/09/propeller-static-dynamic-thrust-equation.html.  I will continue to update and improve the article on my blog as I gain additional insight and understanding in order to improve the equation.


 Please help contribute your thrust data to this project, to help me improve the equation, by clicking here.


Thanks for reading! ***Please rate this article.***


COMMENTS

rljhr on September 18, 2013
EXCELLENT !

I like the mixture of Metric and dark ages Imperial units that plague the aviation industry.
I would like to see the the air speed in Kilometers per hour as I don't come from one of those backward countries ;)
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panther3001 on September 19, 2013
yeah, I myself use mixed units now when I work....at least, to display things so I can relate to them better. mph makes more sense to me for speed, but grams makes more sense to me for thrust, now that I've been working so much with oversees RC airplanes and products. I suppose units can be displayed any way you want, and I chose to do a little mishmash. Next time I'll use units of ergs/meter for thrust, and cubits per fortnight for speed, eh? ;)
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LordVader on September 24, 2013
Great stuff, don't understand it all. But this will make it alittle easier. I am so glad this site has so many people willing to help and show others some much about this hobby. Thank You to all of you, and you know who you are, for helping make this hobby so special and fun!
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panther3001 on September 26, 2013
thanks for your comment!
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tramsgar on September 18, 2013
That was interesting, thanks! I've just read up some on aerodynamics and I agree that props is a very intriguing part of it.
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jawo6015 on August 29, 2014
I am a student at a University in St. Paul Minnesota and was wondering if there was a way I may contact you, panther3001, or for you to contact me via my account info? I am using your formula to calculate theoretical thrust for a propeller for which I only know the propeller dimensions and rpm for which the prop is spinning. I took the geometry for a propeller from CADGrab and used to model a problem in ANSYS and analyzed in CFX. The problem with the model, which I did not catch until after the simulation, is that the prop was generic. I could not find an accurate Thrust coeficient to calculate theoretical thrust and compare to the Thrust results derived by the CFD. Using your equation, I was able to calculate a theoretical that fell within the ranges of percent error noted by yourself and by the CFD. I would like to reference this equation but am not sure how good it will look without knowing your credentials. If you would please contact me or post a link where I may find these I would greatly appreciate it. Thank you very much!!!

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panther3001 on August 29, 2014
jawo6015, I can't seem to find your contact info or profile on FliteTest. To contact me, my website is here: http://electricrcaircraftguy.blogspot.com/. Click on the "Contact Me" link at the top. For some of my credentials, click on the "Old Website" link at the top of my website, and you'll see some university info. at the top of the main, huge document on that main page.

If you email me I can give additional info too.
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panther3001 on August 29, 2014
Also, I recommend you read this: http://electricrcaircraftguy.blogspot.com/2014/04/propeller-static-dynamic-thrust-equation-background.html
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Remzak on September 17, 2015
How can you use this calculation for 3 bladed props? Sorry if I missed something.
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dav1d on December 6, 2016
Thank you for this. I've been searching for an equation for static thrust for a propulsion system employing a ducted fan that does not involve or rely upon parameters for propellers and I think I've been trying to make it more complicated than it really is. I believe that it boils down to Newton's second law. For instance: say you had a ducted fan mounted on a workbench in a fixture such that air exits from the system horizontally, or parallel to the top surface of the workbench. The reaction of the air exiting the system against the fixture is simply Newton's second law: F = m*a, where F is the reaction (thrust), m is the mass flow rate, and a is acceleration or the increment in velocity from inlet to exit (V exit - V inlet), is it not?

My question is: What value should one use for velocity of air entering the inlet? Freestream velocity (zero, since the system is not moving) or the velocity of air entering the inlet (Volume flow rate divided by inlet area)?

Thanks!
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CASEON96 on February 10, 2020
In calculating the mass flow rate, you have used the airspeed downstream of the propeller (=rps*pitch in m) AND the fluid area at the propeller. If the fluid area at the propeller is to be used, the velocity there would be an average of downstream and upstream velocity. I deduced this from https://en.wikipedia.org/wiki/Betz%27s_law. Check for confirmation.
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Propeller Static & Dynamic Thrust Calculation