my journal paper

Original Article

Flow simulation in radial pump impellersand evaluation of slip factor

Mohamed G Khalafallah, Hassan A Elsheshtawy, Abdel-NabyM Ahmed and Ahmed I Abd El-Rahman

Abstract

This work is concerned with the study of the slip phenomenon in centrifugal pumps and the evaluation of its dependence

on the flow rate for a four-bladed pump. The finite volume method is used, and the impeller domain is represented by a

structured grid topology. The calculations assume a rotationally periodic boundary condition, while the frozen rotor

technique is used to model the interaction between the pump impeller and its surrounding volute casing. The simulation

uses an implicit time integration of the dynamic equations and is carried out using the commercial ANSYS CFX-solver.

Results from the simulation are found in reasonable agreement with the pump performance curve with a maximum

relative error of 4% in the range of flow coefficient from 0.8 to 1.2. The calculated values of the slip factor, as a function of

the flow rate, show good agreement with the Qiu’s mathematical model while retaining the default value of the defined

shape factor F¼ 0.52. In this particular study, the results show that although the slip factor improves with the increase of

either the number of blades or splitter length, the corresponding predicted hydraulic efficiency decreases due to the

increasing friction loss.

Keywords

Centrifugal pumps, turbomachinery flow, pump performance/efficiency, slip behavior/factor

Date received: 16 December 2014; accepted: 5 June 2015

Introduction

The slip phenomenon takes place in radial machinesas a result of the induced relative flow circulation, andconsequently, the fluid becomes unable to faithfullyfollow the guiding blades. The slip factor is a measureof such flow deviation and is defined in terms of theexit whirl velocity. Slip leads the flow to leavethe impeller with a mean relative angle �2 less thanthe blade exit angle �20 , as shown in Figure 1. Thisresults in a significant reduction in the work done onthe fluid, and consequently, the pump head is dramat-ically influenced

HE ¼Cu20U2

g¼

U22

g�

U2Q

g�D2b2cot �20 ð1Þ

A pump theoretical head HE is typically representedby the Euler equation (equation (1)), where all termsexcept the flow rate Q are constant. Here, U2 is theimpeller peripheral speed, while D2 and b2 refer to theimpeller diameter and the passage width, respectively,all measured at the exit section. However, due to slipand other losses, the actual head and flow rate (H–Q)characteristics exhibit a significant deviation from thecorresponding Euler behavior. Keeping that in mind,the slip in radial impellers depends on various

parameters, such as the number of blades and theblade geometry, and possibly varies with the pumpflow rate. Although increasing the number of bladeshelps reduce the flow deviation in the impeller exit, italso promotes blockage in the flow channels.Therefore, a careful study on the effect of thenumber of blades on the slip factor, as well as thedeveloped head and the pump efficiency is consideredin this work.

Stodola and Loewenstein1 was the first to develop amathematical expression for the slip factor in centri-fugal pumps. He considered a circular control volumefilling the blade passage near the impeller exit andassumed that the slip velocity is caused by the relativeeddy motion. He derived his equation for the slip vel-ocity CSL in terms of the exit blade angle �20 and the

Department of The Mechanical Power Engineering, Cairo University,

Giza, Egypt

Corresponding author:

Ahmed I Abd El-Rahman, Department of The Mechanical Power

Engineering, Cairo University, Cairo University Rd., Giza 12613, Egypt.

Email: [email protected]

Proc IMechE Part A:

J Power and Energy

0(0) 1–10

! IMechE 2015

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DOI: 10.1177/0957650915594953

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number of blades Z, as follows

CSL ¼� sin �20

ZU2 ð2Þ

Using the potential flow theory, Busemann2 devel-oped a theoretical framework for the estimation ofthe slip velocity and the calculation of the slip factorfor several blade angles and number of blades. Hisresults were plotted as a function of the inlet-to-outlet radius ratio and indicated constant slip behav-ior zone at small inlet-to-outlet radius ratios followedby a sharp reduction in slip at higher values of theinlet-to-outlet radius.

Later, Wiesner3 presented a general review of thevarious prediction methods, developed for the calcu-lation of basic slip factors, applicable for centrifugalimpellers. He concluded the first part of his work bysupporting the validity of the classical theoreticalmethod of Busemann.2 Furthermore, he carefullyexplored the Busemann experimental results and pro-posed the following simpler empirical expression forslip factor estimation, while considering the impellerexit velocity diagram, illustrated in Figure 1

� ¼ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffisin�20

pZ0:7

ð3Þ

Although limited by the impeller inlet-to-outlet radiusratio, Wiesner’s values showed a more accurate fit tothe Busemann test data. Wiesner then carried outextensive comparisons of slip factors, with test data,reported in the literature for more than 60 differentimpeller geometries that further demonstrated its highpotential in describing the slip phenomenon. Thus,Wiesner’s slip model is currently widely accepted forcentrifugal pump design.

Von Backstrom and Theodor4 derived a unifiedcorrelation for the slip factor assuming a single rela-tive eddy centered on the rotor axis in his fluiddynamic model. He ignored other mechanisms

affecting the slip phenomenon such as the blade turn-ing angle and the flow-induced wakes, and proposed amodel to calculate the magnitude of the recirculatingflow caused by the relative eddy. He argued that theeddy-induced slip velocity is dependent on the bladesolidity (c=s2, where c and s2 correspond to the chordand pitch at the blade exit, respectively) and definedthe slip factor in terms of the normalized slip velocity,following the work of Wiesner,3 as introduced inequation (4). In his trial to unify the previouslyderived formulas for the slip factor, Backstrom com-pared his results with other attempts to demonstrateits feasible replacement; however, his model does notshow any dependence on the pump flow rate

� ¼ 1� 1=½1þ 5ðcos�20 Þ0:5ðc=s2Þ� ð4Þ

Memardezfouli and Nourbakhsh5 experimentally stu-died the slip factor in centrifugal pumps at differentflow rates. They compared their results using five dif-ferent industrial pumps with the theoretical slip fac-tors modeled by Wiesner3 and Stodola andLoewenstein.1 They found good agreement at thepump best efficiency point (BEP), whereas a signifi-cant divergence was found at off-design conditions,specifically at low-flow rates operating regimes. Theyfurther defined the local slip factor and illustrated itsnonuniform distribution in the blade-to-blade pas-sage. A relative decrease in the local slip factor wasnoticed, while moving across the streamlines from theblade pressure side (PS) to the blade suction side (SS).Although inconsistent, their values for the mean slipfactors at the impeller exit showed clear dependenceon the flow rates and the number of blades.

Further, Caridad and Kenyery6 carried out a 3Dcomputational fluid dynamics (CFD) simulationusing the commercial CFX-solver package on five dif-ferent impellers of known geometries and specificspeeds. They were able to calculate the slip factorfor both single- and two-phase flow and found a simi-lar linearly decreasing behavior for all slip curves withthe increase of the pump flow coefficient. The pre-dicted slip factor was reported to decrease as thegas-void fraction increases from 10 to 17%. They fur-ther compared their slip results with values producedby Wiesner3 and Stodola and Loewenstein1 correl-ations and showed discrepancies as large as 52%.

Qiu et al.7 were the first to consider the influence ofthe variable pump flow rate in their derivation for aunified slip factor. Qiu et al. distinguished the mech-anism controlling the flow behavior, and thus the slipfactor, within the impeller at the impeller exit intothree components. The first component representedthe radial ‘‘Coriolis’’ effect, evident for typical radialimpellers. The second contribution accounted for theblade-turning rate (d�=dm, where m represents themeridional distance on the Z–R plane) characterizingthe extra loading from the streamline curvature, whilethe third component described the weak effect of the

Figure 1. The slip-influenced (solid lines) and the theoretical

blade-geometry (dashed lines) exit velocity triangles. CSL is the

slip velocity, considered by Stodola and Loewenstein1 in his

definition for the slip factor � ¼ 1� CSL=U2.

2 Proc IMechE Part A: J Power and Energy 0(0)



passage width variation on the back flow and wakesgeneration (d�b=dm, where � is the flow density). Qiuet al.7 defined the slip factor in terms of the slip vel-ocity normalized by the rim rotor speed

� ¼ 1�F� cos�2 sin �2

Z�

Fs2�24 cos�2

d�

dm

� �2

þF�2s2 sin�2

4�2b2

d�b

dm

� �2

ð5Þ

Here, F, �2, and �2 refer to the shape factor, as definedby Qiu et al.,7 the meridional inclination angle, andthe exit flow coefficient, respectively. Furthermore,their model was validated for several case studiesincluding an industrial pump with an actual blade-turning rate d�=dm of 4.92 degrees/m. An adjustablefitting parameter F¼ 0.6 was needed to obtain a goodmatch with the test data. Qiu et al. also comparedtheir slip results with corresponding values, calculatedusing other prediction methods, but specifically indi-cated a close agreement with the Wiesner3 model.

Recently, a CFD simulation, using the commercialsoftware package of STAR-CD, of the flow field incentrifugal compressors was reported by Huang et al.8

They considered the effect of the mass flow rate, theblade exit angle �2, the blade wrap angle � (see Figure2), and the number of blades Z. Considering theBusemann mathematical model,2 they defined theslip factor in terms of exit flow angle as

� ¼ 1� ½ðtan�2 � tan�20 ÞCm2

U2� ð6Þ

For simulation and comparison purposes, they con-sidered the geometry of Eckardt9 ‘‘rotor A’’ impeller.

Huang et al. showed that the onset of slip occurs closeto the exit section at a normalized camberline distanceat which the deviation of the flow angle from theblade angle begins to increase. The calculated totalpressure ratio and slip factor were successfully com-pared to Eckardt9 results. Surprisingly, their slipfactor exhibited a slight rise with the increase of themass flow rate.

Another recent numerical attempt has beenreported by Li10 who performed a CFD analysis ofthe slip factor in a centrifugal pump. The slip factorwas calculated from the velocity triangles at the impel-ler exit and from the impeller theoretical head. Heconcluded that the numerical results from the twomethods were inconsistent. Also, the effect of flowrate on the slip factor was dramatic for bladeshaving small exit angles, as expected, while the viscos-ity had minimal effect on the slip factor.

To the best of our knowledge, no simulation hasbeen developed which provides accurate prediction ofthe slip characteristics of a typical centrifugal pumpand shows plausible comparison with actual perform-ance and empirical relations. The goal of the presentstudy is to develop a finite volume model which helpsto understand the flow behavior within the blade-to-blade passage of the chosen centrifugal pump andreveals the slip variation with the pump flow rate.Also the study aims to verify the relation suggestedby Qiu et al.7 as the authors believe that their devel-oped expression could predict the slip factor at differ-ent flow rates reasonably well. A parametric studyabout the effect of varying the number of blades andsplitters on both the slip behavior and the pump per-formance is presented.

The next section elaborates the generation of thepump geometry along with the meshing details. This is

(a) (b)

Figure 2. The 2D schematic drawing (a) and the full 3D model (b) of the pump geometry.

Khalafallah et al. 3



followed by the ‘‘Setup of the numerical model’’ sec-tion, in which the implementation of the periodicboundary condition is first introduced. This sectionalso presents the model validation by comparing thenumerical values with the corresponding pump char-acteristic curve. In the ‘‘Results’’ section, an illustra-tion of the predicted flow field is presented, followedby the slip factor evaluation and comparison with dif-ferent mathematical models. The dependence of theslip factor and the pump’s hydraulic efficiency onthe number of blades and splitter blades is finallyconsidered.

Pump geometry and domain meshing

In this study, an industrial centrifugal pump with fourblades is chosen for computational modeling andsimulation purposes. To replicate the pump geometry,the CFturbo11 software is used to draw the contoursof both the impeller blade and the volute casing. Theimpeller blade assumes a shape of a fourth-order poly-nomial profile that leads to an exit blade angle �2 of17.5� and inlet blade angle �1 of 20.0

�, measured withrespect to the tangential direction. Here, the angles �and � are measured with respect to the radial direc-tion, as illustrated in Figure 2(a). The volute has atongue angle � of 29� and an inner diameter D3 of397mm, approximately. The main dimensions of thepump geometry are summarized in Table 1.

The impeller and volute data are then passed to theANSYS BladeGen12 software, to produce the 3Dmodel of the centrifugal pump, as shown in

Figure 2(b). The used splitters have three differentlengths of 30, 50, and 70% of the full-blade chord.The ANSYS Turbogrid13 program is used to generatethe 3D finite-volume hexahedral elements for accurateand fast CFD analysis of the flow behavior within theradial impeller. On the other hand, an unstructuredmesh is applied to the volute computational domain,as shown in Figure 3(a). The general grid interface isused to couple the structured impeller grid andunstructured volute grid while the frame changefrozen rotor model is used to model the rotor rotatingdomain relative to the stator stationary domain, asdescribed in detail in the ANSYS CFX14 UserDocumentation.

In this work, the entire domain is first consideredfor validation purpose by including the volute effect inthe simulation to allow comparison of the H–Q char-acteristics. The study then proceeds to predict the slipbehavior within the impeller passage and at the impel-ler exit. For this objective, it is unreasonable tonumerically model the pump entire domain since itcontains a periodically repeating flow field betweeneach two successive blades, instead, a representativecontrol volume is considered and appropriate bound-ary conditions are imposed at its boundaries, as indi-cated in Figure 3(b). To ensure that the encloseddomain behaves as a representative section of theentire impeller, rotationally periodic boundary condi-tions are enforced on the control volume surfaces inthe peripheral direction. This approach also helps toreduce the computational effort.

Setup of the numerical model

In the present model, the atmospheric pressure isimposed at the impeller inlet, while the outlet bound-ary condition is set as a constant mass flow rate. Theno-slip boundary condition is applied at the impeller

Table 1. The pump main dimensions (mm, degrees).

D1 D2 D3 b2 � �

177 360 397 29.5 124 29

Figure 3. (a) illustrates the impeller structured mesh versus the unstructured mesh of the volute casing, while plot (b) shows the

meshing and the imposed boundary conditions of the periodic domain of the impeller.




and volute casing walls. A turbulence intensity of 5%is imposed at the inlet section.

The present model solves the quasi-steadyReynolds-averaged Navier–Stokes equations usingthe two-equation k–e turbulence model and a coupledsolver. It is worth noting here that the k–e model doesnot yield the best accuracy for cases when either stallor reversed flow is present inside the domain.15 Theconvergence criteria settings include the specificationof the maximum number of iterations to 1000 and theresiduals to 10–4, while the imbalances in the govern-ing equations are set to less than 1%. The evolution ofthe total pressure is also monitored as the simulationproceeds to verify and record the attained steady-statesolution. The transient algorithm is employed with aspecified time increment of 60/N, where N refers to theimpeller’s number of revolutions per minute. Here,N¼ 1450 r/min.

Figure 4(a) shows the results of the grid independ-ence study applied to the considered periodic domain.It is clear that a grid size of 366,485 elements is suffi-cient and is thus used in the rest of this study. Thissimulation run is carried out at the pump’s BEP, atwhich the actual flow rate and head are 260m3/h and38m, respectively. Furthermore, to validate thenumerical model, the pump characteristic curve is pre-dicted and compared with the available actual curve.Computations are carried out using the k–e turbu-lence model.

Figure 4(b) compares the present numerical resultsof the H–Q curve with the corresponding actualvalues. Here, both the pump head H and the pumpflow rate Q are normalized with respect to their valuesat the BEP. Good agreement is found in the range ofnormalized flow rate between 0.8 and 1.2 with a max-imum relative error of 4%. However, in relativelyhigh and low flow rate regimes, the difference betweenthe numerical and test results increases up to about13%. Such a discrepancy may be attributed to theinability of the present CFD model to properlyaccount for the expected separated flow regimes,under high heads, far from the design point and theeffect of leakage under high flow rates.

Results

In this section, the numerical slip results are presentedand compared with a few mathematical models, alongwith a description of the flow behavior within theimpeller passage. In the following simulation runs,spatial variations in the flow static pressures andmeridional velocities are presented, followed by adetailed discussion of the effect of increasing thenumber of blades and adding splitters on the evalu-ated slip as a function of the pump flow rate.

Investigation of the flow field

Figure 5 shows the variation of both the meridionalvelocity and the static head along the blade pitch atthree different radial locations, namely those corres-ponding to 25, 50, and 81% of the passage radiallength. The results in Figure 5 refer to the meanflow behavior (i.e. along a plane located midspanbetween the hub and the shroud). Here, the four-bladed impeller operates at its design condition.With such low number of blades, the impeller-dis-charged flow is expected to become nonuniformover the blade pitch. Figure 5(b) shows that at the25% section, the meridional velocity profile is nearlyparabolic along 2/3 of the blade pitch, measured fromthe SS, but then deforms as the flow moves outwardto become strongly nonuniform over the blade pitchat the impeller exit.

It is clear that in the stream-wise flow direction, thevelocity gradually decreases as the flow moves out-ward because of the widening cross-sectional area ofthe flow passage, while a corresponding increase in thestatic head is indicated in Figure 5(d). The evolvementof the recirculating flows nearby the leading edge ontop of the blade PS is shown in Figure 5(a). This istypically generated because of the large blade-turningrate that leads to boundary layer separation. Theinfluence of wakes on the meridional velocity isobserved along the remaining 1/3 of the blade pitch,noting that it becomes less significant at the impellerexit, as shown in Figure 5(b).

(a) (b)

Figure 4. The pump head grid size dependence is presented in (a), while (b) refers to the model validation. The solid dots refer to

the present simulation results, while the solid line represents the pump characteristic curve.




To help to illustrate the induced separation zonewithin the impeller passage, the velocity vectors areplotted in Figure 6. The incoming flow is shown toseparate closely at the leading edge on the blade PS.The portion occupied by the wakes increases to blockabout 30% of the blade pitch and then graduallyshrinks allowing the flow to eventually refill theentire domain.

Slip factor evaluation

In the present study, a separate postprocessingMatLab program is developed to calculate the slipvelocity using the data of the tangential and radialvelocity components (Cu2 and Cm2), together withthe relative exit blade angles �20 , according to equa-tion (7). This is then followed by the direct estimationof the slip factor at the impeller exit. A comparisonbetween the numerical slip result and available empir-ical formulas, at the BEP, is shown in Table 2. Themean slip factor at the impeller exit is calculated to be0.58. The present result is seen to be in close agree-ment with both the Wiesner3 and the Qiu et al.7

models. The adjustable parameter is selected toequal 0.6, as suggested by Qiu et al. in their model

CSL ¼ U2 � Cu2 �Cm2

tan�20ð7Þ

To further illustrate the significant effect of the pumpflow rate on the slip factor, Figure 7(a) shows thevariation of the slip factor with the exit flow coeffi-cient �2, in comparison with the Wiesner and Qiuet al. models. The present simulation shows a linearreduction in the slip factor in the range of the flowcoefficient from 0.075 to 0.105. Consistent results, butunderpredicted by 11.6%, are estimated by the Qiuet al. model in the same range of the exit flow coeffi-cient. The effect of flow rate is represented by the thirdterm of equation (5), in which the blade-turning rate

Figure 5. Contour plots of the meridional-velocity and the static-head distributions between two successive blades at three dif-

ferent radial locations. The normalized arc distance is the ratio of the circumferential distance to the local blade pitch. (a) Contours of

the meridional velocity, (b) meridional velocity profile, (c) contours of the static head, (d) static head profile.

Figure 6. The figure shows a contour plot of the flow abso-

lute velocity, overlapped by the velocity vectors at BEP.




causes extra loadings due to streamline curvature. Thelarger the amount of flow rate, the less controlled theflow guidance and the smaller the slip factor. In thisparticular run, the slip factor varies from 0.6 at�2¼ 0.075 to 0.41 at �2¼ 0.127. The similaritybetween the present CFD results and the calculatedresults according to the Qiu et al. model suggests that,if the fitting parameter F in equation (5) is retained inits default value 0.52 instead of 0.6, a better agreementwith Qiu’s prediction is achieved, as demonstrated inFigure 7(a). Such validation has not been reportedbefore.

The local slip factor is calculated, using the localvelocity components Cu2i and Cm2i, and evaluated atthree different span locations over the blade pitch atthe impeller exit, namely the 0% span (hub), 50%span, and 100% span (shroud). The variation of thelocal slip factor as a function of the normalized arcdistance at the impeller exit is introduced in Figure7(b). Consistent with our understanding of the flowbehavior near the solid boundaries, the local slipfactor exhibits a remarkable increase at the hub sec-tion as compared to those values estimated at themean and shroud sections because of the blade loaddistribution along with the secondary flow effect.Moreover, the comparison reveals a relatively lowerslip factor at 100% spanwise position, correspondingto the impeller top section. This is due to the fact that,in our particular model, the pump is unshrouded. Thisresult supports our knowledge that a shrouded impel-ler helps improving the slip characteristics and that anunshrouded impeller induces a variation of the slipfactor with the flow rate.

Next, the number of blades is varied to study theeffect of their variation on the slip phenomenon, while

retaining the original impeller geometry. Specifically,impellers having six, eight, and 12 blades are con-sidered. In principle, using more blades would resultin larger slip factors because of the improved guidancethe fluid experiences through the impeller. As antici-pated, the present numerical result shows a remark-able increase in the slip factor using more blades, asindicated in Figure 8(a) and (b). Figure 8(b) showsthat as the number of blades increases the slip factorasymptotically approaches a maximum of 0.69. It isimportant to note that equation (5) defines the threemain factors affecting the slip in centrifugal impellers.The contribution by the number of blades is obviousin the second term of the right-hand side, thus, jus-tifying the above effect of the number of blades on theslip behavior.

Two different approaches were reported to enablethe investigation of the energy saving due to splitterblades. Golcu et al.16 considered standard impellershaving different numbers of main blades, fitted withsplitter blades having different lengths. The overallnumber of blades (main and splitter) varies in theirresults. In a recent study, Cavazzini et al.17 comparedthe performance of centrifugal pump both with andwithout splitters. In their work, the overall number ofblades (main and splitter) is preserved and set to eight.Here, we pursue Cavazzini’s approach.

The usefulness of using splitters in a conventionalpump is demonstrated through the comparison pre-sented in Figure 8(c). With respect to the performanceof the eight-bladed standard impeller, the presentCFD simulation captures a relative improvement inthe slip behavior through the replacement of full ori-ginal blades with splitter blades having differentlengths. It is clear that as the splitter length is

Table 2. The present slip factor versus reported correlations.

Z Stodola Backstrom Eckardt Qiu et al. Wiesner Present

4 0.7638 0.7275 0.8146 0.515 0.6288 0.58264

(a) (b)

Figure 7. The figure shows the slip factor variation with the exit flow coefficient as compared to Wiesner and Qiu models (a) and

the change of the local slip factor over the blade pitch (b) at the impeller exit. Here, Z¼ 4.




increased from 30 to 50%, the original slip factor isfairly recovered. However, no further improvement isnoticed using a 70% splitter. Therefore, it is con-cluded that the 50% splitter is sufficient to achieveslip behavior comparable to that obtained using thestandard impeller.

Pump head and hydraulic efficiency

This section presents the pump performance repre-sented by the pump head and the pump hydraulic effi-ciency at different exit flow coefficient �2. Thenumerical values extracted from ANSYS CFX solverenable us to calculate the pump head and efficiency.Contrary to the reduced slip, Figure 9(a) and (b)shows a remarkable decrease in the pump head withless impact on the corresponding pump hydraulic effi-ciency by increasing the number of blades. This is pos-sibly because of the built-up of frictional losses thatovercomes the energy harvested due to slip reduction.

The effect of replacing half of the original fullblades by splitter blades is further examined inFigure 9(c) and (d). The figure shows the effect ofsplitters on the pump head and hydraulic efficiencywhile preserving the overall number of blades (mainand splitter). A remarkable increase in the calculatedpump head along with a limited increase in the cor-responding hydraulic efficiency, particularly below theBEP, is obtained as the splitter length is increasedfrom 30 to 50%. This anomaly in the difference

between the insertion of a whole blade and the inser-tion of a splitter might be due to the blockage and theskin friction effects that further deteriorates the per-formance of a whole blade as compared to a splitter.Thus, we conclude that for the pump under consider-ation replacing half the full original blades with 50%span splitters helps increase the pump head by at least20% with minimal influence in the pump hydraulicefficiency at the pump’s BEP.

Conclusion

Numerical simulation of flow through a centrifugalpump is carried out to study the behavior of theflow through the impeller and to evaluate the slipfactor as function of the flow rate, the number ofblades, and the splitter length. It is shown that thecomputational results are in good agreement withthe pump’s performance curve in the neighborhoodof the BEP but is only in fair agreement with it nearthe neighborhood of the low and high flow rates. Theoverall slip factor is found to change linearly with theflow coefficient in a similar way as the expression byQiu et al.7 Reducing the fitting parameter F given byQui et al. from 0.6 to its default value 0.52 gives verygood agreement. This indicates that the Qui et al. rela-tion for estimating slip, although it is well accepted,needs further investigation to evaluate the fitting par-ameters correctly in its relations to the impellergeometry.

(a) (b)

(c)

Figure 8. The variation of the slip factor as a function of the pump flow rate using a different number of blades is shown in (a),

whereas its dependence on the number of blades at the best efficiency point is presented in (b). Plot (c) shows the influence of the

addition of splitters while preserving the overall number of blades (main and splitter). The blue squares refer to the standard eight-

bladed impeller having no splitters.




The effect of increasing the number of blades to upto 10 for the tested pump, is found to increase the slipfactor to reach saturation at about 0.69. Also, theinsertion of 30 and 50% splitters increases the slipfactor. However, the 70% span splitter does notshow any further improvement in the slip behavior.The application of splitters and the increase of thenumber of blades are also examined in their influenceon both the head and the hydraulic efficiency. Thesplitter blades are found to increase the developedhead by about 20% at the BEP, while a decrease inthe pump head is noticed when using more blades. Itis clear that there is a contrast when applying thesemodifications although both of them perform thesame function of improving the flow guidance. Thereason for this anomaly is that improving flow guid-ance increases the slip factor which in turn increasesthe developed head; however, the higher incurred fric-tion losses and blockage effect due to whole bladesinsertion offset the improvement due to better guid-ance. Using splitters, improvement in the hydraulicefficiency is found with flow rate up toQ=QBEP � 1:0, which is then followed by slightdecrease due to the increased friction losses.

References

1. Stodola A and Loewenstein LC. Steam and gas turbines:with a supplement on the prospects of the thermal prime

mover. New York: Peter Smith, 1945.

2. Busemann A. Das Forderverhaltnis radialerKreiselpumpen mit logarithmisch-spiraligen Schaufeln.ZAMM 1928; 8: 372–384.

3. Wiesner FJ. A review of slip factors for centrifu-gal impellers. J Eng Gas Turbines Power 1967; 89:558–566.

4. von Backstrom and Theodor W. A unified correlation

for slip factor in centrifugal impellers. J Turbomach2005; 128: 1–10.

5. Memardezfouli M and Nourbakhsh A. Experimental

investigation of slip factors in centrifugal pumps. ExpThermal Fluid Sci 2009; 33: 938–945.

6. Caridad JA and Kenyery F. Slip factor for centrifugal

impellers under single and two-phase flow conditions.J Fluids Eng 2005; 127: 317–321.

7. Qiu X, Japikse D, Zhao J, et al. Analysis and validationof a unified slip factor model for impellers at design and

off-design conditions. J Turbomach 2011; 133: 1–9.8. Huang J, Luo K, Chen C, et al. Numerical investiga-

tions of slip phenomena in centrifugal compressor

impellers. Int J Turbo Jet-Engines 2013; 30: 123–132.9. Eckardt D. Flow field analysis of radial and backswept

centrifugal impellers, part I. In: Proceedings of 25th

ASME gas turbine conference and 22nd annual fluidsengineering conference, New Orleans, 1980, New York,NY: American Society of Mechanical Engineers,

�1979; pp.77–86.10. Li W. Effects of flow rate and viscosity on slip factor of

centrifugal pump handling viscous oils. Int J RotatingMach 2013; 2013: 1–12.

11. CFturbo� Turbomachinery Design System, Release 9.0,User’s Guide. Software & Engineering GmbH, Inc.

(a) (b)

(c) (d)

Figure 9. The pump head and hydraulic efficiency are plotted for a different number of blades and splitters lengths. In subplots (c)

and (d), the blue dots refer to the eight-bladed standard impeller having no splitters. The overall number of blades (main and splitter) is

preserved.




12. ANSYS� BladeModeler, Release 14.5, componentsSystem, ANSYS Bladegen User’s Guide. ANSYS, Inc.

13. ANSYS� Turbogrid, Release 14.5, components System,

ANSYS Turbogrid User’s Guide. ANSYS, Inc.14. ANSYS� CFX, Release 14.5, Help System, CFX

Documentation. ANSYS, Inc.

15. Shojaeefard MH, Tahani M, Ehghaghi MB, et al.Numerical study of the effects of some geometric char-acteristics of a centrifugal pump impeller that pumps a

viscous fluid. Comput Fluids 2012; 60: 61–70.

16. Golcu M, Pancar Y and Sekmen Y. Energy saving in adeep well pump with splitter blade. Energy ConversManag 2006; 47: 638–651.

17. Cavazzini G, Pavesi G, Santolin A, et al. Using splitterblades to improve suction performance of centrifugalimpeller pumps. Proc IMechE, Part A: J Power and

Energy 2015; 229: 309–323.




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