Journal of Energy and Power Engineering 9 (2015) 1037-1046 doi: 10.17265/1934-8975/2015.12.001

Comparison of the Convergent and Divergent Runners

for a Low Head Hydraulic Turbine

Zbigniew Krzemianowski1 and Romuald Puzyrewski2

1. Department of Hydropower, The Szewalski Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Fiszera 14, Gdansk

80-231, Poland

2. Department of Energy and Industrial Apparatus, Mechanical Faculty, Gdansk University of Technology, Narutowicza 11/12,

Gdansk 80-233, Poland

Received: September 29, 2015 / Accepted: October 30, 2015 / Published: December 31, 2015. Abstract: The paper presents a method of the runner blades design for simple case of hydraulic turbine. Two differently shaped channels of meridional cross section were examined. The quantitative evaluation was performed by means of 3D algorithm. It has been found that, divergent runner prevails convergent from dissipation point of view. Although, the influence of draft tube has not been analyzed, the presented method is important for the designers of low head hydraulic turbines which are not the matter of standardization due to variety of environmental conditions.

Key words: Hydraulic machinery design, inverse problem, 2D model.

1. Introduction

The paper is devoted to the design of runner blade

cascade supported by 2D and 3D computation models,

for low head hydraulic turbine. The necessity of

developing the method of design comes from the fact

that, planned low head hydraulic turbines are not the

objects for standardization due to variety of

environment, in which such installations are foreseen.

Also from theoretical point of view, the problem is

interesting because it insists to develop the method of

finding the geometry of designed blades. Here, the

attention is focused on two models. Model 2D is

presented in the version of the so-called inverse

problem in which the boundary conditions do not

contain geometry of blades. The model allows creating

such geometry. If geometry is determined from 2D

model, then 3D computation may be the next step to

check the flow field. The aim of the analysis was

focused on finding the geometry where minimum

Corresponding author: Zbigniew Krzemianowski, PhD, research associate, research fields: hydraulic turbines designing and analyzing, hydraulic turbines measurement.

losses are expected. Besides, the influence of turbine

meridional shape in the domain of runner in two

versions: convergent and divergent was investigated. It

was also interesting to find the answer whether the

function of diffuser behind the runner can be partly

overtaken by contour as it is in divergent runner.

Here, the attention is focused on two differently

shaped runners. Convergent shape of runner leads to an

increase of the load of draft tube behind the runner if

the outlet of draft tube is imposed. One has to be aware

of that, the increase of draft tube load may change the

final conclusions.

The main task of the presented paper is the method

of the runner blade design. For 3D computation,

commercial code ANSYS/Fluent 15 was applied. 2D

model used to calculations was described in Ref. [1].

2. Geometrical Boundary Conditions

The geometry of two contours of the flow field

discussed here is shown in Fig. 1.

In both cases, the family of lines (they can be treated

as a wishful order of stream surfaces flowing by a

D DAVID PUBLISHING

1038

Fig. 1 Two and (b) diverg

meridional p

of guide van

lines were us

describing t

internal bor

coordinate

by x(3).

The outer

radius functi

0.1854

and for dive

Comparison

views of flow gent runner.

plane) shown

nes (DV), gap

sed in non-ort

the outer bor

rder denoted

is denoted b

r line for conv

ion f depende

0.008433A

rging domain

of the Conve

(a)

(b) domain: (a) c

n in Fig. 1, co

p (DG) and run

thogonal syste

rder denoted

d by x(1) =

by x(2) and

verging doma

ent on axial c

ArcTan.

n:

ergent and Di

convergent ru

overs the dom

nner (DR). Th

em of coordin

as x(1) = 0

1. The ang

axial coordi

ain is given as

oordinate x(3)

4.421

ivergent Run

unner

mains

hese

nates

and

gular

inate

s the ):

(1)

T

0.06

x(1)

T

intu

qua

In a

ana

flow

T

blad

T

the

exp

blad

of b

3. B

In

para

com

of th

of

velo

geo

Fig.com

ners for a Lo

0.205 0.0

The inner line

6 m. The fam

in the form:

The choice o

uition and ex

antitatively fa

authors’ opini

alysis of a co

w field of turb

The positions

des were also

The number o

geometrical b

perience, it w

des in the run

blades was in

Boundary C

n front of run

ameters were

mputation of

he axial and t

runner casca

ocity distrib

ometry gives t

(a) . 2 Distrib

mponents at inl

ow Head Hydr

08433ArcTa

e (hub) in bot

mily of lines i

of these fun

xperience. T

actor of a con

ion, this moti

onvergence a

bine stage.

of leading an

o chosen intui

of blades in r

boundary con

was taken in

nner domain.

vestigated by

Conditions

nner cascade

e chosen as

guide vane d

tangential vel

ade are show

bution in th

the mass flow

ution of axiaet to the runne

raulic Turbine

an.

4

th cases is equ

s given by th

nction types

These functio

nvergence and

ivation is suff

and divergen

nd trailing edg

itively.

runner domai

nditions. Acc

to account t

The influenc

y means 3D m

for Flow P

, the distribu

it follows fr

domain. The

locity compon

wn in Fig. 2

he chosen

w rate m = 23

al and tangeer domain.

e

4.421 (2)

ual to: finner =

he function of

(3)

is based on

ons represent

d divergence.

ficient for the

nce upon the

ges of runner

in belongs to

cording to the

three or four

ce of number

model.

Parameters

tions of flow

rom previous

distributions

nents in front

2. The axial

example of

5 kg/s.

(b) ential velocity

)

=

f

)

n

t

.

e

e

r

o

e

r

r

s

w

s

s

t

l

f

y

Model 3D

of momentu

where the va

pressure lev

For 2D in

outer contou

pressure from

was given as

where,

;

trailing poin

of non-dimeIt is wort

drop along

pressure dis

domain is sh

8,000 Pa, po

The influe

runner blade

2D and 3D m

For 3D m

formulated

model chose

commercial

will not be d

Fig. 3 Examof the runner

Comparison

D gives the pr

m equation in

alue of pressu

el was set at t

nverse model,

ur line of ru

m inlet to ou

s the function

,

is non-dim

; pinlet is inlet

nt at outer bor

ensional coordth noting tha

the runner

tribution alon

hown in Fig.

utlet = -5,000 P

ence of the p

es was thorou

models.

model, the res

according to

en for the com

code (ANSY

discussed here

mples of pressur domain for 2D

of the Conve

ressure field f

n respect to th

ure is given. T

the outlet to b

it was neces

unner domain

utlet. The pres

n of three para

,

mensional co

t pressure; po

rder of runner

dinate . at, poutlet cont

r cascade. T

ng the outer

3. The param

Pa and -20,00

pressure drop

ughly investig

st of boundar

o the versio

mputation. Th

YS/Fluent 15)

e.

ure distributioD model.

ergent and Di

from the solu

he reference p

The reference

be equal zero

sary to assum

n distribution

ssure distribu

ameters:

oordinate:

utlet is pressur

r and exponen

trols the pres

The example

border of run

meters are: pin

00 Pa, nb = 2.

p on the desig

gated by mean

ry conditions

on of turbule

his is the matte

) and therefo

on at outer bo

ivergent Run

ution

point,

e for

o.

me at

n of

ution

(4)

re in

nt nb

ssure

e of

nner

nlet =

.0.

gned

ns of

s are

ence

er of

ore it

order

4. S

T

mod

deta

solu

the

H

i.e.,

resp

com

and

T

flow

acc

p—

velo

Π—

fun

A

and

are

B

In

t1 =

It

it is

T

The

fun

T

is

traj

dom

T

ners for a Lo

Short Desc

The main po

del are desc

ails of such

ution of mom

determinatio

Having pressu

, meridional a

pectively den

mputed from m

1 ,

d energy cons

The following

w rate,

ount the spac

—pressure,

ocity of runne

—potential e

ction depend

At runner inle

d ec are given

dependent on

Blockage func

,

ntuitively cho

= 0.2, t2 = 0.5,

t generates th

s shown in Fig

The argument

e shape of

ction is show

The line cross

the skeleton

ectory of flu

main.

The suction si

ow Head Hydr

ription of 2

oints of algo

cribed in Re

approach wil

mentum conse

n of pressure

ure determin

and tangentia

noted as

mass conserv

ervation equa

g notations a

, —blo

ce occupied b

ρ—density,

er, e—interna

energy, f—ra

ent on a

et, along the l

n from the bo

n an x(1) coord

ction was cho

1

osen paramet

, t3 = 1.0, t4 =

he distributio

g. 4.

ts x(1) and

profile gene

wn in Fig. 5.

sing the profi

n line which

uid element a

ide of profile

raulic Turbine

2D Inverse

rithm of the

efs. [1-8]. T

ll be omitted

ervation equa

field in doma

ed the rest o

al component

and

vation equatio

ation:

are used:

ckage functio

by the materi

Urot—cir

al energy, ec—

adius which

and .

leading edge,

oundary cond

dinate.

osen as below

1

ers for blocka

= -0.5.

n of blockag

are within

erated by th

file in middle

h is determ

at outer conto

is given by c

e 1039

Model

e inverse 2D

Therefore the

d. Method of

ation leads to

ain of runner.

of parameters

s of velocity,

, can be

on:

(5)

(6)—mass

on takes into

ial of blades,

rcumferential

—total energy,

h is stream

the values m

ditions. They

w:

(7

age factor are

e function as

range <0-1>.

his blockage

e of thickness

mined as the

our of runner

coordinates:

9

D

e

f

o

.

s

,

e

)

) s

o

,

l

,

m

m

y

7)

e:

s

.

e

s

e

r

1040

Fig. 4 Block

Fig. 5 The pmeans of a 2D

and for the p

where, z—th

thickness of

In the en

,

energy. Fro

function oug

of a loss c

prescribe th

paper, autho

coefficient d

Comparison

kage function in

profile of runnD inverse mode

sin

cos

pressure side:

sin

co

he number o

f profile at bot

nergy conserv

appear

m inlet to o

ght to be dete

coefficient. It

he level of lo

ors propose

distribution in

of the Conve

n the runner d

er at outer conel.

2π 1

2π 1

:

n 2π

os 2π

of blades, tp—

th sides of sk

vation equati

rs which repre

outlet of runn

ermined in ad

t is a matter

oss coefficien

the followin

n runner dom

ergent and Di

domain.

ntour generate

/

/

/

/

—the division

keleton line.

ion, the quan

esents the inte

ner domain,

dvance by me

r of intuition

nt. In the pre

ng form of

main for two c

ivergent Run

ed by

(8)

(9)

n of

ntity

ernal

this

eans

n to

esent

loss

cases

diff

In

and

T

for

T

in th

dom

T

it i

incr

5. S

T

was

Aft

by m

In

com

pres

Fig.

anal

ners for a Lo

ferent: (a) and

,

n case (a), pa

w0 = 0.0

w3 = -3.369

d in case (b), p

w0 = 0.00

w3 = -3.3699

They lead to t

both cases in

Then the incr

he reference

main:

Δ

The level of in

s possible to

rease only by

Some Resu

The main goa

s to generate

er that, the ge

means of 3D

n order to d

mputations, it

sented in Tab

. 6 Loss coe

lysed cases.

ow Head Hydr

d (b):

arameters wer

17, w1 = 0.02

992, w4 = 1.98

parameters w

61, w1 = 0.00

92, w4 = 1.98

the loss coeff

n Fig. 6.

rease of intern

of kinetic en

,

nlet kinetic en

o adjust the

y loss coeffici

lts of 2D C

al of 2D inve

e the geomet

enerated blad

computation

diminish the

was assumed

ble 1 below.

efficient distrib

raulic Turbine

re chosen as f

205, w2 = 2.42

83, nl = 3.0, n

were chosen as

0732, w2 = 2.4

83, nl = 3.0, n

fficient distrib

nal energy w

nergy at the in

nergy was kep

value of int

ient distributi

Computatio

erse problem

try of the ru

de geometry w

n model.

number of

d the scheme

bution ,

e

(10)follows:

2842,

ns = 2.0 (11a)

s follows:

42842,

s = 2.0 (11b)

bution shown

was computed

nlet of runner

(12)

pt constant so

ternal energy

ion.

ns

computation

unner blades.

was examined

cases in 2D

computation

for two

)

)

)

n

d

r

)

o

y

n

.

d

D

n

o

Table 1 Var

Pressure poutle

of the runner bparameter)

The refere

abscissa in t

For conve

are shown in

geometries a

To compa

parameters a

were used.

The mean

versus poutle

pressure dro

Fig. 7 Two ((a): poutlet = -5

Comparison

riants of 2D mo

t at upper corneblades (referenc

ence paramet

the most figur

ergent contou

n Fig. 7. For

are shown in

are the compu

averaged ove

n pressure d

et is shown

op is visible fo

examples of g5,000 Pa, (b): po

of the Conve

odel computat

er ce

Convergepoutlet = (-(-5,000, --15,000, -

ter poutlet will

res presented

ur, two examp

divergent con

Fig. 8.

uted cases, th

er the inlet an

drop along th

in Fig. 9. N

or both conto

(a)

(b) geometry for c

outlet = -20,000 P

ergent and Di

ions.

ent contour -20,000)-5,000-6,000, -7,000, -20,000, -23,00

l be plotted a

d below.

mples of geom

ntour, the sim

he mean value

nd outlet surf

he runner bl

No differenc

ours.

convergent conPa).

ivergent Run

Pa -8,000, -11,000

00, -28,000, -30

as an

metry

milar

es of

faces

ades

e in

ntour

Fig.poutle

Fig.cont

ners for a Lo

D

0, -13,000, 0,000)

po

(--2

. 8 Two examet = -5,000 Pa, (

. 9 Pressuretours (converg

ow Head Hydr

ivergent contou

outlet = (-26,000)5,000, -6,000, -

23,426,-28,000,

(a)

(b)mples of geomet

(b): poutlet = -20,

pout

drop along gent and diverg

raulic Turbine

ur )-5,000 Pa -7,111, -12,000-26,000)

)

) try for diverge,000 Pa).

tlet (Pa) the runner bgent).

e 1041

, -18,000,

nt contour ((a)

lades for two

):

o

1042

Fig. 10 Thcontours (con

The outlet

The most

outlet angle

outflow. Th

from a discu

energy at the

all paramete

conversion

domain:

Thus, ene

domain can

Energy co

follows:

where:

is the amoun

is the increa

now can be p

Comparison

he outlet anglnvergent and d

angle of abso

t interesting

value 2 is c

he argument

ussion of ener

e inlet and ou

ers are averag

into mechan

2

ergy conserva

be written as

onservation e

nt of energy c

Δ

se of internal

presented in t

of the Conve

poutlet (Pa) le of absolutedivergent).

lute velocity is

cases are tho

close to 90° w

for the axia

rgy equation.

utlet to runne

ged), which ar

nical energy

2

ation equation

s:

equation can

Δ

converted by

0

l energy due t

the form:

ergent and Di

e velocity for

s shown in Fig

ose in which

what means a

l outflow co

Let us define

er (assuming

re available to

y in the run

(

(

n along the run

n be rewritten

Δ

the runner an

0

to dissipation

ivergent Run

two

g. 10.

h the

axial

omes

e the

that,

o the

nner

13a)

13b)

nner

(14)

n as

(15)

(16)

nd

(17)

n and

T

com

con

give

velo

part

this

inev

sum

as l

why

The

amo

the

O

that

pout

can

such

Fig.two

ners for a Lo

The kinetic en

mponent

nservation equ

en mass flow

ocity compon

t of lost energ

s amount o

vitably will b

m:

osses, which

y behind the r

e existence o

ount of work

sum Le as a

On the basis o

t, the minimu

tlet = -5,000 P

n be conside

h conclusion

. 11 Lost enecontours (con

ow Head Hydr

2

Δ

L

nergy related

/2 is

uation and ca

w rate. The ki

nent

gy (Le). If the

of energy in

be dissipated

has to be min

runner, the ax

of tangential

converted by

function of p

of Fig. 11, the

um Le place

Pa. This point

ered as an a

n should be c

poutlet (ergy Le as a funvergent and di

raulic Turbine

2

to the meridi

s tied up

annot be minim

netic energy

/2 can be t

ere is no mea

nto pressure

d. Then, we

Δ

nimized. This

xial velocity

velocity dim

y the runner. F

poutlet.

e conclusion c

s in the neig

ts out the geo

advisable. N

checked by m

(Pa) unction of presivergent).

e

(18)

ional velocity

with mass

mized for the

of tangential

treated as the

ans to convert

e energy, it

can treat the

(19)

s is the reason

is preferable.

minishes the

Fig. 11 shows

can be drawn

ghborhood of

ometry which

Nevertheless,

means of 3D

ssure poutlet for

)

y

s

e

l

e

t

t

e

)

n

.

e

s

n

f

h

,

D

r

computation

differences b

(1) The f

opposition to

(2) 3D

estimation

assumed in 2

Fig. 12

introduced i

coefficient d

One can n

for two co

coefficient

runner for b

dissipation.

The energ

point of run

into the follo

2

Left hand

values at th

treated as th

of paramete

domain can

described in

has the form

where, Δ is t

It is possi

real solution

This may h

increase of

high. Then w

points in run

points where

Fig. 13 for p

Comparison

n. Two reas

between 3D a

finite numbe

o infinite num

computation

of dissipatio

2D computati

shows the

in 2D compu

distribution (a

notice from F

ntours is of

identically d

both contour

gy equation

ner domain.

owing form:

2 2

+

d side of ab

he runner in

he quadratic

ers are give

n be comput

n Ref. [1]. Th

m:

the discrimin

ible to have t

n for d

happen when

internal ener

we may have

nner domain n

e the solution

poutlet = -5,000

of the Conve

sons will in

and 2D mode

er of blades

mber of blade

n provides

on effect tha

ion.

level of dis

utation for tw

a) and (b).

Fig. 12 that, d

f the same

distributed in

rs leads to th

is valid also

One can rew

Δ

bove equation

nlet. Right ha

equation for

en. Pressure

ed according

he solution o

√nant of quadra

the situation

due to negati

the blockag

rgy due to di

e the situatio

no solution ex

n was sought

0 Pa.

ergent and Di

nfluence on

els:

which stand

es in 2D mode

more real

an it is usu

ssipation en

wo cases of

dissipated en

value. The

n the domain

he same leve

o for an arbit

write this equa

Π

n contains g

and side can

if the

field in run

g to the met

f above equa

√Δ

atic equation.

when there i

ive discrimin

ge factor and

issipation are

on where at s

xists. The grid

for are show

ivergent Run

the

ds in

el;

listic

ually

ergy

loss

ergy

loss

n of

el of

trary

ation

(20)

given

n be

rest

nner

thod

ation

(21)

is no

nant.

d the

e too

ome

ds of

wn in

Fig.com

Fig.by m

T

pres

the

lead

ners for a Lo

. 12 The lemputation by m

. 13 (a) Convmeans of 2D m

The Fig. 13a p

sents diverge

empty regio

ding edge ma

ow Head Hydr

poutlet

evel of dissipmeans of 2D mo

(a)

(b)vergent and (b

model for poutlet

presents conv

ent case. In ca

n of blade (“

ay be observe

raulic Turbine

t (Pa) pation energyodel.

)

) b) divergent bl= -5,000 Pa.

vergent case

ase of converg

“white spot”)

d in which no

e 1043

y assumed in

lades obtained

and Fig. 13b

gent contour,

) close to the

o solution for

3

n

d

b

,

e

r

1044

was o

(Fig. 13b),

runner doma

In the co

section due

velocity

the increase

spot” area,

introduced.

obtained,

neighboring

approximati

the geometr

computation

6. The Domain—Increase Energy Va

The 3D

model k-ε

dissipation

estimate the

turbulence le

the relation

dissipation ε

where, T—a

The seco

equation wri

One can

incompressib

conduction.

written:

and integrat

Comparison

obtained. In t

the solution

ain for compa

onvergent co

to converge

. The decre

of blockage

an additio

To create s

geometrical

area may b

on

ry based on

n may be obta

3D Co—Estimation

by Means alue

computation

provides w

energy ε in

e increase of

et us examine

between entr

ε (m2/s3):

absolute temp

ond relation

itten for inter

n simplify

ble fluid

Combining

ting the incr

of the Conve

the case of d

n was obtaine

arable set of p

ntour, the d

nce increase

ease of cross

e factor. To f

onal assump

surface where

extrapolati

be applied or

is justifi

n 2D inverse

ained.

omputationn of Inte

of Kineti

with versio

with distribu

runner dom

f the internal

e two relation

ropy producti

perature, ρ—d

is the ener

rnal energy:

Δ

the above

and neglec

these equa

rease of inte

ergent and Di

divergent con

ed in the w

parameters.

decrease of c

d the meridi

section manif

fill up the “w

ption should

e no solutio

ion from

r if | |

1,

fied. In this w

e model for

n of Fernal Eneic Dissipat

on of turbule

ution of kin

main. In orde

l energy, du

ns. The first on

ion sm and rat

density.

rgy conserva

Δ

e equation

cting the

ations it can

ernal energy

ivergent Run

ntour

whole

cross

onal

fests

white

d be

on is

the

, the

way,

3D

low ergy tion

ence

netic

er to

ue to

ne is

te of

(22)

ation

(23)

for

heat

n be

(24)

y Δe

(m2

T

T

repr

dom

cho

and

then

T

esti

turb

diss

7. S

F

due

com

T

com

ove

Fig

with

Fig.diss(thr

ners for a Lo

2/s2) is estima

Δ

The mean valu

The interval o

resentative fl

main. As a r

oose the axial

d mean value

n one can esti

The foregoi

imation of th

bulent dissipa

sipation.

Some Resu

Fig. 14 shows

e to dissipat

mputation by

The level of

mputation, a

erestimated fo

. 14. The cas

h the 3D resu

. 14 The leipation compu

ree and four bl

ow Head Hydr

ated as follow

d

ue of let us

of time Δt is r

luid element

representativ

length of the

of axial veloc

imate:

Δ

ing conside

he increase o

ation, which

lts of 3D C

s the level of

ion accordin

means of turb

f energy diss

as it is sh

or the case (a)

se (b) shows

ults.

poutlet

evel of internuted by meanlades runner fo

raulic Turbine

ws:

d

Δ

s estimate as:

related to the

which passe

ve length Δla

e runner at m

city:

Δ

erations for

of internal en

is the prevail

Computatio

f internal ene

ng to the re

bulence mode

sipation assu

hown in Fig

) compared to

a good gues

t (Pa) al energy inc

ns of turbulenor divergent co

e

d

(25)

:

(26)

trajectory of

es the runner

axial, we can

mean diameter

(27)

(28)

rmulate the

nergy due to

ling factor of

ns

ergy increase

esults of 3D

el k-ε.

umed for 2D

g. 12, was

o the values in

ss comparing

crease due tonce model k-εontour).

)

)

f

r

n

r

)

)

e

o

f

e

D

D

s

n

g

o ε

The differ

in runner i

numerical r

for lower

increasing o

Figs. 7 and 8

The sum

shown in F

can be con

the left, g

poutlet = -23,6

for four blad

Fig. 16 s

3-blade runn

pressure at t

outlet pressu

Fig. 15 Sumand four bladcontour.

Comparison

rence of Δe b

is negligible,

esults. The s

values of p

of the profile

8.

of lost energ

ig. 15. Com

ncluded that,

giving the

600 Pa for thre

des.

shows the av

ner domain.

the inlet of th

ure is closer to

p

p

m of Le energy des for (a) diver

of the Conve

between three

, covered by

slight increas

poutlet can b

s length, as i

gy Le from 3D

mparing to da

the minimu

recommend

ee blades and p

veraged pres

Comparing

he runner, one

o zero. This is

poutlet (Pa) (a)

poutlet (Pa) (b)

showing the mrgent contour

ergent and Di

e and four bl

y the scatter

se of dissipa

be explained

it can be see

D computatio

ata in Fig. 1

um is shifted

ded solution

poutlet = -16,15

ssure drop al

to the value

e can notice

s in accordanc

minimum for tand (b) conver

ivergent Run

ades

r of

ation

d by

en in

on is

2, it

d to

n at

50 Pa

long

es of

that,

ce to

three rgent

Fig.dom(3-b

Fig.dom(4-b

the

3D

C

bec

pres

blad

“de

be s

num

mus

inle

sam

In F

3 an

loss

for

ners for a Lo

. 16 Distribumain and presblade runner w

. 17 Distribumain and presblade runner w

boundary co

model.

Comparing th

ame evident

ssure drop d

des. For give

nse”, the hig

seen compari

mbers of runn

st be empha

et to the runn

me was also th

Fig. 18, the p

nd 4-blade ru

s energy, pre

3 and 4-blade

ow Head Hydr

pout

ution of pressussure drop Δp

with divergent c

pout

ution of pressussure drop Δp

with divergent c

ondition for p

he values in F

that, in 2D

due to the fa

en mass flow

gher pressure

ing Fig. 16 to

ner blades (3

asized that, v

ner was kept

he reference p

ressure drop

unner is con

ssure at outle

e runners.

raulic Turbine

tlet (Pa) ure pinlet at inp along the rcontour).

tlet (Pa) ure pinlet at inp along the rcontour).

ressure in co

ig. 16 to data

model, we h

act of infinite

rate, the cas

drop is obtai

o Fig. 17 for

3 and 4-blad

velocity cond

at the same l

pressure behin

for divergent

fronted. Tabl

et and pressu

e 1045

nlet to runnerrunner blades

nlet to runnerrunner blades

omputation of

a in Fig. 10, it

have a higher

e number of

cade is more

ined as it can

two different

de runner). It

ditions at the

level and the

nd the runner.

t contour and

le 2 presents

ure difference

5

r s

r s

f

t

r

f

e

n

t

t

e

e

.

d

s

e

1046

Fig. 18 Comblade number(b) convergen

Table 2 Comand four blad

Parameters

Lemin (m2/s2)

poutlet min (Pa)

Δpstat (Pa)

8. Conclus

(1) 2D in

geometry of

(2) 3D c

runners sho

divergent ru

(3) The in

runner may

Comparison

mparison of prs of the runnent contour.

mparison of 3Ddes runners.

Divergent co

Three blades

Fb

0.3267 0

-23,637 -10,345 ≈ 10,000

1≈

sions

nverse metho

f blades for th

computation

ws that, from

unner is more

ncrease of the

change the

of the Conve

poutlet (Pa) (a)

poutlet (Pa) (b)

pressure dropser with (a) dive

D computation

ontour Conv

Four blades

Threblad

0.3121 0.45

-16,157 -28,210,155 ≈ 10,000

16,5≈ 17

od is effectiv

he runner;

for two di

m dissipation

effective that

draft tube loa

final decision

ergent and Di

s for two diffeergent contour

ns results for t

vergent contour

ee des

Four blades

514 0.5116

235 -21,888547 7,000

17,045≈ 17,00

ve to obtain

fferently sha

n point of v

t convergent

ad for conver

n concerning

ivergent Run

erent r and

three

r

8

00

n the

aped

view,

one;

rgent

g the

cha

pow

prob

Ac

T

Scie

for

Gda

Re

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

ners for a Lo

annel shaping.

wer output (n

blem exceeds

knowledgm

The work wa

ence Centre

The Szewalsk

ansk, Polish A

ferences

Puzyrewski, R

Turbomachine

Mechanical E

Technology P

Puzyrewski, R

of Guide V

Analytical an

Journal of Me

195-202.

Puzyrewski,

Concepts of G

Hydraulic Tur

and Automatio

Krzemianows

Computations

Designed by

Hydraulic Tur

530 (1): 1-8. d

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Puzyrewski, R

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Puzyrewski, R

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anowski, Z. 20

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anowski, Z. 20

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m

s

l

0

n

f

n

f

l

:

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