CFD Modeling Liquid Viscosity Effect on

Centrifugal Pump Performance

Wen-Guang LI

Department of Fluid Machinery, Lanzhou University of Technology

287 Langongping Road, 730050 Lanzhou, P R China

Corresponding author:

Dr Weng-Guang Li

Department of Fluid Machinery

Lanzhou University of Technology

287 Langongping Road

730050 Lanzhou, China

E-mail: liwg38@yahoo.com.cn

1

ABSTRACT

The hydraulic performance of a centrifugal pump handling water and viscous oils was

investigated numerically by using a CFD code based on a steady or unsteady, three-dimensional,

incompressible turbulent flow, where the turbulence effect is involved with the standard

turbulence model combined with the non-equilibrium wall function in this work. The flow passages of

the pump are subject to a particular surface roughness. The effect of liquid viscosity on the impeller

and pump performance was indicated by examining the hydraulic parameters, such as theoretical

head, hydraulic efficiency, slip factor, hydraulic loss coefficient etc in terms of flow rate. The

unsteady flow, wall function and turbulence flow models effects on the performance are also

indicated. The computed pump performance has been compared with the experimental data. The

results showed that the influence of the relative position between volute tongue tip and impeller blade

trailing edge on the performance is less significant. The volute offers increasing influence on the flow

at the impeller exit at low flow rate, causing reduced impeller theoretical head at much low flow rate.

The “sudden-rising head effect” is confirmed to exist and high viscosity and particular large surface

roughness are responsible for this effect.

Key words: centrifugal pump impeller volute viscosity performance roughness CFD

2

1 IntroductionFor recent years, CFD (computation fluid dynamics) has become one vital tool of solving

complicated flow problems in research and engineering communities. As a result, it has found

substantially increasing applications in fluid dynamics of rotor-dynamic pumps. The steady 3D

incompressible turbulent flows in isolated centrifugal impellers have been studied numerically in [1-

5]. The turbulent flows in centrifugal diffuser pumps in [6, 7] and those in centrifugal volute pumps in

[8-15] have been conducted by existing CFD codes, accordingly the performance curves against flow

rate were also established based on the CFD results. In some work, for example, in [10, 11, 13-15],

the computed performance curves have been compared with those measured. Nevertheless, these

contributions shed light on the understanding of fluid dynamics and the performance optimization of

rotor-dynamic pumps. It is very interesting to note that in these numerical investigations, water is

usually used as a working liquid and the roughness of wetted surface of flow passages is specified to

be zero. However, in the practical applications of pump, the liquids handled are in a wide range.

Further, the wetted surface roughness perhaps is rough enough sometimes. Therefore, the

computations of flow in centrifugal pumps, where liquids other than water and rough surface are

involved, should be targeted.

On one hand, for many years, significant contributions have been denoted to the investigation

into the performance and flow in centrifugal pumps handling viscous oils in [16-23]. In more recent

years, the hydraulic characteristics of the centrifugal pumps with respective two different impellers

having 20o and 44 o blade exit angles were measured when water and highly viscous oils were pumped

[24]. The time-averaged turbulent flows of water and viscous oils in the impellers and volute of a

centrifugal pump were investigated experimentally by using LDV in [25, 26].

On the other hand, a laminar steady flow caused from high-viscosity of fluid in an isolated

impeller was examined numerically in [27]. A time-averaged steady turbulent flow of viscous oil in a

centrifugal pump impeller was calculated by means of a CFD code-PHOENICS in [28], the computed

results were compared with those given by using LDV measurements. Unfortunately, because of the

limitation in both computer hardware and code itself, the calculated results were poor. The highly

viscous effect of fluid on pump affinity laws was addressed experimentally and numerically by a

small centrifugal blood pump in [29]. The numerical simulations indicated the Reynolds number is

necessary to insure accurate scaling in the small pump. It should be noticed that these numerical

investigations fail to examine detailed variations of hydraulic performance and internal flow field of

centrifugal pumps in terms of liquid viscosity, causing the relations of hydraulic loss in the impeller

and volute with liquid viscosity are unavailable. Moreover, an effective comparison of performance

between experiment and numerical outcomes isn’t made. So that the numerical computations of

viscous oil flow and corresponding hydraulic performance of centrifugal pumps need to be

investigated in detail. In this paper, we tend to estimate numerically the performance in a centrifugal

pump with an impeller blade exit angle of 20o and a volute when the pump transports water and

3

mechanical oils with various kinematical viscosities and densities by using a CFD code-FLUENT to

investigate effects of the viscosity on the performance on a theoretical basis.

For a centrifugal volute pump, an interaction between impeller blades and volute tongue exists.

Nevertheless, this interaction should be taken into account in the CFD computations. In FLUENT, the

interaction can be handled with both MRF (multiple reference frame) method and sliding-mesh

technique. Currently, two routines can be applied to deal with the interaction: one is steady MRF

method (frozen rotor method in other codes) by which steady flow computations are carried out at a

series of relative position between blade and volute tongue; one is unsteady MRF method by which

unsteady flow calculations are launched with the aid of the sliding-mesh technique. For the

unsteadiness of flow is ignored in the steady MRF method, it is less accurate. In the unsteady MRF

method, however, both the rotational effect of impeller and the steadiness of flow are involved, the

method is more accurate than the former. Further, as a reward, it is able to offer time-dependent flow

variables and pump performance. In general, the time consumed to get a periodical solution with the

unsteady MRF method is as long as about 50 times the steady one [11]. Consequently, the unsteady

MRF method perhaps is applicable in the flow and performance computation of a pump at just a few

operating points. If we are just interested in steady flow and steady performance of a pump, it should

be practical to make use of the steady MRF method in the corresponding computations.

In the paper, we mainly tend to have theoretical effects of liquid viscosity on the performance of

a centrifugal pump with volute in a steady manner. However, we also attempt to a little bit involve

some critical factors, such as flow steadiness, wall function and turbulence model effects to make sure

the viscosity effects have been identified correctly.

2 Computational Models

2.1 Flow Domains

An end-suction, single-stage, centrifugal pump was severed as the computational model, whose

both performance and flow field were investigated experimentally already in [24-26]. The pump

specifications at the design point are: flow rate =25m3/h, head =8m, rotational speed

=1450r/min, specific speed =1317 (USA) =93 (China1). The impeller geometrical parameters are as

follows: eye diameter =62mm, discharge diameter =180mm, number of blades =4, blade

discharge angle =20°, blade warp angle =140°, blade outlet width =18mm. Outside the

impeller, a volute with 40mm wide rectangular cross-section and a discharge nozzle of 50mm

diameter is installed. The volute tongue tip is located at a circle of 190mm diameter (Fig. 1).

As mentioned above, the impeller-volute interaction can be dealt with by either unsteady or

steady MRF method. Since the steady rather than unsteady performance of centrifugal pump is

interesting for us at the fist phase of research, the steady MRF method is applied in this contribution.

1 the specific speed of a pump is defined to be

4

Subsequently, the flow domain geometry of the pump has been built shown Fig. 1(a). The flow

domain consists of a suction pipe, impeller and volute. The domain is created just based on the

dimensions of the solid body presented in the design drawings of the pump, and the variations in

dimension and geometrical shape due to casting process were not taken into account. It should be

noticed that the two spaces between the pump casing and both walls of the impeller shroud and hub

were neglected in the domains.

In the steady MRF method, the pump performances corresponding to a series of the relative

position between impeller blade and volute tongue need to be investigated. Because of the symmetry

of the impeller, even the four blades 1 through 4 illustrated in Fig. 1(b) pass through the tongue

sequentially, it is enough to conduct numerical computations of flow at a series of relative position

between blade 1 and the tongue. The relative position is specified by the circumferential angle

measured from the tip of the tongue to the terminal edge of the pressure surface of blade 1. In the

computations, =90o, 75o, 60o, 45o, 30o and 15o, respectively.

2.2 Physical Properties of Liquids

Four liquids, namely tap water, oil 1, oil 2 and oil 3 have been used experimentally as the

working fluid respectively by the model pump. Their density and kinematical viscosity at 20℃ are

tabulated in Table 1. It was shown that these liquids are Newtonian according to the viscosity-shear

rate relations given by a rotational viscometer. In [24], experiments on the performance of handling

tap water, oil 1, oil 2 and oil 3 were made with the model pump. The 2D steady time-averaged flow

5

1

2 3

4

(a) (b)

Fig. 1 Flow domain of the model pump (a) and the relative position between a blade and the

volute tongue (b)

fields in the impeller and volute of the model pump have been investigated experimentally at both low

flow rate and the best efficiency points in [25, 26] by means of LDV when tap water and oil 2 were

handled by the pump. To compare the performances computed with those measured, these four liquids

were exactly applied in the numerical computations.

2.3 Operating Conditions

The numerical computations were carried out at thirteen operating conditions ranging in a flow

rate of 1-11L/s to cover the entire operating range as close as possible. The corresponding inlet

velocity at the suction pipe was ranged between 0.383m/s-4.209m/s.

2.4 Flow Model

In this contribution, the fluid is assumed to be 3D, incompressible and turbulent inside the model

pump under any operating conditions, further, the time-averaged flow is steady. However, in the

unsteady MRF method, the time-averaged flow is assumed to be unsteady. The fluid in the impeller is

rotated anticlockwise with the impeller at a particular constant angular velocity (this rotation

direction is specified by a view looking from the suction pipe to the impeller eye), this means the fluid

in the impeller is rotational in the absolute reference frame. However, the fluid in both the suction

pipe and volute isn’t rotational in that reference frame. The fluid flow is governed by the time-

averaged continuity and Navier-Stokes equations. Although an attempt was made to investigate

effects of various turbulence models on the results, the well confirmed standard - two equations

turbulence model is mainly applied to indicate the turbulence effects. The non-equilibrium wall

function is chosen to involve the boundary layer effects.

2.5 Computational Method and Mesh

CFD code FLUENT was chosen to compute the 3D, steady/unsteady and incompressible flow in

the model pump. The finite volume method, SIMPLE algorithm and the second-order upwind scheme

were applied to yield the descritization equations of the governing equations.

In the computations, three flow domains have been employed, namely the flow domain of the

suction pipe, the domain of the volute and the domain of the impeller. The first two are stationary,

whilst the last one is rotational. A plane through the mid-span of blade at the impeller outlet and

vertical the central line of the pump shaft is defined as the geometrical datum. An impeller-suction

6

Table 1 Physical properties of liquids

Liquid Water Oil 1 Oil 2 Oil 3

Density, , kg/m3 1000 839 851 858

Kinematical viscosity, , cSt(mm2/s) 1.0 24.47 48.48 60.7

pipe sliding mesh plane is generated at the impeller eye inlet, where is located at 35mm upstream the

datum plane. Likewise, an impeller-volute sliding mesh plane is specified as a cylindrical surface with

a diameter of 92.5mm.

The suction pipe fluid domain was descritized with hexahedron cell, but both the impeller and

volute flow domains were done with tetrahedron cell due to their complicated geometry. It was

confirmed that when the total number of cells is increased up to 910,000 (10,000cells in the suction

pipe, 210,000 cells in the impeller and 520,000 cells in the volute) the pump performance will be

independent on the cell-size. Unless otherwise stated, the computational results were under this

number of cells. The effect of cell size on the pump performance can be found out in the Appendix.

The under-relaxation factors of pressure, velocity, turbulence kinetic energy and its dissipation

rate are given to be 0.3, 0.5, 0.8 and 0.8 respectively. The convergent criterion for all residuals of

pressure, velocity, turbulence kinetic energy and its dissipation rate is chosen to be 1.5×10 -4.

Consequently, the difference of mass flow rate between the pump inlet and outlet is reduced as small

as 8.02×10-5kg/s, so that the continuity equation is much fairly satisfied between flow domains.

2.6 Boundary Conditions

No-slip conditions are applied on wetted solid walls. At the entry of the suction pipe a uniform

inlet absolute velocity is specified, which just has an axial component determined by both known flow

rate and inner cross-sectional area of the suction pipe. The turbulence intensity of flow at the inlet of

the suction pipe (its hydraulic diameter is 62mm) is given to be 5%. A zero static pressure is held at

the volute nozzle exit. The turbulence intensity at its exit (its hydraulic diameter is 50mm) is also

assumed to be 5% for reverse flow. If no reverse flow takes place there, these intensity and diameter

parameters take no effect.

2.7 Wall Roughness Implementation

As a fluid flows over a solid boundary, a boundary layer, where dramatic velocity gradient exists,

is developed on the wall. For a smooth wall, four regions: sublayer, buffer zone, log-law region and

outer layer, can be divided from the wall to the edge of boundary layer. Usually the mesh-size close to

a wall in CFD just can be in the log-law region. Subsequently, the shear stress on the wall will be

estimated approximately by the following expression [30]

(1)

where is Von Karman constant, =0.41; the constant determined by experiments, =5.5; the

normal distance between a node of cell adjacent to the wall and the wall; the kinematical viscosity

of fluid; the density of fluid; the shear tress on the wall. Since the fluid velocity at the point

of distance from the wall has been available, the determination of is fairly straightforward.

7

For a rough wall, the roughness Reynolds number （= ） is defined to indicate the

effects of roughness on the velocity profile and in turn the shear stress on the wall. The roughness

is known as the equivalent sand roughness and is linked to the arithmetic average of absolute values

of the actual roughness of the wall with particular finish by =6 , where =12.5 -50 for a

cast wall. If =50 , then =300 . This sand equivalent roughness estimate is comparable to

250 , a sand equivalent roughness of natural surface of cast iron recommended in [30].

Based on the data of hydraulic experiments made by Nikuradse in 1933 with straight circular

cross-section pipes lined inside with uniform sand grains, Ioselevich and Pilipenko (1974) clarified

that: if , the flow in the boundary layer is in a hydraulic smooth regime and the velocity

profile and shear stress on the wall is unaffected by roughness but by viscosity ; If , the

flow is a transition regime and the profile and stress are impacted by both roughness and viscosity; if

, then the flow is in a hydraulic rough regime and the flow is influenced by roughness only.

The velocity profile involved roughness effects is written as [30]

(2)

where is a function in terms of roughness. As a result of the curve fitting to Nikuradse’s data,

Ioselevich and Pilipenko got an analytical expression of as follows:

(3)

In FLUENT, (3) is applied to involve the effect of surface roughness of a wall on the boundary

layer flow with an adoption by using a roughness coefficient to indicate the uneven property of

rough elements of a practical surface finish and eventually is expressed with the following formula

(4）

where =0.5-1.0 is usual choice. Since the shape and distribution of rough elements of cast surface

more resembled to those of uniform sand grains in hydraulic experiments available, it was to let

=0.75 in the paper. It has been shown that a large results into a low pump head computed. As

is increased from 0.5 to 0.75, the estimated head at the best efficiency point is dropped as low as 2%

8

only. Therefore, seems to have minor influence on the computational results.

Because the treatment of surface roughness effect on boundary layer flows was conducted in

FLUENT by using existing data gained under ideally experimental circumstances on a fundamental

basis of fluid mechanics but nothing new, if a difference of computational flow parameters from

experiment occurs for a flow over rough walls, we shouldn’t be surprised, but consider the case to be

normal as well.

3 Results

3.1 Effect of Relative Position between Blade and Tongue

To illustrate the effect of the relative position between impeller blade and volute tongue on the

pump hydraulic performance, the numerical computations of flow have been carried out at six

positions of blade 1 related to volute tongue, i.e. =90o, 75o, 60o, 45o, 30o and 15o. The working fluid

is water and the surface roughness of both impeller and volute is =50 . Fig. 2 indicates the

impeller theoretical head and hydraulic efficiency , pump head and hydraulic efficiency

in terms of the flow rate at various . The impeller theoretical head is defined as the mass-

averaged total energy head rise of fluid from the impeller inlet to outlet. The impeller hydraulic

efficiency is defined as the ratio of the impeller theoretical head over an ideal theoretical head

corresponding to the consumed shaft-power for developing head rise . This ideal head is specified

by the following expression

(5)

where is the pump volumetric efficiency, =0.865 is given by [24-26]; the total torque

applied to liquid pumped by the impeller blades and the inside shroud and hub, which are calculated

based on CFD results; the impeller rotational angular speed, ; the pump flow rate;

the acceleration of gravity. The pump head is defined as the total energy rise of fluid from the entry

of the suction pipe to the volute nozzle exit. Likewise, the pump hydraulic efficiency is the ratio of

the pump head over the ideal theoretical head .

9

0 2 4 6 8 10 126

7

8

9

10

11

12

13

Q(L/s)

Hi(m

)

data1data2data3data4data5data6data7data8data9

=90o

=75o

=60o

=45o

=30o

=15o

upper margin

mean

lower margin

=1cSt, Ra=50m

0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Q (L/s)

hi

data1data2data3data4data5data6data7data8data9

lower margin

upper margin

mean

=90o

=75o

=60o

=45o

=30o

=15o

=1cSt, Ra=50m

0 2 4 6 8 10 124

5

6

7

8

9

10

11

Q(L/s)

H(m

)

data1data2data3data4data5data6data7data8data9

=90o

=75o

=60o

=45o

=30o

=15o

=1cSt, Ra=50m

upper margin

lower margin

mean

0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Q (L/s)

h

data1data2data3data4data5data6data7data8data9

=90o

=75o

=60o

=45o

=30o

=15o

lower margin

upper margin

mean

=1cSt, Ra=50m

Fig. 2 Impeller theoretical head and hydraulic efficiency, pump head and hydraulic efficiency in terms of flow rate for various angles between blade 1 and volute tongue

In Fig. 2, it is clear that the relative position between blade and the volute tongue does affect the

pump performance. It not only induces the theoretical head and hydraulic efficiency curves’ down or

up displacement but also alters their shape. The theoretical impeller and pump heads are varied in a

range of their mean values 0.2m, whilst the impeller and pump hydraulic efficiencies are confined in

two (upper and lower) margins (mean values 0.02 and 0.03), respectively. At =6L/s, for

instance, the relative variations in the theoretical head and the hydraulic efficiency are just less than

2% and 3.5% respectively. As a result, it is apparent that the influence of the relative position between

blade and tongue is unremarkable. Further, it is obviously noted that the curves, which the mean

values of the theoretical head and hydraulic efficiency line up, are much fairly close to those of

=45o. Consequently, it is necessary to represent the pump performance curves by using those of

=45o. Unless otherwise stated, the performance data just at this angle will be presented in the

following sections.

3.2 Effect of Viscosity on Impeller Performance

Figure 3 demonstrates the impeller theoretical head , hydraulic efficiency , hydraulic loss

coefficient and slip factor in terms of flow rate at the kinematical viscosities of 1cSt, 24cSt,

10

48ct and 60cSt and with wetted surface roughness of =50 . As reference, the theoretical heads

of one-dimensional inviscid fluid flow derived from both Eulerian equation of turbomachinery and the

slip factors of Wiesner and Stodola are also plotted in that figure. Wiesner and Stodola slip factors are

depicted with the following formulae

(6)

Accordingly the impeller theoretical head of one-dimensional inviscid fluid flow is written by

(7)

where is the impeller tip speed, , the impeller exit diameter; the volumetric

efficiency, =0.869 based on the experiments in [24, 25]; the impeller exit area, ,

the blade exit width; the blade blockage coefficient at exit, =1- , the tangential

thickness of blade at exit, =10mm. The slip factor is defined as the ratio of the slip velocity of

fluid at the impeller exit over the impeller tip speed and is mathematically presented by

(8)

Even the slip factors can be estimated by comparing the velocity-triangles at the impeller outlet

constructed with CFD outcomes with those of assumed one-dimensional flow of inviscid fluids, these

factors don’t seem to be linked directly to the impeller theoretical head of viscous fluids because at

the impeller outlet the variable in Euler equation no longer represents the total energy of

viscous fluids. Instead, in the paper, the slip factors will be evaluated by using the impeller theoretical

head of viscous fluids obtained from CFD computations.

Substituting the theoretical head of viscous fluids given by CFD computations into (7), the

slip factors corresponding to these heads of viscous fluids are available via the following equation

(9)

The slip factors calculated just in this way are illustrated in Fig. 3. Note that the impeller

hydraulic loss coefficient showed in that figure is defined as the ratio of the impeller hydraulic loss

over the impeller velocity head based on the impeller tip speed , i.e. .

It is clearly that tremendous difference of the impeller theoretical head between 3D viscous fluid

and 1D inviscid fluid emerges. The impeller theoretical head of viscous fluid no longer increases

continuously with decreasing flow rate, but a peak occurs at a flow rate of 4L/s (around 57% flow rate

11

of the best efficiency point) instead, causing stall operating conditions at less flow rate than 4L/s

which are frequently present in centrifugal compressors at part-load points. The 3D viscous flow

theoretical head no longer increased monotonously with decreasing flow rate agrees well with the

experiments in [31, 32].

The impeller theoretical head of 1D inviscid fluid based on Stodola slip factor is in good

agreement with those of viscous fluid by CFD, specially nearly at =6L/s. The head based on

Wiesner factor is substantially over-predicted. It is implied that Stodola slip factor is more suitable to

the impellers with less number of blades, small blade discharge angle, long and strongly back-curved

blade than Wiesner slip factor.

It should be noticed that since the impeller theoretical head of 3D viscous fluid flow significantly

deviates from the linear theoretical head of 1D inviscid fluid flow except at a flow rate around

=6L/s, the slip factors based on the theoretical heads of 3D viscous fluid no longer are a flow rate-

independent constant, instead are varied with flow rate, subsequently, at both high and low flow rates,

larger slip factors can be found.

Both the impeller theoretical head and slip factor of 3D viscous fluid flow are dependent on

liquid viscosity. Except at a viscosity of =24cSt, the theoretical head of viscous fluid is degraded and

the corresponding slip factors are arisen with increasing liquid viscosity. At =24cSt, however, the

situation is reverse. In this case, as a flow rate is more than 4L/s, the theoretical head of viscous fluid

is higher than that of water ( =1cSt), accordingly the slip factor is less than that of water.

The impeller hydraulic efficiency has a peak between 75% and 90% at a flow rate of =6L/s-

7L/s, in turn the impeller hydraulic loss coefficient has the lowest values at this flow rate range.

The hydraulic loss coefficient is increased rapidly towards low flow rate due to severely reverse flow

developed in the impeller. For example, the loss coefficient at =1L/s is as high as 64 times at

=7L/s. Further, as the flow rate more than 7L/s, the hydraulic loss coefficient is increased but grows

slowly with flow rate because of the significant contribution of skin friction loss.

The impeller hydraulic efficiency and the hydraulic loss coefficient rely on liquid viscosity as

well. In general, the hydraulic efficiency is decreased but the loss coefficient is increased with

increasing liquid viscosity. However, the impeller performance is improved at =24cSt as the flow

rate more than 5L/s since the theoretical head and hydraulic efficiency of oil1 are higher than those of

water ( =1cSt) and the loss coefficient is lower than that of water ( =1cSt).

12

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7

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15

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Q(L/s)

Hi(m

)

data1data2

data3

data4

data5data6

1cSt24cSt48cSt60cSt

1D, Stodola

1D, Wiesner

0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

Q(L/s)

data1data2data3data4data5data6

1cSt24cSt48cSt60cSt

Stodola

Wiesner

0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Q (L/s)

hi

data1

data2

data3

data4

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 1210

-2

10-1

100

101

Q (L/s)

i

data1data2data3data4

1cSt24cSt48Cst60cSt

Fig. 3 Impeller theoretical head, slip factor, hydraulic efficiency and loss coefficient in terms of flow rate at =45o and =50

3.3 Effect of Viscosity on Pump Performance

F igure 4 demonstrates the pump head , pump efficiency , hydraulic efficiency , volute

hydraulic loss coefficient and pump total hydraulic loss coefficient in terms of flow rate at

various viscosities of 1cSt, 24cSt, 48cSt and 60cSt and with a surface roughness of 50 . The volute

hydraulic loss coefficient is defined as the ratio of the hydraulic loss inside the volute over

the velocity head based on the impeller tip speed, i.e. . The computational pump head

isn’t completely consistent with the experimental performance, causing a maximum relative error of

11%, since the slope of the latter is sharper than the former. The mechanism behind this phenomenon

perhaps is a disagreement between practical model and design drawings in the dimensions and/or

shape of volute, especially the area of the volute throat.

In addition, the pump volumetric and mechanical efficiencies were roughly estimated based on

13

the experimental data in [24, 25], then the pump efficiency was calculated by them and the pump

hydraulic efficiency given by CFD. Although the predicted pump efficiency is very comparable to

those measured at =1cSt, 48cSt and 60cSt, a significant deviation from the experiment is found at

=24cSt where the experimental pump efficiency is less than that of water ( =1cSt). The reason for

this is perhaps that the less correct disk loss is involved in the CFD estimate of pump efficiency. The

flow computations tending to include this loss should be worth being conducted in forthcoming

investigations.

The pump hydraulic efficiency is below 75% in all cases. It is the volute that makes the pump

hydraulic efficiency be reduced by at least 15%. In addition, the higher the flow rate, the larger the

reduction. It is more likely that the volute affects the hydraulic losses more considerable than the

impeller does, specially at the high flow rate for this pump.

Unlike the impeller hydraulic loss coefficient, the volute hydraulic loss coefficient is raised with

increasing flow rate. For instance, as the flow rate is increased from 1L/s to 11L/s, the hydraulic loss

coefficient is increased by about 3.5 times.

The pump total hydraulic loss coefficient is minimum at around =6L/s. As the flow rate

becomes low, the total loss coefficient gets significantly large, whereas it rises moderately with

increasing flow rate. Comparing the impeller hydraulic loss coefficient with that of the volute, it is

discovered that the impeller hydraulic loss coefficient is dominated at low flow rate, at high flow rate,

however, the volute loss coefficient is remarked.

14

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12

13

Q(L/s)

H(m

)

data1

data2data3

data4

data5

data6data7

data8

1cSt24cSt48cSt60cSt

Exp

CFD

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Q (L/s)

data1data2data3data4data5data6data7data8

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

CFD

EXP

0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Q (L/s)

h

data1data2data3data4

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 1210

-3

10-2

10-1

Q(L/s)

V

data1

data2data3

data4

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 1210

-2

10-1

100

101

Q (L/s)

i+ V

data1data2data3data4

1cSt24cSt48cSt60cSt

Fig. 4 Pump head and hydraulic efficiency, volute hydraulic loss coefficient and pump total hydraulic loss coefficient in terms of flow rate, the symbols indicate the experimental head and efficiency of the pump

As shown in Fig. 4, even both the pump computed head and hydraulic efficiency are deteriorated

gradually as increasing liquid viscosity, at =24cSt (oil 1), they are improved compared with those at

=1cSt (water). The phenomenon, regarding that the pump performance is improved at a particular

high viscosity of liquid compared to pumping water, is named as “sudden-rising head effect”. It has

been observed that the pump head is improved at =24cSt-60cSt (oil1, oil 2 and oil3) in the

experiments of [24] and indicated the sudden-rising-effect-head is onset at this viscosity. Obviously,

this phenomenon is well predicted by the numerical computations. Therefore, this effect has been

confirmed to exist theoretically and experimentally. Since the effects of flow turbulence and surface

roughness on the boundary layer flows in the pump are estimated approximately, the sudden-rising-

head effect has to merge just at =24cSt in the numerical computations. Nevertheless, it is hopeful

that the analytical prediction may be improved with the development in both turbulence model and

boundary layer flow investigations.

For an absolutely smooth surface ( =0 ), the computed pump head is decreased

15

continuously with increasing liquid viscosity, subsequently, the sudden-rising-head effect is excluded.

For a roughness of =100 , however, the sudden-rising-head effect occurs once more (Fig.5). It

is believed that the surface roughness of flow channels plays a critical role in the occurrence of

sudden-rising-head effect.

Figure 6 shows the averaged skin friction factors applied on the liquid pumped by the wetted

surfaces of the impeller and volute in terms of flow rate. The impeller mean skin friction factor is

defined as the ratio of the averaged shear stress applied by the impeller wetted surfaces on the

liquid over the velocity head , that is . Likewise, the volute mean skin friction

factor is the ratio of the averaged shear stress applied on the liquid by the volute over the

velocity head , . Both mane skin friction factors include the effects of liquid

viscosity and surface roughness on the skin friction loss in the flow passages of impeller and volute.

For the same passage, the larger the mean skin friction factor, the more the skin friction loss.

In Fig. 6, even though the magnitudes of and are comparable, their relations with the

flow rate are quite different. For , it has a minimum value at =4L/s, and then is increased rapidly

with increasing flow rate, , however, grows slowly but continuously from low to high flow rate.

The flow velocity in the impeller channels may be increased by increasing blockage due to the reserve

flow onset at much low flow rate. It should be unsurprised to notice that an increasing can be seen

at a flow rate less than 4L/s. Besides the vortex and shock losses, the increased impeller skin friction

loss is likely responsible for the huge hydraulic loss coefficient of impeller at low flow rate shown in

16

0 2 4 6 8 10 123

4

5

6

7

8

9

10

11

Q(L/s)

H(m

)

data1

data2

data3

data4

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 123

4

5

6

7

8

9

10

11

Q(L/s)

H(m

)

data1data2data3data4

1cSt24cSt48cSt60cSt

Fig. 5 Pump head in terms of flow rate at surface roughness of 0 and 100

Fig. 3.

The mean skin friction factors are raised with increasing liquid viscosity, at =24cSt, however,

both and start to be lower than those at =1cSt (water) respectively from =4.5L/s and

=6.5L/s. The lowered mean skin friction factors correspond to the decreased skin friction loss in the

pump, accordingly the pump performance shows a higher head developed and increased hydraulic

efficiency. It should be concluded that the deceased skin friction loss in both impeller and volute is

responsible for the sudden-rising-head effect.

Based on the experimental outcomes of turbulent flow of the boundary layer over flat rough

plates [30], in the hydraulic rough regime, the skin friction factor is related to the surface roughness

only. As the liquid viscosity increasing, the Reynolds number is decreased, causing the flow regime is

the hydraulic transition zone, where the skin friction factor not only depends on both Reynolds

number and surface roughness but also is decreased with decreasing Reynolds number. As Reynolds

number is decreased further, the flow regime is in the hydraulic smooth zone, in which the factor is

just dependent on Reynolds number but is raised as decreasing Reynolds number. It can be imagined

that the sudden-rising-head effect should take place in the transition zone rather than in the others.

Two critical parameters to decide the sudden-rising-head effect onset are the high enough surface

roughness and properly decreased Reynolds number. If the surface roughness is too small or the

Reynolds number is decreased too much, causing the flow regime is in the hydraulic smooth zone, the

sudden-rising-head effect will unlikely happen. Otherwise, if the surface roughness is too high and

Reynolds number is too large, the effect still won’t occur for the flow regime is in the hydraulic rough

17

0 2 4 6 8 10 121

2

3

4

5

6

7x 10

-3

Q(L/s)

Cfi

data1

data2

data3

data4

1cSt24cSt48cSt60cSt

0 2 4 6 8 10 121

2

3

4

5

6

7x 10

-3

Q(L/s)C

fV

data1

data2data3

data4

1cSt24cSt48cSt60cSt

Fig. 6 Mean skin friction factors of both impeller an volute in terms of flow rate at various liquid

viscosities and =50

zone.

In fact, the sudden-rising head effect was also found with experiments in [33]on the centrifugal

oil pump with a specific speed of 807 (USA) or 57 (China) and confirmed with the estimation of the

friction loss of water and viscous oils flow in the impeller and volute by using the routine boundary

layer theory. At the moment, this effect also exists in the experimental performance of the pump with

the specific speed of 93 and is confirmed with CFD results once again. Therefore, the sudden-rising

head effect may be an essential characteristic of a low specific speed centrifugal pump handling

highly viscous oils.

It should be noticed that even though the sudden-rising head effect occurs, the pump gross

efficiency still is gradually decreased due to the possible increased impeller disk friction loss by the

increasing liquid viscosity [24, 25]. The clear mechanism behind it obviously needs to be debuted

further experimentally and numerically.

3.4 Effect of Volute on Flow

Figure 7 illustrates the impeller static pressure head rises and velocity head versus flow rate at

various liquid viscosities and =50 to clarify the reason why there is a peak onset in the

impeller theoretical head curve at a low flow rate shown in Fig. 3. Further, the results based on the

unsteady turbulent flow computations are also disclosed on the figure to make sure of fully

understanding of the effect of volute. It is observed that the unsteady results are pretty consistent with

those of steady flow. Moreover, the unsteady results show more remarked dependence upon the liquid

viscosity compared to the steady data. The maximum difference of performance between unsteady

and steady flows is less than 4%.

As be seen in Fig. 7, the impeller static pressure head is steadily increased with decreasing

flow rate. Contrarily, the velocity head has a peak at around =4L/s, then a dramatic decrease in

can be found at a flow rate less than 4L/s. Therefore, it is believed this too low velocity head at a

flow rate less than 4L/s should contribute the peak onset on the theoretical head curves.

18

0 2 4 6 8 10 124

5

6

7

8

9

10

Q (L/s)

His

(m)

data1data2data3data4data5data6data7data8

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

steady

unsteady

0 2 4 6 8 10 121

1.5

2

2.5

3

3.5

4

Q (L/s)

HiV

(m)

data1data2data3data4data5data6data7data8

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

steady

unsteady

0 2 4 6 8 10 120.3

0.4

0.5

0.6

0.7

Q (L/s)

Vu2

/u2

data1data2data3data4data5data6data7data8

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

steady

unsteady

Fig. 7 Impeller static pressure head and velocity head rises as well as dimensionless tangential component of absolute velocity of liquid at impeller exit in terms of flow rate at various liquid viscosities and =50

19

1

2

3

4

1

2

3

4

steady

unsteady

1

2

3

4

1

2

3

4

steady

unsteady

=6L/s =3L/sFig. 8 Relative velocity vector and static pressure contour over the impeller blades and hub at a low

flow rate ( =3L/s) and the best efficiency point ( =6L/s) for water ( =1cSt) in the steady and

unsteady cases

To expose the reason for extreme low velocity head occurrence at a low flow rate, the computed

tangential components of absolute velocity of liquid at the impeller discharge have been illustrated in

Fig. 7 too. The tangential component of velocity has been made dimensionless with the impeller tip

speed . It is clearly that a dramatically decreased tangential velocity is caused. It should be noticed

that the LDV measurements also revealed that the decreased velocity happens at the impeller exit with

decreasing flow rate in [26]. The reduction of tangential velocity implies that a volute can apply much

strong impact on the flow in an impeller.

Figure 8 shows the liquid relative velocity vector and static pressure contour over the impeller

blades and hub at a low flow rate ( =3L/s) and the best efficiency point ( =6L/s) when the pump

handling water ( =1cSt) based on both steady and unsteady turbulent flows respectively. The static

pressure is in terms of the liquid columns height, m. At the best efficiency point, the static pressure

and the relative velocity are uniform along the impeller periphery and the separated flow is

20

unavailable. At a low flow rate ( =3L/s), the static pressure at three passages (between blades 4-1, 2-

1 and 2-3) is fairly higher than in the passage between blades 3-4. As a result of this, the relative

velocity in these three channels is very low and subject to server separation. For other high viscosities

near the same phenomenon can be observed.

It should be noticed that a very good agreement is achieved between steady and unsteady

turbulent flow fields, however, the unsteady flow models seem to present more uniform pressure

along the impeller periphery and less separated relative flow in the impeller.

3.5 Effect on Slip Factor

Except by the theoretical head developed in an impeller, the slip factor of the impeller also can be

calculated by using velocity triangles at the impeller discharge [34]. In addition, it was proposed that

the slip factor is estimated by using Euler equation with an actual relative flow angle at the impeller

exit given by CFD computations [35]. In that case, the slip factor is defined the ratio of the impeller

theoretical head over the head of the impeller with infinite number of blades. In this proposal the

radial flow velocity is kept to be unchanged between 3D viscous flow and 1D ideal flow. In fact, 3D

viscous fluid flow showed experimentally an increasing radial velocity at the impeller exit compared

to 1D uniform inviscid flow as indicated in Fig. 9 [36], this is also true for CFD computational results.

In section 3.2, the slip factors estimated by using the impeller theoretical head of 3D viscous fluid

have been illustrated. Here we tend to present those given just by the velocity triangles at the impeller

outlet. In doing so, we have to apply the velocity triangles in Fig. 9 to extract the slip factors.

For a uniform 1D ideal flow, the tangential component of the relative velocity is given by

the following formula

(10)

Based on CFD computational results, the velocity components and are available, the

tangential component of the relative velocity of 3D viscous fluid flow is

(11)

With the slip factor definition (8), the slip factor extracted from the velocity triangles is

21

Fig. 9 Velocity triangle at impeller discharge for estimate slip factor

V2

Vr2

W2

w2v2

vr2

u2

1D3D

Vu2

β2b

expressed by

(12)

The computed slip factors based on both unsteady and steady flows have been shown in Fig. 10

for the impeller with volute at various viscosities and a surface roughness of =50 . The slip

factors are reduced with increasing flow rate. The similar characteristics of slip factor versus flow rate

were revealed by experiments in [36, 37]. Moreover, the slip factors extracted from velocity triangles

considerable differs from those based on the impeller theoretical head by CFD computation (see Fig.

3).

As the flow rate is less than 5L/s, the slip factors are rapidly increased as high as 0.6. Otherwise,

the slip factors asymptotically approach the Wiesner’s slip factor. The slip factors due to the unsteady

flow model much more depend on the liquid viscosity than those due to the steady model. Except at

=24cSt, the slip factor are reduced with increasing liquid viscosity, causing an good agreement with

that was found experimentally in [38]. At =24cSt, although the slip factor is the largest, the highest

static pressure rise is developed across the impeller (Fig. 7). Since the static pressure rise is dominated

compared to the velocity head rise in the impeller, an increased slip factor doesn’t mean a lowered

total head developed by the impeller.

22

0 2 4 6 8 10 120.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

data1data2data3data4data5data6data7data8data9data10

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

steady

unsteady

Stodola

Wiesner

Fig. 10 Slip factor against flow rate at various viscosities and a surface roughness of =50

3.6 Effect on Flow Field

In FLUENT, the fluid physical viscosity rather than the turbulence eddy viscosity in the

Reynolds time-averaged N-S equations is switched off in fully turbulent flow region. Consequently,

the physical viscosity just applies its effect on the main flow through the wall function, for example,

(2) or (3). Fig. 11 shows the absolute velocity and static pressure contours in the plane through the

mid-span of blade at =1cSt, 24cSt, 48cSt and 60cSt at the best efficiency point ( =6L/s) based on

the unsteady turbulent flow. It is very interesting to note that near the same pattern in the pictures has

been found at those viscosities. However, the color of the contours seems to become slightly light

with increasing viscosity, especially the pressure contour at =60cSt to indicate the pump

performance degraded. It is believed that the viscosity effect on the main flow velocity profile rather

than the static pressure isn’t quite considerable. The decreased static pressure rise owing to the

increased skin friction loss in the pump should be responsible to the reduction of performance.

23

=1cSt

=24cSt

=48cSt

=60cSt

Fig. 11 Absolute velocity (left) and static pressure (right) contours in the plane through the impeller blade mid-span at the best efficiency point ( =6L/s) at =1cSt , 24cSt, 48cSt and 60cSt, the velocity is colored by velocity magnitude (m/s), and the static pressure is colored by liquid column height (m)

3.7 Effect of Surface Roughness

Figure 12 illustrates the pump head and hydraulic efficiency in terms of flow rate at various

viscosities and Ra=0 , 50 and 100 . For Ra=0 , the strongest surface roughness effect

can be found. Otherwise, as the liquid viscosity increasing, the roughness effect is suppressed

substantially. Moreover, once the surface roughness Ra 50 , the effect on the performance

becomes less dominated. The experimental data in [39, 40] seem to show the similar effect.

240 2 4 6 8 10 12

4

5

6

7

8

9

10

11

Q (L/s)

H (

m)

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

1cSt24cSt48cSt60cSt1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

Ra=0m

Ra=50m

Ra=100m

0 2 4 6 8 10 120.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Q (L/s)

h

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

1cSt24cSt48cSt60cSt

Ra=0m

Ra=50m

Ra=100m

Fig. 12 Pump head and hydraulic efficiency in terms of flow rate for surface roughness of 0 , 50 and 100 and at various liquid viscosities

Table 2 shows the admissible roughness Ra* of both impeller and volute in terms of Ra at the

25

Table 2 Admissible roughness of impeller and volute

Mean velocity at =6L/sWater=1cSt

Oil l=24cSt

Oil 2=48cSt

Oil 3=60cSt

Impeller, =3.9m/s 4.2 102.6 205.1 256.4

Volute, =4.4m/s 3.8 90.9 181.8 227.3

Ra*= or

0 20 40 60 80 100

8

9

10

11

Ra (m)

H (m

)

data1data2data3data4

1cSt24cSt48cSt60cSt

0 20 40 60 80 1000.5

0.6

0.7

0.8

0.9

Ra (m)

h

data1

data2

data3

data4

1cSt24cSt48cSt60cSt

Fig. 13 Pump head and hydraulic efficiency in terms of roughness at =6L for four viscosity values

best efficiency point =6L/s for various viscosities based on Schlichting formula [39]. Since the

mean relative velocity of flow in the impeller is close to that in the volute, the admissible roughness

Ra* of the impeller is comparable to the volute. The admissible roughness is increased with

increasing liquid viscosity. Fig. 13 illustrates the pump head and hydraulic efficiency in terms of

surface roughness at =6L for four viscosities. At =1cSt, the performance is degraded until Ra

less than 40 . Beyond this roughness the performance is slightly affected. At =24cSt, the

performance is improved as >25 . Based on Table2, the admissible roughness Ra* of both

impeller and volute is around 100 at =24cSt. Consequently, when is between 25 and 100

, the flow regime of boundary layer in the impeller and volute should be in the transition zone

where less skin friction loss is expected. As the viscosity more than 48cSt, the performance is

basically unaffected by the surface roughness because the rough is much less than Ra* and the

corresponding flow regime of boundary layer in the impeller and volute should be in hydraulic

smooth zone.

If a pump is used to transport highly viscous liquids, it may be acceptable that the pump flow

channels maintain a relative large roughness. Otherwise, the surface roughness should be kept as

small as possible.

4 Discussions

4.1 Flow Regime in Section Pipe

In the computations, the Reynolds number at the suction pipe is varied in a range of 3.9 102-3.3

105 from water to oil 3 and from a high flow rate to a low one. The Reynolds number at around 14

operating points is less than 2300, and the flow in the suction piper should be laminar, subsequently,

the flow in the impeller would be either laminar or turbulent at these conditions. Due to lack of

information of flow transition in impellers, even the Reynolds number is so low, the flow in the pump

still is assumed to be turbulent in the paper.

4.2 Wall Function Effect

In the computations, the non-equilibrium wall function is applied to take into account the

pressure gradient along the normal of blade two surfaces. As a result, the computations employing the

standard wall function also were carried out to indicate the wall function effect on the pump

performance. A comparison of wall function effect between standard and non-equilibrium has been

presented in Table 3. At =1cSt, the consistent results have been achieved by both wall functions, at

=60cSt, however, the standard wall function causes an abnormal impeller hydraulic efficiency,

which is even higher than that at =1cSt. Therefore, the standard wall function seems to be

inapplicable in highly viscous fluid flow computations.

26

Table 3 Effect of wall function on performance

Viscosity and flow rate Wall function

(N.m)

(m) (m)

=1cSt, =6L/sStandard 5.822 11.303 9.345 0.849 0.702

Non-equilibrium 5.779 11.415 9.406 0.863 0.711

=60cSt, =6L/sStandard 5.091 11.172 8.482 0.823 0.625

Non-equilibrium 5.392 11.129 8.823 0.774 0.614

4.3 Turbulence Model Effect

The different turbulence models are expected to show strong effect on the pump performance.

Accordingly the turbulence models available in FLUENT, such as the standard , RNG ,

realizable and standard models as well as Spalart-Allmaras models have been switched

on respectively and corresponding results have been summarized in Table 4. At =1cSt, the models

except Spalart-Allmaras give much close performance parameters even the RNG model shows

poor convergence. At =60cSt, the standard , RNG , realizable models show fairly

consistent performance, but the standard and Spalart-Allmaras models are unable to get a

reasonable impeller hydraulic efficiency. The Spalart-Allmaras model shows poorly predicting ability

and should be excluded in the application of internal turbulent flow. Moreover, the standard

should be unsuitable to highly viscous fluid flows. It is apparent that the standard model is

applicable to turbulent flows of liquid with either low or high viscosity and it is unnecessary to apply

other than this model to simulate such flows.

27

Table 4 Effect of turbulence model on performanceViscosity and flow

rateWall function

(N.m)

(m) (m) Convergence

=1cSt, =6L/s

Standard 5.779 11.415 9.406 0.863 0.711 good

RNG 5.778 11.386 9.486 0.861 0.718 poor

Realizable 5.787 11.445 9.542 0.864 0.721 good

Standard 5.830 11.548 9.472 0.866 0.710 good

Spalart-Allmaras 5.913 11.640 9.285 0.860 0.686 good

=60cSt, =6L/s

Standard 5.392 11.129 8.823 0.774 0.614 good

RNG 5.376 11.060 8.705 0.771 0.607 good

Realizable 5.415 11.057 8.522 0.764 0.590 good

Standard 5.269 11.412 8.893 0.812 0.633 poor

Spalart-Allmaras 5.190 11.695 8.944 0.845 0.646 poor

5 Conclusions

The hydraulic performance of a centrifugal pump handling water and viscous oils was

investigated numerically by means of a CFD code-FLUENT. The flow inside the pump is assumed to

be steady or unsteady, turbulent and incompressible. The turbulent effect on the flow is described with

the standard turbulence model and the flow is solved by using the steady and unsteady MRF

methods. Significant attention has been paid on the effect of liquid viscosity on the performance. The

flow channels of the pump are subject to rough surface. The wall function and turbulence model

effects on the performance have been clarified. The computed results have been compared with those

of experiments. It was indicated that the sudden-rising-head effect is present and a decreased impeller

theoretical head is found at much low flow rate. The impeller theoretical head of 3D viscous fluid

shows considerable difference from that of 1D inviscid fluid. The relative position between blade and

volute tongue does affect the pump hydraulic performance but the influence is slight. The sudden-

rising-head effect is revealed analytically and is consistent with the experimental evidence. The flow

regime transition from the rough zone to the hydraulic smooth one is responsible for this effect. An

increasing strong interaction between volute and impeller causes dramatically decreased velocity head

of liquid at the impeller exit, which contributes the peak onset in the impeller theoretical head curve.

The performance and flow field due to the unsteady flow model doesn’t seem to show noticed

difference from those due to the steady flow model. However, the unsteady flow model not only tends

to cause a uniform flow along he impeller periphery but also to indicate stronger dependence on the

liquid viscosity compared to the steady flow model. Although the time-averaged Navier-Stokes

equations with an aid of the standard turbulence model combined with the non-equilibrium wall

function is applicable to handle highly viscous liquid effect on the performance, more accurate results

are still expected to be present in future.

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31

Appendix Effect of Number of Cells on Performance

To check the effect of number of cells in flow domains on the numerical results, the meshes with

respective 540,000 cells, 910,000 cells and 1,290,000 cells have bee generated by using GAMBIT at a

relative position between impeller blade and volute tongue ( =90o), then the numerical computations

of water steady turbulent flow in the model pump were launched with the standard turbulence

model and the non-equilibrium wall function for each mesh. The wetted surfaces of the pump is

subject to a roughness =50 m.

The pump performance curves as a function of flow rate have been illustrated in Fig. 14. The

pump head seems to be independent on all the mesh sizes, but the hydraulic efficiency starts to be

independent on the mesh size when the number of cells is increased up to 910,000. This mesh size is

used in all numerical computations in this paper.

32

0 2 4 6 8 10 124

5

6

7

8

9

10

11

Q(L/s)

H(m

)

data1data2data3

5400009100001290000

0 2 4 6 8 10 12

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Q (L/s)

data1data2data3

5400009100001290000

Fig. 14 Pump head and hydraulic efficiency in terms of flow rate under three different numbers of cells

in the flow domains