How to realistically model propeller static thrust?

Question

I am graphing static thrust vs prop diameter for a 600 kW motor and a prop diameter up to 5 m. I have seen the post 'What is the equation for calculating static thrust?' and using its formula and fixed prop efficiency I get a pretty linear graph. At 5 m the thrust is 19 kN.

Structural factors become increasingly important at larger diameters, but what other aerodynamic factors (perhaps prop efficiency changes with diameter) need to be considered to make the graph more realistic?

Update - I assume the prop pitch can adjust to maximize static thrust, but the twist is optimized for high speed (200mph) flight. The engines full power is also available at the best speed for the size of prop being tested. This Equation to bind velocity, thrust and power simplified by assuming unitary engine efficiency is the same as what I have been using.

$$T_0 = \sqrt[\LARGE{3\:}]{P^2\cdot\eta_{Prop}^2\cdot\pi\cdot \frac{d_P^2}{2}\cdot\rho}$$

Answer 1

This basically depends on the prop loading and RPM. If you are not satisfied with the momentum disk theory based results then the next step would be to use blade element theory based calculation.

If I were you I would certainly run few test cases with a known propeller geometry by scaling the propeller. One of the easiest way I can think of doing this is to use QPROP from Prof. Mark Drela at MIT. This is a free software and on top of this it is a command line program which is very well suited for this kind of an activity because all the parameters could be changed on the go, preferably via a bash or python script.

HTH

Answer 2

A paper to estimate static thrust with a C-172 example is here. Also, a NASA report 447 on calculating static thrust may be of interest. An Excel spreadsheet to calculate static thrust is here. There is a SE discussion here and here.

The formula for thrust is...

Where:
F     = thrust (Newton)
d     = prop dia (inch)
rpm   = rotation (per minute)
pitch = pitch (inch)
Vo    = aircraft speed (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 is hard-coded for a "standard day" density at sea level of 1.225kg/m^3.

The thrust of a propeller is not constant for different flight speeds. Reducing the inflow velocity generally increases the thrust. A reduction of the aircraft speed down to zero tends to increase the thrust even further, but often a rapid loss of thrust can be observed in this regime. That is why the static thrust of a propeller is not such a terribly important number for a propeller - the picture of a propeller, working under static conditions can be distorted and blurred.

As long as an aircraft does not move, its propeller operates under static conditions. There is no air moving towards the propeller due to the flight speed, the propeller creates its own inflow instead. A propeller, with its chord and twist distribution designed for the operating point under flight conditions, does not perform very well under static conditions. As opposed to a larger helicopter rotor, the flow around the relatively small propeller is heavily distorted and even may be partially separated. From the momentum theory of propellers we learn, that the efficiency at lower speeds is strongly dependent on the power loading (power per disk area), and this ratio for a propeller is much higher than that for a helicopter rotor. We are able to achieve about 80-90% of the thrust, as predicted by momentum theory for the design point, but we can reach only 50% or less of the predicted ideal thrust under static conditions.

Static thrust depends also on the inflow, influenced by the environment of the propeller (fuselage, crosswind, ground clearance). Measurements of static thrust can be easily done, but the theoretical treatment is very complicated and only possible with a lower degree of confidence than calculations in the vicinity of the design point. Due to local flow separation, the behavior of propellers under static conditions can be very sensitive with respect to blade angle settings and airfoil shape.

To get a picture of the bandwidth of static thrust, several older NACA reports and some publications from model magazines have been examined. The results are combined in the following graph.

Source

The Source provided a model airplane example calculation that is applicable to full size aircraft.

We have got two different propellers with a blade angle of 10° and 25° respectively. The first one has a diameter of D = 200 mm, the size of the second one is D = 300 mm. Which one would be better suited to build a VTOL aircraft model? How much thrust can we expect using an .60 engine of 2000 W (assuming a suitable gearbox)?

From the diagram above we read a static thrust parameter of 0.32 [kg^(1/3)/m], respectively 0.1 [kg^(1/3)/m] around the center of the blue band. To calculate the thrust we have to multiply these values with the power P [W] and the diameter D [m] to the power of 2/3. Performing the calculation for the first propeller (10° blade angle) yields T = 0.32*54.288 [N] and thus a static thrust of 17.4 N, whereas the second, larger propeller delivers 0.1*71.138 = 7.1 N only. Using the same engine in a helicopter with its large rotor of 1 m diameter and low pitch angles, would give us a lifting force of more than 55 N !

This example shows, that the diameter of a propeller is as important for static thrust, as it is under flight conditions. But, for static thrust the blade angle is also very important - probably even more important than for the design point, where a gearbox can match almost any propeller pitch and flight speed quite well.