PARTICLE TRAJECTORIES NEAR IMPELLER BLADES · PDF filePARTICLE TRAJECTORIES NEAR IMPELLER...

PARTICLE TRAJECTORIES NEAR IMPELLER BLADES IN CENTRIFUGAL

DREDGE PUMPS

ir. C.F. Hofstra1, Dr.ir. S.A. Miedema

2, Prof.Dr.ir. C. Van Rhee

3

ABSTRACT

Centrifugal pumps, as used in the dredging industry, are used to transport mixtures of particles and water. To gain

insight into the processes that take place inside the pump, several studies are underway. In this paper, results of

theoretical and experimental studies of the interaction of particles and the boundary layers inside the pump impeller

are presented. Calculations predict that particle trajectories can be modified at low angles of attack, at higher angles

particles will impact the blade. In certain cases particles are caught in the boundary layer. Experimental data show

that bouncing impacts occur on the pressure side of the impeller, on the suction side some particles are trapped in the

boundary layer.

Keywords: pump, impeller, particles, boundary layer

INTRODUCTION

Centrifugal pumps are used extensively in dredging operations for the transportation of mixtures. Although much is

known about the global characteristics of these pumps such as head and flow, relatively little knowledge is available

about what exactly goes on inside these pumps. To increase understanding and knowledge in this area, a PhD study

is underway at the Delft University of Technology into the physical processes and phenomena that occur inside a

centrifugal dredge pump impeller whilst pumping sand-water mixtures.

Of the various flow processes and phenomena of sand-water mixtures inside a centrifugal pump impeller, the subject

of this paper is the flow near and in the boundary layers along the impeller walls. More specifically in this case, the

interaction of the sand particles with the boundary layers of the flow is studied.

First the flow near and in boundary layers is discussed. This is extended to include the coefficients for the lift and

drag forces acting on particles. Next, predictions of the fluid velocities in the impeller passage were made for several

working points using the commercial CFD code CFX. Based on these predictions, velocity distributions in the

boundary layers were determined from which fluid forces on particles can be derived. Then, using a Lagrangian

model that describes the equations of motion for spherical sand particles as a function their location in the impeller

and the fluid forces on them, these results were used to calculate sand particle trajectories.

Finally, to validate calculation results, laboratory experiments were carried out in a test rig which enables the

visualization of the flow and particle trajectories inside the impeller of a centrifugal dredge pump. The results of the

experiments and the calculated particle trajectories are then compared.

VELOCITY PROFILE IN BOUNDARY LAYER

Flow near and in the boundary layer is determined by a number of factors. Next to the flow velocity and fluid

viscosity, the geometric dimensions (the cross section) of the shape the flow passes along are also important.

Another factor that determines the flow in the boundary layer is the length of the flow channel. If the length is

sufficient, the flow can become fully developed. An example of this type of flow is the flow in a pipe at a distance of

about 40D from the entry.

In a centrifugal pump impeller, the flow does not usually have necessary length to develop fully. Derivations of

velocity profiles for fully developed flow cannot therefore be used. In order to study the flow in boundary layers in

1 C.F. Hofstra, PhD Student, Chair Dredging Engineering, Delft University of Technology, Mekelweg 2, 2628 CD,

Delft, The Netherlands,+31-15-2782879, [email protected], Project manager, MTI Holland BV, Smitweg 6,

2961AW, Kinderdijk, The Netherlands, +31-78-6910364, [email protected]

2 S.A. Miedema, Associate professor, Chair Dredging Engineering, Delft University of Technology, Mekelweg 2,

2628 CD, Delft, The Netherlands,+31-15-2788359, [email protected]

3 C. van Rhee, Professor, Chair Dredging Engineering, Delft University of Technology, Mekelweg 2, 2628 CD,

Delft, The Netherlands,+31-15-2783973, [email protected]

Hofstra, C.F., Miedema, S.A., Rhee, C. van, “Particle trajectories near impeller blades in centrifugal pumps. WODCON XIX, Beijing China, September 2010.

Copyright Dr.ir. S.A. Miedema

such a case, one can start by modeling the flow along the impeller blade as the flow past an immersed plate (see

Gülich (2008)). In that case, the boundary layer thickness will be zero at the trailing edge of the impeller blade, and

will expand along the length of the blade towards the outlet. Initially, a laminar boundary layer will develop. Further

downstream, the boundary layer will become turbulent.

Figure 1: Boundary layer (White (1999))

Figure 1 gives a representation of this process. The thickness of the boundary layer can be determined using:

x

6

1/ 7

5.0 Laminar

x Re

0.16 Turbulent (Re > 10 )

Re

δ→

→ ≈

(1)

x

x URe

⋅=

ν (2)

The velocity profile in the laminar boundary layer, for Rex<106 was determined by Prandtl and is parabolic:

( )2

2y yu y U

= ⋅ − δ δ

(3)

The velocity profile in the turbulent boundary layer can be described using a power law:

( )1/ 7

yu y U

= ⋅

δ (4)

For the flow velocity U, a value of 99% of the developed flow is used.

FLOW PROFILE IN CENTRIFUGAL PUMP

To determine the velocities near the boundary layers in a centrifugal pump impeller, calculations were made using

the commercial CFD-code ANSYS-CFX. The calculations were based on the geometry of an existing model pump

that has been built to study the motion of particles in pump impellers. This pump has an inlet diameter of 100 mm,

an outlet diameter of 220 mm. The impeller has logarithmically spiraled blades with a blade angle of 30° and a

width of 50 mm. Transient calculations were carried out at off design conditions with a flow of 2.8l/s and a pump

speed of 1000 rpm. Figure 2 and Figure 3 show the resulting velocity fields in the rotating reference frame for two

of these calculations (The impeller rotates in the clockwise direction with the pump outlet in the top left corner of

the impeller).



Figure 2 Flow velocities in pump impeller at position 1

Figure 3: Flow velocities in pump impeller at position 2

The results of the calculations show an unstable velocity field in the impeller during the rotation; this is as expected

due to the low number of blades and because the pump is not operating in its design point. The areas of interest in

this study are the flow velocities along the suction and pressure sides of the impeller blade (SS and PS in Figure 2).

On the suction side of the impeller, the flow velocities near the blade are very low, in the order of 0-2 m/s, and they

do not vary significantly during the rotation of the impeller. On the pressure side, velocity variations of 1-5 m/s can

be observed. Also, an area of low velocity can be observed moving along the blade from the leading edge of the

blade towards the trailing edge. This area does not reach the impeller outlet.

The results of the calculations show that the velocity field inside the impeller is unstable. It is therefore unlikely that

single continuous boundary layers exist on either the suction side or the pressure side of the blade impeller.

SS

PS



FORCES AND MOMENTS ON A PARTICLE

The motion of a particle is influenced by several forces, including the drag force, FD, lift force, FL, gravity force, FG,

added mass due to the acceleration of the neighboring fluid, FAM, the Basset history force, FB, the centrifugal force,

FC, the Coriolis force, FCo and the force due to pressure gradients, FPG. Combining these forces gives the differential

equation of motion for a particle in a rotating reference frame:

p

p p D L G AM B C Co PG

dvV F F F F F F F F

dt⋅ ⋅ρ = + + + + + + +

(5)

For the purposes of this study, motion of particles near the boundary layers, as a first approach the particle motion is

studied in a non rotating reference frame with no pressure gradients resulting from fluid acceleration. Equation (5)

reduces to:

p

p p D L G AM

dvV F F F F

dt⋅ ⋅ρ = + + +

(6)

Dividing by Vp gives:

( ) ( ) ( )p pDp f f p f p L p f p f AM p f

p

dv dv3 Cv v v v C v v C g

dt D dt

⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ × − ⋅ ⋅ ⋅ρ = ρ − − + ρ − Ω ρ + ρ − ρ

(7)

Drag coefficient

In 1851 Stokes theoretically derived the drag coefficient for spherical particles in a laminar flow and found that (for

pRe 0.5< ):

D

p

24C

Re= (8)

The drag coefficient CD depends upon the Reynolds number ( p

v DRe

⋅=

ν) according to (to get a smooth

continuous curve the following equations can be applied):

For the laminar region:

pRe 1< D p pp pp

24 3 24C Re ( 0.34) (1-Re )

Re ReRe= ⋅ + + + ⋅ (9)

The transitional region:

p1 Re 10000< < Dp p

24 3C 0.34

Re Re= + + (10)

The turbulent region:

pRe 10000> Dp p pp

10000 24 3 10000C ( 0.34) (1 ) 0.445

Re Re ReRe= ⋅ + + + − ⋅ (11)

For large Reynolds numbers the drag coefficient of spheres is a fixed number for which often the value of 0.445 is

used. In the intermediate range of Reynolds numbers many fit functions are known. A good fit function for the

transitional region has been derived by Turton & Levenspiel (1986), which is a 5 parameter fit function to the data

as shown in Figure 4:

0.657D p 1.09

p p

24 0.413C (1 0.173 Re )

Re 1 16300 Re−

= ⋅ + ⋅ ++ ⋅

(12)



1 10 100 1000 10000 100000

0.1

1

10

100

Drag coefficient of spheres

Re

CD

Turton & Levenspiel Turton & Levenspiel Stokes

Figure 4: Experimental data for drag coefficients of spheres as a function of the Reynolds number (Turton &

Levenspiel, 1986).

The models derived use the drag coefficient of spheres and hardly any discussion about this has been found in

literature, although it is known that for sands and gravels the drag coefficients, especially at large Reynolds

numbers, are larger than the drag coefficient for spheres. Engelund & Hansen (1967) found the following equation

based on measurements and found it best suited for natural sands and gravels (Julien, 1995):

Dp

24C 1.5

Re= + (13)

100

101

102

103

104

105

106

10-1

100

101

102

The drag coefficient for different shape factors

Re

CD

Sf=1.0 Sf=0.80-0.99 Sf=0.6-0.79 Sf=0.4-0.59 Sf=0.20-0.39

Sf=1.0 Sf=0.9 Sf=0.7 Sf=0.5 Sf=0.3

Stokes

Figure 5: Drag coefficient as a function of the particle shape (Wu & Wang, 2006).



It must be noted here that in general the drag coefficients are determined based on the terminal settling velocity of

the particles. Wu & Wang (2006) recently gave an overview of drag coefficients and terminal settling velocities for

different particle Corey shape factors. The result of their research is reflected in Figure 5 and Figure 6. Figure 5

shows the drag coefficients as a function of the Reynolds number and as a function of the Corey shape factor. Figure

6 shows the drag coefficient for natural sands and gravels. The asymptotic value for large Reynolds numbers is

about 1, while equation (13) shows an asymptotic value of 1.5. To emphasise the effect of the natural sands and

gravels, equation (13) will be used in the model for natural sands and gravels, while equation (12) is used for

spheres.

10-3

10-2

10-1

100

101

102

103

10-1

100

101

102

103

104

The drag coefficient of natural sands

Re

CD

Wu & Wang Wu & Wang Stokes Julien

Figure 6: Drag coefficient for natural sediments (Sf=0.7) (Wu & Wang, 2006).

Added mass coefficient

The added mass factor, CAM, which is dependent on the density of the sphere and the surrounding fluid, varies

between 0.5 and 1.05 (Odar and Hamilton (1964)). Given that the particle dimensions and the velocity differentials

lead to high values for the acceleration number 2

c p pA v /(a D )= ⋅ , a value of 1.05 is used.

Lift coefficient

The choice of the lift coefficient is a discussion in many of the models and many different values are found.

Sometimes the lift coefficient is expressed as a fraction of the drag coefficient and sometimes as a constant. In most

models however lift is present in the turbulent flow, but not in the laminar viscous sub layer. In this model also the

choice is made to neglect lift in the laminar region, so for boundary Reynolds numbers below 5. Wiberg & Smith

(1987A), Dey (1999), Pilotti & Menduni (2001), Stevenson, Thorpe & Davidson (2002) and others support this

assumption. For the turbulent region different values are used for the lift coefficient.



100

101

102

103

104

105

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lift coefficient vs the boundary Reynolds number

Re*

Lift C

oe

ffic

ient

CL

Coleman (1967) Bagnold (1974) Davies & Samad (1978) Walters (1971)

Cheng & Clyde (1972) Apperley & Raudkivi (1989) Chepil (1958) Einstein & B-Samni

Figure 7: The lift coefficient as a function of the particle Reynolds number.

Wiberg & Smith (1987A) use a value of 0.2, while using 0.85*CD in (Wiberg & Smith, 1987B) inspired by the

work of Chepil (1958). Marsh, Western & Grayson (2004) compared 4 models, but also evaluated the lift coefficient

as found by a number of researchers as is shown in Figure 7. For large Reynolds numbers an average value of 0.2 is

found, while for small Reynolds numbers the lift coefficient can even become negative. Luckner (2002) found a

relation where the lift coefficient is about 1.9*E*CD (including the effect of turbulence), which matches the findings

of Dittrich, Nestmann & Ergenzinger (1996). For an exposure level of 0.5 this gives 0.95*CD, which is close to the

findings of Chepil (1958).

CALCULATED PARTICLE TRAJECTORIES

The CFD calculations show that along the pressure side of the blade velocities of 1-6 m/s occur. Given this range

and the visible variation, there is no length for full development of the flow. As a first approximation therefore, the

flow along the impeller blade is assumed to resemble the flow along a flat plate. Based on the flow velocities at the

location, boundary layer thickness is calculated using equation (3). The calculations were made using different

particle sizes ranging from 0.5 to 2.5 mm. The figures show the boundary layer and the corrected value y*

(*

py y D / 2= − ) for the particle.

The calculations show that the influence of the angle of attack, the release height and the particle slip (defined as

vpart/vfluid)) on the particle trajectory is significant. If the angle of attack is too large, particles penetrate the boundary

layer and bounce of the wall. If the angle of attack is too small, the particles move parallel to the boundary layer.

The angle of attack that produced results in between was found to vary from 0° to 2° depending on the particle size.

For that reason, it was chosen to show only results of calculations with an angle of attack of 1.43° as these show all

variations of the found results.

The release height is very influential because of the influence on drag on the particle velocity. If the release height is

too large the particle will not reach the boundary layer. It will be carried along by the flow instead. Particle release

points close to or inside the boundary layer were therefore used.

Laminar boundary layer

Figure 8 and 9 show the trajectories of particles entering the laminar boundary layer with a slip of 1 and 0.9

respectively. Figure 8 shows that the larger particles (1.5 and 2 mm) entering the boundary layer bounce of the wall,

the smaller particles are caught in the boundary layer for a certain distance and then expelled. This would not take

place in the model impeller as the blade length is less than the distance needed by the particle to exit the boundary

layer.



0

0.005

0.01

0 0.05 0.1 0.15 0.2 0.25 0.3

Distance along blade [m]

y*

[m

]

0.5 mm

1.0 mm

1.5 mm

2.0 mm

Boundary layer

Figure 8: Particle trajectories – laminar boundary layer (δ=1 mm, U=3 m/s, attack angle 1.4°)

If the slip (=vpart/vfluid) factor of 0.9 is used, the larger particles do not bounce on the wall but are deflected away

from the wall by the boundary layer. Only smallest particle is initially caught in the boundary layer.

The difference in velocity between fluid and particles has a significant effect on the resulting particle trajectory. If

the particle is moving slower than the surrounding fluid, a positive lift force occurs and the particle is deflected

away from the wall. If the particle velocity is higher, the particle trajectory is bent towards the wall.

0

0.005

0.01

0 0.05 0.1 0.15 0.2 0.25 0.3


y*

[m

]

0.5 mm

1.0 mm

1.5 mm

2.0 mm

Boundary layer

Figure 9: Particle trajectories – laminar boundary layer (δ=1 mm, U=3 m/s, attack angle 1.4°)

Turbulent boundary layer

The results show that particles entering the turbulent boundary layer do not reach the wall. The velocity profile of

the boundary layer is such that there is little or no deflection from the original path (Figure 10).

If particles are released inside the boundary layer at a height of 5 mm, the particles are deflected towards the wall



(except for 0.5 mm) but the distance along the blade shows that this would not occur while the particles were inside

the impeller.

0

0.005

0.01

0.015

0 0.05 0.1 0.15 0.2 0.25 0.3


y*

[m

]

0.5 mm

1.0 mm

1.5 mm

2.0 mm

Boundary layer

Figure 10: Particle trajectories – turbulent boundary layer (δ=10 mm, U=3 m/s, attack angle 1.4°) – slip 1.0

0

0.005

0.01

0.015

0 0.05 0.1 0.15 0.2 0.25 0.3


y*

[m

]

0.5 mm

1.0 mm

1.5 mm

2.0 mm

Boundary layer


In this scenario, the particle trajectories are bent towards the wall, but they only make contact a significant distance

downstream. If the particle is initially moving slower than the fluid, the particles are bent towards the wall, but this

motion is so slow that the particles will exit the impeller before contact is made.



0

0.005

0.01

0.015

0 0.05 0.1 0.15 0.2 0.25 0.3


y*

[m

]

0.5 mm

1.0 mm

1.5 mm

2.0 mm

Boundary layer


EXPERIMENTAL RESULTS

To compare the predicted results of the calculations with the actual behavior of particles in a centrifugal pump

impeller, experiments were carried out in an existing test setup at MTI’s laboratory. The setup comprises a model

pump with shrouds and an impeller made of Perspex. A high speed CMOS camera is mounted on the shaft to

capture images of the particle flow inside the impeller.

Figure 13: Test setup

The GigE camera signal is transferred from the rotating frame to the computer using a digital FORJ (Fibre Optic

Rotary Joint). Figure 14 shows an example of images that are captured.



Figure 14: Captured image (example)

At present, the resolution of the camera limits the possibilities of the current setup to particle sizes greater than 1.5

mm. For the purposes of this study, particles of sizes between 1.5 an 2.5 mm were used. Figure 15 gives an

overview of observed particle trajectories in the impeller passage.

Figure 15: Observed particle tracks

At the suction side of the impeller blade, the flow velocities are low (this is inferred from the particle velocities) as

was predicted by the CFD calculations (see Figure 2 and Figure 3). Particles for the most part follow the flow as

described by the CFD-calculations. Some particles are caught in the low speed areas, however, and can be observed

to move slowly along the blade chord towards the impeller outlet.

On the pressure side of the blade, the particles appear to follow the contours of the flow as determined using the

CFD-calculations. They, for the most part, follow the blade geometry, accelerating and decelerating as they pass

through areas of different velocity. It almost appears as if they are being held up by the boundary layer. On occasion,

particles detach from the main flow, possibly as a result of turbulence. Their trajectory bends towards the blade, they

bounce on the blade surface back into the main flow at a larger end height than the original value. This phenomenon

was observed in the second chord half of the blade, but not in the first part.

PARTICLE BOUNDARY LAYER INTERACTION

The calculations show that a laminar boundary will have a larger influence on the particle trajectory than a turbulent

boundary layer. The velocity gradient is large enough to deflect the original particle path if the angle of attack is



small enough. There is also a clear influence of the slip. Particles moving faster than the surrounding fluid are

deflected towards the wall whilst they are deflected away if the velocity is lower. This is also visible in the case of a

turbulent boundary layer, but the effect is limited as the velocity gradient is only steep very near the wall.

Experimental data show that most of the particles do not penetrate the boundary layer at the studied concentration,

most appear to be supported by it. Those particles that do interact with the boundary exhibit behavior dependent on

the side of the impeller blade they approach. The observed particle trajectories can be compared to the results of the

calculations using a laminar boundary layer.

In this study, only the lift force as a result of the vorticity was considered. The effect of particle spin was not

accounted for. While it is expected that particle spin will influence the particle trajectory, the spin speed is not

expected to be high enough to significantly alter the calculation results.

CONCLUSIONS

Calculations of the particle trajectories show that a laminar boundary layer will have a larger influence on the

resulting particle trajectory than a turbulent boundary layer. The ratio of the particle and fluid velocities is also

important. If the particle velocity is less than the fluid velocity, the particle will be deflected away from the wall.

This does depend on the original angle of attack of the particle relative to the wall.

Comparison of the calculated trajectories with experimental observations of actual particle trajectories of particles

with sizes of 1.5 mm to 2.5 mm shows that modeling method can be used to model the trajectories near the impeller

blades.

Although the calculations were carried out using drag coefficients that did not account for the wall effect, the results

appear to justify this approach. This also applies to the size of the particle and size of the boundary layer. The

particle sizes used will have an effect on the velocity of the flow inside the boundary layer. This aspect will be the

subject of further research. This also applies to turbulence, specifically whether or not turbulence in the boundary

layer leads to different particle trajectories.

REFERENCES

Chepil, W. (1958). The use of evenly spaced hemispheres to evaluate aerodynamic force on a soil failure.

Transaction of the American Geophysics Union, Vol. 39(3) , 397-404.

Dey, S. (1999). Sediment threshold. Applied Mathematical Modelling , 399-417.

Dittrich, A., Nestmann, F., & Ergenzinger, P. (1996). Ratio of lift and shear forces over rough surfaces. Coherent

flow structures in open channels. , 126-146.

Engelund, F., & Hansen, E. (1967). A monograph on sediment transport to alluvial streams. Copenhagen: Technik

Vorlag .

Gülich, J. (2008). Centrifugal Pumps. Berlin, Heidelberg, New York: Springer.

Julien, P. (1995). Erosion and sedimentation. Cambridge University Press .

Luckner, T. (2002). Zum Bewegungsbeginn von Sedimenten. Dissertation. Darmstadt, Germany: Technische

Universitat Darmstadt.

Marsh, N. A., Western, A. W., & Grayson, R. B. (2004). Comparison of Methods for Predicting Incipient Motion

for Sand Beds. Journal of Hydraulic Engineering , 130 (No. 7, July 1, 2004)).

Odar, F., & Hamilton, W. (1964). Forces on a sphere accelerating in a viscous fluid. Journal of Fluid Mechanics 18.

, 302-314.

Pilotti, M., & Menduni, G. (2001). Beginning of sediment transport of incoherent grains in shallow shear flows.

Journal of Hydraulic Research, Vol. 39, No. 2. , 115-124.

Stevenson, P., Thorpe, R. B., & Davidson, J. F. (2002). Incipient motion of a small particle in the viscous boundary-

layer at a pipe wall. Chemical Engineering Science , 57, 4505–4520.

Turton, R., & Levenspiel, O. (1986). A short note on the drag correlation for spheres. Powder technology Vol. 47 ,

83-85.



White, F.(1999) Fluid Mechanics, 4th Edition. McGraw Hill.

Wiberg, P. L., & Smith, J. D. (1987A). Calculations of the critical shear stress for motion of uniform and

heterogeneous sediments. Water Resources Research , 23 (8), 1471–1480.

Wiberg, P., & Smith, J. (1987B). Initial motion of coarse sediment in streams of high gradient. Proceedings of the

Corvallis Symposium. IAHS Publication No. 165.

Wu, W., & Wang, S. (2006). Formulas for sediment porosity and settling velocity. Journal of Hydraulic

Engineering, 132(8) , 858-862.



NOMENCLATURE

Ac Acceleration number -

C Coefficient -

CD Drag coefficient -

CL Lift coefficient -

F Force N

g Acceleration of gravity m/s²

D Diameter m

Re Reynolds number -

Sf Shape factor -

t Time s

u Velocity m/s

U Velocity m/s

v Velocity m/s

V Volume m³

x Distance m

y Height m

δ boundary layer thickness m

ν Kinematic viscosity m²/s

ρ Density kg/m³

Ω Vorticity m/s

Subscripts

f Fluid

p Particle

AM Added Mass

B Basset

C Centrifugal

Co Coriolis

D Drag

L Lift



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