Sizing a propeller to match an electric motor

A combustion engine has a power curve that looks something like this (taken from Gerr's Propeller Handbook):


A brushless DC electric motor has an (idealized) power curve that looks something like this:


Does this mean that any electric installation will be best suited by increasing propeller size or pitch until the motor is operating at max power at full speed (when fully loaded, so a little to the right on the curve under normal conditions) and never even approaches no-load RPM?

 

Realistically, an electric motor and propeller set cannot be designed anywhere near maximum efficiency without a lot more information about what operating point you are looking for . This is because, as you may have noticed, the propeller powering curve generally follows the powering curve of an ICE where with a brushless DC electric motor they cross at only one given point. Unless you get the operating point right, or pull some propeller tricks, you will have significantly lower efficiencies and may burn out the motor.

First, propellers are usually described by three critical factors. The advance coefficient (J) which is a function of the propeller advance (Va), diameter (D) and the RPS (n): J=Va/Dn. The Torque coefficient curve (Kq), a function of torque (Q) at a given J and water density (rho): Kq=Q/(rho*n^2*D^5). And the Thrust coefficient curve (Kt), a function of thrust (T) at a given J and water density: Kt=T/(rho*n^2*D^4). the "open water" efficiency (eta O) of a propeller is a measure of the ability of the prop to turn torque into thrust when operating by itself free from any surrounding influences: eta O = (J*Kt)/(2*pi*Kq). For most propellers eta O is about 70% at some discreet J. Propeller curves for Wageningen B-types (a good typical propeller) can be found here: https://deepblue.lib.umich.edu/bits...7/bab2786.0001.001.pdf?sequence=5&isAllowed=y Typically small electric motors use b2.30 (2 blades, 30% blade area ratio) props.

Now the propeller absorbs hp as torque. Torque * rpm / appropriate constant = shaft hp (SHP). Now hp is absorbed by the boat as drag. Drag * speed (V) / appropriate constant = effective hp (EHP). The quasi-propulsive efficiency (eta D) is a measure of the ability of the wheel to change SHP to EHP eta D= EHP/SHP.

As the propeller operates in the wake of the hull, the speed of advance of the prop may not be the same as the speed of advance of hull. This is important as one of the critical factors for propeller/rpm selection is the advance coefficient (J). The Taylor wake fraction corrects vessel speed (V) to propeller advance (Va) by the wake fraction (w) which is the percentage of ship speed the water is moving at the propeller (+ being in the direction of travel): Va=V(1-w). Additionally, the wake may be uneven, which will cause the prop to absorb more or less torque. This is called the relative rotative efficiency (eta R) and is equal to torque behind/torque open water.

Finally, there is the matter of slip and its effect on the wake and drag. In order to generate thrust, the prop must suck water in and push it out faster than the propeller is moving forward. As we have shown above, the prop operates in water of a different speed than the vessel. Therefor there are two slip ratios the real slip ratio: 1-Va/(pitch*n); and the apparent slip ratio: 1-V/(pitch*n). The real slip ratio is the only one we need to worry about but it is very difficult to determine without rigorous testing and is usually estimated from the wake fraction. What this slip does is to speed up the water around the stern of the hull, causing additional drag. Rather than think of it as additional drag, as it is a function of the propeller, it is normally called a thrust deduction (t) so that required thrust = (1-t) delivered thrust.

Now to make the boat go at a given speed, drag must equal thrust, so you can see that SHP and EHP are linked by the ratio of input torque to output thrust at a given speed and rpm for a specific propeller operating behind a specific hull. To write it in efficiency terms:

eta D = (1-t)/(1-w) * eta R *eta O

That said, you can see that the eta D is directly related to the RPS (n) of the propeller by the advance coefficient (J), which is related to the speed of the vessel by wake fraction (w). What this means is that (for a given prop)having too much rpm is as bad as having too little rpm as both can cause the prop to come of its operating point.

Now in your case, if you give more information about the hull, prop, and propulsion arrangement; we may be able to estimate t,w, eta R, and eta O which will not only enable you to size the motor, but give you the proper RPM for most efficient operation also.

 

A good question, and an excellent explanation/answer!

 

Plus 1 from DogCavalry. Excellent

 

I had naively imagined that the electric motor's ability to make torque at low or no rpm would make prop selection easier than it is for an ICE powered vessel.

 

IMO, with the exception of foiling and all out planing electric speed boats, an electric propulsion system should be spec’d and designed for peak efficiency at cruising speed.

The hull design, displacement, and conditions the vessel will be operated in, along with the efficiency of a motor and the type of ESC, all play a role in determining the cruising speed, which in turn dictates the properties of the propellor.

Typically system voltage, battery capacity and or solar area are also limiting factors with regard to propulsion system design, and since a lot of electric propulsion application is tasked with pushing a boat through the water at hull speed, peak efficiency at cruising speed should be the most important design goal.

I took the efficiency data the motor manufacturer provided, and I built my boat around the motors peak efficiency which is about 3/4 “throttle” while the ESC is still Pulse Width Modulating.

I sized everything else around the power draw of the motor at that peak efficiency point -sizing the battery capacity to provide ample range and runtime at that speed/consumption, the solar system to be able to offset all of or a good deal of that power draw, and so on.

With the boat on the water, I turned my focus to further improving efficiency and performance by tuning/sizing the props to the boat and the motors.

By monitoring the power draw under load, I could see that the motors were not pulling their full rated current, this mostly because I run dual motors and have efficient light displacement semi-planing hulls, which said I had the power to swing a bigger/more aggressive prop.

After experimenting with a variety of props, I settled on a pair that allow the motors to pull their max rated current at full throttle, and they have increased top speed, reduced wetted surface area by allowing the hulls to semi-plane, and improving cruising speeds overall without a big efficiency penalty, and they have also improved maneuverability thanks to their added bite.

For the same power draw, the boat actually performs better even though the props are larger and more aggressive and thus put more load on the motors, likely because the original props had more slip.

Now the entire system has been optimized specifically for my use with the primary goal of cruising as efficiently as possible.

What does that mean?

It means I can cruise at 5MPH on sun only at a total draw of 550W which is about 412W of combined propulsive power at the props, or 275W/206W per motor.

My sun only cruising speed was a secondary design target, and at this speed the motors are about 70% efficient.

But since I live in the tropics where we have good sun, my primary target cruising speed was at 3/4 throttle where the motors reach a peak efficiency of 80%, and this is also the speed I matched my prop performance to with regard to motor loading and current draw.

At this speed the motors are using all the solar while drawing some capacity from the batteries, allowing me to cruise in the 7-8mph range.

Peak speed at full amp draw is about 10MPH, but it comes at a cost of efficiency loss since the ESC’s are no longer pulsing.

All that is basically to say that matching a prop to an electric motor is application and component dependent.

One size prop does not perform best or provide the best efficiency for all.

The math will get you in the ballpark, but until you apply the system to the vessel and test it under load in the environment, and most importantly decide what your goal is (cruising, all out speed, etc.) you won’t really know what the perfect prop is until you test and experiment, especially in lightweight applications where small changes can result in big differences.

 

This:

Yes, that's the way I approach it also!
Some other factors, some of which can make a significant negative implication in efficiency some of which have not been alluded to, where Electric Power in particular is concerned:

Battery: Peukert capacity loss, cold ambient battery temperature, battery DOD not at Optimum range..

Propeller: propeller blades thicker than needed, no fairings front and behind the prop, not using low friction gearing at the motor end to lower RPM / increase torque needed for larger and slower turning efficient props, Disturbed water feeding into and out of the prop from forward / rear appendages (rudder, etc) or wake, not continuously adjusting the power delivered to the prop to optimum when at Cruise as outside conditions like wind and waves change..

yes, this has been particularly valuable for my applications.

 

That's what most clients or e.motor suppliers i talk to assume, until informed otherwise!
It is not as straight forward as imaged, as JEH notes in his excellent summary.

 

Thank you jehardiman for the comprehensive reply. I am designing this as an engineering exercise. It will be an unmanned hydrofoil, with a weight of no more than 100 kilograms. I have a donor outboard and will be mating the electric motor to that. The lower unit has a rather dumpy 2.44 to 1 ratio and factory propeller 9.5" diameter and 10" pitch. So, even in factory configuration with no slip this engine would've maxed out at about 20 kn (10 m/s). Assuming 10 percent slip, a loaded RPM of 4900, a pitch of 25" would get me relatively close to my design cavitation limit of 40 kn (20 m/s). According to Gerr, pitch to dimeter ratios of up to 2.5 are generally acceptable.

I have not settled on a motor yet, but it seems my only option is one of these no-name e-vehicle motors that lack documentation or reliable rating. My plan is to run the boat with the factory propeller (which will be enough to comfortably go on foil, but not much speed beyond that) and keep that as a backup for the competition while I make the high speed propeller.

I have a copy of Gerr's Propeller Handbook, and G. Kuiper - The Wageningen Propeller Series, but are there any other texts you would recommend?

 

So as a hydrofoil you will have no thrust deduction, but realistically a typically outboard leg is not up to hydrofoil needs. Additionally, 4 EPH at 40 knts means that total drag could only be 32 lbs which implies 0.324 square feet of wetted surface: so a 9.5x25 prop at 2000 rpm and 40 knts is going to need much more than 4 hp (i.e. the total drag of just turning the prop area will be more than that).

Edit: caught a math error by not squaring V

 

You’ve already identified the two main issues with using an ICE outboard leg for high speed electric propulsion -the lower bevel gear drive ratio and the physical limitation of how big a prop you can swing.

Motor-wise, have a look at what Golden Motor has to offer which is much better quality and will have some performance data. The motors that are used in Hoverboards may also be an option of interest.

Just as important is the controller. This is where most builders of electric outboards experience failures. You must use a high quality ESC with cooling.

While on the subject of sizing propellers and lower gear ratios, interestingly the electric propulsion industry has also now finally recognized these two issues as limiting factors in performance, and are now manufacturing their own lower units with proper ratios and with room to swing more streamlined larger diameter propellers without exhaust hubs, etc.,

 

I have estimated drag of lifting surfaces and struts at 20 m/s (40 kn) as being 84 N (19lbs). This includes a rather poor estimation of the lower unit drag, but does not include prop drag. I would like to do tow tests to find drag numbers for the hull and lower unit. Can I estimate drag of the propeller by letting it freewheel while towing and measuring the RPM, and therefore slip, sort of in reverse?

 

It depends on the RPM rating of the engine. If it has the same power at the same RPM, there is no problem at top speed. However, the OP is looking at matching the propeller curve to the engine power curve. There are many types of electric motors with different torque and speed curves. The curve he shows is for a motor with zero torque at zero RPM and a linear increase. That would make it almost impossible to get a boat going. A hybrid winding may be more adequate.
Also, very importantly, is this for an AC or DC motor?

 

Annoyingly, I picked a graph with the x axis reversed from what you would typically expect. Full speed is on the left, stall is on the right. My interest is in a brushless DC motor, which is really an AC motor with a driver that operates from DC, AFAIK.