Inverse design of a centrifugal pump on the meridional ...

Scientia Iranica B (2020) 27(5), 2478{2488

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Inverse design of a centrifugal pump on the meridionalplane using ball-spine algorithm

H. Fallah-Ardeshira;�, B. Ehghaghia, and M. Nili-Ahmadabadib

a. Department of Mechanical Engineering, University of Tabriz, P.O. Box 51666-14766, Tabriz, Iran.b. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran.

Received 23 February 2019; received in revised form 26 April 2019; accepted 2 June 2019

KEYWORDSInverse design;Centrifugal pump;Meridional plane;Quasi-3D;Impeller;Viscous.

Abstract. This study developed an inverse design algorithm called Ball-Spine Algorithm(BSA) as a quasi-3D method and applied it to the meridional plane of a centrifugal pumpimpeller in an e�ort to improve its performance. In this method, numerical analyses ofviscous ow �eld in the passage between two blades were coupled with BSA to modifythe corresponding hub and shroud geometries. Here, full 3D Navier-Stokes equations weresolved on a thin plane of ow instead of solving inviscid, quasi-3D ow equations on themeridional plane. To demonstrate the validity of the present work, the performance ofa centrifugal pump was numerically investigated �rst and then, it was compared withavailable experimental data. After de�ning a target pressure distribution on the huband shroud surfaces of the ow passage, a new impeller geometry was then obtained inaccordance with the modi�ed pressure distribution. The results indicated a good rateof convergence and desirable stability of BSA in the design of rotating ow passageswith incompressible, viscous ows. Overall, the proposed design method gave rise to thefollowing major improvements: an increase in static pressure along the streamline, a 5%increase in the pump total head, and delay in the onset of ow cavitation inside the impeller.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

Optimization problems related to pump design aregenerally challenging due primarily to the complexityof pump geometry and the corresponding ow patterninside. The optimal design of a geometry on themeridional plane of a pump is of signi�cance, asit directly in uences the ow pattern and velocitydistribution inside the impeller, hence the hydraulicperformance of the pump. In addition, the velocity

*. Corresponding author. Fax: +98 4134754447E-mail address: [email protected] (H.Fallah-Ardeshir)

doi: 10.24200/sci.2019.52991.2989

distribution on the shroud surface of an impeller a�ectsthe onset of ow cavitation and net positive suctionhead of the pump. The inverse design of the owpassage is one of the common approaches to problemsof this nature. This process involves obtaining owboundaries associated with a prescribed wall pressureor velocity pro�le and is generally achieved in twoways: non-iterative (coupled/direct) or iterative (non-coupled). In the former approach, a form of problemformulation can be used, in which the ow border isde�ned either implicitly or explicitly as a dependentvariable and appears in the governing equations. In thelatter method, however, ow and geometrical variablesare independent of each other. These methods use aninitial guess and the process is repeated until conver-gence is achieved. Although iterative methods are more

H. Fallah-Ardeshir et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2478{2488 2479

general and powerful, they are often mathematicallycomplex and computationally intensive. These meth-ods, however, can use black-box analytics to predict ow �eld within the passage, facilitating the design ofmore complex ow passages.

Researchers have previously used di�erent targetfunctions to perform the design procedure [1{4]. Tiowand Zanganeh [5] and Yang and Wu [6] establishedthe aforementioned method by applying virtual ve-locity to blade surfaces. Demeulenaere and Van denBraembussche [7] changed the boundary condition onthe blade walls and modi�ed the geometry by inversedesign code according to target pressure distribution.Virtual velocity on blade camber line and walls hasalso been used previously as a design function [8-12].Yang et al. [13] employed the formulation developedby Mileshin et al. [10] to calculate the induced virtualvelocity.

Poursadegh et al. [14{16] used a ball-spine inversedesign algorithm on the meridional plane of a cen-trifugal compressor impeller and, then, developed anoriginal and progressive algorithm to design the bladecamber line pro�le on the blade-to-blade planes ofimpeller based on blade loading improvement. Madadiet al. [17,18] employed the aforementioned design algo-rithm to design an axial ow compressor on the blade-to-blade surface as well as S-shaped di�users.

Arbabi [19,20] and colleagues developed anaerodynamic inverse design method for the three-dimensional design of viscous ow in blade passages.An aerodynamic inverse shape design approach, fullyconsistent with viscous ow, was successfully developedfor the 3D design of turbomachinery blades. Rama-murthy et al. [21] developed an inverse shape designmethod for steady, two-dimensional viscous internaland external ows over airfoils. In this method,a target pressure distribution was prescribed on theairfoil suction side. The approach was fully consistentwith the viscous ow assumption and was incorporatedinto a time accurate solution of the Reynolds-averagedNavier Stokes equations. The inverse method was �rstvalidated and, then, used to redesign airfoils for inter-nal and external ow cases, where the robustness andusefulness of the inverse method were demonstrated.

Recently, the inverse design algorithms have beenextensively used to design e�cient turbomachines invarious applications. This clearly demonstrates thepotentials of this approach in the �eld [22{29].

In the current work, the ball-spine inverse designalgorithm is developed as a quasi-3D method and, then,applied to the meridional plane of a centrifugal pumpimpeller operating in an incompressible, viscous owregime. In this method, viscous ow simulations areconducted within a thin region between two adjacentblades of an impeller, where the full 3-D Navier-Stokes equations are solved instead of their inviscid,

quasi-3D counterparts. Considering the rotary natureof the frame of reference in this study, a reducedstatic pressure distribution [30] is used as a targetparameter. The performance of the developed designmethod is then examined on the meridional plane of acentrifugal pump. Finally, the ow passage is designedto (a) optimize pressure gradients along the passage,(b) minimize and eliminate potential wakes, and (c)increase the discharge pressure. The designed geometryis then simulated to investigate the realization of theaforementioned targets.

2. Geometry

Impeller is the only rotating part of a pump transfer-ring the supplied energy to the working uid. Figure 1presents the meridional view of a centrifugal pump.As shown earlier, the impeller blades are bounded byhub and shroud surfaces, allowing for the control ofpassages' cross-sectional area. The working uid entersthe impeller in the axial direction and leaves it, whileits direction turns close to 90 degrees.

Table 1 summarizes the hydraulic speci�cation ofthe reference geometry used for the analyses presented.

3. Ball-spine inverse design algorithm

The ball-spine method is fundamentally an iterativeapproach. Figure 2 demonstrates the overall procedureused for geometry modi�cation. As shown, the huband shroud walls are composed of virtual balls with�xed mass that can be freely moved along the speci�eddirections called spine. Fluid ow through the passage

Figure 1. Meridional view of a centrifugal pump impeller.

2480 H. Fallah-Ardeshir et al./Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 2478{2488

Table 1. Features of reference impeller.

Type of pump Radial ow pump witha shrouded impeller

Speci�c speed 25Rotational speed (rpm) 1450Design ow rate (m3/hr) 50Design head (m) 13Blade number 6Impeller eye diameter (mm) 104Impeller exit diameter (mm) 209

Figure 2. Implementation of the inverse designalgorithm.

applies a pressure distribution to the wetted side ofthe boundary. If the target pressure distribution isapplied to the outer side of the boundary, the exiblewalls begin to deform in a way that the actual pressureinside the passage equals the target pressure. In otherwords, the force due to the di�erence between currentand desired pressure pro�les anywhere along the walldisplaces the corresponding balls at any point along thespine. As the desired shape is achieved, the di�erencebetween two pressure pro�les disappears and the ballsstop moving.

Figure 3 demonstrates the free body diagram of

Figure 3. Free body diagram of a ball [16].

Figure 4. Schematic of spines in the meridional plane.

a virtual ball. If the balls move in the directionof applied force, adjacent balls may collide with oravoid each other. To prevent this problem, each ballshould move along the spine during the entire processof deformation.

Figure 4 shows the spines for a ow passage onthe meridional plane. In the current work, the spinesare chosen to be perpendicular to the ow stream. Inorder to obtain a unique solution to the inverse designproblem in hand, the axial length of the ow passageis kept constant.

According to Figure 3, the displacement of ballsin each shape modi�cation step is calculated usingEq. (1). Here, C is an adjustment parameter forthe convergence rate of the BSA method. A smallervalue of this parameter results in a slower convergencerate of the algorithm. Note that if the value of thisadjustment parameter exceeds a certain threshold, thedesign algorithm will diverge [16]. Therefore, allocatingan optimal value to this constant is of signi�cant


importance to minimize the computation costs. Now,assuming that the displacement of the imaginary ballsis only due to the pressure di�erence between the insideand outside of the boundary, we have:

�Si = C � �Pi; (1)

F = �P �A � cos � = ma ) a =�P �A � cos �

m; (2)

�S=12

�P�A�cos ��b�A (�t)2=

(�t)2

2�b��P�cos �; (3)

C =(�t)2

2�b� cos �: (4)

Projecting the displacements into perpendicular hori-zontal and vertical displacements yields:(Xnew = Xold + �S: cos�Ynew = Yold + �S: sin�

(5)

where � denotes the angle of spine from the horizon.In the current method, residual for the proposed

iterative design process is calculated using Eq. (6).A threshold of 10�2 is used here to determine whenconvergence to the target geometry is achieved. Toensure higher accuracy, ANSYS CFX is used to solvethe ow �eld in each iteration.

residual =

NPi=1

�Pi � (Ptarget)i

�NPi=1

�(Ptarget)i

� : (6)

4. 3-D modeling

In this work, a 3D simulation of the centrifugal pump isconducted to examine its performance and the resultsare compared against the experimental data. Thesimulated geometry encompasses three distinct parts:impeller, volute, and outlet pipe. Figure 5 shows theoverall view of the pump.

5. Mesh generation

Structured and unstructured meshes are used for meshgeneration within the simulation domain. Here, anO-grid structured mesh is used in the blade near-wallregion in order to examine the boundary layer better,as shown in Figure 6. In the current study, the averagey+ in the wall region is about 1.5 and its maximumvalue does not exceed 5.

6. Boundary conditions

Table 2 summarizes the boundary conditions used inthis study. Static pressure of the reservoir is used as theinlet boundary condition. Here, an average turbulent

Figure 5. General view of the centrifugal pump model.

Figure 6. Mesh con�guration used for ow analysis.

Table 2. Boundary condition for the numerical study ofcentrifugal pump.

Inlet pressure (bar) 0.15Inlet turbulence intensity (%) 1Outlet mass ow rate (BEP) (kg/s) 15.5Wall roughness (�m) 100Reference pressure (atm) 1Physical time step (sec) 0.001

intensity of about 1% is considered. This is justi�edwhere the turbulent intensity of ow at the inlet isminimized by existing ba�es. A known mass ow rate(or the average velocity) is used at the outlet boundary.Here, the corresponding value at the Best E�ciencyPoint (BEP) of the pump is used. No-slip conditionis enforced at the solid boundaries. The frozen rotormodel is also used to calculate the average tangentialvelocity at the interface of stationary and rotary parts.This model utilizes a quasi-stable algorithm, which


treats the ow from rotor to stator by changing theframe of reference while maintaining their relativeposition. The rotary terms are simulated in themoving frame of reference, but the transient e�ectsare neglected. This leads to an e�ective approach tocalculating the communication between the impellerand volute. Water at 25�C (with a density rate of 997kg/m3 and a dynamic viscosity rate of 0.0010518 Pa.s)is considered as the working uid in this investigation.The shear stress transport model is used to simulateturbulence.

7. Mesh independency

In the current study, the outlet pressure is used asa parameter to investigate the dependency of thesimulations on the used grid sizes. Table 3 summarizesthe results including the impeller, volute, and the outletpipe for the overall simulation domain. Based onthese results, a domain of 2173604 elements is usedto achieve a mesh-independent simulation while beingstill computationally a�ordable.

8. Result validation

In this section, the experimental data obtained fromthe hydraulic test of the main centrifugal pumpare used to validate the simulation results. The

Table 3. Evaluation of the dependency of mesh.

Number of elements Outlet pressure

1842482 180169

2173604 178888

3277226 178261

Figure 8. Comparison of numerical results andexperimental results.

measurements were conducted at the TurbomachineryLaboratory of the University of Tabriz. Figure 7shows a schematic of the centrifugal pump test bed,which is fully equipped with data acquisition andanalysis systems. The geometrical and design pointspeci�cations of the pump used are summarized inTable 1. Figure 8 presents a comparison between themeasured and simulated pump heads in terms of owrates. As shown, the predictions are within 5% of themeasured values and, in fact, the error falls below 1%at the design point of the pump.

9. Algorithm implementation

The inverse design procedure comprises two majorparts: ow solver and shape modi�cation algorithm.Here, ANSYS CFX is used as the ow solver, and thedesign algorithm is implemented through a separate

Figure 7. The schematic of the centrifugal pump test bed located at the Turbomachinery Laboratory of the University ofTabriz.


MATLAB code. The design procedure begins witha comparison between the target and initial pressurepro�les on the hub and shroud boundaries. Thedi�erence between the two pressure distributions re-sults in a change in the geometry of the ow passageat each iteration. Flow �eld associated with themodi�ed geometry is then simulated using CFX, andthe actual pressure pro�le is updated accordingly. Thisprocess is repeated until the di�erence between theactual and target pressure distributions falls below theprede�ned threshold. The current method allows forimplementing di�erent �lters in order to smoothen themodi�ed geometry and utilizing di�erent adjustmentparameters to improve the convergence rate of thealgorithm.

10. Quasi-3D ow passage and boundaryconditions

Considering the symmetrical geometry of the impellerand the incoming ow, the problem is simpli�ed toa quasi-3D case in order to shorten the computationtime. It is important to note that all geometricalchanges occur on the meridional plane; therefore,the passage speci�cations on the blade-to-blade planeare not a�ected. The aforementioned geometry isconstructed in the ICEM CFD environment such thatthe meridional projection rotates by about 2 degreesaround the symmetry axis of the impeller, as shown inFigure 9. No-slip boundary condition is applied to thehub and shroud, while the side walls are consideredto allow free ow slip. The intake of the impelleralso remains una�ected during the design procedure.Note that structured grids are used in the entirecomputational domain.

11. Validation

The performance of the design algorithm has beenextensively evaluated in the literature for various sta-tionary passages and compressible ow con�gurations

Figure 9. Created quasi-3D duct with boundaryconditions.

[14,15,31{33]. Here, a similar evaluation is performedin the case of incompressible ow in a rotating passage.To this end, the reduced static pressure is used asthe target design parameter, which is expected todirectly contribute to boundary layer thickening and ow separation due to adverse gradients [30]. Thereduced pressure is calculated through the followingexpression:

Preduction = P � 12� r2!2: (7)

To perform the validation, the geometry of themain pump is used as a target, and the initial geometryis constructed by applying arbitrary changes to thetarget geometry, as shown in Figure 10. The corre-sponding ow regime is laminar and incompressible,and the ow �eld is simulated by applying a uniforminlet velocity of 1.3 m/s and a relative pressure of zeroat the outlet. Here, 25�50 cells are used for simulation.Despite the rotational nature of the ow passage as wellas ow separation within the initial geometry, designconvergence successfully occurs after 27 iterations, asshown in Figure 11.

To investigate the e�ect of design parameters onthe performance of the algorithm, the static pressureis now considered for this purpose. Starting from thesame initial conditions, the design procedure convergesafter only 56 iterations, as depicted in Figure 12. Thee�ect of grid sizes on the design procedure is alsoinvestigated. For this purpose, the same computationaldomain with coarser (14 � 28) and �ner (45 � 60)elements is considered. Figure 13 compares the con-vergent solution for the coarser domain against thereference case (25 � 50 elements). The results of thecase with �ner elements are not included in this �gure,as the corresponding convergent solution overlaps that

Figure 10. Initial guess and target geometry forvalidation.


Figure 11. Converging process of reduced static pressuredistribution from initial guess geometry to the targetgeometry.

Figure 12. Converging process of static pressuredistribution from the initial guess geometry to the targetgeometry.

of the reference case. This suggests that the designalgorithm becomes independent of element size beyonda certain threshold.

12. Geometry modi�cation

The sharp curvature of impeller on the meridionalplane is combined with the blade geometry resultsin wake and potential ow separation on the suctionside of the shroud surface and is manifested as apressure drop. In addition, high local velocities andunnecessary rapid ow accelerations (or decelerations)also contribute to the overall energy loss. On the otherhand, the high angle of incidence at the ow inlet may

Figure 13. Grid study for design of the meridional planein the rotating frame.

Figure 14. Current and modi�ed reduced pressuredistributions along the hub and shroud.

also result in considerable energy loss, which requiresdesign reconsideration. These design issues can beaddressed on either the meridional or blade-to-bladeplanes of the impeller.

The validated design procedure is now employedto modify the pump geometry on the meridionalplane. Figure 14 shows the current and target pressuredistributions on the hub and shroud surfaces. Thecorresponding geometries, before and after the shapemodi�cation, are shown in Figure 15. The adjustmentparameter C used in the current design procedure is10�8, and the full convergence of the design occursafter 34 iterations. In order to accurately examine


Figure 15. Current and modi�ed geometry of themeridional plane.

the resulting improvements, the full geometry of themodi�ed impeller is constructed and evaluated usingthe 3D solver discussed earlier. The correspondingstatic pressure distribution in Figure 16 shows thatthe ow �eld on the suction side of the modi�edgeometry has improved. Figure 16 shows that thevelocity distribution at the outlet of the impeller hasbecome more uniform for the modi�ed geometry andthe corresponding average value has increased. Table 4summarizes the results of the �nal shape modi�cation.It is important to note that the overall pump head hasincreased by over 5% for the modi�ed geometry at BEP.

Figure 17 demonstrates the pump performance asa function of cavitation for the existing and modi�edpumps. In this graph, head coe�cient is plottedagainst Thomas cavitation coe�cient at BEP. The Hy-

Table 4. Values of dimensionless head and owcoe�cients.

Geometry '

Original 0.577 0.102Modi�ed 0.605 0.107

Figure 17. Cavitation operation curve for optimized andcurrent impellers.

draulic Institute standards have de�ned the cavitationonset as the point where a 3% drop occurs in pumphead. Beyond this point, the local cavitation growsand considerable loss of pump performance is expected.After applying this criterion, Figure 17 shows that thecavitation onset is delayed for the modi�ed geometry,suggesting that the modi�ed pump can successfullyoperate under lower suction pressures.

13. Conclusion

In the current work, the ball-spine inverse designalgorithm was developed as a quasi-3D method and wasapplied on the meridional plane of a centrifugal pump

Figure 16. Static pressure contour on the created quasi-3D conduit: (a) Initial conduit and (b) modi�ed conduit.


impeller operating in an incompressible, viscous owregime. In this method, the viscous ow simulationswere conducted within a thin region between twoadjacent blades of an impeller, where the full 3DNavier-Stokes equations were solved instead of theirinviscid, quasi-3D counterparts. The design algorithmwas implemented using an external MATLAB codecoupled with the ANSYS CFX as the ow solver. Thisapproach further allows for the utilization of di�erentspatial �lters to smoothen the geometry at each step,which in turn allows the adjustment parameter ofthe design algorithm to be enhanced, hence improv-ing the corresponding convergence rate. The resultsdemonstrated that the reduced pressure distributionwas a more physically meaningful design parameterthan static pressure and it improved the convergenceof the design algorithm. It was further found that thealgorithm became independent of the grid size beyonda certain threshold. The developed design algorithmwas �nally applied to an existing pump in an e�ort toimprove its performance by eliminating unnecessary,excessive pressure gradients on the meridional plane ofthe impeller. The simulation of the ow �eld withinthe modi�ed impeller demonstrated an overall increasein pressure along the streamline, resulting in a delay inthe cavitation onset on the suction side of the passageand a 5% improvement in the pump head.

Nomenclature

a Acceleration of the ball (m/s2)A Element area, di�user cross-sectional

area (m2)BSA Ball-Spine AlgorithmC Geometry correction coe�cient (m2 s2

/kg)m Ball mass (kg)

� Density of uid (kg/m3)

�b Density of ball (kg/m3)F Force imposed on the ball (N)P Static pressure (Pa)r Radial coordinate, radius (m)

! Angular velocity (s�1)� Di�erence�t Time step (s)�P Target and computed pressures

di�erence (Pa)' Flow coe�cient Head coe�cientBEP Best e�ciency pointPD Pressure distribution

y+ Non-dimensional wall distance� Cavitation coe�cient

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Biographies

Hadi Fallah-Ardeshir is a PhD Candidate of Me-chanical Faculty of Tabriz University. He received herMaster's degree in Energy Conversion from School ofMechanical Engineering in Iran University of Scienceand Technology. Her research interests include turbo-machinery.

Biyuk Ehghaghi is an Associate Professor and aFaculty Member at the Department of Mechanical


Engineering at Tabriz University of Technology. Hereceived his Master and PhD degrees from TabrizUniversity respectively. His major research interestsinclude the �elds of turbomachinery, experimentalaerodynamics, and manufacturing.

Mahdi Nili-Ahmadabadi is an Associate Professor

and a Faculty Member of Mechanical EngineeringDepartment at Isfahan University of Technology. Hereceived his Master and PhD degrees from SharifUniversity of Technology in 2005 and 2010, respec-tively. His major research interests are inverse design,turbomachinery, experimental aerodynamics, and PIVmeasurement.

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