conceptual optimization of axial-flow hydraulic turbines with non-free vortex design.pdf

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Proceedings of the Institution of Mechanical

http://pia.sagepub.com/content/221/5/713The online version of this article can be foundat:

DOI: 10.1243/09576509JPE394

2007 221: 713Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and EnergyR. B F. Albuquerque, N Manzanares-Filho and W Oliveira

Conceptual optimization of axial-flow hydraulic turbines with non-free vortex design

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713

Conceptual optimization of axial-flow hydraulic turbineswith non-free vortex design

R B F Albuquerque, N Manzanares-Filho, andW Oliveira

Mechanical Engineering Institute, Federal University of Itajub UNIFEI, Itajub, Minas Gerais, Brazil

The manuscript was received on 17 November 2006 and was accepted after revision for publication on 28 March 2007.

DOI: 10.1243/09576509JPE394

Abstract: This paper presents a low cost computational methodology for conceptual designoptimization of axial-flow hydraulic turbines. The flow model away from the blade rows is con-

sidered axisymmetric, steady, and with cylindrical stream surfaces. The flow at the cross-sectionsbehind the distributor and behind the runner is treated by means of the simplified radial equi-librium equation. The flow losses and deviations are assessed by using empirical correlations.

Although simplified, the model allows the consideration of non-free vortex analysis at an earlydesign stage. For reducing the set of design variables to be optimized, the runner blading stagger,camber, and chord-pitch ratio are parameterized in terms of their values at the hub, mean,and tip stations. The optimization problem consists in finding a basic geometry that maxi-mizes the turbine efficiency, given the design flowrate, rotational speed and bounds for thedesign variables and also for the available head. Two optimization techniques have been applied:a standard sequential quadratic programming and a controlled random search algorithm. Anapplication example is presented and discussed for the optimization of a real turbine model. Theoptimal solution is compared with the original turbine design, showing potential performance

improvements.

Keywords: axial-flow hydraulic turbine, non-free vortex, loss and deviation modelling, geometryparameterization, optimal design

1 INTRODUCTION

The development of computers in the second half of20thcenturymadepossibletheuseofcomplexnumer-ical flow simulation techniques for turbomachineanalysis and design. Nowadays, three-dimensional

Euler codes and three-dimensional viscous NavierStokes codes are already standard tools in the devel-opment of new turbomachinery units. Details of flowseparation, loss sources, loss distribution in compo-nents, matching of components at design and off-design, and low pressure levels with risk of cavitationarenowamenableto analysis withcomputational fluiddynamics (CFD) [1].

Corresponding author: Mechanical Engineering Institute, Federal

University of Itajub, Av. BPS 1303, CP 50, Itajub, Minas Gerais,37500-903, Brazil. email: [email protected]

Although three-dimensional NavierStokes codeshave allowed good performance predictions andcontributed for decreasing the costs of turbomachinemodel tests, a considerable computational effort hasstill to be spent with grid generation and with the solu-tion of the flow governing equations in each numerical

investigation. This issue is even more important inthe case of design optimization: when a geometri-cal change is made during the optimization process,complex meshes must be recalculated and the flowsolver must be run again. This high effort preventsthe incorporation of sophisticated NavierStokes sim-ulations throughout the whole design procedure [2].

Actually, the analysis and design of turbomachinesstill require the use of simpler methodologies mainlyin preliminary design phases, when the geometryis not yet completely defined. One example of avery simplified methodology is the mean streamline

analysis for conceptual optimization of mixed-flowpumps [3]. For axial flow gas turbines, it is common

JPE394 IMechE 2007 Proc. IMechE Vol. 221 Part A: J. Power and Energy

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714 R B F Albuquerque, N Manzanares-Filho, and W Oliveira

the useof thesimplified radialequilibrium equation orstreamline curvature methods for evaluating the radialflow variations [4].

For hydraulic turbines, however, the descriptionof this kind of intermediate methodology is appar-

ently scarce in the open literature. On one hand, onecan found theoretical analysis of the overall perfor-mance, which does not account for the effects ofrunner blade geometry changes in the flow field [5].On the other hand, it is not difficult to find modern

works on design optimization using direct CFD anal-ysis, without previous systematic investigations aboutfavourable geometrical configurations [6, 7].

Therefore, it seems desirable to make avail-able intermediate tools for analysis and design of

water turbines. These tools should furnish a reli-able conceptual design, with a simplified but rep-resentative geometry for runners and stators andalso favourable trends towards the optimal flowfield. In the present work, one such methodologyis proposed for axial-flow hydraulic turbines. Theapplication of these low head turbines is expand-ing worldwide because of the progressive exhaustof hydropower resources of high and moderateheads.

The proposed methodology is intended to a con-ceptual design optimization. It couples a fast flowsolver and optimization techniques. The flow solveris based on the simplified radial equilibrium equationand on empirical correlations for flow losses and devi-

ations. Besides allowing fast comparative evaluationsof preliminary configurations, the methodology candeal with non-free vortex analysis at an early designstage. Thus, the use of high computational cost tools,like CFD codes, can be postponed to very final designphases. In this way, the total design effort could besubstantially reduced.

The design optimization problem is stated earlyin section 2. In section 3, the design variablesand the blade geometry parameterization are pre-sented. The flow model for the axial turbine is devel-oped in section 4. A brief explanation about theselected optimization techniques is given in section5. In section 6, the methodology is applied to asmall tube type propeller turbine and the optimizedresults are compared with a previous design forthat turbine. Concluding remarks are presented insection 7.

2 FORMULATION OF THE OPTIMIZATIONPROBLEM

The hydraulic turbine design problem consists insearching some basic geometrical parameters (design

variables) of the guide vanes and runner blades in

order to maximize the turbine efficiency (objectivefunction), given the design rotational speed and vol-umetric flowrate (design point optimization only).The available head should lie within upper and lowerlimits, these being the non-linear constraints of the

problem. There are also lateral constraints for thedesign variables, defining the problem feasible region(design space).

Formally, this can be stated as a constrained mini-mization problem as follows

minimizef(x)

subject togi(x) 0, i=1, . . . , m

x S

where x is the n-dimensional vector of design vari-

ablesxj(j=1, . . . , n). These variables are geometricalparameters, defined in section 3. The searchregion Sisdefined by upper and lower bounds,xUj andx

Lj, respec-

tively, for each coordinate of x: S= {x n :xLj

xj xU

j ,j=1, . . . , n}. The objective function is f(x)=(x), where is the turbine efficiency. gi(x), i=1, . . . , m, are the m = 2 constraint functions, namely,

g1(x)= HL H(x)and g2(x)= H(x) HU, whereH isthe turbine available head andHLand HUare, respec-tively, lower and upper limits, such that HL H HU.The performance of the turbine ( and H) is calcu-lated through an appropriate flow solver. The one tobe applied in this work is developed in section 4.

The optimization problem stated above can besolvedby algorithmsthat treat directly non-linear con-straints. Otherwise, the available head constraints canbe imposed by means of a penalization scheme on theobjective function

f =

+ M(HL H)2, HHU

whereM is a sufficiently large positive number. Again,

the objective is to maximize (minimize ) withHL H HU. The choice of the penalty factorMmustnot drive the optimization process towards a penaltyminimization only, missing objective function maininformation, i.e.the efficiency . Also, the constraintsmust not be violated at the end of the process. Sometests have to be performed to settle suitable values for

Min each particular problem.

3 BLADE GEOMETRY PARAMETERIZATION

The water turbine considered in this study is a tube

type propeller turbine with non-adjustable guide

Proc. IMechE Vol. 221 Part A: J. Power and Energy JPE394 IMechE 2007



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Conceptual optimization of axial-flow hydraulic turbines 715

Fig. 1 Sketch of the propeller turbine water channel

vanes, shown in Fig. 1. The distributor is cylindrical(non-conical) and presents untwisted guide vanes ofconstant chord.Thus, only the runner blade geometry

will be parameterized in this work. The parameteri-zation should lead to a small number of design vari-ables without missing the relevant geometric informa-tion.

The blade profile camber lines are approximatedby arcs of circumference (ARC) of small curvature, anacceptable assumption for slightly cambered profilesas those employed in axial hydraulic turbines [6, 8].The blade thickness is not considered in this work,since no cavitation phenomena or flow separation isaddressed by the adopted modelling. Actually, whenthe blade profiles of a given cascade are thin enough,the thickness does not contribute to the flow turning

angle, which becomes a function just of the angle ofattackand profile camber [9]. Thus the runner cascadegeometry is defined by its stagger angle,, chord-pitchratio, /t, and relative camber at midchord,f/ (Fig. 2).This choice of design parameters is suitable for com-puting the relevant kinematics characteristics at any

Fig. 2 The design parameters in a radial station: (a)

distributor cascade: outlet angle, 2; (b) runner

cascade: stagger angle, , chord length, , camberat midchord,f

radial station (as incidence angles, angles of attack,deviation angles, and flow turning angles).

The runner blading stagger, camber and chord-pitch ratio spanwise variations are parameterizedin terms of their values at the hub, mean, and

tip stations. In this way, the number of runnerdesign variables is reduced to nine. Here, parabolicfunctions of the radius are chosen for parameter-ization. This choice can satisfactorily approximatethe usual geometry found in axial hydraulic tur-bines [7]. Since the guide vanes are untwisted alongthe radius, a single outlet angle, 2, is enough asdesign variable for the distributor (Fig. 2). Thus, onehas a total of ten design variables. These variableshave been chosen in order to easily identify per-formance improvements at an intermediate designphase.

The application example to be presented in thisstudy uses as geometrical reference the tube typepropeller turbine designed and tested by Souza [10].Table 1 shows some basic features of this turbine.The distributor is untwisted and with constant chordlength. The relevant geometric parameters of therunner are reasonably reproduced by the parame-terization here proposed, as shown in Fig. 3. Thisparameterized approximation of the previous geom-etry [10] is referred here as the original design and willbe useful for comparisons with optimization results(section 6). Remark: the original design did notemploy

ARC blades, but Gttingen profiles. Then, equivalent

ARC blades were calculated for the original design sothat the camber angles of the Gttingen profiles havebeen recovered.

It must be emphasized that a conceptual designis book for here, in which the attained flow velocitydistributionpatterns aremore relevantthan thegeom-etry itself. Naturally, the chosen geometric parametersshould be sufficiently representative for the flow vari-ations imparted by the blade rows. They should alsobe relevant for the deviation and loss correlations tobe used in the flow model (section 4). The parame-terization here proposed satisfies these requirements.Further geometrical refinements could be achieved byusing more evolved methodologies out of the scope ofthe present work.

Table 1 Turbine main features [10]

Flowrate 0.267 m3/sRotational speed 1145 r /minHead 3.9 mEfficiency 82%Output 8.3 kW External diameter 280 mmInternal diameter 112 mm

Number of blades 4Guide vanes exit angle 60 (from tangential)




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Fig. 3 Parabolic parameterizations for the runner geometry of the original design

4 FLOW MODEL FOR AN AXIAL-FLOWHYDRAULIC TURBINE

4.1 The simplified radial equilibrium equationwith energy balances through blade rows

It will be considered the assumptions of axisymmet-ric steady incompressible flow in cylindrical streamsurfaces, such that the simplified radial equilibriumequation away from the blade rows becomes the onlyrelevant differentialequationfor the fluid motion. Thisequation relates the meridional (axial) and circumfer-ential velocity components, cm and cu, as functionsjust

of the radius rand can be stated as follows (11, p. 266,equation (4.8))

dhS

dr T

ds

dr =

cu

r

d

dr(rcu)+ cm

dcm

dr (1)

where hSrepresents the stagnation enthalpy (hS =h +c2/2),T the absolute temperature and sthe entropy.Using the Gibbs relation dh= dp/ + Tds and theincompressibility assumption, equation (1) can be

written as

1

dpS

dr

=cu

r

d

dr

(rcu)+ cmdcm

dr

(2)

where pSrepresents the stagnation pressure (pS =p +c2/2) and is the fluid density.

Besides reference [11], the deduction of the simpli-fied radial equilibrium can be found in many othertextbooks [12,13]. However, the differences betweenthe flow conditions behind a stator (where there is no

work transfer) and behind a runner or impeller (wherea spanwise distribution of work transfer occurs) arenot clearly pointed out in thosereferences.Commonly,the actual energy balance in the upstream blade rowsis not explicitly accounted for.

In what follows, suitable formulations of the radialequilibrium equation (2) will be presented for the flow

behind the distributor and behind the runner. Thebasic procedure consists in applying integral energybalances through the corresponding upstream bladecascades and introducing the results in equation (2).It will be assumed that the stream surfaces remaincylindrical along the blade rows. It is possible to showthat radial deviations of the stream surfaces associ-ated with changes in meridional (axial) velocities areindeed negligible in design situations [14]. Iterativeschemes for numerically obtaining the velocity dis-tributions are also proposed. The developments arereferred to in Figs 4 to 7. See also the notation for thevelocities, angles, and subscripts designation.

4.1.1 Radial equilibrium for the flow behind thedistributor

The integral energy equation applied to a distributorcascade at a certain radiusrleads to

pS1 pS2 =YLs (3)

whereYLsis the mechanical energy loss per unit massin the distributor cascade.

The stagnation pressure at the distributor inlet,pS1,is assumed to be constant along the span. There-fore, differentiating equation (3) and substituting in

Fig. 4 Meridional cross-section of turbine water chan-

nel. 1: distributor inlet; 2: distributor outlet;4: runner inlet; 5: runner outlet




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Fig. 5 Instantaneous absolute streamlines in a

cylindrical section

Fig. 6 Velocity components at distributor cascade

Fig. 7 Velocity triangles at runner cascade

equation (2) with subscript 2, one obtains an expres-sion of the radial equilibrium equation suitable for theflow calculation behind the distributor

cu2d(rcu2)

dr + rcm2

dcm2

dr + r

dYLs

dr =0 (4)

Observe in equation (4) that if one neglects theradial loss variation (dYLs/dr=0), the free vortex con-dition rcu2 =const. implies that cm2 =const. and vice-

versa. The free vortex is a classical design approachin axial hydraulic turbines. However, if the guide

vane geometry does not furnish the free vortex atthe enclosure between the distributor and the run-ner, cm2 will not be uniform along the span; thus,the correct radial equilibrium must be evaluated inorder to achieve a realistic velocity profile prediction.

Moreover, the free vortex condition is not necessarilythe optimal one and so a non-free vortex analysis isdesirable in any automatic design system.

The integration of equation (4) leads to the distribu-tion ofcm2in terms of the velocity torque (rcu2)

c2m2(r) c2m2h =2[YLs(rh) YLs(r)]

+

rrh

1

r2d(rcu2)

2

dr dr =Is(r) (5)

or

cm2(r)= c2

m2h+ I

s(r) (6)

As a first approximation, the radial variations ofenergy loss will be considered negligible in compar-ison with the radial variations of meridional kineticenergy in equation (5). However, it must be empha-sized that the losses will be computed for evaluatingthe turbine efficiency, after the flow calculation.

The overall continuity must be imposed in orderto evaluate the meridional velocity at the hub sta-tion (cm2h). Representing the elementary volumetricflowrate by dQ = 2rcmdr, integrating between huband tip and using equation (6), one obtains

rtrh

c2m2h+ Is(r)rdr=Q/2 (7)

The distributions cm2(r) and cu2(r) will be deter-mined by solving equations (6) and (7) iteratively.In any iteration, equation (7) is treated as a non-linear algebraic equation for the unknown cm2h witha given and fixed flowrate. A standard bisection-basedalgorithm was chosen to solve this equation in con-

junction with the Simpson rule for the necessaryintegral evaluations.

The evaluation ofIs(r) in equation (5) requires theprevious knowledge of the velocity torque distribu-tion, rcu2(r). However, the circumferential componentcu2 at a certain cascade depends upon the not yetdetermined meridional componentcm2. Therefore, aniterative scheme is adopted. One first assumes a uni-form distribution to cm2. Thence, some rcu2 valuesare calculated by using cascade relations in N radialstations. These values are fitted to a parabolic distri-bution,rcu2 =K1+ K2r+ K3r

2, by using least-squares.This choice is indeed suitable for axial hydraulic tur-bines, since it is able to satisfactorily reproduce typicalswirl patterns behind the distributor [15] and also

includes the free vortex as a particular case. Thence,the fitted parabolic distribution is used for evaluating




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Is(r) analytically in equation (5) and the meridionalvelocity distribution in equation (6) can be evaluatedafter solving equation (7) forcm2h. This newcm2distri-bution is now used to recalculate the rcu2values in the

Ncascades and the iterations are carried out until the

flow field behind the distributor converges (Fig. 8).

4.1.2 Radial equilibrium for the flow behind therunner

The flow field at the runner inlet is assumed to be sameas that at the distributor outlet, which has just beenevaluated. The integral energy equation applied to arunner cascade at a certain radial coordinate r leadsto

pS4 =pS5+ YLr+ Yblade (8)

where YLr is the mechanical energy loss per unitmass andYbladeis the blade specific work of the run-ner cascade, calculated according to the Euler workequation

Yblade = u(cu4 cu5) (9)

Substituting equation (9) into equation (8), differen-tiating and considering the result in equation (2) withsubscript 5, one obtains an expression of the radialequilibrium equation suitable for the flow behind therunner

(cu5 u)d(rcu5)dr

+ rcm5 dcm5dr

+ rdYLrdr

+ rd(ucu4)dr

=0

(10)

Observe that the distribution ofucu4is the same asthat ofucu2 already calculated. Again, the integrationofequation(10)leadstothedistributionofcm5 intermsof the distribution ofcu5

c2m5(r) c2m5h =2[YLr(rh) YLr(r)]

2

rrh

(cu5 u)

r

d(rcu5)

dr dr

2

ucu4|r ucu4|rh

=Ir(r) (11)

Fig. 8 Iterative scheme for the velocity distributioneval-uation at the exit sections

As a first approximation, the loss variations inequation (11) were also neglected.

Again, the overall flow continuity is imposed forevaluating the meridional velocity at the hub station(cm5h), leading to a non-linear problem analogous to

that one for the distributorrtrh

c2m5h+ Ir(r)rdr =Q/2 (12)

Due to the same considerations made for the flowbehind the distributor, an analogous iterative schemeis adopted for the evaluation of the velocity distribu-tion behind the runner (Fig. 8). Using least-squaresagain, the values of rcu5 in N cascades are now fit-ted to a cubic function(rcu5 =K4+ K5r+ K6r

2 + K7r3)

instead of a parabolic function.This choice has provedto be suitable for reproducing with sufficient accuracythe typical inflections that may occur in cu5spanwisevariation. For the runner, however, a subrelaxationscheme had to be applied for attaining convergence.Each time a new distribution ofcm5 is calculated byequations (12) and (11), leading to correspondingcu5values in theNcascades (from velocity triangles anddeviation correlation), the new values settled to cu5aregiven by

cnewu5 =ccascadeu5 +(1 )c

oldu5 (13)

where is the subrelaxation factor. For starting thisscheme, the first cu5 distribution is equated to zero.The subrelaxation factor has been settled equal to 0.10for a satisfactory convergence rate.

4.2 Loss and deviation correlations; turbineefficiency

The empirical loss correlations used in this study tocalculate the losses through an axial hydraulic tur-bine are summarized in Table 2. The recommendedrange for the empirical coefficients and the adoptedvalues are also indicated. This set of loss models waschosen with the aim of covering the main sourcesof loss in a tube type propeller turbine. For concep-tual optimization purposes the level of accuracy ofthe loss predictions can be considered secondary. Thekey point is the capability of the loss modelling inindicating the correct trends of loss variations dueto geometrical changes, so that different designs canbe judged in a comparative sense [16, 17]. The setof loss models should drive the optimization searchtowards a geometrical configuration with favourablehydrodynamic characteristics.

The flow deviation with regard to the outlet geomet-

rical angle in anycascade is assessed by the correlationof Carter and Hughes [12]. The absolute flow deviation




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Table 2 The set of loss models used for axial hydraulic turbines

Loss mechanism Loss model Reference

Guide vane profile loss (skinfriction loss at stator)

YLs = sc22/2 [12]

Incidence loss (shock loss at

runner inlet)

YLinc =w2inc/2 [18]

= 0.5 to 0.7 (set value = 0.5)

winc =

cm4

tan 4blade+

cm4

tan2f

u

Runner blade profile loss(skin friction loss at runner)

YLr =r w25/2 [12]

Draft tube loss (diffusion andswirl losses)

YLdt =XDmc2m5

2 +XDu

c2u52

[8]

XDm =0.09 or 0.12;XDu =0.20 to 0.40(set value= 0.09) (set value= 0.40)

Mechanical loss (externalloss)

mec =0.95 to 0.99 (set value = 0.95) Adopted

Coefficients for the profile loss (correlation of Soderberg): [12] = (105/Re)

1/4[(1+1)(0.975+0.075b/B)1]1 = 0e

0.01053 , 0 =0.04 to 0.06 (set value = 0.04) =

1f

2for =

4f

5f(flow turning angle)

B/b= radial/axial blade lengths (cascade aspect ratio)Re =VDh/,V =c2orw5, = dynamic viscosity

Dh = 2Btcos 2f/(tcos 2f + B)orDh =2Btcos 5f/(tcos 5f + B)

angle at outlet of a distributor cascade, , is given by

=2f2vane =m

t/ (14)

where 2f is the flow angle, 2vane is the guide vanegeometrical angle, is the profile camber angle ( =

1vane d2vane), is the profile chord, t is the spac-ing, and m is an empirical factor. In reference [12],m is graphically provided as a function of the stag-ger angle, , and the kind of camber line (circular orparabolic). This graph was approximated by the linearfunction m() = 0.210.04(90)/60, for circularcamber lines adopted in this work. The relative flowdeviationat a runner cascade exit is similarlyevaluatedusing the corresponding runner cascade geometry.

In the course of the investigation, it was verified thatthe original correlation, equation (14),underpredictedthe deviation angles for slightly cambered blade pro-files as those used in runners of axial water turbines.The correlation of Carter and Hughes was developedfrom low speed tests of gas turbine cascades with highcambered blades. Then,a simple overprediction basedon the original correlation has been used

=1.5m

t/ (15)

This overprediction was calibrated in order to repro-duce the available head of the turbine tests reportedin reference [10]. For uniformity reasons, it was alsoapplied for the distributor cascades.

The formula in equation (15) and the loss correla-

tions shouldbe used with some care. Although they aresuitable for the conceptual optimization purposes of

thiswork,furtherworkseemsstillnecessarytodevelopreliable empirical correlations for use in hydraulicturbine design.

The losses are evaluated in N radial stations (cas-cades) and these values are adjusted to a cubicpolynomial by using least-squares. Thence, these

regressions are integrated along the span in orderto evaluate the total hydraulic power loss, PL. Therunner blade specific work in equation (9) is alsointegrated along the span for obtaining the totalblade power, Pblade. From these results, one calcu-lates the available head, H = (Pblade+ PL)/(gQ), the

Fig. 9 Overall scheme of the solver code




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hydraulic efficiency, h =Pblade/(Pblade+ PL) and theoverall efficiency, = hmec. The integrals mentionedin this paragraph are carried out in a mass flux basis.The flow chart of the solver described in this section isshown in Fig. 9. It was coded in MatLabTM language.

As described in section 2, the values of efficiency(objective function) and available head (non-linearconstraints) are required for solving the proposedoptimization problem.

4.3 Some illustrative results using the flow solver

In Fig. 10, a comparison is made between the freevortex hypothesis and the results obtained with theapplication of theflow solver just described to theorig-inal design of Souza [10]. Since the guide vanes areuntwisted along the span, the free vortex is not pro-duced at theenclosure between thedistributor andtherunner, thuscm2is not uniform (Fig. 10(a)). As shownin Fig. 10(b), the blade work transfer varies from hubto tip, although the average value is in good agreement

with the free vortex hypothesis. These results empha-size that considering the correct radial equilibrium isindeed important for good predictions of the velocityprofiles downstream and upstream the runner. As therunner blades were not specifically designed for satis-fying these profiles, it seems clear that there is roomfor design improvements. Moreover, the velocity pro-files themselves may not be the most favourable onesfor the given operational conditions (flowrate, rota-

tional speed, and available head). In fact, these issues

have motivated the development of the proposed con-ceptual design optimization tool using the flow solverdescribed in this section.

5 THE CHOSEN OPTIMIZATION METHODS

The conceptual optimization methodology is com-pleted by coupling the flow solver with a suitableoptimization technique. Two optimization methodshave been alternatively testedfor this aim: a sequentialquadratic programming method (SQP) and a con-trolled random search algorithm (CRSA). The SQP isa gradient based method useful for local searchesstarting from a previous design. The CRSA is a pop-ulation set-based direct search algorithm that helpsin exploratory searching throughout the whole designspace.

The SQP method is one of the most effi-cient optimization techniques for solving con-strained non-linear problems being suitable for thepresent application [19]. The fmincon function fromMatLabTM was chosen in this work. This is an efficientimplementation of the standard SQP using the BFGSformula for approximating the Hessian matrix [19]. Inthe present application, the option for evaluating thedirectional derivatives by finite differences was set.

The two main limitations of gradient based methodsare the search for local optimizers only and the need

of a starting guess for the design variables. The success

Fig. 10 Velocity profiles (a) and blade work transfer (b) in the original design




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of the search becomes very dependent of this startingguess and requires that the designer provide an initialconfiguration not too far from an acceptable opti-mum. Moreover, previous investigations have shownthat even slightly different starting points (designs)

can lead to distinct solutions with different values ofefficiency [20].

To try overcoming these limitations, the CRSA hasalso been applied. The CRSA was first proposed byPrice [21] and substantially improved by Ali et al. [22].Like genetic and differential evolution algorithms, theCRSA is a global population set-based algorithm. Itstarts with an initial population of points on thedesignspace and then performs iterative substitutions of

worst points by better points in order to contract thewhole population towards a global optimizer. In CRSA,a single point is replaced per iteration. The CRSA waschosen because of its straightforward implementa-tion, fastness, and good results reported in technicalliterature [23]. Here one applies the algorithm pro-posed by Ali et al. [22] with modifications for avoid-ing ill-conditioning and accelerating the convergence

when solutions lie in the vicinity of the design spaceboundaries [20].

Differently of the SQP, the CRSA does not require acareful starting design. It employs an initial populationrandomly chosen on the design space S. Besides toalleviate the designer effort, the CRSA increases thehope of finding a global solution. A relatively smallnumber of function evaluations for convergence is also

an important feature of CRSA [20].When using the CRSA, the available head constraints

are imposed by means of the penalty scheme alreadydescribed in section 2.

6 APPLICATION EXAMPLE

The original design of Souza [10] is used as a com-parative reference for an application of the design

Table 3 Design point and head

constraints

Flowrate,Q 0.267 m3/sRotational speed,n 1145 r/minLower head, HL 3.0mUpper head, HU 4.0m

methodology proposed in this work. The optimizationruns were performed according to Table 3 for theoperating point and head constraints and Table 4for design variables lateral constraints. The analysisof the original design and the best results obtained

by optimization are compared in Table 5. The valuesmarked with an asterisk correspond to an activatedconstraint.

Several starting points were tried for the SQPmethod. Different solutions have been found, whatdenotes the existence of local minima in this problem.The SQP runs lead to values for the guide vane outletangle, 2, ranging from 51

to 68. This is a main con-cern since the distributor exit flow strongly affects thedesign of the runner blades and, therefore, the over-all turbine performance. The SQP solution shown inTable 5 is the best one among these configurations.Curiously, this solution was obtained usingthe originaldesign as the starting guess.

TheCRSA wasrun 30 times with differentinitial pop-ulations. A penalty factor M =50 m2 was employed.In comparison with the SQP, the obtained guide vaneoutlet angles lied in a narrower range (50 to55) whatsuggests that the concerning global optimizer is withinthis range. The CRSA solution shown in Table 5 is thebest one found. It shall be noted the good agreementbetween the SQP and the CRSA solutions. Therefore,it is hoped that both solutions are close to the actualglobal optimizer. Since the best SQP solution presentsa slightly higher efficiency than that of the best CRSA

solution, the first is treated as the optimal design in thefollowing discussion.

The optimal distributor exit angle,2 = 52.6, yields

a most favorable flow distribution at the runner inlet.In comparison with the original design, this makespossible a more efficient absorption of the flow energyby the runner. Figures 11 and 12 compare the span-

wise variation of the runner blade geometry betweenoriginal and optimal designs. The blades of the opti-mal design have lower stagger angles near the hub andhigher ones near the tip (Fig. 11(a)). They are morecambered near the hub and less cambered along theremaining span (Fig. 11(b)). A sketch of the optimalrunner blade is shown in Fig. 13.

The angular momentum of the optimized runnerinlet flow is higher than that of the original design.This can be seen in Fig. 14(a), cu2 curve. The opti-mized runner produces a more uniform exit flow (cm5

Table 4 Design variables lateral constraints

() /t() f/(%)Design

variable Hub Mean Tip Hub Mean Tip Hub Mean Tip 2()

xLj 40 25 15 1.61 1.08 0.889 0.8 0.5 0.1 50

xUj 55 35 25 1.70 1.20 1.00 6.0 4.0 2.0 70




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Table 5 Comparison between original design, best SQP solution and CRSA solution

Original design [10] SQP (present work) CRSA (present work)Design variables and

resulting quantities Hub Mean Tip Hub Mean Tip Hub Mean Tip

() 49.3 26.2 17.4 45.8 25.9 18.9 49.5 26.6 17.3

/t() 1.61 1.08 0.889 1.61 1.08 0.889 1.64 1.082 0.902f/(%) 4.40 3.06 1.32 5.24 2.35 0.27 4.72 2.15 1.002(

) 60.0 52.6 52.3Blade power (W) 9414 9534 9527Distributor loss (W) 129 151 152Runner+draft tube loss (W) 894 744 745(%) 85.69 86.84 86.82H(m) 4.00 4.00 4.00

Fig. 11 Runner blading (a) stagger angle and (b) camber of original design and optimized solution

curve), decreasing the meridional component of thedraft tube loss. Although the exit swirl is everywherenegative in the original design, the optimal solutionshows a negative swirl near the hub and a positive oneat the tip(Fig.14(a), cu5 curve, andFig. 14(b),5fcurve).

Fig. 12 Comparison between original design geome-

try and optimal one (the thicknesses are onlyillustrative)

This trend is in good agreement with flow measure-ments in well-designed axial hydraulic turbines [24].The optimized flow turning angles in the runner varyfrom 22 at the hub to 1 at the tip (Fig. 14(b)) beinghigher than those of the original design from hub

Fig. 13 Optimal runner blade (the thickness is onlyillustrative)




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Fig. 14 Comparison between original and optimal (a) flow velocity distributions, (b) flow angle

distributions, (c) blade work transfer distribution, and (d) spanwise loss variations

to midspan. However, they still remain below safelimits attainable by axial-flow hydraulic turbines [5].This issue is also important for avoiding cavitationrisk, since the blade loading has a direct impact incavitation phenomena.

Figure 14(c) shows the spanwise distribution of theblade specific work. In comparison with the origi-nal design, the optimal solution increased the blade

specific work from hub to midspan and decreased itfrom midspan to tip. The resulting overall blade power(Pblade)was increased in 1.3 per cent (Table 5).

Figure 14(d) shows the spanwise variation of thespecific losses. The sum of runner and draft tubelosses is also plotted, showing the significant decreaseachieved by the optimization (17 per cent, Table 5).Note, however, that the distributor loss had to be




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increased to accomplish the higher flow deflectionsproduced by the optimal guide vanes. The mainimprovements occur in the incidence and draft tubelosses. This occurs because the optimal distributorand runner geometry together provide favourable

incidence angles at the runner inlet and favourablevelocity profiles for the draft tube inlet flow. This isa key point: the final design of the runner bladesshould be made in order to satisfy the optimal velocitydistributions. More sophisticated design tools couldbe later used to reduce the actual losses to min-imum levels. Thus, among the results obtained bythe conceptual design, one may attribute more sig-nificance to the optimal velocity profiles than to thegeometry itself. Nevertheless, the conceptual opti-mized geometry can be useful as a preliminarydesign.

Finally, it is interesting to make a comment aboutthe computational effort. The runs were on an AMDSempromTM 2.0 GHz processor with MatLabTM ver-sion 5.3 under Windows XPTM platform. The SQP runstake 2 min in average, spending from 200 to 600 func-tion evaluations (flow solver calls). The CRSA runstake 10 min in average (from 600 to 1500 functionevaluations), which can also be considered a rea-sonable time for a conceptual design. Because ofits stochastic features, any CRSA solution should betaken in a statistic sense. Actually, one should runseveral times the CRSA in order to accept a finalsolution. If the flow solver demands high computa-

tional effort and the global optimization algorithmrequires too many function evaluations, it would beprohibitive to perform a comparative study, or evento perform a single optimization run until achieving afair convergence.

7 CONCLUSIONS

A conceptual design optimization methodology foraxial-flow hydraulic turbines has been proposed.

The simplified radial equilibrium equation playsa central role in the present work. In conjunction

with energy balances through blade rows, it allowsa consistent non-free vortex flow analysis behindthe distributor and behind the runner. A geometri-cal parameterization is used to represent the spanwisevariation of the blade geometry, giving a small set ofdesign variables. Loss and deviation correlations com-plete the flow model. This methodology, coupled withnumerical optimization techniques, allows the cal-culation of favourable flow trends at an early designphase.

The methodology was applied to a preliminary

design of a small tube type propeller turbine. Theanalysis of the optimized solution shows potential

performance improvements in comparison with aprevious design.

After applying the proposed design system, a finaldesign of the guide vanes and runner blades could bemade in order to satisfy the obtained velocity distribu-

tions. More sophisticated design tools could be usedto reduce the actual losses to minimum levels. Thus,among the results obtained by the present designmethodology, one may attribute more significance tothe velocity profiles than to the geometry itself. Never-theless, theobtainedoptimal parameters canbe usefulfor reducing the subsequent design effort.

ACKNOWLEDGEMENT

During this work, the first author received financial

support from CAPESCoordenao de Aperfeioa-mento de Pessoal de Nvel Superior, Brazilian Govern-ment Agency.

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APPENDIX

Notation

c absolute velocity (m/s)f objective functionf/ relative cambering of the profileg acceleration due to gravity (m/s2)gi non-linear constraintsh specific enthalpy (J/kg)H turbine available head (m)K1,...,7 coefficients of polynomial fitting

/t chord-pitch ratiom empirical factor for the deviation

correlationM penalty factor (m2)n rotational speed (r/min), number

of variablesN number of cascadesp pressure (Pa)P power (W )Q flowrate (m3/s)r radius (m)s specific entropy (J/K)S search region inn

t spacing (m)T absolute temperature (K)u circumferential velocity (m/s)w relative velocity (m/s)

x vector of design variablesY energy per unit mass (J/kg)

absolute flow angle, guide vaneoutlet angle (from tangential) ()

relative flow angle, stagger angle(from tangential) ()

deviation angle () flow turning angle in a cascade () efficiency subrelaxation factor, loss coeffi-

cient due to incidence dynamic viscosity (Pa s)

loss coefficient due to skin friction density (kg/m3) profile camber angle ()

Subscripts

blade absorbed by the runner bladesf flow h hub radius, hydraulicinc incidence or shock L loss, lower boundm meridional component, mean

radius

mec mechanicS stagnationt tip radiusu circumferential componentU upper bound1 distributor inlet2 distributor outlet4 runner inlet5 runner outlet




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