solution of viscous flow in radial flow pumps

8/13/2019 Solution of Viscous Flow in Radial Flow Pumps

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Transfer inovci 6/2003 2003

207

Kalmr, L.Associate Professor, Ph.D.

Department of Heat and Fluid Engineering,University of Miskolc, Hungary

1. Summary

The aim of this paper is to present anapproximate numerical method to determine themain characteristics of the flow in the bladed spaceof the impeller of a radial-flow pump. The effect ofthe blades of the impeller is represented hydro-dynamically by a field of constrain forces based onthe solution of the inverse problem of the hydro-dynamical cascade theory. In the determination ofthe constrain force field it is supposed that thefrictionless and incompressible fluid flow iscompletely attached to the blade surfaces. The

constrain force field can be calculated by thechange of the moment of momentum in the absolutenon-viscous fluid flow which depends on the stateof the pump. Knowing the constrain force field, theequations of the continuity, motion and energy ofthe viscous relative flow can be solved on the meanstream surface (F) of the impeller and thedistributions of the average relative velocity,pressure and energy loss are determined. Bycalculating the energy loss belonging to differentvolume rates an approximate real head-dischargecharacteristic of the impeller also can be computed[3].

2. Introduction

For obtaining the solution of the flow in thebladed space of the pump impellers it is necessaryto develop an approximate physical model whichfits this task well and satisfies the main conditionsof the physical process taking place in the impeller.The real fluid flows are always frictional, so theeffects of the fluid friction and the turbulence haveto be taken into account in the calculation method.At the same time it is a very important view-pointto find a relatively simple calculation method forthe solution of the task which is directly attainable

for the engineering routine.

The approximate numerical procedure presented inthis paper is applicable for analysing the turbulentflow in the bladed space of the pump impellers. The

effects of the blades, fluid friction and turbulence ofthe flow are taken into consideration separately.The blade effect is represented by a field ofconstrain forces which are perpendicular to thestream surfaces of the relative velocity fieldcongruent to the blade surface of the impeller.

The first step of the calculationis to determinethe change of the moment of momentum of theabsolute non-viscous fluid flow needed to calculatethe components of the constrain force field. Thesecond main step of the calculation is to solve thesystem of the ordinary differential equations basedon the equations of the continuity, motion and

energy of the viscous relative flow on the meanstream surface (F) of the impeller. By knowing thesolutions of the governing equations thedistributions of the relative velocity, pressure andenergy loss can be calculated. The narrowing of theblade channel area due to the blade thickness istaken into consideration by a factor expressed by

the effect of the peripheral thickness of the blades.

3. The Field of Constrain Forces in the Bladed

Space of the Impeller

The basic equations of the calculation methodare formulated in cylindrical co-ordinate system

rotated together with the blades, where the co-ordinates in cyclical order are zr ,, . The Fig.1.

shows the meridional cross-section of the impeller,the shape of the blades viewed from the direction ofthe rotational axis of the pump, the directions of theco-ordinates and the vectors of the absolute velocitycr

, the relative velocity wr

, the peripheral velocityr

u

and the specific constrain force fr

at any arbitrary

point of the blade surface. In the middle of Fig.1.the velocity triangles can be seen at inlet-, outlet-and any arbitrary sections of the impeller.

APPROXIMATE SOLUTION OF VISCOUS FLOW IN RADIAL-FLOW PUMPS

Fig.1


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The blade angle which can be measured between

the tangents of the co-ordinate line =r constantand blade surface at the same point is also shown inFig.1. The blade angle uniquely determines the

normal unit vector nr

of the blade surface.The tangent unit vector of the blade surface can bewritten as: eet r

rrr

cossin = .

Knowing the components of vector tr

, the normal

unit vector nr

of the blade surface can be calculatedas follows:

rz eeten rr

rrr

cossin +== . (1)

Let us summarize the significant assumptions indetermining the field of specific constrain forces f

r

[3]:

The specific constrain force fr

- similarly to

every mechanical constrain force - is a friction-proof effect, in this way the specific constrain

force f

r

is parallel to the normal unit vector n

r

ofthe blade surface : 0

r

r

r

= nf . (2)

Performing the vector multiplication in Equ. (2)yields :

rr nfnf = . (3)

The frictional force is parallel to the wettedwalls of the impeller near to it, so the specificconstrain force and the friction force areperpendicular to each other:

( ) 0= nf rr

, (4)

where is the stress tensor.

Carrying out the multiplications in Equ.(4), we

get the following expression between thecomponents of the specific constrain force,normal unit vector of the blade surface and thestress tensor:

0=+ rrrr nfnf . (5)

Since it is supposed that in the determination ofthe constrain force the fluid is non-viscous, so thespecific constrain force and the relative velocityvectors are perpendicular to each other:

0= wf r

r

. (6)

The mean surface (F) of the meridional channelis a stream surface of the relative flow

consequently the relative velocity component ofzw is equal to zero. At the same time, along this

stream surface (F), the stress vector zr

(it is the

third column of the stress tensor ) and the

relative velocity vector are parallel to each other.So we can write: 0

r

rr

= z

w . (7)

Executing the vector multiplication in Equ.(7), weget:

ww rzrz = . (8)

By using equations (1) and (3) we come to theexpression between the components of the specificconstrain force as follows:

ctgfn

nff rr == . (9)

It is easy to realize that the component rf of the

constrain force can be determined by thegeometrical data of the blades ( is the blade

angle) from Equ.(9) if the component f of the

constrain force is known. The component f of theconstrain force is uniquely determined by thechange of the moment of momentum in the absolutenon-viscous fluid flow.

Because the component zf of the constrain force is

equal to zero, thus by using Equ.(6), we can write :0=+ wfwf rr . (10)

In view of equations (3) and (5) and by using the

fact that the stress tensor is a symmetrical tensor

it is plausible that the component r of the stress

tensor is vanished.

Let us sum up the results obtained up to now. It iseasy to see that along the co-ordinate surfacesdetermined by z = constant the unknown

components rw and w of the relative velocity

and the unknown components rz and z of the

stress tensor depend on the co-ordinates rand z.

The unknown components rf and f of the

constrain force depend on the co-ordinate r.The relative velocity vector and the magnitude ofthis vector on the main surface (F) can be expressedby the components of the relative velocity as

follows :

eweww

rr

rrr

+= ; 22www r+= . (11)

The stress vector and the magnitude of this vectoron the main surface (F) are calculated similarly:

ee zrrzzrrr

+= ; 22zrz += , (12)

where is the resultant shear stress.The equation of the motion relating to the absoluteinviscid fluid flow is as follows [3]:

+=

22

22 uwpfwcrot

r

rr .

Multiplying this equation by the co-ordinate unit

vector er

, we get the relationship for the

calculation of the component f of the constrain

force:

( ) ( )

cr

rr

wcrotwf r

zr ==

r . (13)

The product of cr in Equ.(13) means the change

of the moment of momentum in absolutefrictionless fluid flow. We should remark that the

peripheral component f of the constrain force is

uniquely determined by the meridional velocity rw


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and the change of the moment of momentum cr

in the absolute frictionless fluid flow.By assuming that the number of the blades areinfinite, the energy transfer between the fluid andthe blades is taking place continuously in theannulus domain determined by the interval of

KB rrr . Thus the line integral of the absolutevelocity for a closed curve in this domain, namelythe circulation

r belonging to the closed curve ( rL )

at any arbitrary radius r can be calculated as

follows (see Fig.2.):

( )

( )BBL

r crcrsdc

r

== 2rr . (14)

Fig.2.

By using Equ.(14) it is possible to write down theexpression to calculate the circulation difference

belonging to the radius difference r by thesubtraction of the line integrals for the closedcurves of ( L r r+ ) and ( L r). Neglecting the terms

which are small in second order yields:

( ) ( )

( ) crsdcsdcrrr LL

== +

2rrrr .

The difference of the circulation can beexpressed by the specific circulation as follows:

r= .

Comparing the last two equations, the change of the

moment of momentum cr with respect to r is

expressed by the following equation:

( ) rcr =

2

. (15)

By using the expressions developed above the valueof the circulation

r at any arbitrary radius r in

the interval ofKB rrr is calculated:

( ) ==r

r

BBr

B

drcrcr 2. (16)

Of course, the integral of the specific circulation

with respect to r between the inlet- and outletcross-section of the impeller provides the totalcirculation

BK of the impeller. Considering

equations (13), (14), (15) and (16), we arrive at the

relationship for the peripheral componentf of the

constrain force :

2r

wf r= . (17)

The meridional velocity component rw and the

specific circulation depend on the state of theoperation and so consequently the constrain forcealso depends on the operating conditions of thepump.

Fig.3.

Since in real cases the number of the blades N of apump is finite, the energy transfer between the fluidand the blades is carried out only along the blades,but it isnt taking place continuously in the annulusdomain determined by the interval of

KB rrr . It

is easy to see that blade circulation comes fromonly the specific circulations placed on the bladesand the absolute flow is vortex-free in the domainbetween any two neighbouring blades. Similarly toEqu.(14) the line integral of the absolute velocityfor the closed curve (

rG ) belonging to any arbitrary

radius r(see Fig.3.) can be written down..By using Stokes theory yields:

( ) ( ) ( ) ( )( )

++==

r LLrG sucpreBr

sdcsdcNsdcsdcsdc rrrrrrrrrr

0

.Comparing this equation with Equ.(14), it is evidentthat the circulation

r can be determined by the

line integrals of the absolute velocity along thecurve sections (

sucL ) and ( preL ) placed on the

suction and pressure sides of the blades:

( ) ( )

[ ]( )

( )

sdccNsdcsdcN

rs

rs

presuc

LL

r

Bpresuc

rrrrrrr

=

=

=0

.

The circulationr can also be expressed by a

specific vortex distributionblade placed on the

blade sections as it is well known in the field of thehydro-dynamical cascade theory:


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( )

( )

( )

( ) ( )

=====

rs

blade

rs

rs

t

rs

rs

r dsNdscNsdcN

BB 000

rr . (18)

By using equations (15), (16) and (18) it is possibleto develop the connection between the specificcirculation belonging to infinite number of the

blades and the vortex distribution blade attached tothe finite number N of the blades as follows:

dr

dsN blade= . (19)

The specific vortex distributionblade is obtained

from the solution of the second task of the hydro-dynamical cascade theory relating to the absolutenon-viscous fluid flow [1].For the 2-D flow on the mean surface (F) of theradial-flow impeller (see Fig.1.) the equation ofcontinuity is

( ) 0=+

ccr

r r

. (20)

The equation expressing the condition that theabsolute flow is vortex-free can be written asfollows:

( ) 0=

rccrr

. (21)

To find the unknown distributions of the velocitycomponents ( ),rcr and ( ) ,rc it is necessary to

write down the boundary conditions belonging tothe system of partial differential equations (20) and(21). These boundary conditions can be written at

Brr= (at the inlet section of the impeller) :

Brr cc = ; Bcc = ,and at Krr= (at the outlet sectionof the impeller) :

Krr cc = ; Kcc = .

As it is well known, the specific vortex distribution

blade , which is hydro-dynamically represented the

blade effect, can be determined by applying thefollowing numerical procedure.The 2-D flow on the mean surface (F) of theimpeller can be mapped onto the one on a complexplane. Since the absolute flow remains vortex-freeon this plane, the expressions for the determinationof the velocity distribution can be obtained by thesolution of a Poisson-like partial differentialequation by using the theory of potentials. Thekinematical conditions of the relative flow can alsobe written down, which expresses that the bladesection is a streamline of the relative flow. Thespecific vortex distribution

blade can be looked for

as a form of a trigonometric series. By substitutingthis series into the governing equations andsatisfying the kinematical conditions at discretepoints along the blade sections the unknowncoefficients of the series can be calculated bysolving a system of linear equations. Knowing thesecoefficients, the current vortex distribution, the

components of the constrain force, the velocity andpressure distribution along the blade sections andthe bladed space of the impeller, the characteristics

of optimal state of the impeller and the theoreticalcharacteristics of the impeller can be calculated,belonging to different operating states of theimpeller [3].

4. The Viscous Fluid Flow in the Bladed Space of

the ImpellerFor the relative viscous fluid flow in the bladedspace of the impeller the equation of continuity is

( ) 0=wdiv r

. (22)

The equation of motion in relative system can bewritten as follows :

( )

Divup

fwww1

22

2

+

=+

r

rrrr . (23)

The form of the equation of energy for the relativeflow is

Divwuwp

w =

+

r

r

22

22, (24)

where is factor expressed by the effect of the

peripheral thickness of the blades.By using Equ.(8) we can employ the same notationfor the slope of the relative velocity and the ratiobetween the components of the shear stress asfollows:

rz

z

rw

wt

== . (25)

Let us introduce the following notations for therelative pressure potential

2

2upP = (26)and for the specific relative energy

22

22 uwpE +=

. (27)

By using the notations mentioned above aftercompleting some algebraic computations theequation of energy will have the form:

zt

zr

E zrz

+= . (28)

Considering the equations (12) and (25) we get thefollowing expression:

2222 11 tt

t zrzzrz +=+=+= . (29)

By applying Equ.(29) the previous form of theenergy equation (28) can be rearranged again :

zr

E

t

=

+ 21. (30)

Equ.(30) is directly available to determine thedistribution of the shear stress in the relative flow.Analogously to the flow in a pipe, we introduce adimensionless friction coefficient for thecalculation of the frictional energy loss bysatisfying the following conditions: The frictional elementary energy loss '

Sedin the

direction of the relative streamlines in the bladedspace of the impeller on the elementary length dl


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can be expressed by the dimensionless frictioncoefficient , the average relative velocity w~ referring to the cross-section of the blade channeland the hydraulic diameter

HD of the cross-

section of the blade channel:

2

~2' w

D

dl

edH

S =, (31)

where the hydraulic diameterHD is expressed in

the following form:

r

Nbb

d

N

rb

d

N

rb

DH

2

2

sin

22

sin

2

4+

=

+

=.

(32) The shear velocity

*w can also be interpreted [2]

for the relative viscous fluid flow and itsconnection with the average relative velocity w~

and the dimensionless friction coefficient issimilar to that of the pipe:

2

*~8

=w

w . (33)

The value of the friction coefficient variesalong the blade channel.

Equ.(31) can be rearranged by using Equ.(33), andthe frictional elementary energy loss '

Sed can be

also expressed by the elementary changing of thespecific relative energy E:

dEdrtDww

Ddrdr

wwed

HH

S =+=+=2

2

*

22222

*' 142

~

~8

. (34)Making a comparison between Equ.(30) andEqu.(34) we get the ordinary differential equationfor the determination of the shear stress :

HD

w

z

2*4

= . (35)

In equations (30) and (35) the co-ordinate zis anargument and the other co-ordinate r is a

parameter. By integrating Equ.(35) with respect toz , the distribution of the shear stress can be

calculated along the co-ordinate surfacesdetermined by =r constant. Applying the boundarycondition of ( ) 00 = for the shear stress , thefollowing expression is derived for the distributionof [2]:

zD

w

H

2*4 = . (36)

Knowing the distribution of the shear stress , it ispossible to determine the relative turbulent velocityfield in the meridional channel of the impeller. Theshear stress can be expressed by the Krmn sand Prandtls formula [2]:

z

w

z

wL

2= , (37)

where L is the mixing length. Introducing the

===2

:)2,1( b

zwhereiz

zx i

i

(38)

dimensionless co-ordinate, we will apply thefollowing approximate second order function forthe distribution of the mixing length L :

( ) ( )10;14

2

= xxbL . (39)

In our case 1z 0 is valid (see Fig.1. and

Fig.4.) By using experimental data [2], we will usethe following value : 417.0= .

Fig.4.

Equ.(36) for calculation of the shear stress can berearranged by using the dimensionless co-ordinatesgiven by Equ.(38):

xD

w

z

zb

Hi

i

2*2 = ( , )i= 1 2 . (40)

Let us apply the dimensionless co-ordinates to formEqu.(37) too.Considering the equation obtained this way andEqu.(40), we will get the following differentialequation for the determination of the relativevelocity distribution :

xD

wb

x

wL

H

2*

32

2

2=

( )10 x . (41)

By the solution of the differential equation (41) thedistribution of the dimensionless relative velocitywith respect to z(this direction is perpendicular tothe direction of the main flow), can be determined

by

( )*

max1

*

tan21

1ln

21

w

wx

x

x

D

b

w

w

H

+

+

+

=

, (42)

where :maxw the maximum of the relative

velocity w (the value of the relative velocity atx =0).

*

max

w

w can be calculated by the fitting of the relative

velocity distribution given by Equ.(42) to therelative velocity distribution in the laminar basiclayer at the co-ordinates of the += 1zz and

= 2zz (see Fig.1. and Fig.4.), where is the


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thickness of the laminar basic layer near the wettedwalls of the impeller.Omitting the details of the computations, the

expression for calculating the value of*

max

w

w can be

written as:

+=

22lnln212 *

*

max

bD

bw

D

b

w

w

HH

. (43)

The thickness of the laminar basic layer can beexpressed with a well known formula introduced byPrandtl:

= Bw *

, (44)

where B =11.635 [2] and is the kinematicviscosity of the fluid.The expression for the calculation of the averagerelative velocity ~w referring to the cross-section of

the blade channel can be developed from Equ.(42)by applying equations (43) and (44) indimensionless form:

++= 424

ln21

22

ln21~ *

*

B

D

b

D

bB

bw

D

b

w

w

HHH

.

(45)Knowing the distribution of w~ - by using theanalogy to the flow in a pipe - it is possible todetermine the frictional energy loss in therectangular blade channel. Considering equations(33) and (45) after completing the necessaryrearrangements and the substitution of the values ofB and , the following expression is obtained tocalculate the value of the dimensionless frictionfactor as follows:

( )

+

+= 230.5861.6lnReln199.1

1

HHH D

b

D

b

D

b

, (46)

where Re is the Reynolds number relating to thehydraulic diameter

HD of the cross-section of the

blade channel which will have the form:

HDw~

Re= . (47)

Equ.(46) is applicable to calculate the value of if the inner walls of the impeller wetted by thefluid are hydraulically smooth walls. This case isrealized if the k inequality for the roughness

height kis satisfied.When the inner walls of the impeller wetted by thefluid are hydraulically rough walls, the nextequation - similarly to the formula developed byColebrook - can be used to calculate the value of:

( )

+

++

= 230.5861.6lnReln199.1

715.3

1ln85.0

1

HHHH D

b

D

b

D

b

D

k

, (48)

whereHD

k is the relative roughness height.

The average relative velocity w~ can be obtained bysolving Equ.(22) of the continuity. First let us writedown the connection between the average relativevelocity w~ and its components by using equations(11) and (25):

222 1~~~~ twwww rr +=+= . (49)

Equ.(22) of the continuity and Equ.(49) areavailable to get expression to calculate the value ofw~ as follows:

212

~ tbr

Qw +=

, (50)

where:Q is the volume rate of the flowr

dN

21=

is the factor considered the effect of the bladethickness (51)

d is the blade thickness in the direction of the co-

ordinate

For the solution of the turbulent flow in the bladedspace of the impeller it is necessary to rearrangeEqu.(23) of motion. Applying the notations inequations (25) and (26) and using Equ.(22) ofcontinuity, the Equ.(23) of motion can be writtendown in scalar form:

zwtctgf

r

t

rd

dw

r

P rzrr

12

11 22 ++=

++ ,(52a)

zwf

rd

dwt

r

tw

z

rr

r

12

22 += , (52b)

rrz

P rzrz

11+= . (52c)

Let us see the main steps needed for rearranging ofEqu.(52). First, by Equ.(29) the components of theshear stress are expressed by . Then applyingequations (30), (33) and (34) all the terms inEqu.(52) which contain the slope of the shear stressare also rearranged. The new form of Equ.(52) areapplied on the mean surface (F) where the value of is equal to zero (see Equ.(36)). Weapproximately neglected the slope of the shear

stress component rz with respect to r , which

caused that the slope of the relative pressure

potential with respect to zis neglected too. So wecan write down Equ.(52) of motion on the meansurface (F) of the impeller by substituting theaverage relative velocity w~ in it. Finally we havegot only two scalar equations. The first equation isan ordinary differential equation which serves theslope of the relative pressure potential:

2

2

22

2

1

~

21

2~~1

1

~

t

w

Dt

tuw

r

w

rd

d

t

wctgf

rd

Pd

H

F

+

+++

++=

.

(53)The second equation is also an ordinarydifferential equation which serves the slope of therelative velocity on surface (F):


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222

2

12

1~2

~1

ttD

trw

u

rd

dtf

w

t

rd

td

H

++++

=

,

(54)where u is the peripheral velocity at any arbitraryradius r.Applying equations (30), (33) and (34) thefollowing differential equation can be developed fordetermining the slope of the frictional energy loss:

22'

1~2

twDrd

ed

H

s += . (55)

Equ.(55) is available to calculate the slope of thefrictional energy loss with respect to r. These threeordinary differential equations form a system ofordinary differential equations. For the numericalsolution of the task it is necessary to known thecomponent

f of the constrain force which can be

derived by using equations (17), (19) and (49) asfollows:

21

~

2 t

w

rf

+=

. (56)

The system of the ordinary differential equations(53), (54) and (55) can be solved by Runge-Kuttamethod on the mean surface (F) by using equations(46), (47), (48), (50) and (56). The solution givenby this way serves the distribution of the relativepressure potential ( )rPF , the slope of the averagerelative velocity ( )rt and specific energy loss ( )res

'

with respect to r. Equ.(54) by the limit of 0 isdirectly available to determine the distribution of

the average relative velocity ( )rtid with respect to rin the case of non-viscous fluid.

5. Application of the Calculation Method

Computerised solution as an application of thenumerical method presented above is illustrated bya numerical investigation of the flow in the impellerof a radial-flow pump designed by cylindricalblades with constant thickness. The drawing ofmain cross-section of the pump can be seen in Fig.5.

Fig.5

This pump (noted by P1) was designed,manufactured and experimentally tested by PIDVmeasurement methods in Otto-von-GuerickeUniversitt in Magdeburg. The results of thenumerical investigation are only presented here.

Fig.6. shows the drawing of the back

shroud of the impeller wherein the shape ofthe cylindrical blades and the maindimensions of the impeller can beseen.

Fig.6.

In Fig.7. and Fig.8. the pressure distributionsalong the blades in the case of non-viscous fluidand the average pressure distributions in the case ofviscous fluid in the blade-channel are plottedbelonging to five the same different volume rates.Fig.7. shows the pressure distributions when it issupposed that the wetted walls of the impellerhydraulically smooth. Fig.8. shows the samepressure distributions in case of hydraulically roughwetted walls when the roughness height kis equalto 0.002 mm.

The current volume rate is noted by Q and the

volume rate at the optimal state of the pump byoQ .


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214

Under the title of the figure the current values of theangle

B determined by the direction of the

absolute velocity at inlet section of the impeller andthe optimal volume rate

oQ are plotted. In every

numerically investigated cases the ratio between theactual and the optimal volume rates are plotted onthe bottom of the individual small figures. Underthe title of the figure the current values of theroughness height k and the angle

B are also

plotted. The dashed lines show the pressuredistribution along the suction- and pressure sides ofthe blade in the case of non-viscous fluid. Theaverage pressure distributions in the blade-channelin the case of viscous fluid obtained by the solutionof Equ.(53) are plotted by solid lines.

Fig.7.

Fig.8.

In Fig.9. and in Fig.10. the slopes of the

average relative velocity ( )rt are shown,belonging to the same different volume rates, as inFig.7. and in Fig.8. The structure of all the figuresgiven below are totally similar to the previous twoones. The distributions plotted in Fig.9. belong to

hydraulically smooth walls and the distributionsplotted in Fig.10. belong to hydraulically roughwalls of the impeller. The solid thick lines show theresults in the case of viscous fluid by the solution ofEqu.(54) and the dashed lines are used to show thedistributions in the case of non-viscous fluid. Theslopes of the blade section are plotted by solid thinlines in all sub-figures.

Fig.9.

Fig.10.


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215

In Fig.11. and Fig.12. the relative streamlinesof the flow can be seen determined by numericalintegration of the slopes of the average relative

velocity ( )rt belonging to the same cases as it wasin Fig.9. and in Fig.10. are plotted. In the last twofigures the solid thin lines also show the

distributions belonging to the blade sections.

Fig.11.

Fig.12.

Fig.13. shows the meridional (or in other word

the radial component of the) velocity distributionbelonging to five different states of the pump. The

current volume rate is noted by Q and the volume

rate at the optimal state of the pump byoQ . Under

the title of the figure the current values of the angle

B determined by the direction of the absolute

velocity at inlet section of the impeller and the

optimal volume rate oQ are plotted. The values ofthe ratios between the actual Q and optimal oQ

volume rates also can also be seen under the title of

this figure. The first actual ratio ofo

Q

Q belongs to

the bottom curve and the last actual ratio ofoQ

Q

belongs to the upper curve in Fig.13.

Fig.13.

6. References

[1] CZIBERE, T.: Solution of the Two MainProblems of the Hydrodynamic Cascade Theory bythe Theory of Potentials (in Hungarian), Thesis forD.Sc., Miskolc, 1965.[2] CZIBERE, T.: Die Berechnung der turbulentenStrmung in den Beschaufeelten Rumen vonStrmungsmaschinen, Mitteilungen des Pfleiderer-Instituts fr Strmungsmaschinen, Kolloquium

Strmungsmaschinen, pp. 97-104., Braunschweig,1994.[3]KALMR, L.: Numerical Method for ViscousFlow in Radial-Flow Pumps(in Hungarian), Thesisfor Ph.D., Miskolc, 1997.[4] PAP, E. LANCKE,A. - KALMR, L.:Untersuchung des Einflusses auf das Strmungsfeldim Radialen Pumpenlaufrad bei der Frderung vomGas-Flssigkeitsgemisch, GP 2001/3-4., LII.vfolyam, pp. 43-49., Budapest, 2001.[5] KALMR, L. PAP, E.: Experimental andTheoretical Investigations of the Flow in DifferentRadial Pump Impellers, Conference on ModellingFluid Flow, CMFF03, pp.1016-1023., Budapest,2003.

solution of viscous flow in radial flow pumps

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