Turbines for Wave Energy Devices

Michael Craig

September 8, 2016

1 Introduction

This is a discussion of my study of turbines and their implementation in a waveenergy converter (WEC) deice. The main issue with wave energy devices is cost,efficiency and durability. Here I have only discussed efficiency in any seriousdetail. Having read over various sources I can outline some of the conclusionsI have made in relation to the proposed design of the impulse turbine. Insteadof going down a more rigorous mathematical route, I have chosen to take noteson the current literature concerning the operation of an impulse turbine usedin an wave energy converter. I have also outlined a superficial way to calculatethe desired operational RPM for a Well’s turbine. The studies I have cited alluse computational models and CFD, some alongside experimentation to arriveat their conclusions.

Useful diagram for studying turbine papers!

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2 Aerodynamics

The first theorem studied was the Kutta-Joukowski a theorem which relatesthe lift generated by an airfoil to the speed of the airfoil through the fluid, thedensity of the fluid, and the circulation. This can be demonstrated by physicalintuition or, more formally, using complex analysis[1]. The two key ingredientsare to find the force imparted on blades and consider the Bernoulli equation, toarrive at.

L = ρΓ V

Where, L is the lift, Γ is the circulation, ρ and V are the pressure and velocityupstream of the airfoil. This is a fundamental theorem of aerodynamics.

3 Efficiency

Using the cosine rule and vector analysis we can arrive at an expression forblade efficiency as outlined in one of the handouts.

η = 2(1 + cbcosβu

V)(1 − u

V)

This is, however under the assumption that the angle between the directionof the incident velocity stream and the direction of the blades being 0. Thisis justified in claiming that for most turbines this angle is somewhere between1.7-25◦.

There are many different types of impulse turbine to consider for an OWC,including Wells turbine, impulse turbine with pitch-controlled guide vanes andimpulse turbine with fixed guide vanes The current consensus within the fieldseems is that Wells turbine is deficient in irregular flow conditions, which theOWC is subject to. Pitch-controlled guide vanes have best overall efficiency (Fig1) but the fixed guide vanes mitigate the cost of maintenance and manufacturinginherent to moving parts.[3]

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Fig 1; Efficiencies of different turbines over a range of 1/(Period)(angularvelocity) ; For irregular incident wave frequencies.

We also note the sharp peak of the Wells turbine, which is not optimal. TheWells turbine is also liable to stall quite frequenctly under the bidirectional flow.

The highest promise for wave to wire efficiency from wave energy devicesthat I have seen so far was 50% 3. This efficiency is a product of the OWCefficiency, blade efficiency and inductor generator efficiency.

I am only concerned with the turbine efficiency in this study. The mainparameters of interest when we study turbine design and efficiency are torquecoefficient, CT , input coefficient CA and flow coefficient φ. These are defined asfollows.[4]

∆p Pressure changeQ Air Flow ratera Arc traced by curvature of stator bladeva Axial flow velocityU Circumferential velocityb Blade heightlr Chord lengthz Number of rotorsT Turbine output torquerR Blade radius

CA =2∆pQ

ra(v2a + U2)blrzva

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CT =2T

ra(v2a + U2)blrzrR

φ =vaU

These coefficients are discussed in all design discussions in studies of impulseturbines for OWC. These terms allow us a different, design specific, descriptionof the turbine efficiency.

η =Tω

∆pQ=

CT

CAφ

In most design description of an impulse turbine, these coefficients are dis-cussed, most commonly being graphed against the flow coefficient, as seen inFig 3 taken from[5].

4 Impulse turbine

An impulse turbine is one which gains the majority of the force on it’s vanesvia the change of momentum, without much of a change in pressure. Impulseturbomachines use a stator blade which acts to translate incoming pressure tovelocity. Over the stator velocity increases and enthalpy decreases. Pressure,enthalpy and velocity drop over the rotor are minimal.

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An impulse turbine spins due to the change of direction of air passing overthe rotors. The time rate of change of momentum of the blades in the horizontaldirection is equal to the force in the horizontal direction.

4.1 Possible design modification

The velocity amplitude of OWC devices have been found to be greater on exhala-tion that inhalation. This has lead Lio et al to consider making the bidirectionalturbine asymmetric with a non-zero rotor blade angle.[5] This is shown in fig 2.

Fig 2

The conclusion of the paper was that if the OWC is known to convert mechanicalenergy of the waves to pneumatic energy more effectively on exhalation, a rotorblade angle of 5 degrees is optimal. Below is the flow rate profiles of studiesOWC leading to the interest in this design modification.

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Fig 3; Parameters against flow coefficient for different angles of tilt towardexhalation side of the OWC chamber

5 Wells turbine

In studying the optimum design of a turbine one must become familiar with theactual workings of the proposed turbine in use.

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A Wells turbine is bidirectional through a clever implementation of symme-try and relative velocity.

Fig 4; Schematic of Wells turbine

The main disadvantage of the Wells turbine is that it is only efficient ina very sharp range due to stalling. If the blades spin too slowly the relativevelocity (W in Fig 4) becomes too high and no lift is generated. Therefore thereis an optimum angle of attack (α in the diagram) at all times for a given turbinedesign. This angle of attack can also be related to the flow coefficient as seenbelow.

Tan(α) = φ

This is useful because in most research papers an optimum flow coefficient isshown. Thus for a given flow rate we will know the optimum angle of attack ifwe have such a graph to hand. There is for all Wells turbines a cut-off point forour flow coefficient after which the efficiency drops (this is stalling) as outlinedin figure 5

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Fig 5; Graph of efficiency versus flow coefficient for different Wells turbines Itis clear from the above graph where stalling begins for the turbine designs,

quite an unforgiving device, the Wells turbine!

5.1 Case Study

Using what I have learned about Wells turbines I will outline a method I haveused to discern optimum RPM for given wave energy converting device. Theproposed device is an underwater Wells turbine spinning on a rigid rod attachedto some flywheel and generator. We must know some parameters pertaining tothe device before we can use this method for optimum RPM. Parameters neededare frequency of oscillation (f), amplitude of oscillation (A), radius of the turbineblades (r) and optimum flow coefficient (φ). Given these quantities can vary, Iwill outline the general method before continuing to show a specific example.

Displacement of the blades from the mean position is assumed to be sinu-soidal

y = ASin(2πft)

Thereforey = 2πfACos(2πft)

We now take the maximum velocity, ie 2πfA as our axial flow velocity becausethis is a good approximation to the working speed for the turbine and becauseif anything this overestimates the axial flow velocity which will avoid stallingcaused by high axial flow.

Now, knowing optimum flow coefficient

Tan(α) =ymax

rω=varω

= φ

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ω =varφ

In the case I was to study, the values necessary are given in the table below

f .125 HzA 1mr 1mφ .15 - .3

So for this device the desired RPM is, depending on flow coefficient somewherebetween 25-50rpm. It must be noted that this is optimum for only the maximumwater flow over the turbine so for optimum performance and to keep the angleof attack optimum for longer one might consider lowering the rpm slightly butthis increases the risk of stalling. An adaptable flywheel which monitors thefrequency of oscillation and adjusts accordingly may prove useful in these designsto optimise performance but would of course prove costly. Where the optimumflow coefficient was taken from research papers online[6]

6 Conclusion

In the end there are so many variables to be thought of that one cannot deny thatexperiment is surely the best way to go about testing complex situations suchas these. I have not discussed viscosity, turbulence or stalling of the blades orany number of things. The conclusions outlined from the studies were normallyengineering papers which ran simulations using CFD or ran experiments, theywere not analytical so all there was to do was forward on the findings I hadgleaned from them. There is, however, a lot of literature on this topic, sostudying will prove useful in determining the best course of action. I have alsoshown how to find optimum operating RPM for a specified Wells turbine usingsimple geometry. Ultimately experiment is the best course of action for turbinestudy and implementation in an OWC.

7 References

[1] Rutta-Joukowski wikipedia article.[2] ’A review of impulse turbines for wave energy conversion’ T. Setoguchi

et al.[3] ’A twin unidirectional impulse turbine topology for OWC based wave

energy plants’ V Jayashankar et al.[4] ’Design chart for impulse turbine wave energy extraction using experi-

mental data’ A Thackker et al[5] ’Numerical study on a modified impulse turbine for OWC wave energy

conversion’ Z. Liu et al.

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[6] ’Wells Turbine for Wave Energy Conversion—Improvement of the Per-formance by Means of Impulse Turbine for Bi-Directional Flow ’ S. Okuhara etal.

8 Thanks!

Finally I would like to thank Dr’s John Miller and Laurence Crane for theirguidance and willingness to take me onboard to study this very stimulatingtopic.

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