**van KARMAN **

**INSTITUTE **

### FOR FLUID DYNAMICS

TECHNICAL NOTE 93

THEORETICAL AND REAL SLIP FACTOR IN CENTRIFUGAL PUMPS

A. NOORBAKHSH

DECEMBER 1973

### A1~

### -~O~ RHODE SAINT GENESE BELGIUM

### ~VW

. J"'" • '-" .~'. '..

### .

:THEORETICAL AND REAL SLIP FACTOR IN CENTRIFUGAL PUMPS

**A. ** NOORBAKHSH

blade passages, Considering such a flow, **we ** have defined two slip
factors

- the theoretical slip factor ~Th (FGT) _ the real slip factor ~x (FGR)

The theoretical slip factor will dep end of the unadvoidable slip of the non viscous flow in the impeller channels~ We wil I

make the distinction between two theoretical slip factors FGT : the two dimensional one (FGT2) and the three dimensional one (FGT3) following the type of calculation process used to determine the factor. Varying systematically the main geometrical parameters of the impeller, we have established a general table for FGT2 which can be very useful for the pumps designer.

A detailed experiment al study, covering five different impellers and having involved the development of special instru-mentation, has provided the local and global values of FGR; this latter is strongly influenced by the working conditions. A correl-ation between the real and theoretical slip factors has been de-rived for any working conditions; this allows a more accurate prediction of performance at off-design conditions.

AN - flow distorsion coefficient, dimensionless b - blade height at impeller outlet,

C - absolute velocity

Cu - peripheral component of C Cm - meridional component of C FGT - theoretical slip factor FGR - effective slip factor

g - granitational acceleration H - head of pump

H" - head of flow at the out let of impeller R - radius in impeller P t - total pressure P - statie pressure s Q

### -

flow rateU

### -

impeller peripheral speed W### -

relative velocityZ

### -

number of bladesa

### -

absolute discharge flow angle 8'- impeller discharge blade angle~

### -

c ircumferent ial anglex
~,~ - slip factors
m
m/sec
m/sec
m/sec
m/sec
m H2 0
m H20
m
m *HzO *
m *HzO *
m

### 3/

h m/sec m/sec deg deg deg~ - impeller discharge flow coefficient, dimensionless

SUBSCRIPTS

1 - at the impeller entry 2 - at the impeller exit

~ - stream line

R - measured values th - theoretical

U - uniform

2

1 2 3 4

### 5

### 6

7 8 9 10 11 12 13 14 15(a,b,c,d,e) 16 17### 18

l~(a,b,c,d,e) 20 21(a,b,c,d,e) 22(a,b,c,d,e) 23(a,b,c,d,e) 24(a,b,c,d,e) 25### 26

27Impe1ler discharge velocity diagram Impeller geometrical parameter"s Influence of relative vortex Formation of the dead water Büseman curves

Measured and calculated slip factor (FGR and FGT) Slip factor tab1e

FGT2' FGT_{3 } and FGR comparison

Local slip factor distribution 1n the blade to blade channel, for mean stream surface

Test faci1ity

Experimental arrangement Impellers tested

Section of measurement Triangular probes

Overall performances of pumps

Impeller efficiency versus flow-rate eoefficient

### ,

Maximal efficiency ef impe1ler versus~B2

Head coefficient versus Z at design point

Measured impeller exit absolute flow angle distribution.

Impeller average exit absolute flow angle ; ; versus

B~ and Z

Measured impeller exit static pressure distribution Measured impeller exit tota1 pressûre distribution Measured impeller exit radial veloeity distribution Measured impeller exit slip factor distribution Wall static pressure measurements a10ng the cireum-ference of the scroll

Static pressure distribution along the cireumferenee of the seroll, Q=o

Statie pressure distribution a10ng the eireumferenee of the seroll. Q=QN

30 31 32 33 34 35 36 37 313 39

### 40

### 41

### 42

Slip factor distribution along the circumference of the scroll

MeasureQ values of slip factor (FGR) Boundary layer thickness versus Re

Jet and wake configuration

Return flow, at the impeller entry

Measured impeller exit relative velocity distribution Slip factor versus Re

Slip factor versus flow-rate coefficient Slip factor versus flow-rate coefficient Slip factor versus flow-rate coefficient Slip factor versus flow-rate coefficient

Radial velocity distribution on the design point Slip factor distribution on the design point 43(a,b,c,d,e): Variation of

slip factor

distorsion coefficient and effective

INTRODUCTION

The slip phenomenon which is typical of the radical turbomachines, influences strongly their working conditions. It

is then interesting, from a design point of view as well as from an analysis point of v1ew, to know accurately the parameters

influencing the slip and the calculated slip factor values.

From a theoretical point of view, no valid solution has been g1ven today for calculating the slip factor in the general

case of an arbitrary impeller, Several methods have been ~roposed

by a large number of authors, Büseman, Stodola, Stanitz,

Baljé, Fujié. Wiesner, etc~ Comparisons with experimental results

have been done without showing a clear connection between the slip and the flow in the impeller.

Quite recently, Senoo (ref

### 1)

has tried to link the slipfactor to the viscosity but he did not consider the non uniform character of the flow; moreover his results are only valid for a particular case.

Until now, only Stiefel (ref 2) has proposed for the compressor case , calculation charts and tables which are practi-cal to use. One of our purposes is to establish similar tables applicable in the case of pumps.

From an experimental point of v1ew, different tests have been done on centrifugal pumps and their results have been

published. In most cases, the tested models are specially designed

for the research pur.poses (very long diffusors, specially profiled

The results of such experiments are mainly pedagogical and cannot be used for industrial pump designs. The influence of the volute on the flow is never considered in this work; in reality, however, the fluid is directly entering the volute whose influence is absolutely not negligible.

The experimental study of the slip factor goes through the evaluation of the real velocity triangles at the impeller outlet (velocity magnitude and orientation measurements) in industrial pumps.

Due to the non axisymetrical and unsteady behaviour of the flow, it would be necessary to take the measurements at different stations using devices with response time as short as possible (hot film). In our case, this latter technique was not available and we have used special pressure probes.

Such a systematic research work does not seem to have

been done up today, too many authors having limited théir vork

to the angles measurements only and assuming uniform velocity profile at the impeller outlet.

Work done

The theoretical part consists in a comparative study

be-tween different calculation methods; it resulted in the establishment of slip factor tables usabie in the pump cases. The range of appli-cation of these tables are

- Blade number

- Outlet blade angle values

- Radii ratio Rl/Rl

2 to 16

### 5°

to### 90°

An extension to the mixed flow type pump ~s possible by introducing the cone angle effect, and is in progress.

The generalisation of the theoretical slip factor calcul-ation process has been done by using quasi-three dimensional

calculation procedures (ref

### 3,4).

Such methods of calculation .prövide" local " theoretical slip factor values at any ~int of the flow between the impeller inlet and outlet.

The experimental part has consisted in testing five simi-lartype industrial ACEC impellers. Many measurement points were

considere~ specially designed triangular probes were used. 3-D

measurements (five holes probes) have also been made. The tests have been carried out forall the main working conditions of each impeller.

The analysis of the results put ~n light the prerotation

phenomenon provocated by the outlet flow non uniformity, the outlet flow non axisymetry, the volute influence and the viscosity effects on the slip phenomenon.

The synthesis of the theoretical and experimental studies gives the possibility to link the FGT and FGR. If a good agreement exists at the design point between these two values, a difference appears at off-design conditions. A correlation for predicting

FGR

### =

f (FGT) has been obtained by introducing several coefficients,p~oviding an efficient tooI to predict the pump performances.

Introducing the slip phenomenon

*in *

the theoretical design
• **ij ** . ( t ( !

methodes of radial pump impellers.

I. The slip phenomenon.

Theoretical blade to blade analysis and experimental measurements at the radial and mixed flow impellers outlet have shown a difference between the exit flow angle S2 and~th~ -gêo

to an absolute tangential velocity difference 6Cu and is character-ising the slip phenomenon in the flow,

The loc al slip factor at one point ~ along a streamline (fig

### 1)

is defined b~ : )J ~ 1. Cu.### =

_{.}

_{c-ç--}

~ _{. }

### =

Th\ ACu. 1 _ 1. C'u Th • ( 1 )The mean slip factor (called simply "slip factor") is defined by :

I Cm 21 )( Cu 21 CU2

IJ .

### =

I Cm### 2i

### / C'U

2Thl = -c-t-u-i-T-h'### ëü

_{2 }~s a mass average value; its value depends on the calculation method used - i . e. 2-D or 3-D evaluation).

The 6Cu velocity difference is a function of two types of pa~~ meter a

1 - geometrical paramet~r s 'of the impeller - blade number Z

_{, }

- blade ~utlet angle 82
(fig 2)

- ratio between the impeller inlet and outlet radii Rl/R2

- ratio between the impeller inlet and outlet channel widths b2/bl

- ~~r1dionalshape (o~ ·tàe tmpeller

(cone angle y )

_{, }

- leading edge blade angle

### 61

- space between the casing and the impeller y

2 - flow parameter's _{Cm2 }

- out let flow coefficient ~2 **= **

-- Reynolds number Re

- Mach number Ma

- Compressibility p

- Viscosity \I

Some phenomena (jet and wake, prerotation, non axisymetry etc) may completely modify the theoretically predicted flow behaviour

in the impeller. We will call those phenomena by the general name of "viscosity effect".

We may write 1n a general way

### (3à)

These parametërs are not influency equally the slip

### •

U2

phenomenon. The influences of

### al

and j a r e small and they can beneglected. In the centrifugal pumps case, the Mach number and the compressibility are obviously not to be considered. In radial

impellers y

### =

*900*

_{and may be neglected. In such a case, the }

I-I Theoretical ~nd real slip fa~tors . (F~T ~n~ FGR) Experiments give the values of angles and velocities at the impeller outlet at different flow rates. We can thus calculate the tangential deviation 6ëü2 in function of the flow rate. The variations of 6ëü2 with the flow rate is important and can be different from one impeller to another.

Theoretical 2-D or quasi 3-D methods developped from radial and mixed flow Turbomachine theories give the possibilities of

calculatingtheoretically 6CU2 and FGT for non viscous flows. At the design point the experimental and theoretical

6ëU2 values are generally very close to each other. At off-design conditions and mainly for small flow rates the real flow differs considerably from the theoretical one, due to the viscosity effects

(6ëU 2 will thus take very different values if measured or calcul-edJ41 No theoretical process exists now for calculating such a flow (taking into account the high viscosity effects).

Let us introduce here two tangential deviations : the

theoretical one (6ëU2th) and the real one (6ëU2R ). The firs!_~ne

is calculated at the design point for a non visoous flów. We will thus not speak anymore of the theoretical deviation at off-design conditions. Relation (3f) will thus be simplified and becomes :

-6CU2R ~s obtained from the me~surements results. It is thus

de-pending on ~ëü2th as weIl as on the viscosity effects, the ReynQlds

We can thus write

Let us also define two slip factors; the theoretical one FGT and the real one FGR defined respectively by

11 1;:. 1 -Th

_{'" }

### *

AC~### =

1 - --=r'-C'U2Th (4a)11 Theoretical slip factor (FGT)

In a radial turbomachine, the velocity distribution

equations for an incompressible flow in the blade to blade plane can be written (2 dimensional case) (ref 5)

P L ( 2 -as p

### a

_{(~ }p an p

### ~2)

+### w ::

### =

0### u

2_{w}

2
### - )

+ + 2 Ww 2_{Re }

s along the
n along the
Re radius of
= 0 (6 )
stream
normal
curvature of the blade

For an ideal flow the Bernoulli equation ~n the relative motion will be written

P \(2 _ U2.

(~

### +

### )

### =

ctep 2

we get

Derivating Eq.

### (6)

and replacing### aw

_{= }

-2 ### w

an p s### p

### u

in Eq. (7), (8 )This equation gives the velocity distribution between two blades for a potential theoretical flow. The relative flow is thus not irrational but has a rotation equal to -2w referring to the axis in this plane. The value of the vortex-vector may be written at one point as :

### t =

1### (L _

3W)2 R 3n

c

We can consider the resulting flow as a superposition

*ot *

two partial

flows:-- the first one being an irrotational

to a potential calculation in a non-rotating channel at the considered flow rate;

- the second one being a rotational flow with a -w vQrtex at ~ero flow rate. (ref

### 6)

The circumferential velocity distribution at the impeller outlet corresponding to the resulting flow is not uniform. The velocity will be higher on the suction side of the blade than on the pressure one. The resulting streamlines are bended due to the rotational character of the flow and will thus not follow the blade profile at the impeller outlet. The flow is characterised by a

### e2

flow angle different from the e~ blade angle. Thatcorresponds to the theoretical slip.

Stanitz's analysis (ref

### 7)

for thin blades shows that the streamlines corresponding to a relative irrational flow are pa~llel to the blade profiles. The streamline deflexion is only appe~r~ng when the relative vort ex is considered (fig### 3),

The flow ratecircumferenti~l distribution at the impeller outlet is not any more

uniform. In certain cases, more than 70

### %

of the flow rate is pass-ing through the half of the channel adjacent to the suctiQn side of the blade.The effect of the -w vort ex is to increase the pressure difference between the two faces of the blade. This difference added to the one corresponding to the irrational flow case gives the total theoretical torque transmitted from the blades to the fluid.

11 1. Smal 1 flow rates.

The blade to blade velocity distribution equation can be written as :

### aw

### an

### =

### W

### R

_{c }+ 2w

Decreasing the flow rate induces a mean velocity W decrease.

The value of the

### WiR

term decreases very fast compared to the onec

of the rw term, due to the high value of the curvature radius of the blade. For the backward type of blades, w is negative. The influence of the rw term is thus to increase the velocity along the suction side and to decrease the one along the pressure side

of the blade. (In an ideal flow model). At zero flow rate, .the

velocity along the pressure side becomes negative and the flow direct ion is increased on this face. For weak but different from zero flow rates, the pressure side velocity becomes so small that

a de ad water zone may occur on this side of the blade (fig

### 4).

11 2. High flow rates.

In this case, the

### WiR

term becomes important compared toc

lJ

### =

where

11 3. Two dimensional theoretical slip factor.

It is defined by = 1 -2'ff:/z

### J

### ·

### c

### ·

### u~~

x Cm2i d### ~

### o

(4c)at mid width of the flow passage.

The use of the mean velocity CU2~h' is due to the fact

that in the blade to blade plane, the velocity triangle can be

different from one side to the other of the blade. In the meridional plane, however, for the 2-D case, only one velocity triangle is

considered from the front side of the impeller to the rear one (uniform flow).

The slip factor is sometimes defined by

and also by

(4e)

An important remark worth being done is thus that a

comparison between slip factor values must take into consideration

the different definitions given by different authors. This, as we find out, is not always done in the literature.

In this case, the "slip factor" defined by equation

4d will be called ~Th and the one defined by equation 4e ~ThR.

These two factors are related through the flow coefficient by the relation : 1 - lJ ThR

### =

1 -1 - +2 cotgBi### (10)

For the straight blade case (8~

### =

### 90°),

~Th • ~Th R introducingCU2Th in Euler's equation and neglecting the prerotational effect,

we get :

(ll) .

The manometric head produced by the impeller in the infinite

blade number case (H:) is calculated (ref

### 8)

in function of thegeometrical paramete~. Knowing the theoretical slip factor value,

the theoretical manometric head of Euler can thus be calculated for a finite blade number.

11 4. Calculation method for the theoretical two-dimensional

( **[ ** **( **

slip factor FGT2 •

Several methods for calculating FGT 2 have been proposed

by Büsemann (ref

### 8),

Wiesner (ref 10), Fujié (ref 11), Stanitz(ref

### 7),

Stodola (ref 12), Pfleiderer (ref 13), Baljé (ref 14)and others. We can classify these methods in two different groups.

A) the exact theoretical methods, based on

### a

potential2-D flow analysis in the blade to blade plane. A weIl known

example is the Büsemann method (ref

### 9).

The correspondingcal-culation charts for the slip factor value are rather difficult to use. Two coefficients ho and Cmo (head ratio to the head at zero flow rate and flow rate ratio to the flow rate at zero he ad

respectively) are defined. The Büsemann curves (fig

### 5)

giveho and Cmo in function of zand Rl/R2 fo~ bl~ge ~p'gles 8~ of

50,40,20 and 10°. From these curves, neglecting the prerotation,the slip factor value and manometric head for a finite blade number can be calculated. (cf. appendix). Different authors have derived empirical relations giving slip factor values very close to the Büsemann ones (Wiesner for instance).

B) Approximate methods, based on hypothesis which are of ten valid in particular cases only. Stodola's formula for instance.

The slip factor calculated by such relations does notceorrespónd

exactly to a definition of FGT2' We will call "approximate" formulae for FGT2' For those cases:

1) the formulae are generally only function of the

geometrical parameté~ of the impeller and do not take into

account flow conditions parameters. They are thus theoretical slip factor (FGT) method of calculation.

2) the formulae provide two dimensional slip factors (FGT 2 ), the flow being always assumed to be uniform in the

meridional plane and varying in the blade to blade plane. These formulae will thus give results which will correspond to our con-cept of FGT and FGT2' That is the reason we will call them

"approximate methods for FGT 2 calculation".

Some principles of exact or approximate calculation methods of FGT 2 are given in the appendix.

11 5. Theoretical slip factor calculation tables.

Each of the slip factor calculation methods is valid for some particular cases. None is general and when applying them to the same impeller case, the results obtained are different.

Figure 5 gives the results of the compar~son between the different calculated values for different formulae and a measuremen at the design point. From this figure we may conclude that the

slip factor values calculated by Stanitz for a small blade number case and small outlet angles 6~ (smaller than 50°) are wrong and can give a difference of 50

### %

compared to the experimental values. On the other hand, for a high blade number and 6~### =

90°, theStanitz formula is to be considered as one of the best. Io :the

**tkme ** way, Stodola's formula is only valid in the case of a small

blade number and outlet angles between 20° and 40°. Büsemann's curves may be considered as valid in every case if the RI/R2 value does not go over the above defined limit. In such cases, the error is included between

### 5

and### 15

### %.

The trouble with Büsemann's curves is that they are given only for weIl defined blade numbers and angles and that their use is quite difficult.eas~er~ a slip factor table has been derived from the above mentioned formulae (ref 15). This table comes from a calculation programme

optimizing the different formulae in connection with the experimental results. The tables are covering ranges of blade numbers from

### 2

to### 15,

outlet blade angles from### 5°

to*90°, *

ratios RI/R2 from
.2 to

### .8.

The programme selects the best formulas in the differentcases. An abstract of this table is given in figure

### 7.

The slipfactor considered in this table is defined by :

lJ_{ThR }

It ~s obvious that the values given ~n this table are

different from the experimentalones : effectively the formulae are considering two dimensional and non-viscous flows. However the difference is small at the design point and will increase when going far from this design point.

We will try to define the connecting coefficients to be applied to table values (FGT) for getting the real slip factor values (FGR). These connecting factors will be defined in function of the flow rat es • ";""

Such coefficients come from experimental results and will only then be strictly valid for the same family of impellers as the tested one. We will discuss in more detail these correct ion coefficients.

It ~s worth noting that in the slip factor tables, some

values are independent of the parameters RI/R2. The reason is

that for such impellers; the" outlet flow is not influenced by the

inlet flow. From the inpeller design point of view, it wil 1 thus be interesting to select the slip factor values in such constant

value zones. For example, for an impeller with Rl/R 2

### =.4

and*8!=4oo, *

In conclusion, the three parametres (b1ade number, blade angle, radii ratio) are stongly interconnected and the choice of the slip factor constitutes and optimisation criterion of these parameters.

11

### 6.

Three dimensional sliE factor FGT3The extension of the theoretical slip factor calculation

to a general impeller has been made by the use of quasi 3-D

calculation process. Such methods provide the velocity field at every point of the flow in a turbomachine. They álso fix the streamline positions, generating axisymetrical stream surfaces.

The use of a quas~ 3-D method makes it possible to

cal-culate the globa1 slip factor value in an impeller; moreover,

the local slip factor value can also be deduced giving interesting

data for the flow analysis in the impe11er. The globa1 ca1culated

slip factor ~s more accurate than the one obtained by the

traditional methods based on two dimensional ca1culations. In our

calculations, the method of ref

### 3,16

has been used.11

### 6-1

Global 3-D slip factor (FGT3)The manometric head of Euler can be defined by the following equation :

H" . =

### ~

x### ~

### .

### J J

52

G flow rate ~n Kg/sec

**IJ **

U
**IJ**

1 x C1i

X _{c05a1iX }

### d~

Sl

becoming when neg1ecting the prerotation

H"

### =Og'

### Q'

### ,

*IJ *

U2. x C2i x cosa.2i x dQ"
s2.
We have a1so
dQ" ### =

thus . b2 Q"### =

### zJ'

### ,

### o

### Ri

x Cm~~ x dl) x d t*21(/z*

*I *

R:_{2 x Cm2i x }

### o

db, x dtRep1ac ing the Q and dQ val ues in (12) ~ ]!.e p:et :.

b2, *21(.1 *

### z

### · Jo Jo

### H"

### = -

, °X _ ,---~~---1 " g b2.o 2tr / z### .

### t

### Jo

### Cm.i·

x db x d'tConsidering the velocity triangle

C2 cosa.2.

### =

U2 - W2 cos~2. (12' )### (14)

Cm2i x Cosa.2i### (16)

### (16' )

thus

### H"

### = -

1 x g 1*tr*

*iz *

### o

0 (12")For an infinite number of blade, the Eu~.r equation can be written

### H

tt . ~ 1### J

b~.*\1*

*2 *

sin62 db
### o

*·r *

### o

(12"'JIn this case, the velocity in the meridional plane may vary but is constant in the circumferential plane. Due to this, the value of W2sin 82 will change from the hub to the tip of the blade and must thus stay under the integral sign.

In the equation 12" and 12"', W2i, 82., W2 , and 82, are _{l. }
known giving the possibility to solve the equations. The three
dimensional FGT3 can thus De defined as :

The so calculated FGT 3 has been compared to the two dimensional one obtained by the different methods and to the experimentally measured one (fig

### 8).

We can see that the global three dimensional slip factor value is slightly lower (1 to 2### %)

that the two dimensional one (tables) and the experimentalone.

The advantage of the use of a quasi three dimensional

process for the FGT3 determination is that this method is absolutely general (valid for any type of impeller). The disadvantage of this method is the length and the cost of the computer calculations. For this reason we have preferred to write the tables of slip factor and to use the quasi three dimensional process as a check of the table values (for classical radial impellers).

11 6-2. Local slip factor

It can be defined by Cu.

### u. -

W. cotga. 11.### = -

l.'I1b l. 1. 1. l. C'u.,### u. -

### w'

cotga' l.'Ih 1.### (18)

It will thus be determined at each point of the set constitued by the streamlines and the normals in the flow path between two

blades fr om the inlet to the outlet of the impeller (three

dimensional slip factor distribution) • An example of such:a dis-tribution is given in fig

### 9.

We can see that :1) in the blade to blade passage, the slip factor distribution at the impeller outlet is non uniform, lts local value is higher

along the pressure side than along the suction side

### (.88

to### .77);

2) at the impeller inlet, the FGT variation is very small and a choked flow is shown at the leading egge of the blade;

3) on streamlines close to the suction side, near the inlet, the local slip factor value can be higher than 1 and the streamlines are bended in the opposite direction as compared to the usual case. A dead water zone appears on this side.

111 Real slip factor (FGR) determination by experimental methods.

111 1. Pump test facilitl'

The VKI pump test facility (fig 10) ~s a closed loop circuit.

The actual measuring devices are designed for centrifugal pump tests

at a maximum rotational speed of 1500 RPM; the maximal outer

diamter of the impeller is about:O.3m.

The pump is driven by a

### De

motor of 25 KW under 220 V. Avariable resistance allows a continuous variation of the rotational speed; a valve is controlling the water flow rate (fig 11).

The flow rate can be measured by a venturi or a diaphragm. Pressure tappings are directly connected to a multimanometer or through a manual valve selector to special pressure transducers for water tests. An electronic counter gives the RPM; the water temperature is measured in the settling chamber by a mercury ther-mometer.The shaft power can be measured by a torquemeter or a

voltmeter and ammeter and the use of the calibration curves of the electrical motor.

111 2 Tested impellers - Local performances measurements.

Five industrial impellers have been studied on this test

facility (fig 12). They were designed in order to vary two of the

three main parameters (z and B~)~ The radius ratio has been kept

constant for practical reasons, as well as the meridional shape of the impeller.

The slip factor table has shown that for the RI/R2 va~~es

of the different impellers considered; the slip factor value is independent of this parameter.

Th ough the blade inlet angle was kept constant, the blade profiles differ from one impeller to another.

At the impeller outlet, measurements have been made at a

constant distance of

### 8

mm from the impeller periphery; they weremade at three circumferential locations (45°. 135°, 225° from the horizontal line) (fig 13).

The tests were made at constant rotational speed for different flow rates included between the zero flow rate and

the maximum one. A 13 point investigation is obtained by traversing probes through the impeller width.

At each position, the statie, total pressures and flow angles are measured.

At the outlet, triangular probes designed by the author and manufacturer at VKI , ref.(17) fig (14) are used; pressure signals are sent to pressure transducers.

The measurement accuracy is pretty good;the difference of

measured flow rate by probes or venturi is about 5 to

### 6

### %.

Theaccuracy on the manometric head is about 1

### %.

A maximum variationof 5

### %

has been measured on the rotational speed of the motot.At the impeller outlet, the absolute flow is axisymetrical only for flow rates around the design point value. The experiments must thus be carried out considering the non axisymetrical behaviour of the absolute flow at the impeller outlet.

111 3 Measurements results.

_{, }

Firgure 15 shows the characteristic curves of the five

impellers : head coefficient(gH) power (P,KW) and efficiency

U 2 , - ' .

( n ) in function of the flow 60efficient

### (~2

### =

### Cm~

).The design point position does not vary much from one impeller to the otheri i t corresponds to & flow coefficient of about 14. We may deduce that the impeller 111 (8~ = 30°, z

### =

### 6)

gives a higher efficiency on a larger zone than the other impellers. The working condition of this pump is satisfactory around the

nominal point. Decreasing the flow rate increases the head, passes through a maximum and stays constant. The maximal efficiency zone covers the 1.1 ~N to •

### 65

~N range.111 3-1 "Impeller performances

It is quite interesting to compare the different impellers performances. As the scroll is the same for the five impellers, it is difficult to judge of the impeller performance on the basis of fficiency alone. The n

R efficiencies (n

### R

### =

(Pt, -### V

x### I

Ptl)X q ) x n### E

and the head at the design point for the different impellerf have shown that, fig

### (16) :

### 1)

the design point for the five impellers 15 identical onthe efficiency curve and 1S located at ~

### =

.14. We may concludethat the two geometrical parameters z and 8~ do not influence the design point location.

2) maximal efficiencies for the three impellers with different blade numbers and same 8~ are quasi equalsi we may then say that z has no large influence on n

R at the design point (increasing the blade number increases the slip factor values

but decreases the rotor hydraulic efficiency). We may thus suppress one or two blades 1n a pump impeller which can represent some

3) the blade shape, ~.e. the angle value variation between

the leading and trailing edges has a large influence on n at the

### R

design point. This efficiency can vary about

### 6

### %

from an impellerto another.

The maximal rotor efficiency variation in function of the

e~ value is given in figure 17. The n(e~) curve is a straight

line with a 1.1 slope. The wheel maximal head versus blade number

curve is given on the fig 18 plot; it is also a straight line.

III 3-2 Flow outlet parameters measurements.

Flow direction.

The absolute flow angles a2 have been measured at the

impeller outlet and are given ~n figure 19. The measurements were

made for different flow rates between the zero flow rate and the maximum one. At each flow rate three curves were obtained,

corresponding to the three measured points at the impeller periphery The absclssa represents the impeller width (tip:O, hub: 1).

Before and af ter these limits abscissa, measures taken in the space between impeller and c asings are given.

The absolute flow angle a2 distribution versus the outlet width b2 is generally not uniform.

- At the maximum flow rate : the a2 variation at the per-ipheral station I is important. For the five impellers, the

difference between the extreme values of a2 is about 25°. The curves obtained for each of the three peripheral stations are of different types and that, for each of the five impellers. The conclusion is that the flow is absolutely not axisymetrical at maximum flow rate.

- Around the design point the a2 (bz) variations are weak.

the three eurves eoineide and

- At small flow rates: the az (bz) eurve is quite

different and a2 is higher at the tip than at the hub. Close to the zero flow rate, az (bz) is like a parabola with a minimum point rear .5b z ; az may be negative at this point, indieating the existenee of a baek flow.

Let us eonsider the mean value of az, at design point, ~n

eaeh impeller, along the impeller width :

We can see in fig 20 that aZ 15 funetion of the outlet blade angle

, . , . ( ' ) . b 1 t

*Sz *

and z. Deereas~ng *Sz *

~nereases aZ; aZ Sz 15 the hyper 0 a ype.
The law of influence of z on aZ shows the the same trend but is

linear.

Statie pressure.

lts distribution ~s quasi uniform along the impeller width

(fig 21).

For high flow rates, P is max~mum at the peripheral point

5

I (farest from the volute edge); its minimum value happens at point

111 (elosest to the volute edge). At the design point, the statie

pressures at point land 111 tend to a eommon value equal to the

statie pressure at point 11. For low flow rates, the situation is

Total pressure

In opposition to what happens to the statie pressure, the total one varies widely at the impeller outlet. This fluctuation could give an explanation to the possibility of dead water zones at the impeller outlet with practically uniform statie pressures along the rotor width. Total pressures distribution are shown in fig 22.

We can notice that the total pressure value ~s generally

higher at the blade tip than at the hub.

Meridional velocity component.

lts variation along the impeller width influences directly the slip factor value. Several authors have assumed this distributio curve to be of a sinusoidal or parabolic type. Our measurements

show that the Cm2 (b2) curve sl'ape depends on the flow rate and

cannot be represented by a general equation. I n the next chapter

we will come back to this problem and to the definition of the distribution curve at nominal working conditions. Figures 23 give different Cm 2 (b2) curves coming from our experiments.

Slip factor.

From the measured values of Q2, Ps, Pt, we can draw the real velocity triangle at each point of the impeller width and at the three peripheral positions, Comparing this triangle to the theoretical one, we can calculate through the CU2Tht and CU2

values, the slip factor value, We can then obtain lJ(b 2 ) curves such as the ones represented in figs, 24, Let us remark that for all the tested impellers, the slip factor at the tip is higher than at the hub and that for all flow rates.

Statie pressure along the scroll fig.

### 25

This parameter influences the operating conditions of the impeller, considering the fact that a uniform distribution of Ps around the volute will correspond to a power saving.

The statie pressure distribution is not uniform at the rotor outlet and its relative variation can be of about 200

### %

at zero flow rate, fig.### 26.

At the design point, Ps lS quasi uniform and the flow

lS axisymetrical at the impeller outlet, fig

### 27.

For higher flow rates, Ps lS again not uniform; its distribution curve is the inverse of the one obtained at zero flow rate, fig. 20. The reason lS shown when calculating the

degree of reaction A, fig. 29.

### ( A

### =

~Ps rotor ~Ps pumpFor high flow rates the degree of reaction is higher than unity and the statie pressure at the volute outlet lS smaller than the one at the impeller out let : the flow is thus accelerated in the volute. In an opposite way, for low flow rates, the degree of reaction is smaller than one and the flow is decelerating thro~Bh the volute.

The slip factor variation at different flow rates, along the rotor periphery are given in fig. 30 (impeller I). The

juxtaposition of the laws ~(b2) ~nd ~(t) gives the spatial variation of the slip factor (which is calculated from the absolute flow).

111

### 4

Real slip factor (FGR) determination.As seen above, the measured slip factor varies"along the

impeller width as weIl as circumferentially in an absolute

reference system. In order to estimate the link with the classical
concept of slip factor, we define the mean FGR_{t } taking into

account the possible non axisymetr1

*ot *

the absolute flow :
FGR

where

x

### =

lJCm~

### dP

x dtAt the design point, the flow is axisymetrical, thus

b

### J

02 CU2 x Cm2 db 11 R### R

### "

Cu### =

*" *

*J:2 *

n2
Cm2R dl:>
( 20t
The values of FGR for the nominal point of the different
impellers are given in fig. _{31. } _{We can see that FGR depends on a2 }

### ,

and z, but these parameters do not influence the slip factor in the same way. Reducing the blade angle value from 30 to 18° increases the slip factor of about### 5

### %

but reducing the blade number fromWe will thus conclude that the blade number is the most

_{, }

~ :

important parameter acting on the slip factor.

The comparison between the measured and calculated values (Wiesner and Stodola's formulae) shows that these formulae are the best to use for this particular type of application. On the other

hand, Stanitz's formula gives an error of about 30.

### s.

Let us compare now the measured values to the table values. We can see in such a case that the maximum difference between

V. Relation between the theoretical (FOT) and real (FOR)

(

slip factors.

The real value of the slip factor does not only depend on the rotational character of the flow. This chapter tends mainly to show the influences of the parameters which differenciate the real slip factor from the theoretical one. The purpose is to

obtain aprediction correlation of the FOR starting from the FOT.

V 1. Influence of the real effects on the slip factor.

The real slip factor has been defined as

### (21)

It is very interesting to know the function ~x at any

flow rates. That ~ould permit us to calculate at off-design

conditions. It is obvious that the three parameters (v,Re,~) are

interconnected. To simplify the procedure, we vill try to study each of them seperately.

V 1-1. Viscosity effect v

As said before, we will group under the name ·viscosity effects", all the real phenomena susceptible to modify the

potential flow behaviour.

a) Boundary layer thickness

The wall boundary layer thickness effect is more important when the impeller width is smaller. Figure 1 shows the absolute tangential velocity decreases, reducing the slip factor value, if the impeller width becomes smaller. In the same case, the meridion velocity is increasing.

Let E be the boundary layer thickness along an impeller vall,

the total blockage of the boundary layer, referred to the impeller vidth vil be : x

### =

### "

2E b2 Hl= x g (22~,-The manometric head viII then become

+ (1 - x) H" . (23)

It viII thus be reduced by a certain amount; so for the slip factor vhich is

H"

~

### =

### H"'"

### ...

As the boundary layer thickness is a function of the Reynolds number and the radius, ve viII be able to vrite :

x

### =

f(Re) x It..l.b2.

### (

24~-The f(Re) function can be calculated by the Poulain's curve (ref 5) fig. 32.

Due to the relatively impQrtant vidth of the tested impellers;

~n our case, the influence coming from this curve viII be nearly

b) Jet and wake.

Hot wire measurements (Dean and Senoo ref 18) have shown that in some cases, outlet velocity distributions are exactly opposite to those predicted by the potential model. The velocity on the suction side is smaller than the one on the pressure side fig. 33. On the other hand, we have seen that the effective value of the slip factor depends on the impeller outlet velocity profile •

.. : The two slip factor values corresponding to the theoretical and

real flow models will thus not be the same.

c) Prerotation. back flow.

At quasi zero flow rate, the back flow phenomenon appears fig. 34. This back flow, leaving the impeller with a rotational

speed

### w,

tends due to the viscosi~~,to take the ma~n flow with i t .The consequence will be that the flow will have a rotational

component ahead of the impeller inlet. This is called the prerotation

It is thus provoked _ by a back flow and can only be observed at

low flow rates.

At the impeller outlet, the fig. 35 (W(b2)) shows at low flow rates, a parabolic type of velocity distribution, the minimum being at mid impeller width. A flow reduction decreases this

minimum giving negative values of W showing that the flow goes back into the impeller at this placet In the inlet pipe, however, the fluid flows along the walls and enters the impeller at the center of the inlet channel.

From the Cm2(b2) curves, we can calculate the back flow rate 3

In the impeller I case for example, at a 59 m /h flow rate, fig~23

shows a negative Cm2 between .25b2 and ,Y5b 2 (- .15m/sec). As b 2 =40mm and b'=20mm is the back flow width

The outgoing flow rate will thus be

Q total

### =

Q+Q back flow### =

### 68.6

m3 /hIn this case, the back flow rate will thus be equal to 15

### %

of the total flow rate. The back flow phenomena appears at the

out let of the impeller and could change completely the distribution of velocity calculated by the potential method. Consequently the slip factor measured is different from the theoretical one. In this case the peripheral component of absolute velocity at the impeller entry, CUl is not zero and the value of the slip factor is different

from the ration of H" •

~

~

V 1-2 Reynolds number effect.

Watanabe's tests (ref 19, fig

### 35)

show that the Reynoldsnumber may influence the slip factor. Reducing the Reynoldsnumber

leads to a slip ifactor decrease (passing from Re

### =

.5105 to 1. 10 6corresponds to an increase of

### 3.5

### %

for ~x ).The Reynolds number is defined as Re

### =

b2XV,VO

In our case, Re is identicalfor the five impellers and is

equal to

### .88

### Ie?

### v

1-3 Flow rate### t

effect~Test results of different authors checked by ~ur own

measurements indicate that th~ radial impellers can be differenciated

x

by the ~

### (f)

curve types1. increasing curve with decreasing ~ (group A)

2. decreasing curve with decreasing ~ (group B)

### 3.

horizontal curve (group C)a) Group A

This group includes the straight blade compressors

### (s1

### =

*90°)*

characterised by the jet and wake phenomenon.

Stiefel's impellers are ~n this group (ref 2) as Dean's

ones (ref 18) (separated flows) fig.

### 37.

Physically, we can feel in advance that for an increasing curve, the flow will be separated. If this is the case, a part of the space in the blade passage is filled by a wake crossed by a small part of the flow rate. The rest of the flow rate passes through the jet zone. The impeller could then be replaced by a model such as the wake zone is assumed to be part of the blade.

The active part of the channel narrows thus. Applying, for instance, Wiesner's formula we will get a higher slip factor than in the

origi impeller case. We may say also that the flow in such a

### cas~

will be better guided, corresponding to an increasing slip factor

### I

when decreasing the flow rate.

b) Group B

Some backward blades impellers constitute the second group.

It ~s however very difficult to predict which one, due to the fact

that several parameters are to be.taken into account.

Moreover, we have not enough experimental checks to make a definitive classification. In the largest part of these cases, we may say that backward blade impellers with a small outlet-inlet

areas rat ion and a outlet blade angle of about *20° * are in this

group. The jet and wake configuration has not yet been visualised for this group. The outlet-inlet areas ratios are smaller in the back flow blade impeller than in the straight blades ones. Their

channels are thus less divergent. Tests done by Fo~ler (ref 20'

on a very big impeller have shown that for backward blades, at

low flow rates, the separation point is located far trom the leading

Physically, we can imagine that by decreasing the flow rate, the flow is less guided and that a decreased slip factor value will occur.

In figure 38 we give the FGR variation ~n tunction of the

flow rate,for our impellers;we can see that the impellers IV

_{, }

_{, }

*-(B2 * = 18

_{, }

z=~) and,II *(B2*

### =

24, x=4) are ~n this group. The Kasai's (ref 21) impellers 6,7 and 10a are also ~n this group (fig 39).c) Group C

In the practical case, a tew impellers are in this group. We find, for example, the K-260-48, b 2 /d2 .059 (fig 40) impeller of Livshits (ref 21). lts lJx(tP) curve is quasi fl.n hor·izontal.

d) Group D

Nearly all the backward blade industrial impellers for centrifugal pumps are in this group. The the three impellers I

( B ~ = 2

### 4,

z = 6),I I( B ~ = 2### 4,

z = 8) a n d I I I (B ~### =

300, z = 6)

are in this group.

When decreasing the flow rate, the FGR increases, then passes

through a max~mum and stays th en horizontal until the design point;

it decreases then (fig 38). Referring to the conclusions given for the other group, we can feel the flow evolution. For high flow rates where the FGR increases, an important flow separation could be

expected on the blade pressure aide; this tendancy will decrease as the flow rate is increasing. When the curve is passing through its maximum, the boundary layer could reattach. In the horizontal part of the curve, the flow is uniform and when the curve is

de-creasing, the flow channels are not well filled and the flow guidance

Comparing the experimental results and observing the ~x(~) curves shows that the above explanations are not too far from

reality. For example, ~n the case of the impeller 111, in the

reg ion between ~

### =

~ max and .198, ~x(~) is increasing. In thisregion, the total pressure and a2 distribution curves are highly

non uniform fig. 22. In the region between ~

### =

.198 and ~ i~sign,### ~x(~)

is horizontal and we see that the total, statie pressureand a2 distribution curves are quasi-uniform. Under ~ design to

### ~

### =

0,### ~x(~)

decreases and Ps, Pt and a2 are again non-uniform.V 2 FGT-FGR correlation definition.

The real slip factor has been defined as

### (21)

The FGT-FGR relation will be defined in connexion with the

different influences of the above mentioned parameters. The actual

state of the art of knowledge in radial turbomachines flows g~ves

no possibility to theoretically calculating the viscosity effect. We will thus limit ourselves to considering the problem due to the flow non-uniformity. For this purpose, we will try to transform the real flow model (non-uniform one) to a uniform flow one with fixed distorsion coefficients : this simplified flow model uses the two dimensional theoretical slip factor FGT 2 (calculated in the tables). The distorsion coefficients will be

circumferential plane correction coefficient.

meridional plane " "

non axisymetry

_{" }

_{" }

### -

### 37

-From this relation, we can get the manometric head produced

by the impeller for off-design conditions. AN l cannot be actually

determined due to the lack of measurement techniques for measuring

the circumferential plane velocity distribution; we will thus assume

ANl eq~al to unity.

Several authors have shown that the flow velocity distribution

curve in the outlet meridional plane has a large influence on the

manometric head (reduction of the latter for highly non-uniform

distribution). This phenomenon can easily be shown by the following

considerations Manometric head b

### JO

### ·

2 2 cotg6 2 d:t> Cm2 )(### "

### U~

U2 H### =

### -

_{g }

### -

-_{g }x

_{(b2 }

### Jo

Cm2 d.b ( 25 )-Assuming a constant blade angle

*B2 *

at the impeller outlet fr om hub
to tip, the equation becomes :

### "

### H

### =

_{g }

b 2

### Jo

Taking into account the Schwartz inequality

b

### L'

where

b b

### 10'

### C

K.> 1and replacing ~n equation 26', we get

( 26!~) g

where

The manometric head decreases for a non-uniform outlet velocity disrtibution; the slip factor will also be smaller than in the uniform distribution case.

From the impeller outlet velocity distribution, we can integrate (26') obtaining the slip factor value for a non-uniform flow. Let us notice that the constant 82 hypothesis is rarely checke in practical cases.

We may then define

### AN

_{2 }

_{as }

(27) where b

*f02 *

CU2 x Cm2 db
R R .
Cu =
n2
*: *

*J:2 *

Cm2R db .
(28)
Considering the velocity triangle, the equation (28) becomes

C'u

### =

n2

For a uniform flow, the Cu 2 tangential component will be

u
Cu = _{Cm2 } x _{cotga 2 }
u2 _{(29) }
(213! )
where

### C

and b 2**Jo **

cotga2 dl:
Cm~ db
Cm2. = cotga2. =
b2 b 2
The AN3 distorsion coefficient cannot be calculated

directly. We must first define the tangential component for a non-uniform and non axisymetrical curve in an ahso1}de svstem

### J

b

02 Cm~ dP x d~

Giving the AN product defined as

AN· with

### (31)

~### C

~u c:### C

m2 x co gal t UZAN

### =

where### J

211'### J

b.2### =

0 0 Cm2R x db x d~ 2 x 1f x b2 2 x 1f xb2 providing the AN 3 value.The general form of AN will be

*f211' *

fb2 . _{Cm2n }

_{x }

_{db }

_{x }

_{d~ }

_{I }

_{I }

_{f:11' }

_{f:11' }

*f:*

*2*Cu~ x 2

_{o }

0
(2x1fxb2) X
( *f211' *

_{o }

Jb_{0 }2

_{Cm~ }

_{db d }

_{~) }

_{x }

_{( f:11' }

Jb_{( f:11' }

_{o }

2 _{cotga2 }Cm2 db x d~ R db d~)

In practical cases, the AN value will be used, which he1ps avoid the AN3 calculation. Finally we get :

### (33)

We first have to define the AN(~) law.

V 2-1 AN(p) at design point.

All the parameters of integral (28) are measured

*tor *

different flow rates; AN(~) is th us defined at any flow rate.

We can now make a generalisation to different pumps of the

x

same family at design point; this is done by calculating Cm2R' ~

and a2.

The variation curves of Cm2R • f(b 2 ) for the different
impellers tested are given in figure 41. We can see that the
dis-tribution is very similar (2 _{maximum points at b • 0.15b 2 and }

0.85 b2 and one minimum at b • 0.5 b2~~ Looking for a mathematical expression defining these curves leads to :

with the coefficients given 1n table

### I.

We can see that the A,B,C, values for the five curves are rather small and may be neglected. We obtain then a fourth degree p1ynom to define the variation law of the flow velocity 1n function of the impeller width ( for the impeller family we have considered)

In this equation b is expressed in

### %

and Cm2 is non dim-ensionalised referred to### om .

The corresponding variation curves are given 1n figure 42 for the five tested impellers. These curves are quite similar and their mathematical representation wi11 be a ninth degree plynom

(except for impeller 111 which is a ninth degree function)

### (35)

The coefficients are given in table 11.

We see that the A,B,C,D,E and F coefficients are rather

small and may be neglected; PR(b 2 ) becomes then a cubic law :

### ( 35' )

where b is expressed ~n

### %.

Following the same process, we get a fourth degree law

At the design point, the flow being axisymetrical, AN3

will be 1 and considering the above defined relations, AN_{2 } can be

calculated by the equation

### (26").

The### K

coefficient of this equatioDcan also be calculated. The corresponding results for the five

tested impellers are given in table IlI.

ROUE K AN2· I

### 1.012

### .985

11### 1.035

### .986

### .III

### 1.014

### .988

### IV

### 1 .013

### .995

### V

### 1.030

### .915

TABLE 111We can see that the

### AN2

values are very close to 1 and thatFGT 2 decreases lor2

### %

at design point due to the non uniformity.V 2 - 2

### AN(p)

at off-design conditions.In such a case,

### AN

can be calculated by(36)

For the five tested impellers, the AN(~) law is given figure

### 43.

The FGT 2 calculation at off-design condition is possible by using

the following equation :

We notice that the " pseudo-theoretical" slip factor, cal-culated by this way does not vary very much with the flow rate, except at the limits (increase at low flow rates and hasardous

oscillations at high flow rates; the-latter being due to measurement difficulties).

### v

3 Influence of the outlet angle*B, *

### ,

on AN(~l.The flow non uniformity which influences strongly the AN(~)

curves differs from one impeller to another.

### ,

The

*B2 *

parameter has an increasing influence on AN(~) when
decreasing its value. This fact can be shown by the following conditions

### =

### =

---CUu2 U2 Cm2 x eotgS~{K - 1)### =

1 -U2 - Cm~ x cotg62 U~ AN~### =

1 - (K - 1) x (_{CU}2 - 1)

For 8

### =

_{90°, U2 is very close to Cut, we thus have }

### AN

2### =

1;for very small va1ues of

*B, *

Uz becomes much 1arger than Cuz and
### AN2

becomes then very small.Range of application of

### AN(p)

Due to the lack of systematic experimental resu1ts on the (

flow at impeller outlet, it is impossible to get a general expression

for

### AN.

We must thus limit our coefficient to the fami1y of pumpswe have tested.

### v

### 4

Ca1culation methods of the FGR.Other methods have been proposed by different authors; we

present these methods in appendix 11.

VI Comparison between measured and calculated data at whee1

### •

oü't1et'~

The distribution of the measured velocity at the wheel outlet Cm2Th as veIl as tha,t of the relative outlet velocity WZTh , calcu1ate

by the quasi

### 3-D

method is gi ven in fig .### 43,

which 8,lso presents theOne can see in fig, 43a that the mean W_{2Th } value is some

### 8

### %

lower than the measured one, This difference has an effect on the slip factor. From the velocity triangle, one can see that the tangential component of the theoretical absolute velocity becomes larger than the realone, and thus, the theoretical slip factor is higher than the real one.~The evolution of W2Th on the wheel width is almost linear. lts value is higher on the hub than on the casing (~ 15

### %),

which is the inverse .of the experimental evolution.

The theoretical meridional velocity distribution is also linear with its maximum value near the casing. This value is 20

### %

above the hub side value. The measured meridional velocity is higher than the theoretical one in the vicinity (15

### %

_{of b 2 ) of }the hub side, while the theoretical values are higher near the casing

### (85

### %

_{of b 2 ). On average the theoretical values are higher }

Conclusion.

We may consider two slip factors in radial machines ; the theoretical one (FGT) and the real one (FGR).

The FGT is defined from the theoretical flow of an inviscid fluid and the FGR from the real flow.

To simplify the pump design, a theoretical slip factor table has been obtained from different calcula~on processes. This table covers the complete range of the centrifugal impellers.

Theuse; of a quasi three dimensional method provides a three dimensional theoretical slip factor for any kind of impeller. AloeaJ

slip factor can be obtained by such a method. The slip factor calculated in this way can be locally higher than 1.

A measurement technique has permitted to check that

- the impeller absolute outlet flow is generally non

axisymetrical and non uniform, except close to the design conditions (maximal efficiency);

- for low flow rates, a partial back flow occurs and can induce a prerotation in the inlet pipe with ~ non axisymetric character.

The experimental results have provided the FGR distribution along the impeller width at different flow rates.

The difference between the FGT (table) and the FGR at the design point is not very important. It becomes very important at off-design conditions, especially at very low flow ratesr

The real slip factor at any working conditions is calculated by introducing distorsion coefficients~ The~ latter ar~ caleulated from the experimental results.

The range of application for these coefficients is not yet well defined due to the lack of information in the literature,

FaT calculation methods.

A 1-1 Exact theoretica1 methods (aroup A).

A. Büseman.

This method was deve10pped in 1928 and is s t i l l the best one if we compare its results to·the experimentalones (ref 9). The calculation is only made at design point.

Between the different geometrical parameters, Büseman chose

### •

z,82 and RI/R2; he th us calculated :

Rypothesis

1) constant impeller width .(two dimensional study)

### .

'

2) logarithmic type of blades (82

### =

cte)3) ideal fluid; absolute irrotational flow.

By a conformal mapp~ng, the impeller is ~ransferred into a

### •

cascade with constant *B2' *

### •

Büseman made the calculation for 82 angles between 5 and 90°

for different blade numbers and for radius ratios RI/R2 between 0.2 and 0.8. He defined two coefficients ho and Cmo. Ris curves

(fig