Modeling to Reduce Shaking Cyberiad 23 Feb 2012

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Whether blades are straight, helical or troposkein, Darrieus HKTs have some advantages and some

drawbacks compared with axial flow HKTs. These have been summarized in [1]. Parameters of particular

interest to designers of Darrieus HKTs, aside from the obvious ones of efficiency and cost-effectiveness, are

(i) adequate starting torque and (ii) smooth operation. Fixed pitch Darrieus HKTs typically fail to self-start

under load, exhibit low efficiency and shake violently due to cyclically varying angle of attack and hencefluid dynamic forces on blades, with each blade experiencing two peaks in both radial and tangential force

per revolution. The upstream peaks are typically much larger than the downstream peaks, especially at

higher tipspeed ratios as shown in Figs.2 and 3, so a 3 blade turbine will typically experience 3 major peaks

per revolution.

Fig.2. (left) Nondimensional tangential thrust F*Ton a fixed pitch blade as a function of and .Fig.3. (right) Nondimensional radial force F*Ron a fixed pitch blade as a function of and .

This problem is not related to mass balance of the rotor. The variation in tangential force, commonly

referred to as torque ripple, affects the transmission and load, while the variation in radial force affects the

support structure, and if this frequency coincides with the natural frequency of the support structure it can be

destructive. During tests in Canada reported in [2], the fixed pitch straight blade Darrieus HKT shown inFig.1 was found to shake so violently due to cyclical hydrodynamic forces on blades that tests on it had to be

cut short for fear that it would shake itself or its mounting to pieces.


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2. Modeling fixed pitch blade forces

The VAWTEX-B model has been described in [3]. It is a momentum model based on Paraschivoius DMST

(double multiple streamtube) approach [4], developed originally to model Darrieus wind turbines, and

further developed to model passive variable pitch and incorporating dynamic stall. In the present study,

dimensions were based on those of the turbines tested in Canada, reported in [2], i.e. 3 blades of NACA0021 section, 1.25 m long, 0.14m chord 0.5m turbine radius, and V = 2 m/s except where otherwise

indicated below.

2.1Validation of VAWTEX-B model against fixed pitch turbine experimental data

The model was used to predict performance of the

straight blade fixed pitch turbine shown in Fig.1,

and Fig. 4 shows good agreement between

measured and modeled torque coefficient CQ and

performance coefficient CP without allowance for

dynamic stall. It is not clear why the agreement is

not as good when dynamic stall is taken into

account, as shown in Figs.5 and 6.

As explained in [2], it was not practicable to

measure CQ and CP at below maximum CPbecause the band brake used in the tests exerted a

more or less constant torque at a given setting,

causing the turbine to stall at low . Althoughexperimental data are available at V > 3 m/s,

these showed evidence of loss of performance,

probably due to cavitation and/or the onset of

supercritical flow and resulting surface effects, and

sites with such high flow velocities are extremely

rare, so these data are not shown in Figs. 4-6.

Left. Fig.4. Measured CP and CQ for a fixed

pitch HKT compared with predictions using

VAWTEX B without dynamic stall.

2.2 Modeling shaking forces for fixed pitch turbines

Having validated VAWTEX-B model predictions against experimental data for fixed pitch HKTs, the model

was then used to model the variation of radial and tangential force on a single fixed pitch blade with 0.1m

chord, but otherwise identical to the turbine shown in Fig.1, as functions of azimuth angle and tipspeed ratio.

Predictions are shown in Figs.2 and 3. It is clear that both force components experience large variations with

azimuth angle at all tipspeed ratios. When the contributions of 3 blades at 120 deg intervals are

superimposed there is still considerable force fluctuation at = 2, as shown in Fig.7. High solidity turbines

achieve maximum efficiency at ~ 2, as shown in Figs 4-6, so the region ~ 2 is of particular interest.


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Above left: Fig.5. Measured CPand CQfixed pitch HKT compared with predictions using VAWTEX

B with full dynamic stall, right: Fig.6. Measured CP and CQ for a fixed pitch HKT compared with

predictions using VAWTEX B using Cardonas modified dynamic stall model.

It is this force fluctuation that must be minimized, and various strategies were investigated to achieve this

aim. As a first step, it might be expected that increasing the number of blades would help, and some

commercial turbines have 4 or 5 blades. It appears from Fig.8 that this helps to a limited extent, but the

fluctuation is still large, and increasing the number of blades n further increases solidity = nc/r unless

chord length c is reduced or turbine radius r is increased, and reduced c in turn reduces structural strength

which could lead to structural failure, and also reduces blade chord Reynolds number Re, which adversely

affects performance by reducing stall angle, maximum lift coefficient and lift to drag ratio. Helical blades,

provided they cover the complete 360, e.g. with 3 blades each wrapped around a 120 arc, effectivelysmear the torque fluctuations evenly around the circumference, but the radial forces at opposite ends are

still unbalanced so helical turbines cannot completely overcome the shaking problem, nor can they

overcome the problem of low starting torque. Increasing to 3 shows a marked reduction in shaking, evenwith a 3 blade turbine, as seen in Fig.9. This is a result of reduced blade stall. However, increasing requires lower , i.e. more slender blades, which again reduces structural strength and Re. But given that no

fixed pitch turbine can produce adequate starting torque, whether it has straight or helical blades, it was

decided to model shaking of a variable pitch turbine to see if shaking could be reduced to an acceptable level

while maintaining high starting torque and efficiency.


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Fig.7. Nondimensional radial (shaking) force F*Ras a function of azimuth angle for 3 blade fixed pitchHKT at = 2.

Fig.8. Nondimensional radial (shaking) force F*Ras a function of azimuth angle for 6 blade

fixed pitch HKT at = 2.

Fig.9. Nondimensional radial (shaking) force F*Ras a function of azimuth angle for 3 blade

fixed pitch HKT at = 3.


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3. Modeling the performance of a turbine with basic passive pitch control

Pitch control mechanisms can be either active or passive. Various researchers have modeled the

performance of variable pitch Darrieus wind turbines (also known as giromills or cycloturbines) with active

pitch control, i.e. pitch is driven by a mechanism according to some pitching regime assumed to be optimum

[5-9], but as far as we are aware, only [3] and [11] have modeled turbines with passive (self-acting) variablepitch in which pitch is determined by a combination of fluid dynamic and inertial forces. This is a more

challenging task. Given that there is no general agreement as to what constitutes an optimum angle of

attack or pitch angle, and given the mechanical complexities of active mechanisms, it is our view that it is

more productive to focus on simple passive pitch control systems which aim not for optimum performance,

but for performance much improved over fixed pitch systems without great complexity.

Fig.10. Cyclical pitching with blade leading edges pivoting upstream to avert stall, after [10] (no pitch

control system shown).


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Left, Fig.11. Pitch angles required to limit to 10. Right, Fig.12. Torque coefficient CQ and

performance coefficient CPvs. for various pitch angle limits.

4. The Memetic Algorithm to minimize shaking

It is apparent from the foregoing discussion that a passive pitch control mechanism that limits the angle of

attack as described above can greatly improve performance over an equivalent fixed pitch turbine, assuming

that the physical mechanism can achieve the pitch regime assumed in the model, but the issue of shaking

remains. To assess this factor, downstream and side forces were calculated for all and a range of from and corresponding CL and CD, and hence turbine downstream and side force coefficients were calculated.

Two shaking" parameters were defined respectively as

D= variation of downstream force coefficient around the meanS= variation of sideforce coefficient around the mean

A Memetic Algorithm was developed to search for optimum pitching parameters to reduce shaking while

maintaining strong starting torque and high peak efficiency. Five optimization objectives were used within

the program. The first objective was to maximize the starting torque CQ0. The second objective was tomaximize the peak performance coefficient CPmax. The third and fourth objectives were to minimize,

respectively, the downstream shaking" parameter Dand the side force parameter S. As an attempt at


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defining some combined figure of merit, a fifth objective was to minimize a function f combining these

variables, defined as best fitness f = (D2+ S

2)/Cp max.

Many engineering design tasks can be cast in the form of optimization problems. One formulation, of

sufficient generality for many applications, is to minimize a real-valued function f(x1; x2; ; xn), with each

real parameter xisubject to (domain) constraints aixibifor some real constants aiand bi. In the contextof the present study, f is either the performance coefficient CPor some other quantity such as the shaking

parameters Dand S. Constraints can be imposed on, among other quantities, the pitch angle limits, thelength of radial arms, position of the blade pivot, the blade polar moment of inertia, etc.

Many techniques exist for solving optimization problems such as the one described above, but they vary

greatly in efficiency and in the quality of the final solution for a given number of function evaluations. No

single technique is best for all design problems. Gradient-based methods work well with smooth, unimodal

functions, but may yield local optima for multimodal functions. Heuristic algorithms can increase search

efficiency, but at the expense of guaranteed optimality - they do not always find the global optimum.

Genetic Algorithms (GA) are adaptive search methods that use heuristics inspired by natural population

dynamics and the evolution of life. According to [12], they differ from other search and optimization

schemes in four main respects:

- Search proceeds from a population of points, not from a single point

- They use a coding of the parameters, not the parameters themselves

- Objective function values guide the search process - they do not use gradients or other problem-specific

information

- State transition rules are probabilistic, not deterministic.

While they are quite effective techniques for global optimization, GA can sometimes take an intolerably

long time to converge. In short, both GA and local search techniques can spend disproportionate amounts of

effort examining infeasible regions of the (often enormous) search space. GA sometimes search too broadly,

local search methods too narrowly.

Many different methods have been devised combining GA with other search techniques in an attempt to

improve their overall performance. The evolutionary biologist Richard Dawkins coined the word meme as

a term for non-material entities that are capable of transmission or imitation, such as ideas, tunes, and catch-

phrases [13]. Memetic algorithms are GA that incorporate local search techniques.

4.1 Computer Techniques

VAWTEX, Author 1's computer program, implements a non-traditional GA similar to Eshelman's [14]

CHC algorithm. The system has been enhanced with additional heuristics such as cataclysmic restarts, and

incest prevention, described in Eshelman and Schaffer [15]. VAWTEX's general operation can be described

quite succintly: create and evaluate new (candidate) designs until some termination criterion is met. In the

present study, a run is terminated when a certain fixed number of designs have been evaluated. There are no

theoretical results that can be used to determine the optimum number of evaluations that should be

performed. VAWTEX begins the optimization process by creating an initial population of real-valued design

vectors and calculating CP and other quantities for each design. In the present study, initial designs arerandomly generated. Genetic operators and other heuristics are used to create candidate designs. Genetic

operators create new (offspring) vectors from two parent vectors in the population, using heuristics inspired

by the recombination of DNA. The primary genetic operator in VAWTEX is based on an operator gleaned


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from fuzzy set theory as described in Voigt et al [16]. VAWTEX uses binary tournament selection to

select parent vectors from the population. In this method, two individuals are selected without replacement

from the population. The individual with the higher CPbecomes the first parent. A second binary tournament

determines the other parent. After evaluating the CPof the offspring, VAWTEX replaces another individual

in the population with the offspring if the offspring's CPis larger. This replacement strategy guarantees that

the best individual in the population is never replaced by an inferior individual.

Stochastic search techniques can place great demands on computer resources. It is important to run a

sufficient number of cases, each time varying only the initial population, in order to enable some confidence

that an optimal solution has been found, or at the least that it has been well-approximated. Just as

importantly, each run must be allowed to continue for long enough to be able to find the optimal solution,

but not for so long that resources are wasted. Unfortunately, there are no theoretical results available that

allow reasonable estimates to be made for the required duration, nor of the number of runs required to

guarantee that an optimal solution has been found. To put the size of the present design problem into

perspective, imagine that we wanted to do an exhaustive search to find the six angles defining the presentpitching system, each to within 0:1. This brute force method would require evaluation of about 300 300 160 160 160 160 6 1013individual designs. At 1 second per design evaluation on a 3.2GHz PC, thatwould take about two million years.

4.2 Initial optimization results

The principal physical and model parameters are shown in Table 1. and the constraints on pitching

parameters are shown in Table 2.

Parameter Value

Environ-

mental

Fluid density (kg.m-3) 1000

Kinematic viscosity (10-6m

2s

-1)

1.14

Ambient velocity V(m.s-1

) 2

Turbine Number of blades n 3

Radius r (m) 0.5

Height (m) 1.2

Blades profile NACA 0021

Chord length c (m) 0.10

Model Dynamic stall Yes

Parasitic drag No

Table 1: Principal parameters (held fixed) during optimization.

The optimum pitching parameters found using the algorithm are shown in Table 3, and the corresponding

values of the objective function are shown in Table 4 (these values apply only to 0 < < 3.0, the range oftipspeed ratios used during the search). Fig.13 shows CQand CPvs for the 5 optimization criteria. It willbe seen that optimization for any single criterion alone involves considerable sacrifices to either starting

torque or CPmax, but the best fitness curves show only a modest loss in performance for all up to about 3.Fig.14 shows how the downstream force and sideforce coefficients CDS and CSF vary with at = 2. Asmall variation indicates reduced shaking. There is a considerable reduction in variation of CDS and CSF

compared to fixed pitch, and a significant reduction compared to the good CP max curve.

Minimum Maximum

min(deg) -30 0

max(deg) 0 30

p-(deg) -16 0

p+(deg) 0 16

Table 2: Constraints on pitching

parameters.


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Objective min(deg) max(deg) p-(deg) p

+(deg)

Maximize CQ0 -30.0 30.0 -13.4 2.4

Maximize CP -16.0 13.0 -7.0 5.0

Minimize D -18.0 30.0 -3.0 2.0

Minimize S -26.9 17.7 -0.9 0.3

Minimize fitness -6.5 28.2 -1.0 5.2

Table 3: Pitching parameters found using the algorithm.

Objective CQ0 CP max D S f

Maximize CQ0 0.157 0.417 0.140 0.316 0.829

Maximize CP 0.092 0.470 0.141 0.276 0.659

Minimize D 0.131 0.312 0.072 0.245 0.818

Minimize S 0.121 0.166 0.156 0.135 1.246

Minimize fitness f 0.103 0.426 0.112 0.217 0.574

Table 4: Values of the objective function found using the algorithm within the range 0 < < 3.0.Optima shown underlined.

Fig.13 (left). CQand CPvs. for the 5 optimisation criteria.Fig.14 (right). Variation of downstream force and sideforce coefficients CDSand CSFwith .


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4.4 Minimizing shaking while retaining high Cp max

The aim in the next optimization study was to determine whether shaking can be reduced significantly by

allowing a slight reduction in CPmax. A different objective function was used, incorporating Cpmaxat all times

Optimization was done for the range of tipspeed ratios 1.75 3, since peak efficiency occurs in thisrange. The first objective was to maximize CP. The pitching parameters and output quantities found for this

objective are identical to those given in Tables 5 and 6. The second and third objectives were to minimizethe resultant" shaking force parameter Rsubject to constraints on CPmax. The best" variable pitch turbinefound in the first optimization has CPmax= 0.47, although it should be noted that parasitic drag has not been

allowed for and this will reduce actual CPsignificantly. The first constraint imposed demands that CPmax0:95 0.47, i.e. we will accept a 5% reduction in CPmax, as long as we reduce R. The second constraintdemands that CPmax 0:9 0.47, i.e. we are prepared to trade a 10% reduction in CPmax for improvedshaking characteristics.

The values of the pitching parameters found using the algorithm to minimize Rsubject to constraints on

CPmaxare shown in Table 7.Objective min(deg) max(deg) p

-(deg) p

+(deg) p0

-(deg) p0

+(deg)

Maximize CP -18.4 10.9 -7.6 5.1 0.0 0.0

Minimize R: 0.95 -10.5 14.4 -9.8 8.3 -2.6 3.5

Minimize R : 0.90 -11.8 14.2 -9.4 9.3 -3.9 4.5

Table 7: Pitching parameters found using the algorithm to minimize R subject to constraints onCPmax.

The resulting pitch angles are shown in Fig.16, and the values of the three objective functions found during

the search are summarized in Table 8.

Objective CP max D S RMaximize CP 0.471 0.187 0.262 0.322

Minimize R:0.95

0.452 0.124 0.147 0.192

Minimize R :0.90

0.429 0.115 0.123 0.168

Table 8: Values of the objective function found using the algorithm to minimize R subject toconstraints on CPmax. Optima shown underlined.

The underlined values correspond with the particular objective function. Thus, for example, the underlined

value in the first row is the value of CPmaxfound when the objective was to maximize CP, etc. The other two

objectives were to minimize R. The results shows that they are each better than the Rachieved whenusing the CPmaxobjective, but that they have a worse CPmax(by about 5% and 10%, respectively), as shown

in Fig.17, in which curves are labeled according to their respective objective functions. For example, the

turbine labelled R: 0.95" was optimized for minimum Rsubject to the constraint CPmax0.95 0:471.


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Fig.16 (above left). Angle of attack vs for extended pitch control model to minimize shaking.Fig.17 (above right). CQand CPfor optimization to minimize the shaking force parameter Rsubjectto constraints on CPmax.

Figure 18 on left shows D and S (it is moreinformative to plot these quantities rather than just the

resultant R.) The top graph shows that the differencesbetween the two R -optimized turbines are relativelyminor and that both are better than the CPmaxvariant for

tipspeeds below about 2.5 and slightly increased,

although still less than 1/3 of the fixed pitch turbine, at = 2.75. The impact of minimizing Ris more significanton the sideforce coefficient S than on the downstreamforce coefficient D. For tipspeeds greater than about1.0, the sideforce is about half that of the CPmax variant,

and about 1/3 of the fixed pitch turbine at = 2.75.


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Left, Fig.19. Downstream force coefficient CDS. Right, Fig.20. Sideforce coefficient CSF.

The effect of pitching on the downstream and side force coefficients CDSand CSFare shown in Figs. 19 and

20. The improvements over the fixed pitch turbine are significant when 5% of CP maxis sacrificed.

Figs 21 and 22 show, respectively, the (non-dimensional) tangential force and the (non-dimensional) radial

force for all three blades combined. The variable pitch systems do an excellent job at smoothing out the

oscillations apparent in the curves of F*Tfor the fixed pitch variant. Reductions are apparent, but not quite

so spectacular for F*R.


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Fig.21 (left). Non-dimensional tangential force coefficient F*Tfor all 3 blades combined.Fig.22 (right). Non-dimensional radial force coefficient F*Rfor all 3 blades combined.

4.5 Modeling passive eccentric pitch

The foregoing indicates that a turbine with stabilized free pitch can in principle achieve much better

starting torque, efficiency and smoothness of operation than an equivalent fixed pitch turbine. However at

the time of writing this has not yet been achieved in practice, and it was decided to model a turbine with

sinusoidal pitch control, as it appeared that this would be easier to build. Since a cam can in principle

produce any pitch regime, an optimization was performed for pitching including up to 5 harmonics rather

than limiting the study to a simple sinusoidal pitch variation. Optimization was done for V= 2.0 ms-1and

= 2.5. The pitch system is defined by

() = c1+ M

j = 2cjsin[(j 1) ]

where the ciare constants to be found by the optimization process and results are computed for values of M

from 2 to 6. Thus, for M = 2 we have

() = c1+ c2sin

which is identical to the system examined in [5]. Results are shown for pitching systems optimized for

maximum CPmaxat tipspeed ratio = 2.5. These are labeled S2, S3, S6 in the graph legends. The last digitis the value of M, i.e. the number of free parameters. Results labeled P2, P3, P6 are optimized for


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minimum Rwith 5% loss of CPmax. These can again be considered as compromise solutions where we aretrading maximum performance for lower shaking. Figs.23 and 24 show CQand CPfor maximum CPmaxand

for minimum R with 5% loss of CPmax. Although CQ0 is less for this form of pitch control than for thestabilized free pitch system shown in Fig.17 (about 0.05 compared with 0.08), it is still much better than

the fixed pitch turbine, and CPmax is about the same. Additional harmonics appear to offer little if any

advantage over a simple first order sinusoidal pitch regime shown in bright red, so the higher orders will notbe discussed further.

Fig.23 (left). Sinusoidal pitch: CQ for maximum CPmax (top) and for minimum R with 5% loss ofCPmax(bottom).

Fig.24 (right). Sinusoidal pitch: CP for maximum CPmax (top) and for minimum Rwith 5% loss ofCPmax(bottom).

Figs. 25 and 26 show downstream and side force shaking parameters Dand Sfor maximum CPmaxand forminimum R with 5% loss of CPmax. Comparing the 5% loss of CPmax curves for the 2 pitch controlsystems shown in Fig.18 and Figs. 25 and 26, Dis significantly higher (i.e. worse) in the sinusoidal system,but still much better than for fixed pitch (about half), and Sin the range 2 < < 3 is marginally better forthe sinusoidal system.


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Fig.25 (left). Sinusoidal pitch: downstream shaking parameter Dfor maximum CPmax (top) and forminimum Rwith 5% loss of CPmax(bottom).Fig.26 (right). Sinusoidal pitch: sideforce shaking parameter S for maximum CPmax (top) and forminimum Rwith 5% loss of CPmax(bottom).

Conclusions

Modeling using VAWTEX B and a memetic algorithm to optimize pitch parameters for two basically

different passive pitch control systems predicts that performance can be greatly increased over that of anequivalent fixed pitch turbine. Peak efficiency is predicted to be about 50% higher, starting torque several

times higher and shaking can be reduced by factors of 2 to 3. It remains to be seen if these improvements

can be achieved in practice.

References

1. Kirke, B.K. and Lazauskas, L. Limitations of fixed pitch Darrieus hydrokinetic turbines and the

challenge of variable pitch.Renewable Energy36 (2011) 893-897.

2.

Kirke, B.K.,"Tests on ducted and bare helical and straight blade Darrieus hydrokinetic turbines."Submitted toRenewable EnergyDec 2010.

3. Lazauskas, L. and Kirke, B.K., Performance optimization of a Self-acting Variable Pitch Vertical

Axis Wind turbine, Wind Engineering 16 No.1, 1992, 10-26.


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4. Paraschivoiu, I. Wind Turbine Design, with emphasis on Darrieus concept. Presses Internationales

Polytechnique, 2002.

5. Grylls, W., Dale, B. and Sarre, P.-E. (1978). A Theoretical and Experimental Investigation into the

Variable Pitch Vertical Axis Wind Turbine. Proc. 2nd

Int. Symposium on Wind Energy Systems,

Amsterdam, Oct 3-6, pp.E9-101-118.

6. Natuvetty, V. and Gunkel, W.W. (1982). Theoretical Performance of a Straight-bladed Cycloturbine

Under Four Different Operating Conditions. Wind EngineeringVol.6, pp.110-130.7. Zervos, A., Dessipiris, S, and Athanassiadis, N. (1984). Optimisation of the Performance of the

Variable Pitch Vertical Axis Wind Turbine. Proc. EWEC Conf, Hamburg, FRG, 22-26 Oct., pp.411-

416.

8. Vandenberghe, D. and Dick, E. (1987). Optimum Pitch Control for Vertical Axis Wind Turbines.

Wind EngineeringVol.11, pp.237-247.

9. Paraschivoiu, I. and Trifu, O, H-Darrieus wind turbine with blade pitch control, 12thInt

Symposium on Transport Phenomena and dynamics of rotating machinery, Honolulu, Hawaii, Feb 17-22, 2008.

10.Salter, S.H. and Taylor, J.R.M. (2007). Vertical-axis tidal-current generators and the Pentland Firth.

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Thesis, University of Calgary, Alberta, May.

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13.Dawkins, R., The Selfish Gene", Oxford University Press, 1976.

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San Mateo, CA, Morgan Kaufmann, 1991, pp. 265-283.

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(eds): Evolution and Biocomputation, Lecture Notes in Computer Science 899, pp. 123-141, Springer,Berlin, 1995.

Notation

Ab Blade planform area

c Blade chord length

CDS Turbine downstream force coefficient

CL Blade lift coefficient

CP Turbine performance coefficient = P/(0.5AtV3)

CPmax Best value of CP

CQ Turbine torque coefficient = Cp/CQ0 Turbine starting torque coefficient, i.e. CQat = 0CSF Turbine side force coefficient

CT Coefficient of tangential thrust = CLsin - CDcos

f best fitness parameter (to be minimized) = (D2+ S

2)/Cp max.

FD Drag force = CDqrelAb

FL Lift force = CLqrelAb

FN Centripetal hydrodynamic force = CNqrelAb

F*R Non-dimensional blade radial force = FN/qAb)


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Modeling to Reduce Shaking Cyberiad 23 Feb 2012