1-s2.0-S0307904X10002271-main

Applied Mathematical Modelling 35 (2011) 242–249

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

The flow simulation of a low-specific-speed high-speed centrifugal pump

B. Jafarzadeh a,*, A. Hajari b, M.M. Alishahi a, M.H. Akbari a

a High Performance Computing Center (HPCC), Department of Mechanical Engineering, School of Engineering, Shiraz University, Shiraz, Iranb Advanced Materials Research Center (AMRC), Materials & Metallurgical Engineering Department, Iran University of Science & Technology (IUST),Tehran 16844-1314, Iran

a r t i c l e i n f o

Article history:Received 5 July 2009Received in revised form 2 May 2010Accepted 25 May 2010Available online 31 May 2010

Keywords:Centrifugal pumpTurbulence modelingCFDInducer

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.05.021

* Corresponding author.E-mail address: [email protected] (B. Jafarza

a b s t r a c t

In this paper a general three-dimensional simulation of turbulent fluid flow is presented topredict velocity and pressure fields for a centrifugal pump. A commercial CFD code wasused to solve the governing equations of the flow field. In order to study the most suitableturbulence model, three known turbulence models of standard k–e, RNG and RSM wereapplied. The complex flow configuration required us to use around 5,800,000 cells, and12 computational nodes (processors) for parallel computing. Simulation results in the formof characteristic curves were compared with available experimental data, and an accept-able agreement was obtained. Additionally, effect of number of blades on the efficiencyof pump was studied. The number of blades was changed from 5 to 7. The results show thatthe impeller with 7 blades has the highest head coefficient. Finally, it was observed alsothat the position of blades with respect to the tongue of volute has great effect on the startof the separation. Thus, to analyze the effect of blade number on the characteristics of thepump, the position of blade and tongue should be similar to each other. Investigations ofthis kind may help to reduce the required experimental work for the development anddesign of such devices.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Centrifugal pumps are prevalent for many different applications in the industrial and other sectors. Nevertheless, theirdesign and performance prediction process is still a difficult task, mainly due to the great number of free geometric param-eters involved. On the other hand the significant cost and time of the trial-and-error process by constructing and testingphysical prototypes reduces the profit margins of the pump manufacturers. For this reason, CFD analysis is currently beingused in hydrodynamic design for many different pump types [1–3].

Numerical simulations can provide quite accurate information on the fluid behavior in the machine, and thus help theengineer to obtain a thorough performance evaluation of a particular design. However, the challenge of improving thehydraulic efficiency requires an inverse design process, in which a significant number of alternative designs must be eval-uated. Despite the great progress in recent years, even CFD analysis remains rather expensive for the industry, and the needfor faster mesh generators and solvers is imperative [4]. Some of the recent investigations in this field are mentioned in thefollowing.

Guleren and Pinarbasi [5] analyzed a centrifugal pump by solving Navier–Stokes equations, coupled with the standard k–eturbulence model. Their pump consisted of an impeller having five blades and a low rotating speed of 890 rpm. Numericalsimulations were performed on a commercial FLUENT package assuming steady flow. Asuaje et al. [6] performed a 3D-CFD

. All rights reserved.

deh).

B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249 243

simulation of impeller and volute of a centrifugal pump using CFX code with a specific speed of 32. In this simulation, struc-tured grid was used in the impeller and unstructured grid in the volute. k–e, k–x and SST turbulent models were used. Theyfound velocity and pressure fields for different flow rates and radial thrust on the pump shaft. Cui et al. [7] investigated theeffect of number of splitting blades for long, mid and short blades using a one-equation turbulent model. Their computationswere performed using commercial FINE/TURBO 6.2 at a specific speed of 18. Their results show that the bulk flow in theimpeller has an important influence on the pump performance. Anagnostopoulos [8] simulated 3D turbulent flow in a radialpump impeller for a constant rotational speed of 1500 rpm based on the solution of the RANS equations. The flow equationswere discretized using the control volume approach, and the standard k–e model was adopted for the turbulence closure.The computations for the steady flow field in a particular impeller were presented. The characteristic performance curvesfor the entire load range of the impeller were constructed, and their pattern was found reasonable and in agreement withtheory. None of the previous works includes study of 3D modeling within a full domain considering interaction between ro-tor and stator of a high centrifugal pump using various turbulence models.

In the current study the effect of various turbulence models (k–e, RNG and RSM) on the flow field and efficiency of a high-speed centrifugal pump has been carried out. Using the most suitable model the effect of blade number on the specific char-acteristic of the pump has been investigated.

2. Pump specifications

One of the significant methods to prevent cavitation is to make use of inducers in high-speed pumps. The simulated pumpincludes a two-way inducer and a 6-blade impeller with Dimpeller/Dinlet of 1.53 and Doutlet/Dinlet of 0.56 (Dimpeller, Dinlet andDoutlet are, respectively, diameter of impeller, diameter of volute’s inlet and diameter of diffuser’s outlet). The specific speedis defined as:

Ns ¼XQ 0:5

H0:75 ; ð1Þ

where Q is volume flow rate and H is the pump head. For the pump studied in the current work Ns is 16.32 with rotationalspeed over 13,000 rpm. The three-dimensional configuration of pump and its main parts are shown in Fig. 1.

3. Governing equations

Since the fluid surrounding the impeller rotates around the axis of the pump the equations must be organized in two ref-erence frames, stationary and rotating reference frames. To accomplish this, the Multiple Reference Frame (MRF) model hasbeen used. In this approach, the governing equations are set in a rotating reference frame, and Coriolis and centrifugal forcesare added as source terms. Continuity and momentum equations of an incompressible flow are as the following:

r:u ¼ 0; ð2Þr:ðquuÞ ¼ �rp þrs þ s: ð3Þ

In the above equations u is the relative velocity of fluid, s the stress tensor and s is the source term, which consists ofCoriolis and centrifugal forces

s ¼ �2qX� u� qX� ðX� rÞ: ð4Þ

Here X is rotational speed and r position vector.

Fig. 1. Different parts of the simulated pump.

Fig. 2. Structured grids around impeller blades and inducer helix.

244 B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249

4. Grid generation

The pump is divided into three regions, inlet, rotary and outlet. Each region is discretized independently; structured gridsare used for inlet and rotary regions, but a mixture of structured and unstructured grids are used for the outlet region. InFig. 2 the structured grids around blades and inducer are shown. In the present study, four sets of grids were used for gridstudy, and considering the results the case with 5,835,589 cells was selected for the final simulation. Table 1 shows the setsof grids which were used for grid study.

The non-dimensional head and power coefficients in Table 1 are defined as:

Table 1Grid stu

Cond

1234

Head coefficient w ¼ gH

X2r2; ð5Þ

where w is head coefficient, H is the head of the pump, X is rotational speed and r is outer radius of the impeller.Power coefficient

Power coefficient Pc ¼Psh

qX3r5; ð6Þ

where Psh is the consumed shaft power and q is density. Psh also is defined as:

Psh ¼ M �X; ð7Þ

where M is the measured torque.

5. Boundary conditions

In the present study, volume flow rate and pressure outlet boundary conditions were used for the inlet and outlet, respec-tively. Outer walls were stationary but the inner walls were rotational. There were interfaces between the stationary androtational regions. Non slip boundary conditions have been imposed over the impeller blades and walls, the volute casingand the inlet wall and the roughness of all walls is considered 100 lm. The turbulence intensity at the inlet totally dependson the upstream history of flow. Since the fluid in the suction tank is undisturbed, the turbulence intensity for all conditionsis considered 1%. Water was used as a working fluid in ambient condition.

6. Numerical scheme

6.1. Solver

In order to calculate the flow field a commercial CFD code, FLUENT, was used. The governing integral equations for theconservation of mass, momentum and when appropriate, energy and other scalars such as turbulence were solved. Twonumerical solvers of segregated and coupled employ a similar discretization process, but the approach used for linearizing

dy data.

ition Number of cells Number of nodes Head coefficient Power coefficient

1,952,914 5 0.705 0.03113,558,151 7 0.673 0.02865,835,589 12 0.637 0.02747,251,216 14 0.629 0.0269

B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249 245

and solving the discretized equations is different. The segregated solver solves the governing equations sequentially, whilethe coupled solves them simultaneously [9]. In the present analysis, the segregated solver was used since the coupled solveris usually used in high compressible flows in which the flow and energy equations are coupled, and this method often resultsin a faster solution convergence. A trade-off involved in the use of the coupled solver is that it requires more memory (1.5–2times) than the segregated solver.

The Pressure-velocity coupling methods recommended for steady-state calculations are SIMPLE or SIMPLEC [10,11]. Forrelatively uncomplicated problems in which convergence is limited by the pressure-velocity coupling, the convergence couldbe achieved more quickly using SIMPLEC. With SIMPLEC, the pressure-correction under-relaxation factor is generally set to1.0, which aids in the convergence speed-up. In some problems, however, increasing the pressure-correction under-relaxa-tion to 1.0 can lead to instability due to the high grid skewness. In the present simulation, SIMPLE algorithm was preferredconsidering the complexity of the flow and grid qualities.

6.2. Modeling of the rotary region

FLUENT provides a powerful set of features for solving problems in which fluid rotates around an axis, such as flows insideturbomachineries in different methods. Some of these methods include multiple reference frames (MRF), mixing plane andsliding mesh models. Each method has a different accuracy and computational expenses. The first and second models areappropriate for steady flows and for cases in which the interactions between rotor and stator are negligible. For instance,for a pump with bladeless stators the MRF can be used as a suitable approach. The sliding mesh model is appropriate wherethe interaction between rotor and stators is noticeable and the unsteadiness of problem is supposed to be reproduced. Sincein the present problem, the stator has no blade and the unsteadiness of the problem can be ignored, the MRF model wasused.

7. Results and discussion

7.1. Modeling of turbulence

In the present study, we intend to investigate the effect of number of blades on the pump efficiency. Additionally we wantto choose the most efficient turbulence model for the problem. Considering the various turbulence models, three known tur-bulence models of standard k–e, RNG k–e and RSM are utilized at a constant blade number of 6 for the pump. In order toshow the results in a more practical order non-dimensional parameters defined in reference [12] are used and for the sakeof generality the data are reported in non-dimensional form.

Fig. 3 shows the head coefficient vs. flow coefficient for the three turbulence models compared with available experimen-tal data. The curves show that with increasing the flow coefficient, the head coefficient is decreased. Comparing various tur-bulence models data with experimental data, it has been concluded that each of these turbulence models provide acceptableresults, but two models RNG k–e and RSM show better agreement than the standard k–e model.

In the above figure flow coefficient is defined as:

Q c ¼Q

Xr3 ; ð8Þ

where Q is volume flow rate and the efficiency of the pump is defined as:

Fig. 3. Head coefficient vs. flow coefficient with three different turbulence models and one available experimental data.

Fig. 4. Power coefficient vs. flow coefficient with three different turbulence models.

246 B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249

g ¼ qQHgPsh

� 100: ð9Þ

The power coefficient and efficiency curves vs. flow coefficient are plotted in Figs. 4 and 5, respectively. As shown in Fig. 4in low and high mass flow rates different turbulence models approach each other. In Fig. 5, in addition to low and high massflow rates, in mid mass flow rates, difference between various turbulence models are low because efficiency considers theeffect of both head and power.

In most of the previous models of centrifugal pumps, the impeller and volute were simulated separately, and the inter-action between them was ignored. In present work, however, all of such interactions are taken into account.

Fig. 6 indicates static pressure distribution around the impeller and volute in three turbulence models (standard k–e mod-el, RNG k–e model and RSM). The results show that the pressure distribution is predicted similarly using the three turbulencemodels, moreover it appears that RNG and RSM models predict a lower minimum pressure at the entrance of the impeller(approximately 2.59e+05 pascal in standard k–e model, 9.36e+04 pascal in RNG k–e model and 9.09e+04 pascal RSM). It isfound out that these models can predict cavitation phenomenon sooner than standard k–e model. In these counters withincreasing the radius of impeller and obtaining energy of pump the velocity of the fluid increases and in entrance of diffuser,the velocity is converted to pressure thus the velocity exits with high pressure.

In the case of k–e model, two additional transport equations are solved, but in the RSM model, seven additional transportequations must be solved in 3D [13,14]. Since RNG k–e has lower number of transport equations to solve, therefore it needslower memory compared to RSM, we select RNG k–e model to continue of other parts.

Of course computations with the RNG k–e model, due to the extra terms and functions in the governing equations and agreater degree of non-linearity of standard k–e model tend to take 10–15% more CPU time.

Compared with the k–e model, the RSM requires additional memory and CPU time due to the increased number of thetransport equations for Reynolds stresses. However, efficient programming in FLUENT has reduced the CPU time per iterationsignificantly. On average, the RSM in FLUENT requires 50–60% more CPU time per iteration compared to the k–e model. Fur-thermore, 15–20% more memory is needed [9].

Fig. 5. Efficiency vs. flow coefficient with three different turbulence models.

Fig. 6. Contours of static pressure (pascal) with different turbulence models.

Fig. 7. Head coefficient vs. flow coefficient with different number of blades.

Fig. 8. Power coefficient vs. flow coefficient with different number of blades.

B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249 247

Fig. 9. Efficiency vs. flow coefficient with different number of blades.

248 B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249

Aside from the time per iteration, the choice of turbulence model can affect the ability of FLUENT to obtain a convergedsolution. For example, the standard k–e model is known to be slightly over-diffusive in certain situations, while the RNG k–emodel is designed such that the turbulent viscosity is reduced in response to high rates of strain. Since diffusion has a sta-bilizing effect on the numerics, the RNG model is more likely to be susceptible to instability in steady-state solutions. How-

Fig. 10. Velocity magnitude (m/s) in the volute tongue with different number of blades.

B. Jafarzadeh et al. / Applied Mathematical Modelling 35 (2011) 242–249 249

ever, this should not necessarily be seen as a disadvantage of the RNG model, since these characteristics make it moreresponsive to important physical instabilities such as time-dependent turbulent vortex shedding. Similarly, the RSM maytake more iterations to converge than the k–e model due to the strong coupling between the Reynolds stresses and the meanflow [9].

7.2. Effect of blade number on the pump characteristics

In this section, the effect of number of blades on the pump efficiency will be analyzed through three blade number of 5, 6and 7.

Fig. 7 shows the head coefficient curves vs. flow coefficient for three cases of 5, 6 and 7-blade pumps. In low flow rates,head coefficient for 6 and 7-blade pumps are very close and higher than 5-blade pump. With increasing the flow rate, thehead coefficient for the 5-blade pump does not change but this characteristic for 6 and 7-blade pumps decreases. The 6-blade, however, has a steeper slope than the 7-blade pump and shows a strong decrease of head coefficient at higher flowrates. Generally, it is clear that the impeller with 7 blades has the highest head coefficient when compared with 5 and 6-blade pumps at all ranges.

Power coefficient and efficiency curves are shown in Figs. 8 and 9, respectively. Fig. 8 indicates that all the cases are closetogether particularly at high flow coefficients. For the efficiency, it is expected for the curves correspond to the 6-blade case(Fig. 9) to locate between the two other curves corresponding to the 5 and 7-blade cases. However, because of a differentpositioning of the 6-blade impeller with respect to the tongue, the simulation results for this case are not as expected.

Noting the fact that the problem is solved in steady-state situation and the position of the first blade (Fig. 10) is the samein all the pumps and moreover, other blades have been considered relative to this blade and the pumps impellers rotateequally but the angles between two consecutive blades are different (72� in 5-blade pump, 60� in 6-blade pump and51.43� in 7-blade pump), therefore the distances between the nearest blade before the tongue and tongue – denoted byparameter d, Fig. 10 – are different in three pumps.

Fig. 10 shows that this distance has a great influence on the flow field. 6-blade pump has a larger distance of blade withthe tongue; therefore flow separation has begun later than the other two cases. The unique behavior of the 6-blade pump isrelated to its large distance of blade with the tongue. Larger distance has stronger backflow effects.

8. Conclusions

In the present investigation numerical simulation of a high-speed centrifugal pump was performed. At first the optimumturbulence model for the problem was found. Considering the available experimental data, the best result appears to be ob-tained by RNG k–e model. Investigation on the effect of number of blades on the efficiency head coefficient as the selectioncriteria shows that the impeller with 7 blades has the highest head coefficient when compared with 5 and 6-blade pumps atall ranges.

Finally, it was observed also that the position of blades with respect to the tongue of volute has great effect on the start ofthe separation. Thus, to analyze the effect of blade number on the characteristics of the pump, all pumps should have similarpositioning of the blades with respect to the tongue.

References

[1] C. Hornsby, CFD – driving pump design forward, World Pumps 2002 (2002) 18–22.[2] S. Cao, G. Peng, Z. Yu, Hydrodynamic design of rotodynamic pump impeller for multiphase pumping by combined approach of inverse design and CFD

analysis, ASME Trans. J. Fluids Eng. 127 (2005) 330–338.[3] F.A. Muggli, P. Holbein, CFD calculation of a mixed flow pump characteristic from shutoff to maximum flow, ASME Trans. J. Fluids Eng. 124 (2002) 798–

802.[4] M. Asuaje, F. Bakir, S. Kouidri, R. Rey, Inverse design method for centrifugal impellers and comparison with numerical simulation tools, Int. J. Comput.

Fluid Dyn. 18 (2) (2004) 101–110.[5] K.M. Guleren, A. Pinarbasi, Numerical simulation of the stalled flow within a vaned centrifugal pump, J. Mech. Eng. Sci. 218 (2004) 425–435.[6] M. Asuaje, F. Bakir, S. Kouidri, F. Kenyery, R. Rey, Numerical modelization of the flow in centrifugal pump: volute influence in velocity and pressure

fields, Int. J. Rotat. Mach. 3 (2005) 244–255.[7] Boaling Cui, Zuchao Zhu, Jianci Zhang, Ying Chen, The flow simulation and experimental study of low-specific-speed high-speed complex centrifugal

impellers, Chin. J. Chem. Eng. 14 (4) (2006) 435–441.[8] J.S. Anagnostopoulos, Numerical calculation of the flow in a centrifugal pump impeller using Cartesian grid, in: Proceedings of the Second WSEAS

International Conference on Applied and Theoretical Mechanics, Venice, Italy, November 2006.[9] Fluent, Release 6.2.16, 2004, Documentations.

[10] H.K. Versteeg, W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, Loughborough, 1995.[11] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor & Francis Group, New York, 1980.[12] I.J. Karrasik, W.C. Krutzsch, W.H. Fraser, J.P. Messina, Pump Handbook, McGraw Hill, New York, 2003.[13] H. Schlichting, Boundary Layer Theory, McGraw Hill, New York, 1999.[14] T. Cebeci, T. Cousteix, Modeling and Computation of Boundary-layer Flows, Indian Wells, 1998.