design of the rotor blades of a mini hydraulic bulb-turbine

7222019 Design of the Rotor Blades of a Mini Hydraulic Bulb-turbine

httpslidepdfcomreaderfulldesign-of-the-rotor-blades-of-a-mini-hydraulic-bulb-turbine 19

Design of the rotor blades of a mini hydraulic bulb-turbine

LMC Ferro ab LMC Gato b AFO Falcatildeo b

a Department of Mechanical Engineering Escola Superior de Tecnologia de Setuacutebal Polytechnic Institute of Setuacutebal Campus do IPS Estefanilha 2910-761 Setuacutebal Portugalb IDMEC Instituto Superior Teacutecnico Technical University of Lisbon Av Rovisco Pais 1049-001 Lisboa Portugal

a r t i c l e i n f o

Article history

Received 26 July 2010

Accepted 16 January 2011

Available online 21 March 2011

Keywords

Axial-1047298ow turbine

Design method

Rotor bladesSingularity method

FLUENT code

a b s t r a c t

The rotor blades of a mini hydraulic turbine were designed using a quasi-three-dimensional method The

meridional 1047298

ow is computed by a streamline curvature method and the blade-to-blade 1047298

ow bya singularity method The rotor blade sections are the NACA 66 (MOD) with a frac14 08 meanline The

camber and the stagger angle of the blade sections are adjusted to ful1047297l the prescribed angularmomentum distributions at the rotor inlet and outlet sections and zero-incidence 1047298ow angle at the

blade leading edgeTurbine head and ef 1047297ciency versus 1047298ow rate curves were obtained for different rotor blade stagger

angles at constant rotational speed Radial distributions of time-averaged velocity and pressuremeasured at the rotor exit section with a 1047297ve-hole probe are also presented

The design and the experimental results are compared with three-dimensional inviscid1047298ow numericalresults computed by the FLUENT code The domain is discretized by unstructured meshes witha maximum of 25 106 elements The numerical results show good agreement with the design values

and the experimental results validating the design hypothesis of small radial velocity in the 1047298ow throughthe rotor

2011 Elsevier Ltd All rights reserved

1 Introduction

The contribution of electricity from renewable energy to totalelectricity consumption has increased in Europe (EU-27) in lastdecade from 131 in 1997 to 156 at 2007 [1] Hydropoweraccount for about 512 of the electricity generated from renewable

energy in 2008 with a production of 359 TWh from an installedcapacity of 1023 GW Production from small hydro plants in 2008was 427 TWh for an installed capacity of 126 GW (Mini-hydropower systems were responsible for 102 TWh with a capacity of

30 GW)Small hydro plants not only represent a good technological

solution to provide electrical energy to disperse communities

specially in developing countries but also can be a good solution inindustrialized countries due to their relatively small environ-mental impact and to the fact that most of the large scale hydroresource is already explored

In the European strategy a special effort is to be devoted to the

development of low-head plants (and axial-1047298ow turbines) becauseof their large potential and their relatively small environmentalimpact [2]

A design method for the inlet guide vanes for mini hydraulicbulb-turbines was presented by the authors in [3] The present

paper presents a design method for low-cost mini hydraulic bulb-turbine rotor blades The rotor blades are bounded by two coaxialcylindrical surfaces The shape of the blade is de1047297ned using a quasi-three-dimensional design method by prescribing constant angular

momentum distribution along the span at the rotor inlet and exitsections The meridional through-1047298ow is computed by a streamlinecurvature method and the blade-to-blade 1047298ow by a singularitysurface method

The blade sections are the NACA 66 (MOD) with a frac14 08meanline as described by Brockett [4] Maximum blade relativethickness is prescribed at two sections oneclose to the hub and the

other close to the blade tip Maximum thickness of the othersections is obtained from linear interpolation along the radius Theblade section meanline camber and the blade stagger angle arecomputed to ful1047297l the required design circulation and zero-inci-dence 1047298ow at the leading edge

A 05 m diameter turbine rotor with four blades was manu-factured and tested in an air1047298ow rig to validate the designmethod Measurement of the turbine head and power versus 1047298owrate were performed for different rotational speeds and rotor

blade stagger angles Traversing measurements of velocity andpressure along radial directions were also made at the exit sectionof the rotor blades with a 1047297ve-hole probe The experimental

Corresponding author Tel thorn351 265790000 fax thorn351 265721869

E-mail addresses lferrohidrolistutlpt (LMC Ferro) luisgatoistutlpt

(LMC Gato) falcaohidrolistutlp (AFO Falcatildeo)

Contents lists available at ScienceDirect

Renewable Energy

j o u r n a l h o m e p a g e w w w e l s e v i e r c o m l o c a t e r e n e n e

0960-1481$ e see front matter 2011 Elsevier Ltd All rights reserved

doi101016jrenene201101037

Renewable Energy 36 (2011) 2395e2403



results for the velocity and pressure distributions are comparedwith the prescribed design conditions and with numerical resultsfrom three-dimensional analysis with the FLUENT code for

inviscid 1047298ow

2 Design of rotor blades

The through-1047298ow analysis approach as described by the authors

in [3] is applied to the design of a mini hydraulic turbine rotor withfour blades A streamline curvature method is used for the solutionof the meridional 1047298ow and a singularity method is utilized for theblade-to-blade 1047298ow

The four rotor blades are radially set between two coaxial

cylindrical wall surfaces The outer-diameter is D frac14 05 m and theinner diameter is DH frac14 0214 m

Three steps need to be considered in the design of a turbine (i)

speci1047297cationof thedesign variables1047298ow rate Q h available head H hand rotational speed U (ii) de1047297nition of the velocity diagramsupstream and downstream of the blade rows (iii) calculation of the

blade geometry camber and thickness distributions and cascadechord-to-pitch ratio c s One can start byspecifying oneof the threedesign variables (eg H h) and then compute the other two fromdimensionless values of the tip speed velocity kU frac14 U (2 gH h)12 and

the meridional velocity kV m frac14 V m=eth2 gH hTHORN1=2 obtained in the liter-ature from the experience of different manufacturers of hydraulicturbines [56] Theradial distribution of theangular momentumrV qat the inlet section of the rotor can then be calculated from the

known values of H h Q h andU

21 Computation of the meridional velocity 1047297eld

The meridional 1047298ow 1047297eld through the turbine is computed by

the streamline curvature method as described in [7] Two quasi-orthogonal lines were considered at the rotor one at the bladeleading edge and the other at the trailing edge A constant spanwisedistribution of the angular momentum rV q was prescribed at the

leading and trailing edges with the maximum (design prescribed)value at the leading edge and zero at the tailing edge The radialcoordinates of the meridional streamlines and the axial velocitycomponent distributions at the inlet and the outlet sections of the

rotor blades are used as input data for the computation of theblade-to-blade 1047298ow as explained in Section 23

22 Computation of the blade-to-blade velocity 1047297eld

The blade-to-blade 1047298ow is represented by the 1047298ow througha rectilinear cascade of blades computed by a panel method [8] Theairfoil contour is discretized by N linear elements with constant

strength source distribution on each element The circulation ismodeled by a vortex distribution on the airfoil meanline

The complex velocity w frac14 V xiV y induced at z frac14 x thorn iy bya source of strength s at z frac14 x thorn ih is given by [9] w( z ) frac14 s( Z z) and

the complex velocity induced by a cascade of sources of pitch s is

weth z THORN frac14 sp

s coth

peth z zTHORN

s (1)

Nomenclature

A cross sectional area

c blade chord

C L lift coef 1047297cient

C tot total pressure coef 1047297cient

D diameter tip runner diameter

DH hub runner diameter

g acceleration due to gravity

H available head

h linear displacement of the probe measured from thehub wall

ku dimensionless tip speed velocity

kV m dimensionless meridional velocity

N number of elements of discretized airfoil contour

N p number of points at each measurement line of 1047297ve-

hole probe

p static pressure

P tot total pressure

Q 1047298ow rate

r dimensionless radius 2r D

(r q z ) cylindrical coordinate systems circumferential pitch

S 1 blade-to-blade surface

S 2 meridional surface

T torque

t blade thickness

U runner tip velocity V vector of absolute 1047298ow velocity

V absolute 1047298ow velocity

V m meridional 1047298ow velocity component

V r V qV z absolute velocity components along r q z directions

V ref reference velocity 4Q =frac12pethD2 D2H THORN

V dimensionless velocity W vector of relative 1047298ow velocity

W relative 1047298ow velocity

w complex velocity

( x y z ) cartesian coordinate system

Z number of rotor blades

Greek symbol

a absolute 1047298ow angle measured from meridionaldirection arctan (V qV m) also angle of attack

a p relaxation factor for pressure

av relaxation factor for momemtum

b absolute 1047298ow angle between meridional and radialdirections arctan (V r V m)

G vortex circulation intensity also swirl number

h turbine ef 1047297ciency

l stagger anglen kinematic viscosity

P power coef 1047297cient

r density

s intensity of source distribution also swirl number

F dimensionless 1047298ow coef 1047297cientJ dimensionless head

U rotor angular speed

Subscript 12 inlet and outlet respectively

ax axialhyd hydraulic

m meridionalmax maximum

min minimumh relative to design conditions

LMC Ferro et al Renewable Energy 36 (2011) 2395e 24032396



The 1047297nal equation for the velocity due to a constant distributionon a panel is obtainedby integration of Eq (1) The 1047297nal result is [10]

V x iV y frac14 seiblnsinhfrac12peth z z2THORN=s

sinhfrac12peth z z1THORN=s (2)

where z1 and z2 are the position vectors of the end points of the

panel and b is the angle of the panel with the x-axis Eq (2) can besimpli1047297ed when the point z is the midpoint of the panel The 1047297nalequation is

V x iV y frac14 ipseib frac14 spethsin b thorn icosbTHORN (3)

The velocity induced by a vortex is computed replacing s in Eq(2) by the vortex circulation intensity G and multiplying the result

by the imaginary unit iThe strength intensity of the source on each panel and of the

vortex distribution on the meanline are computed from the solu-tion of a linear system of N thorn 1 equations obtained from the

application of the impermeability condition at the midpoint of each

panel and of the Kutta condition equal velocity magnitude at thetwo control points close to the trailing edge The resulting linear

system of equations is solved either by a Gauss direct method or bya GausseSeidel iterative method

23 Computation of the aerodynamic parameters

At the design condition the angular momentum is constantalong the spanwise direction (radial direction) at the inlet and theoutlet sections with prescribed non-zero value at the inlet and zeroat the outlet Therefore the radial component V r of the inviscid 1047298owis zero [11] and the inviscid 1047298ow may be assumed as irrotational

Nevertheless the velocity 1047298ow 1047297eld obtained from the solution

of the meridional 1047298ow is incompatible with the two-dimensionalcascade 1047298ow assumptions since the axial velocity is not constantalong a streamline through the rotor For design purposes the

average axial velocity and average radial coordinate of the merid-ional streamlines between the rotor inlet and outlet sectionsare selected as input to the cascade 1047298ow computation The

-Cp( min)

α

0 1 2 3 4 5-1 2

-1 0

-8

-6

-4

-2

0

2

4

6

Design r

=0518

Design r=0962

r =0 962r =0 518

α

x

c

-1 0 -5 0 5 1000

01

02

03

04

05

06

07

08

09

10 r =0518

r =0962

ab

Fig 1 Aerodynamic properties of the two reference rotor blade sections (a) minimum pressure envelope (b) suction peak coordinates xc versus angle of attack a

xc ()

- C p

0 20 40 60 80 100

-10

-05

00

05

10

15

20 r = 0428

r = 0518

r = 0666

r = 0776

r = 0876

r = 0962

r = 1000

xc()

- C p

-1 0 1 2 3 4 5-10

-08

-06

-04

-02

00

02

04

06

08

10

xc ()

- C p

95 96 97 98 99 100-02

-01

00

01

02

03

04

05

06

Fig 2 Design pressure coef 1047297

cient distributions for the rotor blade sections

LMC Ferro et al Renewable Energy 36 (2011) 2395e 2403 2397



circumferential velocity component at the inlet section is V q frac14 K r where K is the speci1047297ed valueof the angular momentum at the inlet

section At the outlet section the circumferential velocity compo-nent is zero

The blade sections are the NACA (MOD) 66 type series with

a frac14 08 as described by Brockett [4] These sections exhibit good

cavitation properties with almost constant pressure differencebetween the lower and the upper airfoil surfaces till 08c A linearvariation of the maximum thickness t max along the radius is

prescribed with t c frac14 012 for a section close to the hub ( r frac14 0518)and t c frac14 003 for a section close to the rotor tip (r frac14 0962) with

r frac14 2r D The cascade chord-to-pitch ratio c s for those tworeference sections are computed from (see Ref [11])

C L frac14 2s

c

DW qW N

(4)

with C L frac14 12 for r frac14 0518 and C L frac14 04 for r frac14 0968 In Eq (4)

W N frac14 ( W 1thorn W 2)2 is the mean relative velocity between the inlet

and the outlet relative velocities W 1 and W 2 and D W q frac14 W 1 W 2A panel method code based on Eqs (2) and (3) is used to opti-

mize the pressure distribution over the blade surface by modifyingthe camber and the stagger angle of the two reference sections The

method uses straight elements with source distributions of constant intensity s at each panel The circulation is generated bya vortex distribution along the airfoil meanline The meanline isdiscretized into straight elements with constant vortex distribu-

tion The intensity g of the vortex distribution at midpoint of eachpanel is given by g frac14 g0sm where sm is the distance from themidpoint of each panel to the trailing edge measured along themeanline and g0 is a constant to be computed from the Kutta

condition The stagger angle l and the airfoil camber areadjusted toful1047297l the prescribed D(rV q) distribution between the inlet and outlet

sections ie the circulation G frac14 D(rV q)(2p Z ) around the bladesections zero-incidence 1047298ow angle at the leading edge and smoothpressure distributions along the airfoil contour with minimumpressure observed far from the leading edge The stagger angle and

the camber are de1047297ned using two different iterative cycles In theinternal cycle the camber of the airfoil is speci1047297ed and the staggerangle is computed by an iterative method to satisfy the prescribed

circulation around the airfoil In the external cycle the value of themaximum camber of the airfoil is changed so that the pressuredistribution on the suction surface is smooth and the suction peakis located far from the leading edge Fig 1 shows the minimum

pressure envelope and the coordinates xc of the suction peakversus the angle of attack a for the two reference sections Theairfoil boundary is discretized into 320 panels and the meanlineinto 160 panels Ten more sections were speci1047297ed at different radiiafter the computation of the two reference sections The centers of

these sections are placed at the radial line connecting the center of the two reference sections The airfoil chord is obtained assumingstraight leading and trailing edges at the meridional plane The

section stagger angle and the camber arecomputed as described for

the two reference sections Pressure distributions for different radiiare shown in Fig 2 The main characteristics of the pro1047297les arepresented in Table 1 and three views of the rotor blade are plotted

in Fig 3

3 Experimental facility

The experimental facility is described in detail in Ref [12] Air isthe working 1047298uid The experimental rig includes the turbine a DCgenerator connected to the turbine shaft a centrifugal fana plenum chamber downstream of the turbine diffuser and a cali-

brated nozzle to measure the 1047298ow rate downstream the plenumchamber The turbine has a 950 mm outer-diameter upstreamannular duct an inlet conical guide vane system a 05 m tip-

diameter rotor and a 2 m long 6 angle diffuserThe rotor has four blades Fig 4 The blades are made of carbon

1047297ber and are connected to the hub by a rod The setting angle of therotor blades can be changed The hub and the shroud of the rotorare spherical to allow the variation of the blade setting anglewithout change in clearance

The measurement of the velocityand pressure distributions at therotor outlet sections was performed with a 1047297ve-hole probe onecentral hole and four equidistant lateral holes (two holes with thecenter on a planenormalto theprobeaxise right and leftholese and

Table 1

Characteristics of the rotor blade cascades

r (mm) r c (mm) c s t maxc f maxc l ( ) a ( )

1070 0428 2230 1327 0143 0103 3454 387

1180 0472 2280 1230 0132 0085 4129 370

1295 0518 2339 1150 0120 0068 4730 296

1485 0594 2451 1051 0102 0052 5408 234

1650 0660 2561 0988 0086 0042 5853 169

1805 0722 2674 0943 0073 0035 6176 1201945 0778 2782 0910 0062 0031 6403 093

2065 0826 2879 0888 0053 0028 6564 070

2190 0876 2984 0867 0044 00255 6704 045

2300 0920 3080 0852 0037 00235 6814 0262405 0962 3173 0840 0030 0022 6906 015

2500 1000 3260 0830 0024 00205 6977 003

Trailing edgeLeading edge

z

y

Trailing edgeL eading edge

y

x

z

xa c

Fig 3 Geometry of the rotor blades (a) view ZY (b) view XY (c) view ZX




two others on the probeaxise upper and lower holes) [12] The probehas two degrees of freedom radial displacement and rotation around

the probe axis The radial and yaw movements of the probe areproduced by a mechanical system with a step displacement of

001 mm and an angular step of 01 The probe axis is positionedalong the radial direction The probe is rotated until the pressuredifference between the left and right holes vanishes The velocitydirection (two angles) the dynamic pressure and the total pressure

areobtained by linear interpolation from theprobe calibration curvesThe experimental facility was also designed to measure the

turbine performance variables The available head H is computed

from the pressure difference between the atmosphere and theplenum chamber The 1047298ow rate Q is measured by the pressuredifference between the plenum chamber and the calibrated nozzlethe torque T by an inductive transducer placed between the rotor

and the DC generator the angular speed U by a photoelectric

transducer and pressure by differential manometers

4 Computation of inviscid three-dimensional 1047298ow using the

FLUENT code

The FLUENT 62 [13] code was used to compute the three-dimensional inviscid 1047298ow through the rotor The contour of each

section is described by 602 points The domain is divided intodiscrete volumes by a computational mesh generated using theGAMBIT 22 code [14] with the turbo option The numerical results

presented were computed with an unstructured tetrahedral meshwith 2572942 elements

The numerical calculations used the segregated method withPISO [15] pressureevelocity coupling algorithm QUICK [16] inter-

polation scheme for momentum and a second-order upwindscheme for the pressure were applied

The equations of motion (continuity and momentum) were

solved in a rotating frame using the absolute velocity vector V asindependent variable

The boundary conditions used are the velocity-inlet typecondition at inlet section prescribing the radial distribution of the

velocity vector (absolute value and direction) and the out 1047298owcondition at the outlet section The velocity pro1047297le at the inletsection is given by

V z frac14 Q

A V q frac14

ethrV qTHORNhr

and V r frac14 0 (5)

where A is the inlet section area Periodic boundary conditions are

used at the periodic boundary surfaces

Fig 5 Experimental results for dimensionless head J and ef 1047297ciency h versus 1047298ow

rate coef 1047297cient F for l frac14 85 70 and 60

Fig 4 Turbine rotor




5 Results

Turbine performance curves were obtained for six rotor bladesetting angles de1047297ned by the tip section stagger angle l frac14 85 8075 70 65 and 60 The variables measured in each test are thestatic pressure at the plenum chamber (turbine exit pressure) the

static pressure at the rotor outlet section the rotor angular speed

U the 1047298ow rate Q and the shaft torque T Tests were performedfor 1138 rpm U 2561 rpm corresponding to Reynolds numberRe frac14 UD2v between 20 106 and 44 106 Results for the

dimensionless head J and ef 1047297ciency h versus the 1047298ow ratecoef 1047297cient F for l frac14 85 70 and 60 are plotted in Fig 5 Isolinesof ef 1047297ciency h on planes (F l) and (F J) are plotted in Fig 6They show maximum ef 1047297ciency h frac14 0877 for F frac14 0117 and

l frac14 749 Values of the available head and power coef 1047297cients formaximum ef 1047297ciency are J frac14 00677 and P frac14 000653respectively

Measurements with the 1047297ve-hole probe described in Section 3

were made at the rotor outlet section The probe moves along theradial direction on a plane 142 mm downstream of the planede1047297ned by the rotor blades axis At each traversing line the span of

the probe displacement is 172 mm A total of N p frac14 23 points are

considered at each measurement line with a cosine lawdistributionfor the probe position

hi frac14 h1 thorn hmax

2cos20

cos20

cosf0i

(6)

where h is the probe linear displacement along the radial directionmeasured from the hub wall h1 frac14 19 mm and hmax frac14 1681 mmThe last point with h frac14 170 mm is 2 mm away from the shroudwall The values of f0

i are given by

f0i frac14 f0

1 thorn180

2 20

N p 1 ethi 1THORN (7)

with f01 frac14 20

and i frac14 1 N p

Measurements were made for two setting angles of the rotorblades l frac14 70 and l frac14 74 For each angle measurements were

made at different 1047298ow rates and constant rotor angular speed Thevariables measured at each test are the nozzle differential pressurethe differential pressure between the right and left probe holes thepressure at the plenum chamber the pressures on the 1047297ve-hole

probe and the torque The velocity is non-dimensionalized by the

mean-velocity at the rotor inlet section

V ref frac14 4Q

p

D2 D2

H

(8)

The dimensionless velocity pro1047297le at the inlet section can becomputed from Eq (5) and is given by

V z frac14 10 V q frac14ethrV qTHORNhV ref r


The angular momentum is made non-dimensional by V ref D2The experimental and the design (axisymmetric) radial distri-

butions for the axial velocity component V z the radial velocitycomponent V r and the angular momentum (rV q) for l frac14 70 and

l frac14 74 are plotted in Fig 7 for 1047297ve different 1047298ow rates Goodagreement between the experimental results and the design values

for the axial and radial velocity components is observed (except inthe vicinity of the hub due to the shapemodi1047297cation of the blade sothat it can turn around its axis and the conical shape of the hub

downstream of the rotor) The rV q distributions for 1047298ow rate coef-

1047297cients close to the design value are almost invariant with theradial coordinate

The computational domain for the three-dimensional 1047298ow is

composed by an annular cylindrical duct with inside diameter DH

and outside diameter D The total length is 06 m (ie 326c ax at thehub and 531c ax at the shroud where c ax is the chord axial length)The distance between the inlet section and the leading edge is D z

c ax frac14 079 at the hub and D z c ax frac14 162 at the shroud The corre-

sponding values for the distance from the trailing edge to thedomain exit section are 148 and 268 The blade surface iscomputed from the coordinates of the twelve sections used in thede1047297nition of the blade geometry as described in Table 1

To stabilize the iterative process a 1047297rst approximation of thesolution was obtained for U frac14 0 Then the rotor rotational speed

was increased till the design rotational speed U

The procedurechosen for the present test cases was (i) angularspeedU frac14 0 linearinterpolation for pressure and momentum with relaxation factors

a p frac14 03 for the momentum and av frac14 07 for the velocity (ii)angular speed U frac14 0 second-order interpolation for pressure with

a p frac14 03 and QUICK for the momentum av frac14 07 and (iii)

Fig 6 Isolines of the ef 1047297

ciency h (a) plane (F l) (b) plane (F J)




acceleration of the rotor till the speci1047297ed rotational speed U with

a p frac14 03 and av frac14 04 The evolution of the residuals in thenumerical solution of the equations of continuity and of the three-

components of the momentum are shown in Fig 8Fig 9 shows distributions of the axial and the radial velocity

components and the angular momentum for (i) the design valuesgiven by the solution of the meridional 1047298ow (ii) the experimentally

obtained values and (iii) the circumferential mass average of

numerical results computed by FLUENT for inviscid 1047298owand (iv) forviscous 1047298ow [17] A fairly good agreement is observed betweennumerical results for the inviscid 1047298ow and the design values for the

axial and radial velocity components the computed value for theangular momentum of the inviscid 1047298ow at the outlet section isnegative but very small with a swirl number G [13] de1047297ned as theratio of the axial 1047298ux of angular momentum to the axial 1047298uxof axial

momentum

a

a

a

b

b

b

Fig 7 Experimental results for radial distributions of V z V r and (rV q) at the rotor outlet section (a) l frac14 70 (b) l frac14 74




G frac14

Z A

rV qeth V $nTHORNd A

r hid

Z A

V z eth V $nTHORNd Afrac14 00067 (10)

where A is the cross section area n is the unit vector of the outside

normal and r hyd is the hydraulic radius de1047297ned by

r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)

The area-averaged 1047298ow angle with the circumferential direction is

a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)

Fig 8 Evolution of the residuals of continuity and momentum components equations

for the grid of 2572942 elements

a b c

Fig 9 Comparison of numerical experimental and design velocity and angular momentum distributions at the rotor exit section (a) axial velocity V z (b) radial velocity V r and (c)

angular momentum (rV q)

Fig 10 Isolines of numerical results for velocity angles a and b of angular momentum and of total pressure coef 1047297

cient at the rotor exit section




Axial and radial velocity distributions obtained by the solutionof inviscid and viscous1047298ow are similar except close to the rotor huband the turbine shroud The agreement is poorer for the circum-

ferential velocity due to the viscous effect on the turbine shroudand on the hub surface

Experimental results show poorer agreement with design

values than with the numerical results Due to experimentalconstraints the measurement plane is located inside the diffuser142 mm downstream of the cross section de1047297ned by the rotor bladeaxis where the probe displacement span is 172 mm ie greaterthan DDH frac14 143 mm On the other hand the computational

domain is an annular cylindrical duct with Dr frac14 143 mm withoutany conical zone downstream of the rotor The experimental axialvelocity is smaller than the numerical one due to the increase incross section area A qualitative agreement is observed for the

angular momentum distributions of the experimental and thenumerical results

Isolinesof velocityanglesafrac14 arctan (V qV m)andbfrac14 arctan(V r V m)of theangularmomentum andof thetotal pressure coef 1047297cient C tot frac14

R A

ptotd A=eth05rV 2ref ATHORN at the exit section of the domain are plotted in

Fig 10 The distributions of the four variables are almost uniform attheexit sectionwith values closeto design ones Howeverthe isolinesof the circumferential angle a show that the de1047298ection of the 1047298ow is

greater than the design condition for the sections close to the hubwhereas the opposite occurs close to the tip

The design pressure coef 1047297cient distributions for the relative

1047298ow C rel p around the contour for the sections at r frac14 0518 close to

the hub r frac14 0778 at middle section and r frac14 0962 close to thetip are plotted in Fig 11 Also shown in the same 1047297gure are theFLUENT code numerical results for three-dimensional inviscid 1047298owThe relative pressure coef 1047297cient is de1047297ned by

C rel p frac14

p pN12rW 2

N

(13)

where pN frac14 ( pinletthorn poutlet)2 Good agreement between the

numerical and the design values is observed for the three sectionsSmall differences close to the trailing edge for 095 lt xc lt 1 mightbe explained by three-dimensional effects

6 Conclusions

The rotor of a mini hydraulic turbine was designed using thethrough-1047298ow analysis approach combining the streamline curva-

ture method for the solution of the meridional 1047298ow and a panelmethod for the blade-to-blade 1047298ow The designed rotor was man-ufactured and tested in an air1047298ow rig Measurement of the turbinehead 1047298ow rate and power was performed for different rotational

speeds and rotor setting angles Turbine maximum ef 1047297ciency of

877 for l frac14 749 and F frac14 0117 was obtained from the measuredvalues Numerical results of the three-dimensional inviscid 1047298owcalculations obtained with FLUENT at exit section as well as pres-

sure distribution for the relative 1047298ow on the blade contour showvery good agreement with design values

The experimental results show that the rotor blades produce

with very good approximation the desired turbine head at thedesign condition of zero angular momentum at the exit section

Acknowledgments

This work was supported by the Portuguese Foundation forScience and Technology under contract PTDCEME-MFE666082006

References

[1] EUROPA e Eurostat e environment and energy European CommissionAvailable at httpeppeurostateceuropaeuportalpageportalenergydatadatabase June 2010

[2] Thematic Network on Small Hydropower (TN SHP) Proposals for a Europeanstrategy of research development and demonstration (RDampD) for renewableenergy for small hydropower Available at httpwwweshabe 2005 Tech-nical report MHyLab and ESHA Brussels

[3] Ferro LMC Gato LMC Falcatildeo AFO Design and experimental validation of theinlet guide vane system of a mini hydraulic bulb-turbine Renew Energ201035(9)1920e8

[4] Brockett T Minimum pressure envelopes for modi1047297ed NACA-66 sections withNACA a frac14 08 camber and buships type I and II sections Washington DCDTMB 1966 N 1780

[5] Raabe J Hydro power Duumlsseldorf VDI-Verlag 1985[6] Nechleba N Hydraulyc turbines their design and equipment Prague Artie

1957[7] Denton JD Through1047298ow calculations for transonic axial 1047298ow turbines J Eng

Power Trans ASME 197897549e60[8] Hess JL Smith AMO Calculation of potential 1047298ow about arbitrary bodies Prog

Aerosp Sci 196781e138[9] Lamb H Hydrodynamics Cambridge University Press 1932

[10] Geising JP Extension of the Douglas Neumann program to problems of lifting

in1047297nite cascades Douglas Aircraf Company Report 31653 1964[11] Lakshminarayana B Fluid dynamics and heat transfer of turbomachinery John

Wiley amp Sons Inc Artie 1996[12] Ferro LMC Numerical and experimental study of the 1047298ow through an axial

hydraulic turbine (in Portuguese) PhD thesis Instituto Superior TeacutecnicoPortugal Technical University of Lisbon 2009

[13] Fluent Incorporated Fluent 62 userrsquos guide USA Centerra Resource Park2005

[14] Fluent Incorporated Gambit 22 userrsquos guide USA Centerra Resource Park2005

[15] Issa RI Solution of implicitly discretized 1047298uid 1047298ow equations by operatorsplitting J Comput Phys 198662(1)40e65

[16] Leonard BP Mokhtari S ULTRA-SHARP nonoscillatory convection schemes forhigh-speed steady multidimensional 1047298ow e NASA TM 1-2568 (ICOMP-90-12)Technical report USA NASA Lewis Research Center 1990

[17] Tiago AF Ferro LMC Eccedila L Gato LMC Computation of viscous 1047298ow through anaxial turbine rotor (in Portuguese) II Conferecircncia nacional de meacutetodosnumeacutericos em mecacircnica de 1047298uidos e termodinacircmicarsquo08 Portugal UnivAveiro 2008

a b c

Fig 11 Pressure coef 1047297cient distribution around rotor blade sections (a) r frac14 0518 (b) r frac14 0778 and (c) r frac14 0962




results for the velocity and pressure distributions are comparedwith the prescribed design conditions and with numerical resultsfrom three-dimensional analysis with the FLUENT code for

inviscid 1047298ow

2 Design of rotor blades

The through-1047298ow analysis approach as described by the authors

in [3] is applied to the design of a mini hydraulic turbine rotor withfour blades A streamline curvature method is used for the solutionof the meridional 1047298ow and a singularity method is utilized for theblade-to-blade 1047298ow

The four rotor blades are radially set between two coaxial

cylindrical wall surfaces The outer-diameter is D frac14 05 m and theinner diameter is DH frac14 0214 m

Three steps need to be considered in the design of a turbine (i)

speci1047297cationof thedesign variables1047298ow rate Q h available head H hand rotational speed U (ii) de1047297nition of the velocity diagramsupstream and downstream of the blade rows (iii) calculation of the

blade geometry camber and thickness distributions and cascadechord-to-pitch ratio c s One can start byspecifying oneof the threedesign variables (eg H h) and then compute the other two fromdimensionless values of the tip speed velocity kU frac14 U (2 gH h)12 and

the meridional velocity kV m frac14 V m=eth2 gH hTHORN1=2 obtained in the liter-ature from the experience of different manufacturers of hydraulicturbines [56] Theradial distribution of theangular momentumrV qat the inlet section of the rotor can then be calculated from the

known values of H h Q h andU

21 Computation of the meridional velocity 1047297eld

The meridional 1047298ow 1047297eld through the turbine is computed by

the streamline curvature method as described in [7] Two quasi-orthogonal lines were considered at the rotor one at the bladeleading edge and the other at the trailing edge A constant spanwisedistribution of the angular momentum rV q was prescribed at the

leading and trailing edges with the maximum (design prescribed)value at the leading edge and zero at the tailing edge The radialcoordinates of the meridional streamlines and the axial velocitycomponent distributions at the inlet and the outlet sections of the

rotor blades are used as input data for the computation of theblade-to-blade 1047298ow as explained in Section 23

22 Computation of the blade-to-blade velocity 1047297eld

The blade-to-blade 1047298ow is represented by the 1047298ow througha rectilinear cascade of blades computed by a panel method [8] Theairfoil contour is discretized by N linear elements with constant

strength source distribution on each element The circulation ismodeled by a vortex distribution on the airfoil meanline

The complex velocity w frac14 V xiV y induced at z frac14 x thorn iy bya source of strength s at z frac14 x thorn ih is given by [9] w( z ) frac14 s( Z z) and

the complex velocity induced by a cascade of sources of pitch s is

weth z THORN frac14 sp

s coth

peth z zTHORN

s (1)

Nomenclature

A cross sectional area

c blade chord

C L lift coef 1047297cient

C tot total pressure coef 1047297cient

D diameter tip runner diameter

DH hub runner diameter

g acceleration due to gravity

H available head

h linear displacement of the probe measured from thehub wall

ku dimensionless tip speed velocity

kV m dimensionless meridional velocity

N number of elements of discretized airfoil contour

N p number of points at each measurement line of 1047297ve-

hole probe

p static pressure

P tot total pressure

Q 1047298ow rate

r dimensionless radius 2r D

(r q z ) cylindrical coordinate systems circumferential pitch

S 1 blade-to-blade surface

S 2 meridional surface

T torque

t blade thickness

U runner tip velocity V vector of absolute 1047298ow velocity

V absolute 1047298ow velocity

V m meridional 1047298ow velocity component

V r V qV z absolute velocity components along r q z directions

V ref reference velocity 4Q =frac12pethD2 D2H THORN

V dimensionless velocity W vector of relative 1047298ow velocity

W relative 1047298ow velocity

w complex velocity

( x y z ) cartesian coordinate system

Z number of rotor blades

Greek symbol

a absolute 1047298ow angle measured from meridionaldirection arctan (V qV m) also angle of attack

a p relaxation factor for pressure

av relaxation factor for momemtum

b absolute 1047298ow angle between meridional and radialdirections arctan (V r V m)

G vortex circulation intensity also swirl number

h turbine ef 1047297ciency

l stagger anglen kinematic viscosity

P power coef 1047297cient

r density

s intensity of source distribution also swirl number

F dimensionless 1047298ow coef 1047297cientJ dimensionless head

U rotor angular speed

Subscript 12 inlet and outlet respectively

ax axialhyd hydraulic

m meridionalmax maximum

min minimumh relative to design conditions





















-Cp( min)

α

0 1 2 3 4 5-1 2

-1 0

-8

-6

-4

-2

0

2

4

6

Design r

=0518

Design r=0962

r =0 962r =0 518

α

x

c

-1 0 -5 0 5 1000

01

02

03

04

05

06

07

08

09

10 r =0518

r =0962

ab


xc ()

- C p

0 20 40 60 80 100

-10

-05

00

05

10

15

20 r = 0428

r = 0518

r = 0666

r = 0776

r = 0876

r = 0962

r = 1000

xc()

- C p

-1 0 1 2 3 4 5-10

-08

-06

-04

-02

00

02

04

06

08

10

xc ()

- C p

95 96 97 98 99 100-02

-01

00

01

02

03

04

05

06













C L frac14 2s

c

DW qW N

(4)















in Fig 3







Table 1



1070 0428 2230 1327 0143 0103 3454 387

1180 0472 2280 1230 0132 0085 4129 370

1295 0518 2339 1150 0120 0068 4730 296

1485 0594 2451 1051 0102 0052 5408 234

1650 0660 2561 0988 0086 0042 5853 169

1805 0722 2674 0943 0073 0035 6176 1201945 0778 2782 0910 0062 0031 6403 093

2065 0826 2879 0888 0053 0028 6564 070

2190 0876 2984 0867 0044 00255 6704 045

2300 0920 3080 0852 0037 00235 6814 0262405 0962 3173 0840 0030 0022 6906 015

2500 1000 3260 0830 0024 00205 6977 003


z

y


y

x

z

xa c














FLUENT code










V z frac14 Q

A V q frac14

ethrV qTHORNhr






Fig 4 Turbine rotor




5 Results











2cos20

cos20

cosf0i

(6)


i are given by

f0i frac14 f0

1 thorn180

2 20


with f01 frac14 20

and i frac14 1 N p





V ref frac14 4Q

p

D2 D2

H

(8)
































momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c






















-Cp( min)

α

0 1 2 3 4 5-1 2

-1 0

-8

-6

-4

-2

0

2

4

6

Design r

=0518

Design r=0962

r =0 962r =0 518

α

x

c

-1 0 -5 0 5 1000

01

02

03

04

05

06

07

08

09

10 r =0518

r =0962

ab


xc ()

- C p

0 20 40 60 80 100

-10

-05

00

05

10

15

20 r = 0428

r = 0518

r = 0666

r = 0776

r = 0876

r = 0962

r = 1000

xc()

- C p

-1 0 1 2 3 4 5-10

-08

-06

-04

-02

00

02

04

06

08

10

xc ()

- C p

95 96 97 98 99 100-02

-01

00

01

02

03

04

05

06













C L frac14 2s

c

DW qW N

(4)















in Fig 3







Table 1



1070 0428 2230 1327 0143 0103 3454 387

1180 0472 2280 1230 0132 0085 4129 370

1295 0518 2339 1150 0120 0068 4730 296

1485 0594 2451 1051 0102 0052 5408 234

1650 0660 2561 0988 0086 0042 5853 169

1805 0722 2674 0943 0073 0035 6176 1201945 0778 2782 0910 0062 0031 6403 093

2065 0826 2879 0888 0053 0028 6564 070

2190 0876 2984 0867 0044 00255 6704 045

2300 0920 3080 0852 0037 00235 6814 0262405 0962 3173 0840 0030 0022 6906 015

2500 1000 3260 0830 0024 00205 6977 003


z

y


y

x

z

xa c














FLUENT code










V z frac14 Q

A V q frac14

ethrV qTHORNhr






Fig 4 Turbine rotor




5 Results











2cos20

cos20

cosf0i

(6)


i are given by

f0i frac14 f0

1 thorn180

2 20


with f01 frac14 20

and i frac14 1 N p





V ref frac14 4Q

p

D2 D2

H

(8)
































momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c












C L frac14 2s

c

DW qW N

(4)















in Fig 3







Table 1



1070 0428 2230 1327 0143 0103 3454 387

1180 0472 2280 1230 0132 0085 4129 370

1295 0518 2339 1150 0120 0068 4730 296

1485 0594 2451 1051 0102 0052 5408 234

1650 0660 2561 0988 0086 0042 5853 169

1805 0722 2674 0943 0073 0035 6176 1201945 0778 2782 0910 0062 0031 6403 093

2065 0826 2879 0888 0053 0028 6564 070

2190 0876 2984 0867 0044 00255 6704 045

2300 0920 3080 0852 0037 00235 6814 0262405 0962 3173 0840 0030 0022 6906 015

2500 1000 3260 0830 0024 00205 6977 003


z

y


y

x

z

xa c














FLUENT code










V z frac14 Q

A V q frac14

ethrV qTHORNhr






Fig 4 Turbine rotor




5 Results











2cos20

cos20

cosf0i

(6)


i are given by

f0i frac14 f0

1 thorn180

2 20


with f01 frac14 20

and i frac14 1 N p





V ref frac14 4Q

p

D2 D2

H

(8)
































momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c














FLUENT code










V z frac14 Q

A V q frac14

ethrV qTHORNhr






Fig 4 Turbine rotor




5 Results











2cos20

cos20

cosf0i

(6)


i are given by

f0i frac14 f0

1 thorn180

2 20


with f01 frac14 20

and i frac14 1 N p





V ref frac14 4Q

p

D2 D2

H

(8)
































momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c





5 Results











2cos20

cos20

cosf0i

(6)


i are given by

f0i frac14 f0

1 thorn180

2 20


with f01 frac14 20

and i frac14 1 N p





V ref frac14 4Q

p

D2 D2

H

(8)
































momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c












momentum

a

a

a

b

b

b





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c





G frac14

Z A


r hid

Z A




r hyd frac14 1

A

Z A

r d A frac14 2

3

r 3max r 3min

r 2max r 2min

(11)


a frac14 arctan

0BBBBB

Z A

V qdA

Z A

V z dA

1CCCCCA frac14 051

(12)



a b c















R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c












R A







C rel p frac14

p pN12rW 2

N

(13)



6 Conclusions








Acknowledgments


References


















a b c



design of the rotor blades of a mini hydraulic bulb-turbine

Documents