## Wednesday, September 18, 2013

### Propeller Static & Dynamic Thrust Calculation - Part 1 of 2

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Propeller Static & Dynamic Thrust Calculation

By: Gabriel Staples
Written: 16 July 2013
Last Updated: 13 April 2014
--made a minor correction to an example, & added a plot
--added many more statistical details in order to better explain the accuracy of the equation.
--added Simplified form of eqn. too. - 15 Oct. 2013
--added links to Part 2 of this article - 13 Apr. 2014

Other Articles:

Propeller Thrust Equation, & Downloadable Excel Spreadsheet Thrust Calculator:

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

Figure 1: 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 to the left).  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 to help you put the equation into a calculator or Excel spreadsheet easier. Eqn. 1: Thrust Equations, expanded and simplified forms.

F 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)

Figure 2: Dynamic thrust for a 10x6 propeller at 10,500 RPM.

Here is a thrust example, to demonstrate the use of the equation above.  Refer to the plot just above, copied from my spreadsheet thrust calculator, whose link is just above the plot.  The example is as follows: an airplane has a 10x6 propeller (10 in. diameter, 6 in. pitch), spinning at 10,500 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?
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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 the thrust calculation to give you a thrust value accurate to within +/- 26% of the actual thrust for 95% of all cases, and accurate to within +/- 13% for 68% of the cases.  Slow Fly (SF) props are the least accurate, and usually produce much more static thrust (up to 67% more) than the equation estimates.  Therefore, the worst-case scenario is that your actual thrust will be 1.67 times the value of the calculated thrust, and of the 149 cases I looked at, this only occurred this dramatically in one of the SF prop cases.

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 to increase the accuracy and confidence of my estimate.

Long answer:
Static Thrust:
Once I got my analytical form of the equation, with many big assumptions made, I took 149 data points of actual, measured static thrusts from various propellers from various online sources, and I did an empirical fit to get an empirical correction constant.  The prop sizes ranged from as small as 5x5 to as large as 17x8.  After applying the correction factor, here are my results.  Blue diamonds are actual, measured static thrusts.  Red squares are my calculated results from the equation above, with V0 = 0.  As you can see, my semi-empirical thrust calculation is pretty good, despite its simplicity.  All it takes into account, aside from atmospheric conditions (air density) is propeller RPMs, diameter, and pitch.

For the below plot, across all 149 data points checked, the calculated thrust was at most 30% higher than the actual thrust, and no lower than 40% less than the actual thrust, with a standard deviation of 13%.

For 68% of the cases, or 101 of the 149 cases looked at, my thrust calculation was accurate to within +/- 13% (one standard deviation) of the actual thrust, and for 95% of the cases, or 142 of the 149 cases looked at, my thrust calculation was accurate to within +/-26% (two standard deviations) of the actual thrust.

For 58% of the 149 cases, the calculation was a slight overestimate, and for the remaining 42% of the cases, the calculation was a slight underestimate.  The underestimates were most dramatic for Slow Fly (SF) style props.  My equation is most accurate for standard props, and least accurate for SF style props, as SF props tend to generate significantly more static thrust (due to their wider blades) than my calculation predicts.

Figure 3: 149 Data points of actual vs. calculated static thrust, for propellers ranging from 5x5 ~ 17x8.

Dynamic Thrust:
Whereas I've had 148 data points to compare to for static thrust, as of today (18 Sept. 2013), I have only had one data point to compare to for dynamic thrust.  Therefore, I still have a lot of work to do.  Here is that data point below, graciously obtained from Matthew McCrink of Ohio State University.

As the dynamic thrust plot below shows, the results from my equation show the same basic trends as the experimental data.  The experimental thrust vs. freestream velocity curve, in blue, is nearly linear, and the calculated result for a constant propeller RPM is linear.  RPMs do increase with freestream velocity (see RPM plot  farther below, also obtained from that experimental data point), as I also hypothesized--presumably because drag on the blades (tangential to their rotation) decreases since incident angle of attack to the propeller blades decreases with increased inflow velocity (see propeller diagram below).  One other thing to consider:  my dynamic thrust equation assumes that the zero-thrust freestream velocity is equal to the propeller’s “pitch speed,” Vpitch, since the propeller will have an incident angle of attack of 0 deg when Vpitch = V0.  In other words, the value of the freestream velocity where my calculated thrust curves intersect the 0 thrust line (y = 0) is equal to the propeller’s pitch speed at that RPM.  However, on a real propeller, with a cambered airfoil, thrust will still be produced even with a 0 deg incident angle of attack.  Perhaps this explains why my thrust calculation is an underestimation when compared to the experimental results:  0 thrust from the propeller will actually occur when V0 > Vpitch, and the blade α is some slightly negative value.  Again, this fact will make the experimental dynamic thrust results higher than my calculated dynamic thrust results, with my calculated dynamic thrust results having a steeper negative slope than actuality.
Figure 4: Dynamic thrust wind tunnel data vs. estimates using Equation 1.

Figure 5: RPM vs. tunnel velocity for the wind tunnel data above.

Figure 6: Propeller diagram, showing how the blade strikes the air, and how its apparent angle of attack, α, decreases with increasing forward airspeed.

How Did I Come Up With This Equation?

Read here: Propeller Static & Dynamic Thrust Calculation - Part 2 of 2 - How Did I Come Up With This Equation?

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

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#### 95 comments:

1. Hi, If I wanted to use this to calculate the thrust for a 3-bladed prop, could I just scale up by a factor of 3/2?

1. Sorry, it's not that simple, and I don't think it scales linearly with blade #. Honestly I don't have the answer right now for how to get this calculation to work for a 3-bladed prop. I'd need a bunch of data points so that I could tweak the correction factor for this equation in order to make it applicable to a 3-bladed prop.

2. Thinqer, since it's very good questions like yours that drive me to learn and think and explorer more, let me give a more lengthy explanation. Since a propeller blade is like a wing, it is nice to think of them as wings. This is a very good analogy, and is correct to think of them as such. One way of analyzing a propeller blade even involves slicing the blade into many many cross-sectional elements of a small, but finite depth, then analyzing them each individually as 2-D airfoils, multiplying the results by the depth to get 3-D lift forces, and summing the results of all segments to get the thrust (summed lift) of the entire propeller. This is called Blade Element Theory.

Now, it may seem logical to extend this comparison of propeller blades to wings in an incorrect manner as well. For example, if you have a wing of a given size, at a given angle of attack, and at a given airspeed, it will produce a certain amount of lift, L. If you have two of these wings separated by enough distance that they are each getting clean airflow, you will get a lift equal to 2L. Three wings will produce 3L units of lift, etc. However, this linear scaling does NOT apply to propeller blades. Five blades of a given size, pitch, and angular velocity DO produce more than 4 blades, which DO produce more than 3 blades, which DO produce more than 2 blades, etc, but the scaling is not linear. Therefore, according to my understanding (but I’d like to see hard data), the efficiency of the propeller decreases each time you add more blades. Ex: a 2-bladed prop might be 70% efficient, but the same prop in a 3-bladed version might be only 66% efficient, a 4-bladed 60%, etc. The reason thrust (or lift) scaling is nonlinear is because the blades must occupy a finite amount of space, equal to the frontal, circular area of the spinning propeller, and the more area the blades occupy, the less area that air has to pass through the blades. At some point, adding more blades will produce LESS thrust, not more (though this would have to be many many blades, blocking much of the frontal area).

Ok, so let’s call this frontal area the propeller “disc.” The area of the disc is equal to pi*(d^2)/4, where d is the diameter of the propeller. I derived my equation knowing that thrust is a factor of this disc area, as well as the velocity of the air moving through this area. Note that the air moves from the front of the airplane, perpendicularly through the propeller disc, and towards the tail of the airplane. I call the velocity of this air the “induced velocity,” or “exit velocity.” There is a theoretical maximum induced velocity that the air can travel through the blades, however, at a given propeller pitch and RPM, and no matter how many extra blades you add to the prop, the actual velocity of the air can never exceed this theoretical velocity. It can only approach it asymptotically.

To make my point, let’s call this maximum theoretical “induced velocity,” or “exit velocity,” Ve_max. For this example, let’s arbitrarily assume that a 1-bladed prop (properly balanced of course) at a given RPM produces an exit velocity of 0.5Ve_max. A 2-bladed prop with the same pitch and diameter and at the same RPM might produce an exit velocity of 0.7Ve_max (1.4x [the “x” is read as “times” here] the exit velocity of a 1-bladed prop), a 3-bladed prop might do 0.8Ve_max (1.6x that of a 1-bladed prop), and a 4-bladed prop might produce 0.85Ve_max (1.7x that of a 1-bladed prop).

Now, to complicate matters, static thrust is proportional to the exit velocity squared, so in this example, the 2-bladed prop would produce 1.4^2 = 1.96x as much thrust as the 1-bladed prop, the 3-bladed prop would produce 1.6^2 = 2.56x as much thrust, and the 4-bladed prop would produce 1.7^2 = 2.89x as much thrust.

3. See how non-linear this is? If it were linear, a 3-bladed prop would produce 3x as much thrust as a 1-bladed prop, and a 4-bladed prop would produce 4x as much, etc. Now, in actuality, I do not know these exact relations, and I made up the numbers in this example to make my point. I need more data to get real numbers. If you, or anyone else has a thrust stand and wants to take data on 3 or 4-bladed props, please do! You can click the link in my document above to add your results to my spreadsheet.

Ok, so if you read this far, you are amazing! If you’d like the explanation of how I got my equation, please let me know and I’ll take the time to post it in the next week or two.

2. Hi Gabriel,

Great work! I'm looking forward to digging into this in more detail over the coming weeks (it's pretty dense). Have you looked at the corollary for prop drag (i.e. motor torque) at a given RPM? This bit is required for calculating overall system efficiency (grams thrust / Watt electrical power).

1. Thanks for reading it! I have not looked into motor torque much yet, mostly because I don't have a dynamometer, and motor torque data is hard to find. Motor torque would be really interesting to have, however, as output power (W) is simply the quantity of torque (N-m) x angular velocity (radians/sec). Input power is measured through a power meter as VxA, and then efficiency is equal to Power_output(W)/Power_input(W). Efficiency is indeed something that interests me, but I don't have it yet. Very good point!

2. Hi, several years late to the party!
Correct me if I'm wrong but I'm pretty sure motor torque = current / Kv. So if you're collecting data using a motor with a known Kv, finding torque is as simple as reading current draw, no?

3. Hi, I've been coming up with some r/c model designs for a university assignment, and I came across this beautiful spreadsheet of yours. Now, through motor testing, my team has determined the static thrust of a specific propeller is a certain number(at about 50% thrust of the motor). Using this number and this table (http://www.flybrushless.com/tools/thrustCalc), I got an estimate of the RPM at its static thrust. I slapped the RPM into your spreadsheet along with the dimensions and it gave me a figure of static thrust that is within your error range for the tested static thrust. I looked down the table to find an airspeed I would've wanted at this RPM and took the dynamic thrust of that +/- the error. Did I utilise your spreadsheet in an acceptable way? I'm about 80% sure that I have. Also, Looking at your equation, I was trying to figure out where some of the numbers come from. I'm kind of struggling where you got 3.29546 from. I'm fairly sure it's a conversion factor. Can you tell me pls?

1. Ryan, I need to post my follow-up post to explain where I got this equation. I"ll try to get to that in the next couple of weeks. The 3.29546 is not a conversion, it is an empirical correction factor, as is the 1.5 power.

Next, let me make sure I understand:
What is it that you are trying to find out? I'm having a hard time seeing what you are trying to get. Do you want to know how fast your plane can go? Perhaps you have a drag coefficient for your plane and want to know when (at what airspeed) the Thrust = the Drag, in order to estimate max airspeed? Perhaps you want to know what prop is best? I'm a bit confused what you are trying to get to.

It looks like you are doing things in a bit of a round-about way, and you could be more direct. Ex: if you have measured static thrust, why not just use my equation to estimate RPM? My equation requires RPM, prop diameter, prop pitch, and airspeed. For static thrust, airspeed = 0, and you know what prop you used, so you can use my equation alone to get RPM. The diameter and pitch is written on the front of the prop.

Now that you have estimated your static thrust RPM (which you could have also measured by the way; though backing it out after measuring thrust is good because it is basically a way to “calibrate” my equation to your exact setup), you can use my equation to hold diam, pitch, and RPM constant, while varying forward airspeed. This will get you a linear curve with a negative slope (like Figure 2). The point where the line crosses zero is the zero-thrust airspeed. If my equation were accurate you could know how fast your plane can fly by seeing at what airspeed Thrust = aircraft Drag (assuming you know your aircraft drag coefficient, since D = CdqS [S = planform area]). Since my equation, when holding RPM constant, estimates the zero thrust value to be much lower than in real-life, however (see Figure 4), you might just guestimate that the airspeed at which my eqn shows zero thrust is somewhat indicative of the max airspeed you can expect from your vehicle. In other words, even though it says it is the “zero-thrust” value, since we know it’s a low estimate (as Figure 4 shows), you might just say that it’s a “low thrust” value, at which point Thrust ≈ Drag.

Again, I’m not sure what you are trying to get to, but here’s another approach you can try, to see how propellers affect your vehicle’s performance:

*******************************
How to use my equation to choose a propeller for your design, WHEN YOU ARE CAPABLE OF MEASURING STATIC THRUST:

1) Measure static thrust with a variety of propellers. BE SURE TO USE A WATT-METER in-line with your power supply TO ENSURE YOU ARE NOT EXCEEDING MOTOR/ESC POWER AND CURRENT SPECS BY USING A PROPELLER OR VOLTAGE COMBINATION WHICH IS TOO LARGE.
2) Back out the RPM using my equation.
3) Using my spreadsheet, hold the RPM constant (use the value calculated just above in step 2), and create plots like Figure 2.
4) Overlay all of the plots, to see where the dynamic thrust curves intersect.
5) Choose a propeller to use: if you want maximum thrust and acceleration at cruise speed, choose the prop which has the greatest thrust at your estimated cruise speed; if you want maximum top speed, choose the prop which has the greatest zero-thrust airspeed.

-if you do the above experiment. I’d love to see your results, so please communicate with me to figure out how I can get them, and I’d be happy to post them on my blog, with your permission.

2. ***********************
ONE MORE IMPORTANT NOTE about my test method recommended above: in order to make the propeller comparisons fair, you must use the same motor, and the same input Power (W) to the motor. To do this, you will have to vary input Voltage. If you (mistakenly) keep Voltage constant, props with larger diameter and/or pitch will draw more power, and props with smaller diameter and/or pitch will draw less power. This is not a fair comparison.

Here's an example of what you'll need to do:

A given motor is rated for 300W max power (full throttle, sustained for 30 sec. max or motor will overheat). Let's say with a 3S LiPo battery (11.1V nominal), and an 8x6 prop, it draws 300W. If you put on a 9x6 prop, but keep the input voltage at 11.1V, it will try to draw perhaps 375W. Therefore, you must LOWER THE INPUT VOLTAGE to keep the input power constant, at the 300W max test condition, even with this larger prop. Now, with a 7x5 prop and 11.1V input voltage it might only draw 200W. Again, you must keep power constant. In this case INCREASE THE INPUT VOLTAGE to get it back up to 300W input power. When you are all done, you will have good propeller performance comparisons, with all of them drawing the same 300W input power.

3. Empirical correction? I see! Thanks! I was getting all confused about it! Right, I initially wanted to use your spreadsheet to pick out a prop, basically. We've been given the static thrust of two propellers at both 50% and 100% throttle and I actually overlooked the fact that I can get RPM out of that using your equation, sorry! Seems I made my work more complicated than it needed to be. Now, I'm trying to use your spreadsheet to determine the dynamic thrust of our chosen prop at a fixed cruising airpseed, which we've decided to be 3 m/s. I chose the RPM at 50% throttle (which I have now found through your equation to be around 11000 rpm) and then picked out a 3 m/s airpseed then took the dynamic thrust from that which is around 219 grams. So this shows me that my thrust for our propeller dimensions at 50% throttle on the cruise speed we want of 3 m/s, we'll get an estimate of 219 grams dynamic thrust. Is this correct?

About the testing method, my lecturers did the testing while we observed and recorded so I didn't know if they adjusted the input power, I'm just working off the figures I got. We didn't use a watt-meter either. We're only at preliminary design stage at the moment so I reckon we'll be able to get back to that at a later time.

4. What you've described sounds correct. Keeping constant input power is only important if comparing different props. For a given prop, and varying throttle settings, the input power will certainly vary. I'm also very curious as to what size plane this is. It sounds like it must be a tiny little airplane with perhaps a 2-ft wingspan max, and a little brushed in-runner motor with a 3x2.5 prop or something very small. Is it a Parkzone Vapor perhaps? I'd like to verify your specs if you don't mind. What are the test results for 50% & 100% throttle? I'll need static thrust & prop dimensions of course.

5. Ok I just used my spreadsheet really quickly to see that my previous guess seems to be way off. My spreadsheet shows a dynamic thrust (at 3 m/s airspeed) of 220.6g with a 6x4.5 prop (as one solution) at 11,000 RPMs. Anyway, I'll have to see what you respond though. I'm curious.

4. Oh yes, right! So basically, I'm a university student for Aerospace Systems Engineering and my entire year has been divided into different groups assigned to design and build our own RC model given a few predetermined components such as a selection of batteries and props, an ESC, a collection of servos, and a transmitter-receiver pair. Our group has been divided into a further 2 smaller groups to come up with some preliminary designs. The aircraft itself has a universal weight limit of around 800 grams but my smaller group has calculated ours to be no more than 600 grams, and it is more than likely that we'll have less than that. You have calculated correctly, we'll have a 6x4E prop. It's a pretty small thing (wing-to-wing is like 60 cm and fuselage is 40cm long) so wow, you're pretty right on the money there, good sir. Also, our motor is actually a brushless outrunner with 1920 Kv (it's labelled a KD2213 which made me assume it was a KEDA motor but as it turns out, it's actually one of those cheap Turnigy ones). Our static thrusts for our chosen prop are 247 grams at 50% throttle and 520 grams at 100% throttle. Sorry, I read my dynamic thrust wrong, my eyes must've been tired. Our dynamic thrust was calculated at about 208 grams, and our RPM was actually 11100. GIven the error margin you've mentioned, I've calculated the actual dynamic thrust of the aircraft to be between 239 grams and 270 grams at 3m/s which means our thrust to weight ratio on our cruise speed at our maximum calculated flight weight would be between 40% and 45%.

5. Also, if you don't mind sir, I need to jot you down in my list of references for the handouts we're currently making right now (our preliminary design presentation is today at 1 pm UK time). And if you also don't mind again, I'd like to refer to you should I have future enquiries. Obviously, it is not your priority to answer my questions so I'll perfectly understand if you decline.

1. That sounds fine. You have my full name, and my email is rcflyyer [ a t ] gmail [d o t] com if you wish to email me further questions. You may also post them here if you like, of course. A few more notes I have though: what error margin are you using to get 239g~270g estimated thrust? And can your plane even fly as slow as 3m/s? That is just above a brisk walking speed.

Also, what battery are you using? And do you choose the motor or was that chosen for you? A 1920 Kv motor is a good mid-range kv for a 2S Lipo, but it is a very high kv for a 3S Lipo.
--For slow flying, you want a large diameter prop with low pitch, and a low kv motor. This will maximize your *static* thrust while giving you a very steep negative slope on your dynamic thrust plot (see Figure 2). So, you will have a low top speed.
--For fast flying you want a small diameter prop with a large pitch, and a high kv motor. This will give you a very shallow negative slope on your dynamic thrust plot (see Figure 2), drastically reducing your static thrust, but dramatically improving your high-speed dynamic thrust. Therefore, you will have a much higher top speed.

PS. I frequently use Turnigy motors. They work fine. Perhaps not top-of-the-line, but plenty good for my purposes. And for their price, they are excellent.

6. Thank you very much! I'm very new to this whole designing a model thing, and could use an experienced mentor when dealing with stuff like this. Didn't you mention that the dynamic thrust using your equation is an underestimate of between 15-30%? That's where I got 239-270 from. Did I misunderstand that point? The 3 m/s cruise speed at about 50% throttle is just a desired theoretical fixed airspeed, we haven't dabbled too deeply into it as we are still only in preliminary design. We'll get to discuss more about it once the finalised design gets decided upon by our lecturers and our small groups reconvene into the big group again. Saying that, my lecturers seemed to not have many major qualms about my small group's design, not on the bits that I did, anyway. They had quite a lot of questions about stability and if the plane would be capable of leaving the ground in the first place though. We were given a choice between three 3s LiPos of varying capacities: 300maH, 800maH, and 1300maH(all have a 20C discharge rate). My group decided to go for the 800maH as it was more in the middle of things when it came to weight and capacity which we felt would be sufficient enough for our design. I do know that we needed large diameter, low pitch prop with a low kv motor but we've had to work with what was given us. 3 m/s is definitely ambitious for the set-up we've come up with, but I think it's just a matter of calculation and design changes. Again, once the lecturers pick between the designs of the two smaller groups and we can re-merge into our proper big group, we'll be able to delve deeper into things. But thank you very much for hearing me out. I know I sound like I have no idea what I'm doing so I'm quite glad that you're willing to provide some advice.

7. Hello,

I am trying to use your equations to design a rc airplane and choose a propeller based on the highest dynamic thrust. Could you tell me how you got that equation or where you got it from?

1. I came up with the equation. I hope to get the chance over the Christmas Holiday to post how I did so.

8. OK I have skimmed here and I see you people are trying to solve something different than I am.. I need some input though so I will ask. I am building a model p-51 It is 37" wingspan, and I have sheeted the whole thing. I am trying to get it to go over 100 mph. Prop selection of course is very important to me and I want to make sure I am thinking straight. I have a 425 watt 1,400 kv motor on it. The static thrust calculators say that I will draw substantial wattage with an 8x8 statically and I have proven that. Given that I will be going in the range of 100mph Will I be drawing substantially less power at speed? I am thinking that I will be drawing much less and that as long as I don't spend a lot of time climbing vertically or trying to hoover, I will make most efficient use of my motor with a prop that overloads it at 0 mph.... is this correct?

1. Cole, that is a fantastic question....one that I first asked myself a couple years back. I have finally built up the skills to test this, but I still have to build up the infrastructure (products, programming code, sensors, etc) to do so. So, the answer is *I do not know*---but I *think* you are right on. My prediction is the same as yours, but I have not yet had/made the time to conclusively find out. I have a 31.5" HobbyKing radjet with an 875W 1800kv motor and I plan on using an 8x8 prop with a 3S and maybe 4S LiPo, but I have not done so yet. I have a feeling that even if you overload a motor at 0mph, the additional cooling at 100mph may also help to keep it from being damaged, in addition to the prop unloading which I also predict (but do not know for sure) will occur. One way to find out is to grab an Frsky telemetry system and a current/power meter, and monitor the power draw from the ground while you do some test flights. That is also something I plan to someday do. Currently I am learning Arduino microcontroller programming and circuits like crazy so that I can build my own telemetry system, current, RPM, and power sensors, and datalogging. I have many of the pieces of the puzzle partially worked out already, but it is a very complicated and slow process. As you do testing, please share your results with me too as again, my prediction is the same as yours, and I'd really like to know!

2. Thanks for the response. I will really think this is dependant on drag. Since excleration of air is the motivating force, clearly a prop with a pitch speed of say 100 cannot accelerate the air as much while traveling at 50as it can 0. I suspect a Fairley linear curve, and I suspect that a prop in a wind tunnel would draw very little power in a wind nearly it'spitch speed for the given rpm. The part I am confused about is drag. I am thinking if I could predict thrust requirement at100 mph. .... never mind. I wasn't thinking straight.

3. Remember, at any given *constant* airspeed and altitude (not climbing or accelerating), Thrust = Drag. And Drag is proportional to Velocity^2, so a vehicle at 100mph has 4x the drag as it does at 50mph (every time you double the airspeed, you have to quadruple the thrust, since drag quadruples too). Therefore, the thrust must be 4x as much at 100mph as it is at 50mph.

4. ....and all my equation does, as shown in my post above, is help predict thrust. Read my whole post and you'll see thrust for a fixed-pitch prop *is* linearly dependent on airspeed. My equation predicts that, and the one dataset I have from a wind tunnel shows an approximately linear relationship too. The bow in one of my calculated dynamic thrust curves (the red line) is bowed due to changing RPM. However, at a fixed RPM, the thrust is certainly linearly dependent on forward airspeed.

5. Great thread and discussion, thanks Gabriel! I wanted to ask for the equation of thrust. How have you get this expression? Theoretically, thrust is mass of air per second * (Vp – Vo) , where Vp is pitch speed and Vo is airspeed related to the aircraft. I’m I missing anything? It looks like you are calculating mass of air per second * (Vp^2 – VpVo)*(d/3.29546.pitch)^1.5

Where the term (d/3.29546*pitch)^1.5 comes from? I’ll appreciate your feedback.

I’m right now testing static thrust of my motors and props with a device I made with a kitchen scale and an engine mount in “L”. On stable situation, horizontal flight, dynamic thrust will be equal to drag at a done aircraft speed. That’s why I think it’s not so useful the linear relation between RPM and airspeed measured in a wind tunnel. Only one certain airspeed will happen on real flight and it depends on dynamic thrust and drag. This is only one point. The rest of the graphic is not a real situation on cruise speed.
I wonder if following approximation could be appropriate to estimate dynamic thrust: If airspeed is zero, then dynamic thrust is same as static thrust. If airspeed is pitch speed, then dynamic thrust is zero. Let’s consider linear relation between related airspeed and dynamic thrust (this means we can conect these 2 points with a straight line). This is not necessary true, but we could use triangulation to roughly estimate dynamic thrust:

Ts/Vp = Td/(Vp-Vo) so that Td = Ts*(Vp-Vo)/Vp
Where Ts is static thrust, Td is dynamic thrust, Vp is pitch speed and Vo is airspeed.
This is easier to explain with a picture but I don’t know how to upload one :-)

6. Augustin, as you state, theoretical thrust is m_dot*(Vp - Vo), where m_dot is mass flow rate. The squared velocity terms come out by means of getting mass flow rate, which is m_dot = rho*Vel*Area. Vel from m_dot *Vp and Vo --> the squared terms.

Throughout this discussion, don’t forget that dynamic thrust simply refers to any thrust where Vo does *not* = zero. Dynamic thrust does *not* have to be specifically the full-throttle, max airspeed thrust, though this is one of the infinite dynamic thrust possibilities.

The (d/(3.29546*pitch))^1.5 is purely an empirical correction factor that corrects my STATIC THRUST ESTIMATE ONLY (*not* my dynamic thrust estimate) for all things imperfect, such as imperfect mass flow rates and varying efficiencies/effectivenesses of props due to different diameter/pitch ratios, etc. I really need to post my detailed explanation of how I got this equation, as you are not the 1st to ask.

As you state, in steady, level, unaccelerated flight (SLUF conditions, as I call it), Thrust = Drag. To counter what you state about the usefulness of the wind tunnel data, however, here is why the wind tunnel data is so useful 1) It validates my dynamic thrust equation, as well as your comments below about the 2-points thing (static thrust and pitch speed), and 2) it allows someone to know the exact drag on their aircraft (assuming they have the same prop and RPM as what was used in the wind tunnel) at any given airspeed! That’s great information, as this one single plot can apply to any aircraft with this prop and at this RPM. Since in SLUF, Thrust = Drag, look at the plot and follow me with these examples I will make:
-Ex 1: Let’s say you have a plane with a 10x6 prop and it is at ~10,400RPM at full throttle (ie: approximately the same conditions as the wind tunnel plot). You run it up to full throttle (10,400RPM) and fly it and it only goes 40mph max. What’s your drag coefficient? Looking at the wind tunnel data, your thrust = drag = ~0.75kg at 40mph, so you have enough info. to get your drag coefficient, C_D, by using the formula D = C_D*1/2*rho*V^2*planform_area. Solve for C_D.
-Ex 2: Same prop and RPM as before, and your plane goes 60mph at full throttle. You can again, use the plot to see thrust = drag = ~0.25kg. Use this to solve for C_D.
-Ex 3: Same prop and RPM as before, and your cardboard flying kite, which has horrible contours and is very rough and not streamlined, goes only 30mph max. Thrust = drag = ~0.92kg. Use this to solve for C_D.

So, as you can see, this one plot can help anyone with a plane under these conditions know how aerodynamic their plane is, by helping them calculate their drag coefficient, C_D, so that they can make aerodynamic improvements to their plane and also see how good their drag coefficient is compared to other planes.

It is very useful. Also, it helps to validate my equation and your point about the 2-points thing.
So, it looks like your equation at the bottom of your statement is off, but your idea is correct. To correct your equation, simply look back at your original equation: Thrust = m_dot*(Vp-Vo). This already does what you are saying. It already incorporates the “2-point” concept. If Vo = Vp, then you get a thrust of zero since (Vp-Vo) = (Vp-Vp) = zero. Therefore, thrust = m_dot*0 = 0. It’s already taken care of right here. Now, this equation also shows that thrust will linearly drop off with increasing aircraft speed, Vo. So, if Vo = 0 you get thrust = m_dot*Vp, which is static thrust. If Vo = Vp you get the airspeed at which thrust = 0. My equation doesn’t yet have a correction factor for this portion of the equation, so it simply assumes that the airspeed at which thrust = 0 is the pitch speed. This is the intersection of the curves in the wind tunnel plot with the x-axis.
(continued in next comment)….

7. …(continued from previous comment)
Now, as you stated, you have 2 points, where one is the static thrust (x = 0 and y = static thrust) and one is the zero-thrust airspeed (x = zero-thrust airspeed = pitch speed, and y = 0). Draw a straight line between these 2 points and you get my equation. Do the calculation for pitch speed and you will see that my equation will show that thrust = 0 when Vo = pitch speed.

Now, to jump back to the correction factors: First, I must state: my equation DOES correct for static thrust, to make its estimate of static thrust quite accurate. Without the empirical correction factor, it is waaaaaaaaaaay off. It does NOT, however, at this time, correct for the zero-thrust airspeed point (where x = zero-thrust airspeed = pitch speed, and y = 0). This makes my equation less accurate than I’d like it to be, for any dynamic thrust (where Vo does *not* = zero), since that 2nd point has not been corrected. In order to correct that point, I need more wind tunnel data that can be used to find the second point (again, the x-intersect where x = zero-thrust airspeed = pitch speed, and y = 0). Once I get more wind tunnel data (or flight test data) that I can use to find the x-intersect, I can come up with another empirical correction factor and get my equation to be more accurate for dynamic thrust too.

8. Let me add just one more comment to this: the 2nd point, assuming that thrust = 0 where Vo = pitch speed, is wrong. In theory it seems right, but it is not right. It is a half-decent estimate in this case, so my equation is currently using this assumption, but it is wrong. Let me give you one reason why it is wrong: camber. Cambered airfoils still produce lift at a zero degree angle of attack (AoA). Similarly, cambered props still produce thrust at a zero degree AoA. Since the effective AoA on the prop will be zero degrees when Vo = Vpitch, we are assuming that the props do not produce thrust at this condition, so we incorrectly assume this to be the x-intercept, where thrust = zero. This is wrong.

Also, most props are cambered, so this is going to make that assumption cause some error. This is just one reason, though in pretty much any calculation, there are always other seen and unseen, known and unknown factors to make our theory give us an answer that is a little off. The goal is to capture enough of the factors in our equations that we can get "close enough," and where we can't do it analytically, we use correction factors which we determined experimentally. We call these "empirical correction factors," and they help to correct for unusual phenomenon we can't always capture with theory.

9. Thank you for your great explanation and clarifications, Gabriel. I was wrong with my last equation. Air mass flow * (Vp-Vo) is already giving the linear relation between RPM and thrust, and actually it's a very usefull graphic. I messed it all up! :-)

Interesting how can you estimate the drag coeficient of your plane, thanks for that. Anyway, I guess it's not so easy to know Vo, which is relative speed between air and plane. Maybe in an absolutely calm day... even up there... ?

I've just received a tachometer I ordered some weeks ago and I'm starting to measure it all. For now I've seen quite real linear relation between RPM and thrust when RPM are above certain value. I have to check theoretical values and compare it, but in the almost-linear zone, I would say it should be easy to find out the eficiency of the prop.

I was surprised with the low current needed at full throttle (about 3 Amps, GWS8040, 6000 RPM), which is quite far from motor limits -GWS8040 is recomended prop-. I will try larger props to achieve a better usage of the motor. In fact, this is the main reason I want to understand how this works, theoretically and on real usage. I will share anything of interest I may find, although I see you have done an accurate and great job already. Cheers!

1. Augustin, please do post plots and anything of interest. You may share links to plots in the comments here by putting them as images on photobucket.com, then posting the direct link, or by posting docs on Google Drive (drive.google.com) then sharing them and posting the shared link here.

As for finding the aircraft speed, Vo, there are many ways to do so. See my document here (https://sites.google.com/site/electricrcaircraftguru/), pgs. 65-79, for just a few methods, including one FREE one using the Doppler Effect (which requires analysis of sound waves, of the plane flying by, from a standard digital camera, for instance).

2. Hi...I see thay I have derived the static thrust equation...that is only for hovering like for a helicopter... but when it comes to airships ane airplanes which move forward with a velocity... we have to use the dynamic thrust equation....
Bit one big doubt ...the pitch...
Is pitch already defined for a propeller when we purchase it...
If pitch is the distance traveled by the propeller within one rotation...then the pitch will vary with respect to the velocity of the blimp...
Or is it a charateristic of the propeller itself?

10. Hi Gabriel.. I really liked your post.... you see even iam also trying to derive a relation between propeller thrust and rpm, pitch...
You see what really confuses me is this:
Consider the propeller only ...in that case the pitch of the propeller is the distance travellled by the propeller in one rotation...and for that if I calculate the thrust...it is
Thrust=pi×d^2×p^2×r^2×rpm^2/3600 all units s.I.units
r is radius of the propeller..p is pitch...d is density of air..
Now if I attach this propeller to the bottom of an airship... the distance moved by the propeller or the airship in one rotation is not gonna be the
same as the pitch of the propeller considered previously... It depends on the velocity of the airship...
So what really is pitch..
Can I know how u arrived at ur equation?...
Thanks

11. Gabriel, I'm playing with the plots and I tried to find out the "equivalent prop" that under ideal conditions will have the same thrust as the real one. In other words: I have tested prop 8x4 and the plot for Static Thrust vs RPM is the same as ideal prop 6.4x3.2 using theoretical formula (mass flow x Vp). So, in this case, a real prop 8x4 behaves like ideal one 6.4x3.2 which means 20% less diameter and 20% less pitch.
This is just a first approaching but it also applies for a 9x5 prop I tested as well. Performance is very low because thrust is proportional to pitch^2*D^2. This means that 80% pitch and 80% diameter creates 0.8^4 = 41% of theoretical thrust. Anyway, this is acceptable cause the device I made to measure thrust is not exactly ideal :-). Performance of the prop should be higher than that 41%.

I wanted to ask you, if you know a simple way to measure air speed created by a prop. With this value I would know the equivalent prop's pitch and I could confirm if the rates for equivalent prop are 80%pitch and 80% diameter or any other combination. This would be a start point to measure and calculate dynamic thrust with this “equivalent prop” approaching, which is at least for me more intuitive.

@Smiling Buddha: Pitch is the distance -RELATED TO AIR- travelled by the propeller in one rotation. Imagine the prop as a screw through the air. In ideal conditions the shape of the prop itself will determine the quantity of air pushed back for every cycle. A prop with 20cm pitch will move the air 20cm back every cycle (theoretically). If the air would be kinda static gelatin, then it would be the prop the one who is moving forward 20cm every cycle. If this prop is rotating 10 times per second, it will move 200 cm of air per second. If it's attached to an airship moving 30cm per second -RELATED TO AIR- the propeller will push air back anyway at 200cm per second, but this means that the airflow created by the prop RELATED TO AIR is 200-30 = 170cm per second. Of course this is theoretical model. Real situation is "a bit" more complicated. Hope this clarifies your question :-)

12. So are you saying that static thrust is equal to m(dot)*Vp?
Where Vp is the Velocity of the air at the propeller?

1. Yes, at least theoretically. This is the highest value of static thrust you could get, but in real situation the turbulences and other wastes on prop's performance reduce the real thrust. Typical value for real thrust is around 60 - 70% of m(dot)*Vp

13. To all: I just added an article about how I came up with this equation. It is called "Propeller Static & Dynamic Thrust Calculation - Part 2 of 2 - How Did I Come Up With This Equation?" There is a link to it at the top of this article now in case you are interested. It is quite the in-depth article for you folks who had a lot of questions.

14. hi...it is possible for hexacopter??

1. Yes, this equation works with any 2-bladed propeller. For a hexacopter, the total thrust will be equal to the sum of the individual thrusts from each motor.

15. Hi: I am working on a project for a fairly large model (about 8' wingspan). We recently tested our prop: Wooden 2 bladed 29/12 on a 245kv Turnigy using Lipo 48v battery pack. Our calculated static thrust should be 17-18 lb at around 3000 rpm, our test yielded thrust no where near that. Closer to 10-11 lbs. (outside any of the margins for error I can think of). Can you make any suggestion as to why it may be so low?

1. Can you send me a link to your prop? Also, what was your altitude and temp at the time? (rough guess is ok).

16. Re: Link to prop site. Unfortunately this was a custom made prop (by Bolt) so I can't send you to a link. It looks similar to this one: http://www.hobbyking.com/hobbyking/store/__43817__Turnigy_3D_Gas_wood_propeller_28x10.html
But is 29x10. I can measure the cord tonight, but can't give you the airfoil number. We tested it at sea level and about 70F. Anything else I can measure to help figure it out let me know... Thanks for the help.

1. My thrust eqn is heavily empirical--meaning that its coefficients are based on experimental data. My experimental data's *lowest* RPM value was 6000, so you are well below that. Also, the largest diameter I had thrust data for was 17", so you are well above that. Being outside the empirical data by this much, in two areas, means that accuracy is going to decrease. We just don't know what to expect outside my data-set. However, if you spend a bunch of time and give me thrust data for a whole bunch of prop sizes and RPMs, for your region of interest (prop size & RPMs), I can customize the equation to fit your data specifically, and dramatically improve the equation just for the area you are looking at. That's the best answer I can give you. Seriously though, if you send me data, I'll be more than happy to find new empirical coefficients just for your data, and then you should be much better off. I don't have any data in the size and RPM you are speaking of. Place any data you have into my spreadsheet (link at bottom of my article). Currently, it's empty. If you can give me a dozen or so datapoints (the more the better), I'll give you a custom equation.

2. I've entered the first data point for you. How accurate was your RPM measurement, too? At this low of an RPM, with this large of a prop, your RPM measurement becomes very important, since small errors in RPM can result in large errors in predicted thrust. Also, is the prop 29x10 or 29x12? You mention both above.

17. Hello Gabriel,

I am a student from Memorial University in Newfoundland and part of the robotics team. We are interested in building quadcopters. I stumbled across your website and was fascinated by your formula and read everything you posted on it and the above comments. I think you have the only non cfd approach for dynamic thrust I know of that can be readily used by broad public.

In one of the paragraphs of your explanation for the derivation of your formula you state the following "First, I logically deduced (or rather hypothesized) that both the thrust, and also efficiency, of a prop must be related to its diameter to pitch ratio, since the diameter of a prop directly affects the incident (tangential) velocity (and hence also dynamic pressure and Reynold’s number) of air striking the blade at any given location of radial distance, r, from the hub. "

I am curious how you deduced from the above that propeller efficiency is related to the prop diameter. I do know this is true from momentum theory and Bernoulli Equation.

I am studying the use of your formula with the performance of the copters and if I can get more data points I will certainly provide those to you.

Best regards,
Nicolai

1. Nicolai, I think you may be correct when you say, "I think you have the only non cfd approach for dynamic thrust I know of that can be readily used by broad public."

If you can provide data points I'd certainly be interested.

To answer your question: How did I know that prop efficiency is related to prop diameter? I suppose I cannot say that initially I *knew*, but it more or less surfaced over time as I learned more. Here's a few factors in helping me hypothesize this: 1) experience. I fly a lot of RC airplanes. It is commonly known that slow-flying and hovering planes should use large-diameter, low-pitch, high-torque/low-RPM power setups to maximize static thrust and efficiency during hover 2) As diameter increases, even if you hold the blade area constant by making the blade narrower (shorter chord), you can still decrease RPMs while maintaining the same thrust since thrust of any blade segment will increase as the segment is moved farther from the axis of rotation, due to increased incident velocity. In other words, incident velocity (tangential velocity) at any point along a prop is equal to 2*pi*r*n [distance/minute], where r is the radial location [any units you choose], and n is RPMs. Lift = C_L * 1/2 * rho* V^2 * Area, so if you increase Vel, V, you increase Lift according to V^2, yet if you make the blade slimmer, you only decrease lift proportional to the reduction of Area. 3) As diameter increases, in actuality, blade area increases, so again, you can decrease RPMs. Also, the distribution of tangential velocity increases linearly with the increase of diameter (again, V = 2*pi*r*n [distance/minute]), but the distribution of lift (Thrust) and drag increase with the square of tangential velocity (again, L = C_L*1/2*rho*V^2), which indicated to me a *probable* change of efficiency (again, I was guessing here) since you're changing RPMs while maintaining a fixed thrust. 4) Also, as diameter changes, assuming constant RPMs, tangential velocity increases, which increases Reynold's number (Re = rho*V*x/mu), and as Re increases, the flow is more likely to trip from laminar to turbulent, and we know from experiments that turbulent airflow can turn corners easier, without separating, thereby probably slightly delaying stall on the blade, and probably increasing efficiency. 5) As you increase diameter, you get a greater area of the blade that has a low angle of attack. Since the blade is helical (ie: it is twisted, with a high AoA near the root and a low AoA near the tip), its AoA will be greater than the stalled AoA near the hub and < the stalled AoA as you move towards the tip. Stall is inefficient, so larger diameter means less stalled surface area and greater Re and tangential velocity on the unstalled portions of the blade. 6) I later learned about Advance Ratio, J (see here: http://en.wikipedia.org/wiki/Advance_ratio), which perfectly supports my hypothesis. J = V/(nD), where V is related to pitch speed, which is directly related to prop pitch, and D is the diameter, and n is the RPMs. Notice that the denominator, nD, looks surprisingly similar to 2*pi*r*n, or pi*d*n. Since pi is a constant, you can just take it out, and you are left with d*n, or nD, just like in Advance Ratio. Also, Advance ratio is well-known to be related to prop efficiency. See, for example, Figure 12 here: http://m-selig.ae.illinois.edu/pubs/BrandtSelig-2011-AIAA-2011-1255-LRN-Propellers.pdf

So, in summary: I used logical thought and guessed well. That's as good of an explanation as I know how to give. Some of it just seemed intuitive to me, albeit unclear, when supported with much logical thought over the course of many days while I pondered the topic, whether driving or lying in bed at night.

2. Sorry I can't give a better explanation than that. It's a tough topic.

18. Hello Gabriel,

Thank you for the above reply.

My next question is why are you concerned with efficiency in your thought process if you want to determine thrust? You start with a momentum theory approach and made the assumption of exit speed being approx pitch speed. (did you consider slippage corrections?) Then you plotted a set of data points on a graph. To find the empirical correction factor. Why is efficiency important. If you have those data points, and you know the pitch/d ratio for all the props, just find the constant for them that will minimize their standard deviation. What am I missing?

Thank you,
Nicolai

19. Hi, I have downloaded your template. However, when i wanted to find the thrust of my RC aircarft at the given velocity, the thrust is negative. Do you have any justification on this? Because it is not logic. thanks!

1. Great question. yes, this is likely correct. At high velocities (or at some velocity, but with very low pitch or RPM values), thrust will be negative. From the aircraft's perspective, negative thrust is simply considered to be drag. It simply means that the pitch is too low, and/or the RPM is too low, for the blade to be accelerating air. Rather, it is decelerating air, causing drag (or negative thrust) on the aircraft. See Fig 2 above, for instance. On that figure, any speed greater than approx. 60mph will result in a negative thrust.

2. Hi, my concern is because i got my velocity from fight test when running at full throttle. I used stroboscope to get my RPM during static test at full throttle. When i entered the propeller diameter n pitch and the RPM, i gt my thrust negative at the cruising velocity...which doesn't make sense.

3. Hamtaro, it's because my equation's thrust estimate is wrong for your case. Look at the right-most plot of Fig 1. Notice that my estimate (the green line) shows a much lower zero-thrust velocity than real-life. My estimate is simply wrong. It's somewhat close, but it is still a pretty significant underestimate of zero-thrust velocity. I need more dynamic thrust data to correct it. My equation's zero-thrust estimate occurs at a lower velocity than in real-life...and your data helps confirm that. In reality the thrust at full-throttle, full speed, is probably *near zero* for your case (the lower your vehicle's drag the closer to zero the actual thrust will be at full speed), but in real life it certainly is positive. As I get more data (and time to work on this), I'll see if I can correct my equation to make it more accurate so that it does a better job correctly predicting the zero-thrust velocity. Again, refer to the right plot of Figure 1 for more clarity.

20. Gabriel, you could try to use the aerodynamic pitch instead the geometric pitch to calculate the cero thrust velocity, as it is a little bigger. I mean the pitch resulting from the angle of attack for CERO LIFT, i.e. between the plane of rotation and the cero lift chord.
Thanks for your work
Sincerely
Damián (hobbyst)

21. Gabriel, you could try to use the aerodynamic pitch instead the geometric pitch to calculate the cero thrust velocity, as it is a little bigger. I mean the pitch resulting from the angle of attack for CERO LIFT, i.e. between the plane of rotation and the cero lift chord.
Thanks for your work
Sincerely
Damián (hobbyst)

1. Good comment. I've been planning on this for a couple yrs but still haven't gotten around to it. :)

Note to self: TODO: Look into correcting the dynamic pitch portion of the equation more. You need to spend more time to find a correction factor to estimate the zero-thrust aircraft speed, which actually occurs for a cambered airfoil when the blade AoA is slightly negative. See here for starters, including posts #12 & 13: http://www.rcgroups.com/forums/showthread.php?t=288091#post29499497

22. Hello Gabriel, i used the simplified formula to find the dynamic thrust and even recalculated by taking the same values from the thrust table which is uploaded but im not arriving at the right solution. Please help me out in this.
And also i want to know if we could somehow use dynamic thrust to calculate take-off distance of any rc model plane .

1. 1) "but im not arriving at the right solution. Please help me out in this." Yes, I am willing to help, but you have not provided enough information for me to help you.

2) "And also i want to know if we could somehow use dynamic thrust to calculate take-off distance of any rc model plane." Yes, you could iteratively come up with an estimate. It would go something like this: First, determine the take-off speed of an aircraft. This can be determined experimentally or calculated based on the wing size, lift coefficient, dynamic pressure (a function of air density and velocity), and weight of a vehicle. 2nd: iterate over small time increments to determine how much time and distance occurs before take-off speed is reached. You'll need to know your vehicle's drag coefficient, C_D. Using physics & aerodynamics: F = m*a. v = a*t. dist = vt. Drag = C_D*q*S, where q = 1/2*rho*v^2 and S = planform area. All thrust greater than drag will accelerate the vehicle, according to its mass, so you can use F=m*a to calculate a. The F you will use is Thrust - Drag. For the first iteration, Drag = 0 since the vehicle is not moving. Iterate over a small time to get the vehicle's new velocity. Calculate the distance the vehicle moved assuming an average velocity over that time period. Calculate a new drag based on this velocity. Calculate a new thrust at this velocity. The new F to use is Thrust - Drag. Use this in the new calculation of accel, based on F=ma. Repeat, etc etc. Iteratively, in this manner, you will be able to calculate the distance the vehicle has moved before takeoff velocity has been reached. This will be your takeoff distance! :) If you're not sure how to do the above, do a lot of research, learn a lot of equations, go to a lot of school, become an engineer, and MOST IMPORTANTLY, NEVER GIVE UP!

2. Oh, and the final step: once you have the answer, create a website like mine (or use Instructables.com or something similar), and write a nice, step-by-step article to share your knowledge and help others too! Help advance the world a little at a time, to make others just as capable as you. :) Good luck. This is an interesting problem, but I just have too many other ones I'm working on right now.

23. I really appreciate the second article which has entire derivation !!!!
Regard,
Krupal Vithlani

1. Good. I'm glad you appreciate it. :) It took forever to write.

24. Its really great one to calculate the dynamic thrust equations. Could you post the full form of the derivation

1. The thrust equation is
T= pi/4(D^2 p V (v2-v1))

25. Hello and good day sir..

Can we calculate propeller efficiency from static thrust experiment?

And how?

TQ.

26. Hello sir,
I'm a student and preparing for science project ( i mean a quadcopter) so after using given equation for thrust i can get a thrust of 15 kg. I want to lift a load of 10 kg. So my question is that the load of 10 kg could be lifted by that 15 kg thrust? I mean this weight of 10kg is more than the limit of ordinary quadcopters.

1. Total thrust from all 4 motors is 15 kg

2. By definition, so long as your 15kg of thrust is being directed downwards, and is not being restricted by striking the vehicle itself, and is greater than the weight of the vehicle, the vehicle will fly. You need a little extra "room" (ie: extra thrust) for maneuvering and accelerating, but yes, it will fly. How did you determine RPM? If you experimentally measured RPM by running the motors up, why didn't you experimentally measure thrust too--ex: by using a scale?

3. Actually i haven't built the quadcopter yet, i wanted to gather all the information before building it. As i don't want to spend my all money on wrong parts of quadcopter, so i decided to do some paperwork first (i mean calculations!) . Another question :- does higher rpm will damage the propeller?. I mean from 51000 to 100000 rpm? And thanks for this helpful article,thanks for your feedback on my comment, thanks for your help :)

4. How are you calculating RPM? I don't even know how to do that yet. I measure experimentally. Yes, too high RPM will cause propellers to break apart. APC posts RPM limits for their props, for example, here. Also, in addition ot my equation, take a look at these calculators here: http://rcplanes.000webhostapp.com/calc_motor.htm (motor & prop calc) and http://rcplanes.000webhostapp.com/calc_thrust.htm (static thrust calc). They may prove useful.

5. https://www.amazon.com/Coreless-51000RPM-Helicopter-Aircraft-magnetic/dp/B01B1DVWHE link is here. I am going to use these motors. This motor is coreless brushless dc motor with 51000 rpm!

6. amazon.com/Coreless-51000RPM-Helicopter-Aircraft-magnetic/dp/B01B1DVWHE

7. I can promise you there is no way in this world one of those motors is going to produce 3750g (15kg/4) of thrust. You are off by probably more than a factor of 100. 51000 RPM? I don't know--maybe--when completely unloaded. Claiming that motor can produce 3750g of thrust is probably equivalent to claiming my 4-cylinder car has a 33000 Hp engine. Nope...it doesn't. It doesn't go 2000 mph either. :) Take a look at HobbyKing's brushless motors. Look for ones that have thrust and power specs and recommended props listed, to get a feel for what's required to produce 3750g of thrust, then go from there. Good luck!

8. My propeller is 10*4.5" (dia 10" pitch 4.5") and if i assume these motors are really moving at 51000 rpm then, according to your calculator. I get a thrust of 33 kg! So is it wrong? I mean the value i got ( thrust of 33 kg) is unreal?

27. Hi gabriel, do you recall the source you extracted the dynamic thrust equation?

1. Yeah, I came up with it. Here's where I describe my process: Propeller Static & Dynamic Thrust Calculation - Part 2 of 2 - How Did I Come Up With This Equation?. I still have some questions about it myself but haven't had time to re-attack the problem for the last several years. Someday I'd like to put some more research and work into it, as I have some more things to learn about propellers and thrust still myself.

2. It's my own equation. I didn't extract it from anywhere/anyone else.

3. Coooooooool!!! I liked your thinking very much Gabriel. Also special thanks for replying in just 10 minutes :)

4. By the way, I am doing my graduation project on calculating and measuring the thrust force of propellers... If there is something I could do for you, I'd gladly do

5. Yes, actually there is! Please provide me links and data to your results and papers you write or publish. I'd like to see what you conclude from your work, especially if you do any dynamic thrust work in a wind tunnel or via flight test or something. You can find my email via the "Contact Me" tab at the top of my website.

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29. Hi Gabriel! Thanks a lot for this beautiful research! Its a piece of art! It is guiding me through my college's senior design project. I do have one question however: Could you guide me as to how to calculate the thrust for a 3-blade prop? My project involves a air-water convertible drone. The back prop is a 3 blade rc boat prop with 3"x5.5.

1. 30. I need help for underwater propeller calculations. Do you have any material regarding it?

1. No, sorry, but if you read my Part 2 article you'll see that the 1.225 in the equation is the density of air in kg/m^3. Maybe you could try replacing that with the density of water, but I haven't done any experiments to see if that gives good results. Remember also, my equation assumes you *know* the RPMs. In other words, I originally planned on using it in-flight where you can measure live RPMs but don't have a thrust stand.

31. Have you done any journal paper relating to this. I would like to cite your work in my senior college design project.

1. No, sorry, I haven't written any journal papers on this. I did it all outside of school and normal work while I was a researcher in an indoor flight lab, but it hasn't been published in any journal. You are welcome to use it obviously but you'll have to just cite my name and website since there's no journal publication to cite for it.

32. One of the early steps of your derivation has a flaw. Propeller thrust should be derived as
F = 0.5 * A * p * (Ve^2 - Vac^2)

Sources:
https://www.grc.nasa.gov/WWW/k-12/airplane/propanl.html
https://www.grc.nasa.gov/WWW/K-12/airplane/propth.html

1. *I believe one of your early steps has a flaw

2. The links you provide show both are correct.

33. I need to know more about the equation. Can i get your mailid?

34. What does that negative dynamic thrust means ??

35. @Unknown, simply search this page's comments for my previous answer to someone: "From the aircraft's perspective, negative thrust is simply considered to be drag."

36. Hi sir
Is there any paper which use equation with diameter, pitch, and rpm as an input to find the thrust?

1. You mean, did I write a paper? No, just these posts online here. Are there academic papers referencing my work here? Yes, dozens I believe. Here are a few of them which show up on Google Scholar search results, for instance: Google Scholar search for '"gabriel staples" propeller thrust'.

37. do u have an email that can be contacted? thanks

1. Yes, see this comment here. Also, see the "Feedback/Corrections/Contact me" link at the very top of this site.

38. Thanks for your comment or question! If it is a question, I will try to get back to you quickly. Notice to spammers: I personally remove all spam promptly and report spammers to Google, so please don't do it.

Note: some HTML tags are allowed in your comments. To learn how to add bold (<b>...</b>), italics (<i>...</i>), or hyperlinks (<a href="URL">NAME</a>) to your comments, read here.

~Sincerely, Gabriel Staples.

P.S. Yo hablo español también. Je parle français aussi. (I speak Spanish & French too).