# Maximum Efficiency of a turbine on a moving vessel

Discussion in 'Hydrodynamics and Aerodynamics' started by MicMac, Nov 16, 2022.

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### MicMacNew Member

Hi all,

Imagine a traditional propeller shaft arrangement on board a vessel, with the vessel moving forward using sail power. If we inverted the propeller or replaced it with one designed to generate rather than propel, what could the maximum efficiency of this 'turbine' be?

Initially I was under the impression it would be limited by Betz law like a wind turbine, but now I am not sure this is the case as in this scenario we could assume the water is stationary and all the kinetic energy is in the vessel. A perfect (impossible) system would leave the water in front and behind the turbine stationary so does that mean it is not restricted by betz law ~59%?

My question is, what is the maximum possible efficiency of a turbine being pulled through the water by a boat?

EDITED to clarify qw

Last edited: Nov 16, 2022
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### gonzoSenior Member

Kinetic energy is relative. You can pick a frame of reference to make the math easier, but the absolute values don't change. As I understand your question, the water is stationary. However, you refer to a "generating turbine rather than a propulsor" which indicates a stationary turbine and moving water. Bets law won't change regardless of your frame of reference.

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### MicMacNew Member

Okay so practically speaking, if I have a boat moving through still water and want to use a turbine where a propeller is usually mounted so that I can generate some power. Then the maximum efficiency will still be governed by Betz law?

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### gonzoSenior Member

Yes, minus all the losses due to friction, inefficiencies in the design, etc.

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### jehardimanSenior Member

The efficiency of the turbine? Or the overall efficiency?

For the turbine itself it is wash; overall efficiency is much less. I'd have to revisit my notes, but it shouldn't be theoretically more than the complement of the Betz law, or ~41%; and based on 4-quadrant propeller data, much less due to wake fraction. This is because you have to pay the energy penalty for the moving turbine as well as the acceleration of the water. From the wiki page...
And see the wiki or threads on sailing directly upwind for the energy balance equations (if you can get through them all).
The Wind Powered Sail-less Boat https://www.boatdesign.net/threads/the-wind-powered-sail-less-boat.24669/
Finding the best way to sail directly upwind https://www.boatdesign.net/threads/finding-the-best-way-to-sail-directly-upwind.67271/

Generally, for a sailing vessel it is better to have a wind generator than a water generator, especially downwind.

(EDIT: X-post with gonzo)

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### gonzoSenior Member

Another source of losses for a submerged turbine is due to the pitching, rolling, etc. of the vessel which creates turbulence.

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### philSweetSenior Member

The Betz limit is based on the idea that the base of the propeller is fixed to the earth, and that the cost of the pilon is zero, or at least that the cost doesn't increase with increasing tipping moment from the prop. It only relates to an optimum propeller when drag is a non-issue.

With a mobile platform, where you are trying to transport the hull, but want to use a hydro-turbine to generate some power, drag is never a zero-cost item. So you have to come up with your own metric for what optimum performance looks like. This is, as they say, a nontrivial problem.

By far the cheapest and most reliable way is to use a diesel generator for electric power. Solar power can be competitive, but is heavier and either draggier or more expensive. It looks a bit better on very long durations because of the fuel load of a diesel genset. Sail power is just about the most expensive form of propulsion you can imagine, both in terms of initial expense and operating expense. It costs more to maintain the sail systems than it does to fuel and maintain an equivalent diesel propulsion engine, and the initial cost of the boat is much more than a diesel boat with the same accommodations. If you want a hydro-generator, you need a bigger sail plan and a hull with a greater RM. These are both really expensive. RM will always increase hull drag. Thus the effect on the sail plan gets compounded. So you see, the thing to do if you can run a hydro-generator is to not run it at all, ever, and instead reduce the sail plan and run a diesel genset. You will get there at the same time and will save lot of money that way. Pretty much anything you can do to reduce sail area and sailing time will save money.

The bottom line is that if you figure the cost of sailing in the overall metric, the performance metric of a hydrogenerator will be negative compared to virtually any other form of electrical generation. They have historically been used on the Vendee Globe and other ultra endurance high-speed boats. But these were rules limited events and had crazy power requirements compared to crew size. Some boats used pedal generators even then. The one thing that works out nicely is that the power generated by a hydro-generator will closely track the power demand of the autohelm over a wide speed range. This lets you operate on somewhat of a schedule for the generator.

But to try to answer your question - the Betz limit is about maximum power extraction without regard to forces. What you want is the maximum power extraction compared to some value measure of lost propulsive power due to the hydro-generator's drag. So we know which direction Betz optimized prop will be pushed - it will be pushed towards lower drag and lower power and lower wake production. A big pilon-mounted wind turbine might have a design Cl of around 0.6. But we don't use that high of Cl for keels or foils or anything wet on a boat. A Cl of 0.1 is more typical for keels and rudders under design conditions. So you might suspect a Cl of 0.1 would be a good starting point for a hydro-generator operating point. Now you just have to get a realistic drag count for a machine with that Cl and solve for the corresponding tip speed ratio which provides optimum performance as you have chosen to define it. This is normally done with a bunch of computers. There are some rather tedious hand crank methods that can get decent results in about a week.

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### DCockeySenior Member

What definition of Cl are your using? Do you mean the Coefficient of Lift of the blades, presumably the blade sections? Or something else?

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### philSweetSenior Member

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### DCockeySenior Member

The sectional lift coefficients at design condition can be easily adjusted by changing the blade chord and the twist. Increase the blade chord and decrease the twist so that the lift remains constant and lift coefficient decreases. Inverse to increase the lift coefficient. For any section there is a coefficient of lift which in the minimum drag.

Note that the sectional lift coefficient may vary along the span of an axial flow turbine or propeller.

This may have been true many decades ago. Either lifting line or lifting surface calculations can easily and quickly run on many current PC's.

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### seandepagnierJunior Member

diesel power is not only unreliable compared to other sources of power, but extremely expensive to everyone.
The betz limit has nothing to do with efficiency of extraction of energy, but only to how much potential energy in the water that the turbine interacts with that can be extracted.

You can therefore extract 90% or even better of the energy into the turbine for the drag incurred. Of course a propeller will not achieve this, it needs to be a purposefully designed turbine for generating. I have 3d printed mine, but they have layer lines which increases friction, I am looking to try with non-planar slicing. I also painted them with epoxy to make them smooth.

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### MicMacNew Member

Some excellent inputs from you all! I can see some claims of 90% theoretical extraction, as well as claims that BETZ law would be a good place to start.
Assuming I don't want to use diesel or other combustion engines, or for that matter will carry any fuel on board and we are just looking at in water turbine extraction.

Would anyone be able to point me towards an equation or spreadsheet that I can use to work this out?
My current thinking is very rudimentary at this stage, in that power offtake is a function of turbine drag. For Example:
One portion of the drag is useful drag (eg turning the turbine and generating power) and the remaining portion is wasted drag. Pessimistically lets say 50% power drag 50% wasted drag (100% total added drag of turbine)

Ignoring appendage drag for now and just looking at the blade area, keeping it academic for now!

The remaining effort then goes into sizing a rig and hull that can achieve the desired speed whilst including the total added drag of the turbine.
At this high level, and to start some rough estimates, this is why I am hoping to get a more accurate extraction efficiency because obeying Betz law means a whole lot of extra drag for my power requirements.

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### seandepagnierJunior Member

again, the betz law only states you can only extract 59% of available energy the turbine sweeps with a single turbine.

It doesnt mean the actual energy extracted is at this level of efficiency which would mean considerable drag. Not the case, turbines/propellers can exceed 90% efficiency

You will find at higher speeds, even substantial turbines do not slow the boat very much. My 33ft boat for example needs at least 15hp worth of sail power to get up to 12 knots of boat speed. If I am extracting 50 watts (4 amps 12 volts) this is not even half a percent of the total power.

wetted surface drag is related to speed squared, so ignoring wave drag, extraction 0.4% of the sail power would yield 1/(12- (1-.004)^.5)*12) or 1/40th of a knot reduction in speed. At lower speeds, the boat speed reduction is more and more for the same amount of power extracted.

The betz law does not mean extra drag!! it only means not all the energy in the water is extracted (or it would be stopped relative to the boat which would never work) in practice you will never get anywhere near to the betz limit for a homemade turbine of small scale and there is no reason to try to. It is more important to reduce drag vs power, and use a larger turbine if needed and not try to attempt to extract all the energy that whatever particular turbine sweeps.

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### gonzoSenior Member

In other words, you believe that a diesel that start with the turn of a key and powers a boat is unreliable compared to a sailing rig that depends on the vagaries of the weather. Further, that a diesel engine is "extremely expensive" compared to sailing rigs or solar panels/batteries systems. Any data to substantiate the claim?

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### philSweetSenior Member

In remote areas, such as forward areas in a military conflict, it costs more than \$100 per gallon to stock petroleum. But we still use diesel because it is just that good. Even at \$100 gal, it is difficult to find substitutes as a primary power source. It is, for the time being, the absolute best thing going, and nothing else really comes close. This is why it has been universally adopted the world over. In the US, you practically can't give coal or natural gas away. Texas flares enough natural gas at the well head each year to heat every home in Texas. Ironically, they get in trouble for leaky pipes, but not for no pipes. Methane is a lot worse than CO2, of course, but flares aren't that great at full combustion, either.

There is absolutely nothing more reliable than diesel. A diesel engine will outlast anything else. You can walk away from one for thirty years, come back, crank it, and drive away. There isn't anything else that can do that.

As a friend of mine likes to remind people "the stone age didn't end because they ran out of stones, and the petroleum age won't end because we run out of petroleum." But I'm not so sure about that. It's a good thing they didn't run out of stones, because they may start looking good again.

... which is the default definition of efficiency of extraction of energy wrt turbines. Feel free to craft your own definition, but please tell us what it is and how you plan to use it. This is different than the default definition of efficiency used for propellers, for instance.

Turbine Efficiency
The turbine efficiency η, also called power coefficient, is the ratio of the turbine power output Pt to the power of either the water head for traditional design or unconstrained water current Pw, i.e. η=Pt/Pw.

From: Encyclopedia of Ocean Sciences (Second Edition), 2001

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