NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID

FLOW IN THREE-DIMENSIONAL IMPELLER

AND IMPELLER-VOLUTE OF A CENTRIFUGAL PUMP

Massinissa Djerroud, Guyh Dituba Ngoma and Walid Ghie

University of Quebec in Abitibi-Témiscamingue, Department of Applied Sciences

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

Keywords: Centrifugal Pump, Impeller, Volute, Navier-Stokes, Computational Fluid Dynamics, Modeling and

Simulation.

Abstract: In this study, the effects that the blade width, the blade number, and the impeller diameter have on the

steady state liquid flow in three-dimensional impeller, and combined impeller and volute were investigated.

The continuity and Navier-Stokes equations with the k-ε turbulence model and the standard wall functions

were used by mean of ANSYS-CFX code taking into account of the suction pressure variation as a function

of the valve volume flow rate. The achieved results reveal that the selected key design parameters have an

impact on the head, the brake horsepower and the overall efficiency of the centrifugal pump. To valid the

developed approach, the results of numerical simulation were compared with the experimental results

considering a special case of combined impeller and diffuser.

1 INTRODUCTION

Nowadays, centrifugal pumps are widely used in

industrial and mining enterprises. One of the most

important components of a centrifugal pump is the

impeller (Peng, W. W., 2008). The performance

characteristics related to the pump rely a great deal

on the impeller. To achieve better performance for a

centrifugal pump, design parameters such as the

number of blades, blade angles, the blade width, the

blade height, the impeller diameter and the volute

radius must be accurately determined, due to the

complex liquid flow through a centrifugal pump.

This liquid flow is three-dimensional and turbulent.

It is therefore important to be aware of the liquid

flow’s behavior when traveling through an impeller.

This can be done by accounting for the volute in the

planning, design, and optimization phases at

conditions of design and off-design. Many studies

have been carried out on the liquid flow through a

centrifugal pump (Zhou, W. et al., 2003;

Derakhshan, S., et al., 2008; Spence, R., et al., 2008;

Amaral-Teixeira, J., et al., 2008; Cheah, K.W.,

2007; Lee, T. S., et al., 2007; Wen-Guang, L., et al,

2002; LIU, H., et al., 2010; González, J., et al.,

2007; Asuaje, M., et al., 2005; Kaupert, K, et al.,

1999), where the effects of the number of impeller

blades on the pump’s performance were examined

experimentally by Wen-Guang, L., et al, 2002 and

Liu, H., et al., 2010. González, J., et al., 2007 had

numerically investigated the dynamic effects due to

the impeller-volute interaction within a centrifugal

pump, whereas the effects of the volute on velocity

and pressure fields were examined by Asuaje, M., et

al., 2005 and Kaupert, K, et al., 1999. The analysis

of previous works clearly demonstrated that research

results obtained are specific to the centrifugal pump

design parameter values and thus cannot be

generalized. In this work therefore a numerical study

was performed using a finite volume method

according to the CFX code (Ansys inc., 2008) to

gain further insight into the characteristics of the

three-dimensional turbulent liquid flow through an

impeller and a combined impeller and volute

accounting for suction pressure variation as a

function of the valve volume flow rate, while also

considering various flow conditions and pump

design parameters: blade width, blade number and

impeller outer diameters.

2 GOVERNING EQUATIONS

The models selected for the liquid flow in an

Djerroud M., Dituba Ngoma G. and Ghie W..

NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID FLOW IN THREE-DIMENSIONAL IMPELLER AND IMPELLER-VOLUTE OF A

CENTRIFUGAL PUMP.

DOI: 10.5220/0003566800930102

In Proceedings of 1st International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2011), pages

93-102

ISBN: 978-989-8425-78-2

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

impeller and a combined impeller and volute are

depicted in Fig. 1, placing greater emphasis on the

fluid domain.

Fluid domain

a b

Figure 1: Models of a) impeller and b) impeller-volute.

In the governing equations for liquid flow in the

centrifugal pump components, the following

assumptions were made: (i) a steady state, three-

dimensional and turbulence flow using the k-ε

model; (ii) it was an incompressible liquid; (iii) it

was a Newtonian liquid; and (iv) the liquid’s

thermophysical properties were constant with

temperature.

To account for these assumptions, the theoretical

analysis of the liquid flow in an impeller, and a

combined impeller and volute, was based on the

continuity and Navier-Stokes equations (Ansys inc,

2008). For the three-dimensional liquid flow through

the components of a centrifugal pump as shown in

Fig. 1, the continuity equations are expressed by:

0U. =∇

(1)

and the Navier–Stokes equations are given by:



B))U(U.(p)UU.(

eff

+∇+∇∇μ+−∇=⊗∇ρ

GGGG

(2)

where

()()()()

z,y,xw,z,y,xv,z,y,xuUU

is the liquid

flow velocity vector, p is the pressure, ρ is the

density, μ

eff

is the effective viscosity accounting for

turbulence,⊗ is a tensor product and B is the source

term. More particularly, for flows in an impeller

rotating at a constant speed ω, the source term can

be written as follows:

()

(

)

rxxUx2B

ωω+ωρ−=

(3)

where

is the location vector.

In addition, μ

eff

is defined as:

teff

μ=μ

(4)

where μ is the dynamic viscosity and μ

is the

turbulence viscosity.

According to the k-ε turbulence model, μ

is linked

to turbulence kinetic energy κ and dissipation ε via

the relationship:

−

ερ=μ

(5)

where C

is a constant.

The values for κ and ε come directly from the

differential transport equations for turbulence kinetic

energy and turbulence dissipation rates:

ρε−+∇

+μ∇=ρ∇

p]k).[()kU.(

(6)

)CpC(

]).[()U.(

2k1

ρε−

+ε∇

+μ∇=ερ∇

εε

(7)

where C

ε1

, C

ε2

and σ

are constants. p

is the

turbulence production due to viscous and buoyancy

forces, which is modeled using:

kbt

p)kU.3(U.

)UU.(Up +ρ+∇μ∇−∇+∇∇μ=

(8)

ρ∇

ρσ

−=

.gp

(9)

where p

can be neglected for the k-ε turbulence

model.

Moreover, for the modeling of flow near the

wall, the logarithmic wall function is used to model

the viscous sub-layer (Ansys Inc., 2008).

2.1 Impeller

Three velocity types are involved when considering

the flow through a centrifugal pump impeller: the

tangential velocity U = r ω, the relative velocity W,

and the absolute velocity V. The last is expressed in

vector format as follows:

WUV

(10)

Fig. 2 shows the velocity triangles at the impeller

inlet and outlet at the design conditions where the

liquid enters and leaves the impeller at the blade

angles β

and β

. The components of

and

the direction of

are V

(swirl velocity), and W

respectively, while those normal to

are V

and W

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

Figure 2: Velocity triangles.

Moreover, according to the Euler equation (Peng,

W. W., 2008), the energy transfer per unit mass of

flow for a centrifugal pump can be formulated as:

1u12u2i

VUVUgH

−

(11)

where H

is the ideal pump total head.

Neglecting the swirl velocity at the impeller inlet,

Eq. 11 can be expressed as follows

2u2i

VUgH =

(12)

When accounting for the hydraulic efficiency, η

the actual pump head rise is given by:

HH η=

(13)

Also, the hydraulic efficiency can be calculated

using the following empirical formula (Peng, W. W.,

2008):

25.0

)Q03.15859( 8.01

−

−=η

(14)

where Q is the volume flow rate in m

/s. It is given

by Q = V

A with A as the flow passage area normal

to the meridional direction.

Since in reality the flow through a centrifugal

pump is turbulent and three dimensional, the actual

relative flow direction at the impeller exit is

different from that of the blade angle. As depicted in

Fig. 3, the flow angle β

is always less than the

blade angle β

b2.

. This can lead to secondary flows in

the flow passage, from the pump inlet through to

discharge (Peng, W. W., 2008).

Figure 3: Flow angle and blade angle.

As such, the slip factor μ

is used to take into

account the difference between β

and β

, which is

formulated as:

=μ

(15)

where V

is the actual swirl flow velocity at the

impeller exit and

is the ideal swirl flow velocity

at the impeller exit.

In addition, the slip velocity is given by:

2u2u

V −=−=Δ

(16)

Taking into account the slip factor, Eq. 13 can be

expressed as:

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

μη=

tanA

(17)

Moreover, to account for the leakage flow from the

impeller, the volumetric efficiency is defined by:

QQ +

=η

(18)

where Q

is the leakage flow from the impeller exit

back to the inlet through the clearance.

In addition, the pump’s mechanical efficiency is

formulated as follows:

imp

=η

(19)

where P

is the brake horsepower and P

imp

the power

delivered by the impeller to the fluid.

is globally expressed by:

ω=

CPPPPPPs

dfmLfh

(20)

where C is the pump shaft torque, P

is the

centrifugal pump horsepower. It is expressed as:

QgHP

(21)

is the loss power due to the friction, which is

given by:

(

)

HHQgP

−

(22)

is the loss power due to leakage, which is defined

as:

iLL

gHQP

(23)

is the friction loss power in bearings and seals

and P

is the disk friction power due to impeller

shrouds.

NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID FLOW IN THREE-DIMENSIONAL IMPELLER AND

IMPELLER-VOLUTE OF A CENTRIFUGAL PUMP

imp

in Eq. 19 can be formulated as follows:

dfmsimp

PPPP −−=

(24)

Furthermore, Eq. 24 can be rewritten as :

()

iLimp

gHQQP +ρ=



(25)

Accounting for Eq. 24, Eq.19 can be expressed as:

dfms

PPP −−

=η



(26)

Thus, the overall efficiency of a centrifugal pump

can be formulated as:

=η

(27)

Finally, the overall efficiency can also be formulated

in terms of the other efficiencies as:

mvh

ηη=η

(28)

2.2 Volute Parameters

Fig. 4 shows the parameters of a volute defined by

the radius of volute basic circle r

, the radius of

volute cut water circle r

, the volute angle α

, the

volute cross-sectional area A

,which depends of the

angle ϴ, and the volute outlet cross-sectional area A

(Peng, W. W., 2008).

Figure 4: Impeller-volute.

The average flow velocity at the volute outlet is

given by:

gH2KV

(29)

where the volute velocity constant K

is an empirical

parameter correlated with the specific speed, as

shown in Fig. 5 along with other volute parameters

such as the volute angle α

and the volute basic

circle diameter D

Figure 5: Volute velocity constant, volute angle and

diameter of volute basic circle versus specific speed.

In addition, the volute cross-sectional area A

can be

formulated as:

CL2

(30)

where r

is the centroid radius of the volute cross-

sectional area, L is the angular momentum of flow at

the impeller outlet which can be expressed by

2u2

VrL

. C≅ 0.95 to account for friction loss.

To solve Eqs. 1 and 2 numerically while

accounting for the boundary conditions and the

turbulence model κ-ε, the computational fluid

dynamics ANSYS-CFX code, based on the finite

volume method, was used to obtain the liquid flow

velocity and the pressure distributions. In the cases

examined involving the impeller, and combined

impeller and volute, the boundary conditions were

formulated as follows: the static pressure provided

was given at the inlet, while the flow rate provided

was specified at the outlet. The frozen rotor

condition was used for the impeller-volute interface.

A no-slip condition was set for the flow at the wall

boundaries. Fig. 6 shows the inlet, outlet and

interface domains for the selected centrifugal pump

components.

Figure 6: Domains of inlet, outlet and interface.

Furthermore, the ANSYS-CFX code comprises by

geometry (DesignModeler), CFX-pre, CFX-solver

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

and CFX-post modules. According to the applied

ANSYS-CFX code, Fig. 7 depicts the steps

specifically used to obtain the numerical simulation

results from the geometry models to the numerical

models for the impeller, and the combined impeller

and volute.

Figure 7: Steps from 3D geometry model to numerical

simulation results.

3 RESULTS AND DISCUSSION

Water was used as the working liquid for all

simulations run and for use in this study considered

to have the following reference values: temperature

of 25 °C for water, density of ρ = 997 kg/m³ and

dynamic viscosity of μ = 8.899 x 10

-4

Pa s. The main

data for the reference impeller and volute are given

in Tabs. 1 (Technosub Inc) and 2 (Peng, W. W.,

2008).

Table 1: Main data of the reference impeller.

Inlet diameter [mm] 145

Outlet diameter [mm] 320

Inlet blade angle [°] 11.69

Outlet blade angle [°] 28

Inlet blade width [mm] 12

Blade thickness [mm] 4

Number of blades 7

Rotating speed [rpm] 1800

Table 2: Main data of the reference volute.

Volute

angle [°]

Volute

radius

[mm]

Volute

angle [°]

Volute

radius

[mm]

0 165 225 278.96

45 183.79 270 302.76

90 207.58 315 326.55

135 231.38 360 350.35

180 255.17

Accounting for the fact that the pump rotating speed

was constant, the volume flow rate was controlled

by a regulator valve, which had an influence on the

pressure at the pump inlet, as shown in Fig. 8

(Technosub Inc.). This was accounted for in the

numerical simulations performed.

Figure 8: Pressure at the pump inlet versus valve volume

flow rate regulation.

3.1 Case Studies

Three key design parameters of a centrifugal pump

were selected for an examination of their effects

mainly on the pump performance: impeller blade

width without volute, impeller blade number with

volute, impeller diameter with volute.

3.1.1 Effect of Impeller Blade Width

To investigate the effect that the impeller blade

width has on the pump head, the pump brake

horsepower and the pump overall efficiency, the

blade widths of 4 mm, 10 mm and 15 mm were

selected, while the other parameters were keep

constant. Fig. 9 shows the pump head as a function

of the volume flow rate, illustrating that the pump

head decreases with increased blade width. This is

due augmenting the liquid pressure drop with

increasing blade width. Also, the required pump

brake horsepower decreases when the blade width

rises, as indicated in Fig. 10. The corresponding

overall efficiency curves are shown in Fig. 11,

illustrating that the blade width’s impact on the

overall efficiency is more pronounced in at high

volume flow rates. In other words, the overall

efficiencies for the three blade widths decrease

rapidly to the right side of the best efficiency point

(BEP) and the lowest overall efficiency is obtained

when e = 15 mm.

NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID FLOW IN THREE-DIMENSIONAL IMPELLER AND

IMPELLER-VOLUTE OF A CENTRIFUGAL PUMP

Figure 9: Pump head versus volume flow rate (blade

width).

Figure 10: Pump brake horsepower versus Volume flow

rate (blade width).

Figure 11: Overall efficiency versus volume flow rate

(blade width).

In addition, Figs. 12 and 13 show the static

pressure contour and the liquid flow velocity vector

for Q = 0.065 m

/s, illustrating that the static

pressure difference between the impeller outlet and

inlet decreases with increasing blade width, due to

the increase in liquid flow velocity at the impeller

outlet. The average liquid flow velocities at the

impeller outlet are 15.92 m/s, 19.10 m/s and 20.57

m/s for e = 4 mm, e = 10 mm and e = 15 mm,

respectively.

a) e = 4mm b) e = 10 mm

∆P= 3,292 10

Pa ∆P= 2,547 10

c) e =15 mm

∆P= 2,056 10

Figure 12: Static pressure contour (blade width).

a) e = 4 mm b) e =10 mm

c) e = 15mm

Figure 13: Liquid flow velocity vector (blade width).

3.1.2 Effect of Impeller Blade Number

When Accounting for Volute

To analyze the effect of the impeller blade number

on the pump head, the pump brake horsepower and

the overall pump efficiency, three impellers whose

blade number were 5, 7 and 9 were selected, while

the other parameters were kept constant. Fig. 14

shows the pump head as a function of the volume

flow rate, illustrating that the pump head increases

with a greater blade number. This is explained by

the decrease in the liquid pressure drop in the flow

passage with an augmented impeller blade number,

keeping the same total volume flow rate. Also, as

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

shown in Fig. 15, the pump brake horsepower

increases relative with the augmented blade number.

This is due to the increase in the request pump shaft

torque, as the pump blade number also increases.

Figure 14: Pump head versus volume flow rate (parameter:

impeller blade number).

Figure 15: Brake horsepower versus volume flow rate

(impeller blade number).

In addition, Fig. 16 shows the overall efficiency

curves, showing that the impeller having 5 blades

has the lowest overall efficiency.

Figure 16: Overall efficient versus blade Number (blade

number).

Moreover, Figs. 17 and 18 depict the corresponding

static pressure contour and liquid flow velocity

vector for Q = 0.065 m

/s, respectively. These

figures thus clearly show the increased static

pressure difference between the volute outlet and the

impeller inlet relative to the increasing blade

number. This confirms the reduction in the liquid

flow velocity at the impeller outlet relative to the

greater blade number, as represented in Fig. 18

where the average liquid flow velocities at the

impeller outlet were 16.06 m/s, 15.40 m/s et 12.53

m/s for 5 blades, 7 blades et 9 blades, respectively.

a) 5 blades b) 7 blades

∆P = 3,096 10

Pa ∆P = 3,60510

Pa

c) 9 blades

∆P= 4,223 10

Figure 17: Static pressure contour (impeller blade

number).

a) 5 blades b) 7 blades

c) 9 blades

Figure 18: Vectors of liquid flow velocity contour

(impeller blade number).

NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID FLOW IN THREE-DIMENSIONAL IMPELLER AND

IMPELLER-VOLUTE OF A CENTRIFUGAL PUMP

3.1.3 Effect of Impeller Diameter

The impeller outlet diameter values of 285 mm and

320 mm were selected to investigate their effects on

pump performance when keeping the other

parameters constant. Fig. 19 shows that the pump

head increases with increasing impeller diameter,

which can be explained by the fact that the liquid

static pressure drop in impeller decreases with

increasing impeller diameter. In other words, for a

given volume flow rate, the pressure difference

between the volute outlet and the impeller inlet is

higher for an impeller with a greater diameter. In

addition, Fig. 20 shows that the brake horsepower

increases relative to the increasing impeller

diameter, due to the requested augmented impeller

shaft torque relative to the size of the impeller

diameter.

Figure 19: Pump head versus volume flow rate (impeller

diameter).

Figure 20: Brake horsepower versus volume flow rate

(parameter: impeller diameter).

Moreover, the corresponding overall efficiency

curves shown in Fig. 21 indicate that the impeller

having a great diameter has better overall efficiency

with volume flow rates greater than 0.02 m

/s.

Figure 21: Overall efficiency versus volume flow rate

(impeller diameter).

Additionally, Figs. 22 and 23 depict the static

pressure contour and the liquid flow velocity vector.

Fig. 22 thus clearly shows the correlation between

the increase in static pressure difference between the

volute outlet and the impeller inlet, and the increase

in the impeller diameter. The average liquid flow

velocities reached at the impeller outlet are 12.51

m/s and 15.40 m/s for D

= 285 mm and D

=320

mm, as shown in Fig. 23, respectively.

a) D = 285 mm b) D=320mm

∆P= 2,691 10

Pa ∆P= 3,386 10

Figure 22: Static pressure contour (impeller diameter).

a) D = 285 mm b) D=320mm

Figure 23: Vectors of liquid flow velocity (impeller

diameter).

3.2 Model Comparison

Since the experimental results of the case of a

combined impeller and diffuser were available from

Technosub Inc., the developed numerical approach

was validated transforming the case of a combined

impeller and volute to a combined impeller and

SIMULTECH 2011 - 1st International Conference on Simulation and Modeling Methodologies, Technologies and

Applications

100

diffuser. When taking into account of experimental

boundary conditions for the numerical simulations

run, Figs. 24-26 show the comparison between the

experimental and the numerical results for the pump

head, the brake horsepower and the overall

efficiency. The discrepancies observed in these

figures could be explained by the fact that lost

mechanical power, power lost due to leakage and the

pump casing were not taken into account in the

numerical simulations carried out. The brake

horsepower for experimental pump brake was

therefore higher than the numerical brake

horsepower obtained, as illustrated in Fig 25.

Figure 24: Pump head versus volume flow rate (numerical

and experimental).

Figure 25: Brake horsepower versus volume flow rate

(numerical and experimental).

Figure 26: Overall efficiency versus volume flow rate

(numerical and experimental).

4 CONCLUSIONS

In this research work, a steady state liquid flow in

three-dimensional impeller, and combined impeller

and volute of a centrifugal pump was numerically

investigated using the ANSYS-CFX code. The

obtained results demonstrate, among others, that the

pump head and the brake horsepower increase with

increasing impeller blade number, while they

decrease with increasing impeller blade width. Also,

the interaction between the impeller and the volute

reveals that the decrease of the impeller outer

diameter keeping the volute dimensions constant

leads to the reduction of the pump head and the

brake horsepower. The pump overall efficiency is

also influenced by the selected parameter. A

relatively good agreement was observed comparing

the developed numerical approach with the

experimental results for special case of the combined

impeller and diffuser. Further research is planned to

develop a generalized numerical approach for

optimizing of combined impeller, diffuser and

volute, accounting for experimental data from pump

manufacturers.

ACKNOWLEDGEMENTS

The authors are grateful to the Foundation of

University of Quebec in Abitibi-Témiscamingue

(FUQAT) and the company Technosub inc.

REFERENCES

Peng, W. W., 2008, Fundamentals of turbomachinery,

John Wiley and Sons. Hoboken, New Jersey.

Zhou, W., Zhao, Z., Lee, T. S. and Winoto, S. H., 2003,

Investigation of Flow Through Centrifugal Pump

Impellers Using Computational Fluid Dynamics.

Intern. J. of Rotating Machinery, 9(1): 49–61.

Derakhshan, S., Nourbakhsh, A., 2008, Theoretical,

numerical and experimental investigation of

centrifugal pumps in reverse operation. Exper.

Thermal and Fluid Sc. 32, 1620–1627.

Spence, R., Amaral-Teixeira, J., 2008, Investigation into

pressure pulsations in a centrifugal pump using

numerical methods supported by industrial tests.

Computers and Fluids 37, 690–704.

Cheah, K. W., Lee, T. S., Winoto, S. H., and Zhao, Z. M.,

2007, Numerical Flow Simulation in a Centrifugal

Pump at Design and Off-Design Conditions. Hindawi

Publishing Corporation International Journal of

Rotating Machinery, Volume 2007, Article ID 83641,

8 pages.

NUMERICAL PARAMETRIC STUDY OF COMPLEX LIQUID FLOW IN THREE-DIMENSIONAL IMPELLER AND

IMPELLER-VOLUTE OF A CENTRIFUGAL PUMP

101

Wen-Guang, L., Fa-Zhang, S. and Cong, X., 2002,

Influence of the number of impeller blades on the

performance of centrifugal oil pumps. World Pumps,

Volume 2002, Issue 427, Pages 32-35.

Liu, H., Wang, Y., Yuan, S., Tan, M., and Wang, K.,

2010, Effects of Blade Number on Characteristics of

Centrifugal Pumps, Chinese j. of mech. eng., Vol. 23,

No. 6.

González, J., Fernández-Francos, J., Blanco, E. and

Santolaria-Morros, C. 2002, Numerical simulation of

the dynamic effects due to impeller-volute interaction

in a centrifugal pump. Transactions of the ASME, J. of

Fluids Engineering, vol. 124, no. 2, pp. 348–355.

Asuaje, M., Bakir, F., Kouidri, S., Kenyery, F., Rey, R.,

2005, Numerical Modelization of the Flow in

Centrifugal Pump: Volute Influence in Velocity and

Pressure Fields. Intern. J. of Rotating Machinery: 3,

244–255.

Kaupert, K. A., Staubli, T. 1999, The Unsteady Pressure

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of the ASME, J. of Fluids Engineering, Vol. 121, 621-

626.

Ansys inc., 2008, ANSYS-CFX, User Manual, USA.

Technosub Inc., http://technosub.net/.

APPENDIX: NOMENCLATURE

B source term (Nm

-3

)

b height (m)

d diameter (m)

e width (m)

g acceleration of gravity (ms

-2

)

H head (m)

P power (W)

p pressure (Nm

-2

)

turbulence production due to viscous and

buoyancy forces

Q volume flow rate (m

-1

)

r radial coordinate (m)

V absolute velocity (ms

-1

)

v flow velocity in y direction (ms

-1

)

U velocity or tangential velocity (ms

-1

)

u flow velocity in x direction (ms

-1

)

W relative velocity (ms

-1

)

w flow velocity in z direction (ms

-1

)

x x-coordinate (m)

y y-coordinate (m)

z z-coordinate (m)

Greek symbols

α angle between V and U (°)

β blade angle between W and U (°)

Δ difference

ε turbulence dissipation (m

-3

)

η efficiency

κ turbulence kinetic energy (kg m

-2

)

θ angle (°)

ρ fluid density (kg m

-3

)

μ dynamic viscosity (Pa s)

eff

effective viscosity (Pa s)

slip factor

turbulence viscosity (Pa s)

ω angular velocity (rad s

-1

)

Subscripts

1 inlet

2 outlet

3 volute outlet

b blade

df disk friction

f flow

h hydraulic

i inlet or ideal

imp impeller to fluid

L leakage

m mechanical

o outlet

r radial or perpendicular to the vector U

s shaft or slip

u direction of vector U

v volumetric or volute

w wall

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Applications

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