Numerical Study of the Axial and Radial Forces,

the Stresses and the Strains in a High Pressure

Multistage Centrifugal Pump

Mohand-Amokrane Abdelouahab

1

, Guyh Dituba Ngoma

1

, Fouad Erchiqui

1

and Python Kabeya

2

1

University of Quebec in Abitibi-Témiscamingue, School of Engineering’s Department,

445, Boulevard de l’Université, Rouyn-Noranda, Quebec, J9X 5E4, Canada

2

University of Kinshasa, Department of Mechanical Engineering, Kinshasa, Democratic Republic of the Congo

Keywords: Multistage Centrifugal Pump, Axial and Radial Forces, Stress, Strain, CFX, Static Structural Analysis.

Abstract: This paper deals with the numerical study of the axial and radial forces, the stresses and the strains induced

by a liquid flow in a high pressure multistage centrifugal pump to improve their performances account for the

material of the shaft and the impellers. Indeed, a model of a 4-stage centrifugal pump is developed considering

a design point. The continuity and the Navier-Stokes equations are used for the liquid flow through the pump.

Additionally, the equations of the stresses and the strains are applied for the shaft and the impellers. The

obtained system of equations is solved by means of the CFX-code and the static structural analysis to

determinate the fields of velocity and pressure, the axial and radial forces, the stress and the strain on the

pump shaft. Moreover, numerical simulations are carried out to analyze the shaft behavior in terms of the

induced axial and radial forces, the stresses and the strains. The diffuser in the last pump stage is used as

parameter. Good trends are achieved comparing the obtained results with these found in the literature and

these calculated using the classical equations of the stresses and the strains.

1 INTRODUCTION

The growing technological development in the field

of liquid transportation in the mining sector is leading

pump manufacturers to design and develop high

pressure multistage centrifugal pumps that are

continually adapted to the industrial needs. Its

operation and its adequate life expectancy greatly

depend on the design and manufacture of its

components, such as, inter alia, the shaft, the

bearings, the impellers and the diffusers.

To better design and optimize the pump shaft and

the bearings of the multistage centrifugal pumps, the

axial and the radial forces acting on the impellers

must be known with accuracy to determinate the

bearing loads, the shafts stresses and the shaft

deflection accounting for the entire flow range. This

allows to avoid, mainly, the excessive bending of the

pump shaft, too high stresses on the pump shaft, and

the overloading of the axial and radial bearings. In

fact, several theoretical and experimental research

works have been done on the multistage centrifugal

pumps in the terms of the axial and radial forces, the

stresses and the strains acting on the pump shaft

(Karassik and McGuire, 1998; Gülich, 2010; Wang et

al., 2013; Watanabe, 2019; Gantar et al., 2002;

Bolade and Madki, 2015; TM.P. S.p.A.

Termomeccanica Pompe, 2003; Karassik et al., 2008;

Wang et al., 2014; and Suke et al., 2015).

In most previous study on the multistage

centrifugal pumps, the effects of the induced axial and

radial loads, the stresses and the strains accounting for

the diffuser return vanes and the diffuser in the last

pump stage in relation to the pump head, the brake

horsepower and the efficiency have been less

investigated (

Jino, T., 1980). Since the pump

performances of the multistage centrifugal pumps

depend, among other things, on the design of the

diffuser including the return vanes (Miyano et al.,

2008; and La Roche-Carrier et al., 2013), this is

essential to consider also the impact of the diffuser on

the axial and radial forces, the stresses and the strains

on the pump shaft.

Therefore, this research is focused to the

development of reliable and accurate numerical

models of a high pressure multistage centrifugal

Abdelouahab, M., Ngoma, G., Erchiqui, F. and Kabeya, P.

Numerical Study of the Axial and Radial Forces, the Stresses and the Strains in a High Pressure Multistage Centrifugal Pump.

DOI: 10.5220/0009783001810188

In Proceedings of the 10th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2020), pages 181-188

ISBN: 978-989-758-444-2

Copyright

c

2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

181

pump in order to study in-depth the axial and radial

forces, the stresses and the strains on the pump shaft

due to the liquid flow through the pump at varying

operating conditions using the 4-stage centrifugal

pump and the diffuser in the last pump stage as

parameter in relation to the pump performances.

The achieved results using two diffuser types for

the pump head, the brake horsepower and the

efficiency are compared. Furthermore, considering

the first pump stage, a good trend is obtained

validating the numerical results for the stresses on the

pump shaft from the determined axial and radial

forces by means of the classical equations of the

strength of materials and the mechanical design of

machine elements.

2 MODEL DESCRIPTION

Fig. 1 illustrates the components of the reference 4-

stage centrifugal pump considered in this study. It is

mainly composed of a shaft, four impellers and four

diffusers. The solid and fluid models of the 4-stage

centrifugal are shown in Fig. 2.

a) Shaft, impellers and diffuser with return vanes of type 1.

b) Pump stages: impeller and diffuser combined.

Figure 1: Components of a 4-stage centrifugal pump

(School of Engineering’s Department, Turbomachinery

laboratory, E-216, www.uqat.ca).

a) Solid model

b) Fluid model

Figure 2: 4-stage centrifugal pump models (Abdelouahab

M.-A., 2018).

Furthermore, relating to the pump diffuser, there are

some multistage centrifugal pumps that use another

type of the diffuser (La Roche-Carrier et al., 2013) as

indicated in Fig. 3. The performance comparison of

both diffuser types are done in this work.

a) Technosub Inc., www.technosub.net.

b) Diffuser solid model.

Figure 3: Diffuser with return vanes of type 2.

SIMULTECH 2020 - 10th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

182

3 GOVERNING EQUATIONS

To determinate the fields of liquid flow velocity

and pressure in a multistage centrifugal pump, the

following hypotheses are considered (La Roche-

Carrier et al., 2013): (a) a steady state, three-

dimensional and turbulence flow using the k-

model is assumed; (b) the liquid is an

incompressible liquid; (c) it is a newtonian liquid;

and (d) the liquid’s thermophysical properties are

constant with the temperature.

Additionally, for the equations that govern the

solid mechanics for the calculation of the stresses

and the strains in the multistage centrifugal pump,

the applied assumptions are formulated as follows

(

Popov, 1999)

: (i) the material is considered

continuous, doesn't have cracks, nor cavities; (ii)

the material is homogeneous and presents the same

properties in all points; (iii) the material is

considered as isotropic; and (iv) no internal force

acts in the material before the application of the

external loads.

3.1 Liquid Flow Velocity and Pressure

The fields of liquid flow velocity and pressure are

found resolving the equations of the continuity and

the Navier-Stokes accounting for the corresponding

assumptions and using the ANSYS CFX-code

(

ANSYS inc.), based on the finite volume method (

La

Roche-Carrier et al., 2013)

. The boundary

conditions are formulated as follows: the static

pressure is given at the pump inlet, while the flow rate

is specified at the pump outlet. The frozen rotor

condition is used for the impeller-diffuser interface

per pump stage. A no-slip condition is set for the flow

at the wall boundaries.

Applying the coordinate system, the equations of

the continuity are given by:

0

z

w

y

v

x

u

(1)

where u(x,y,z), v(x,y,z) et w(x,y,z) are components of

the liquid flow velocity in the direction x, y and z.

Moreover, the equations of the Navier-Stokes are

expressed by:

222

222

2

222

222

( 2 )

eff

zx z

eff

uu u uuu

uvw

xy z

xyz

p

rv

x

vv v vvv

uvw

xy z

xyz

2

222

222

( 2 )

zy z

eff

p

ru

y

ww w www

uvw

xy z

xyz

p

z

(2)

where p is the pressure; is the density;

eff

is the

effective viscosity accounting for turbulence, it is

defined as

teff

. is the dynamic viscosity and

t

is the turbulence viscosity. It is linked to turbulence

kinetic energy k and dissipation ε.

The k- turbulence model is used in this work due

to the better convergence than with other turbulence

models.

To determinate the performance parameters of the

multistage centrifugal pump, the pump head is

described as H = (p

to

– p

ti

) / g, where p

ti

is the total

pressure at the pump inlet and p

to

the total pressure at

the pump outlet. The hydraulic power of the pump is

formulated as P

h

= QgH, where Q is the flow rate.

Moreover, the brake horsepower of the pump is given

by P

s

= T, where T is the impeller torque and is

the angular velocity. From the hydraulic power and

the brake horsepower, the efficiency of the pump can

be expressed as = P

h

/ P

s

.

3.2

Axial and Radial Forces

The axial forces is the result of unbalanced impeller

forces acting in the shaft axial direction. It is

composed of the force on the impeller’s front shroud

and hub shroud due to the static pressures on the

surface areas of the shrouds, and the momentum force

due to the change in direction of the liquid flow

through the impeller.

Furthermore, the radial force on the impeller

results from a non-uniform distribution of pressure on

the circumference of the impeller. The non-uniform

pressure distribution can be caused by: the

geometrical form of the diffuser for the multistage

centrifugal pumps; the non-symmetrical impeller

inflow; or the pump operating regime. It is to

highlight that the radial force depends on the time. Its

components are the static radial force and the

dynamic radial force. Generally, the static radial force

is greater than the dynamic radial force (Karassik and

McGuire, 1998; Gülich, 2010; Wang et al., 2013;

Numerical Study of the Axial and Radial Forces, the Stresses and the Strains in a High Pressure Multistage Centrifugal Pump

183

Watanabe, 2019; Gantar et al., 2002; Bolade and

Madki, 2015;TM.P. S.p.A. Termomeccanica Pompe,

2003; Karassik et al., 2008;

Jino, T., 1980; Abdelouahab

M.-A.,2018).

In this research, the axial and radial forces due to

the liquid flow through the multistage centrifugal

pump are determined using the ANSYS CFX-code.

3.3

Stress and Strains

The fundamental principles of the strength of

materials are accounted for the resolution of the

solid mechanics equations (

Popov, 1999

): the

superposition principle and the Saint-Venant

principle. The method of resolution of the

structural problems in the solid mechanics consists

of three steps: (i) the force analysis forces and the

equilibrium conditions; (2) the study of the

displacements and the geometric accounting; and

(iii) the application of the relations of forces and

displacements.

Neglecting the forces per unit of volume, the

equilibrium equations of elasticity in terms of three

normal and three shear stress components are

expressed by:

0

0

0

yx

xzx

xy y zy

yz

xz

z

xyz

xyz

xyz

(3)

Furthermore, the principal stresses, designed as

1

,

2

and

3

, are calculated from the

x

,

y

,

z

,

xy

,

xz

and

yz

. They are the invariants. Its values do not

depend on the orientation of the part with respect to

the xyz coordinate system. It is to highlight that the

principal stresses are available as individual result.

They are ordered such that

1

>

2

>

3

. These can be

given by:

1

1.5

2

22

10 3

1

1.5

2

22

20 3

1

1.5

2

22

30 3

1

2 cos arccos 0.5

33 3

1

2 cos arccos 0.5

33 3 3

1

2 cos arccos 0.5

33 3

JJ

J

JJ

J

JJ

J

3

(4)

where

0

222

2

222

3

000

1

3

; ;

xyz

xy yz zx xy yz zx

x y z x yz y zx z xy

xx yy zz

Jssssss

Jssssss

sss

(5)

Moreover, concerning the normal strains (

x

,

y

and

z

) and the shear strains (

xy

,

yz

and

zx

), they

can be written as:

; ;

; ;

xyz

xy yz zx

uvw

xzz

uu wv uw

x

yyzzx

(6)

where u, v and w are the displacements respectively

in the directions of x, y and z.

In addition, for the relationships between the

stresses and the strains, the generalized Hook’s law

for isotropic materials is used:

1

()

1

()

1

()

; ;

xxyz

yyzx

zzxy

xy yz

zx

xy yz zx

E

E

E

GGG

(7)

where E is the modulus of elasticity, G is the shear

modulus and is the Poisson’s ratio.

Inversely, the stresses can be expressed by:

(1 ) ( )

(1 )(1 2 )

(1 ) ( )

(1 )(1 2 )

(1 ) ( )

(1 )(1 2 )

; ;

xxyz

yyzx

zzxy

xy xy yz yz zx zx

E

E

E

GGG

(8)

The stress according to von Mises is selected for

the yield criteria. This stress is written as follows:

222

12 23 31

1

2

(9)

For the calculation of the stresses and strains,

the ANSYS-code for the structural static analysis

is used.

4 RESULTS AND DISCUSSION

Two cases are selected for examination of the shaft

behavior of a multistage centrifugal pump with

particular emphasis on the axial and radial forces, the

SIMULTECH 2020 - 10th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

184

stresses and the strains due to the liquid flow through

the pump. The impact of the diffuser in the last pump

stage on the shaft is analyzed in different operating

conditions in terms of the flow rates.

Tabs. 1-4 indicate the reference data used for the

impeller, the diffuser and the shaft. The rotation speed

of the shaft is 1800 rpm and the flow rate range is

from 300 m³/h to 900 m³/h.

Table 1: Reference data of the impeller.

Inlet blade height b

1

[mm] 56

Outlet blade hei

g

ht b

2

[mm] 44,25

Hub diameter D

h1

[mm] 84,84

Inlet diameter D

h2

[mm] 194,95

Outlet diameter D

2

[mm] 401,3

Inlet blade angle β

b1

[°] 18

Outlet blade angle β

b2

[°] 22,5

Blade thickness e [mm] 7,94

Blade number Z

b

6

Table 2: Reference data of the diffuser (front side).

Inlet blade height b

3

[mm] 41,98

Outlet blade height b

4

[mm] 68,4

Inlet diameter D

3

[mm] 401,3

Outlet diameter D

4

[mm] 459

Inlet blade angle α

3b

[°] 16,44

Blade thickness e

3

[mm] 6,04

Blade numbe

r

Z

Le

11

Table 3: Reference data of the diffuser (rear side).

Return vane number Z

R

11

Outlet return vane height b

5

[mm]

24,4

Diameter at the inlet of the return

vane D

3

[mm]

459

Blade angle at the inlet of the

return vane α

5

[°]

17,04

Blade angle at the outlet of the

return vane α

6

[°]

93,84

Blade thickness of the return vane

e

3

[mm]

6,04

Table 4: Reference data of the shaft.

Length L [mm]

1000

Diameter d [mm] 71,12

Furthermore, the properties of the standard steel

and the water considered are indicated in Tabs. 5 and

6.

Table 5: Properties of the standard steel.

Module of the Young [Pa] 2x10

11

Poisson ratio 0,3

Com

p

ressibilit

y

module [Pa] 1,6667x10

11

Shear module [Pa] 7,6923x10

11

Resistance coefficient [Pa] 9,2x10

8

Ductility coefficient [Pa] 10

9

Yield stren

g

th [Pa] 2,5x10

8

Ultimate tensile stren

g

th [Pa] 4,6x10

8

Densit

y

[k

g

/m

3

] 7850

Table 6: Properties of water in 25 °C.

Density

[kg/m

3

]

Thermal

expansion

coefficient [

K

-1

]

Kinematic

viscosity

[m

2

/s]

997 2,57x10

-1

0,884x10

-6

4.1 Case Study

4.1.1 Effect of the Flow Rate

To analyze the effect of the flow rate on the radial

forces, the stresses and the strains on the shaft of the

4-stage centrifugal pump, the flow rate from 300 m³/h

to 900 m³/h are selected keeping the reference data as

constant.

Figs. 4-9 show the pump head, the brake

horsepower, the efficiency, the radial force, the stress

and the strain as a function of the flow rate. From

these figures, it is observed that the radial force

decreases and increases with growing flow rate. It is

the lowest in the best efficiency point (BEP) or in the

best efficiency zone, as indicated in Fig. 4. This can

be explain by the fact that the impeller is designed for

constant velocity near the best efficiency flow rate,

which yields an uniform static pressure around the

periphery of the impeller, at this flow rate. However,

as the flow rate moves away from the BEP, the

pressure distribution around the impeller changes,

resulting in high radial loads on the pump bearings.

Relating to the stress on the pump shaft, it

decreases, then it increases with rising flow rate. The

strain follows the same trend as the stress. At the low

flow rate and the high flow rate, the radial load on the

pump shaft is greater, this can lead to important stress

and the strain on the pump shaft. Thus, Figs. 4-9 are

relevant to better understanding the relationship

between the pump performances and the radial forces,

the stresses and the strains on the pump shat.

Numerical Study of the Axial and Radial Forces, the Stresses and the Strains in a High Pressure Multistage Centrifugal Pump

185

Figure 4: Head, stress, and radial force versus flow rate.

Figure 5: Brake horsepower, stress, and radial force versus

flow rate.

Figure 6: Efficiency, stress, and radial force versus flow

rate.

Figure 7: Head, strain, and radial force versus flow rate.

Figure 8: Brake horsepower, strain, and radial force versus

flow rate.

Figure 9: efficiency, strain, and radial force versus flow

rate.

4.1.2 Effect of the Diffuser in the Last Pump

Stage

To analyze the effect of the diffuser in the last pump

stage on the axial and radial forces on the shaft, two

models of the 4-stage centrifugal pump are selected

whose one having a diffuser in the last pump stage.

The reference data are kept as constant. Figs. 10-

14 present the pump head, the brake horsepower, the

efficiency, and the axial force and radial forces as a

function of the flow rate.

Fig. 10 indicates that the pump head with a

diffuser in the last pump stage is greater than the

without diffuser case. This can be explained by the

fact that a diffuser provides greater static pressure.

Moreover, it can be observed in Fig. 11 that the brake

horsepower for both cases is practically identical,

whereas the efficiency for the case of the last pump

stage with the diffuser is relatively higher than the

without diffuser case as shown in Fig. 12.

Relating to the axial force, it can be seen that the

use of the last pump stage without diffuser leads to

the greater axial force as illustrated in Fig. 13. In

addition, Fig. 14 presents the fact that the radial force

for the case of the last pump stage with the diffuser is

higher than the without diffuser case from 300 m³/h

to about 460 m³/h, whereas the radial force for the

case with the diffuser is lower than the without

diffuser case for the flow rate more than 460 m³/h.

Thus, the in-depth knowledge of the resulting

axial and radial forces acting on the shaft impellers of

the multistage centrifugal pump is essential to better

design and to optimize the shat bearings accounting

for the last pump stage configuration.

Figure 10: Head versus flow rate.

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186

Figure 11: Brake horsepower versus flow rate.

Figure 12: Efficiency versus flow rate.

Figure 13: Axial force versus flow rate.

Figure 14: Radial force versus flow rate.

4.2 Comparison of the Results

4.2.1 Pump Performances According to the

Diffuser Type

The results of the pump head, the brake horsepower

and the efficiency relating to the diffuser of type 1 in

Fig. 1 are compared with the obtained results using

the diffuser of type 2 in Fig. 3.

Fig. 15 shows that the head achieved of the

diffuser in Fig. 1 is higher than that obtained with the

diffuser in Fig. 2 for the flow rate from 300 m³/h to

about 660 m³/h. Furthermore, Fig. 16 indicates that

the brake horsepower for the flow rate from 300 m³/h

to 900 m³/h for the diffuser in Fig. 1 are the highest.

In addition, the efficiency for the diffuser in Fig.

2 is better as depicted in Fig. 17. This comparison is

relevant to illustrate that the performances of a

multistage centrifugal pump can depend, inter alia, on

the used diffuser type.

Figure 15: Head versus flow rate.

Figure 16: Brake horsepower versus flow rate.

Figure 17: Efficiency versus flow rate.

4.2.2 Stress Results

The results of numerical simulations for the stresses

in this study are compared with the results for the

stresses using the classical equation for the first pump

stage. While confronting the curves in Fig. 18, the

trend of the classical and the numerical results are

similar. The gaps between the curves can be justified

by the different hypotheses used in terms of classical

calculations and numerical simulations.

Numerical Study of the Axial and Radial Forces, the Stresses and the Strains in a High Pressure Multistage Centrifugal Pump

187

Figure 18: Stress versus flow rate.

5 CONCLUSION

This study focused on the numerical investigation of

the axial and radial forces, the stresses and the strains

on the shaft of a multistage centrifugal pump due to

the liquid flow through this pump. The continuity and

the Navier-Stokes equations are used for the liquid

flow in the pump. Moreover, the equations of the

stresses and the strains are applied for the shaft and

the impellers. The ANSYS CFX-code and the static

structural analysis code are used to determinate,

respectively, the fields of velocity and pressure for the

liquid flow, and the axial and radial forces, the

stresses and the strains for the shaft. Numerical

simulations are accomplished to examine the shaft

behavior in terms of the axial and radial forces acting

in the impellers, the stresses and the strains on the

shaft. For these simulations, inter alia, the diffuser in

the last pump stage is considered as parameter. This

is to find a relevant relationship between the pump

performances, the axial and radial loads, the stresses

and the strain on the shaft. The numerical results for

the pump performances using two diffuser types are

compared. In addition, the obtained numerical results

for the stresses on the shaft are validated considering

the first pump stage by means of the results found

applying the classical equations.

ACKNOWLEDGMENTS

The authors are grateful to the Technosub Inc.,

Industrial pumps manufacturing and distribution

(Rouyn-Noranda, Quebec, Canada).

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