HAO BING

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published online 9 July 2013 2013 227: 567 originallyProceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy

Hao Bing and Shuliang CaoThree-dimensional design method for mixed-flow pump blades with controllable blade wrap angle

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Original Article

Three-dimensional design method formixed-flow pump blades with controllableblade wrap angle

Hao Bing and Shuliang Cao

Abstract

In this article, a direct and inverse iteration design method for mixed-flow pumps was introduced, and a three-

dimensional design platform for mixed-flow pumps was established. Through iteration calculation of two kinds of

stream surfaces to solve both the continuity equation and motion equation of the fluid at the same time, and

therefore obtaining the quasi-three-dimensional flow field inside the mixed-flow pump, the accuracy of the flow

field calculation was improved. By repeating iteration of the direct calculation and inverse design, influence of blade

shape on flow field calculation was fully considered, thus assuring that the final design of blade camber line complies

with actual flow pattern. After comparing and analyzing the meridional velocity distribution, as well as the relative

circulation distribution and expelling coefficient in the blade zone, it was proved that compared with others designed

by conventional methods, the impeller designed with three-dimensional design platform has a better hydraulic per-

formance, especially the blade energy conversion capacity that witnessed a significant improvement. With this three-

dimensional design platform, a parametrization was applied to velocity moment distribution and blade leading and

trailing edges positions. The influence of velocity moment distribution parameters P and a0, and the blade leading and

trailing edges positions parameters �h and �t on blade wrap angle and the internal flow of the impeller were analyzed.

By curve fitting, the functional relationship between blade wrap angle ’ and parameters �t, P and a0 was obtained. The

blade wrap angle control strategy during the process of mixed-flow pump impeller design was then put forward, thus

realizing three-dimensional design of mixed-flow pump blades with controllable blade wrap angle. The performance

test for the mixed-flow pump model suggested that the three-dimensional design method with controllable blade wrap

angle, by controlling blade wrap angle size and adopting direct and inverse iteration design method, ensures a better

hydraulic and cavitation performance for the mixed-flow pump impeller designed with three-dimensional design

platform.

Keywords

Mixed-flow pumps, three-dimensional design, controllable blade wrap angle, loading distribution, flow analysis

Date received: 8 January 2013; accepted: 3 April 2013

Introduction

Mixed-flow pumps have the advantages of a wideapplication and a wide range of high efficiency oper-ation, and are extensively used in agricultural irriga-tion, water diversion, flood prevention and waterlogdrainage, and water circulation process. In recentyears, scholars have conducted many research workson three-dimensional design,1,2 performance predic-tion,3–5 performance analysis6–8 and optimization9,10

of mixed-flow pumps. Based on experiments andnumerical calculation, research works on param-eters11–13 that significantly influence mixed-flowpump performance have been carried out, producingmany important results, which effectively improvedimpeller internal flow, efficiency and suctionperformance.

Based on conventional design methods, this articleintroduced a direct and inverse iteration designmethod (DIIDM), and established a three-dimensional design platform (TDP) for mixed-flowpump impellers by using FORTRAN programminglanguage. By parametrization, parameters that dir-ectly control blade wrap angle variation duringdesign process were obtained, and the influence ofthese parameters on blade wrap angle and impeller

State Key Laboratory of Hydroscience and Engineering, Tsinghua

University, Beijing, People’s Republic of China

Corresponding author:

Hao Bing, State Key Laboratory of Hydroscience and Engineering,

Tsinghua University, Beijing 100084, People’s Republic of China.

Email: [email protected]

Proc IMechE Part A:

J Power and Energy

227(5) 567–584

! IMechE 2013

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internal flow was analyzed. Based on these results, theblade wrap angle control strategy was then put for-ward, thus realizing blade wrap angle control ofmixed-flow pump during design.

Three-dimensional design platform

In mixed-flow pump designs based on conventionalone-dimensional or two-dimensional flow assump-tion, the impeller is assumed to be constituted by aninfinite number of infinitively thin blades, and itsinternal flow is axisymmetric. As a result, the merid-ional velocity obtained by these design methods canonly satisfy fluid continuity equation. To improve thedesign method, this article introduced DIIDM andbuilt a TDP for mixed-flow pump impellers.

During calculation and design, TDP assumes thatthe fluid is incompressible, inviscid and flows steadily.By using FORTRAN programming language, the fol-lowing steps were completed: (1) draw the meridionalflow passage according to the given impeller param-eters; (2) obtain meridional flow field by direct calcu-lation; (3) obtain three-dimensional impeller model byinverse design; (4) iterate the direct calculation andthe inverse design and (5) after iteration converges,produce wood model data, three-dimensional coord-inates and model for the blade.

The meridional flow passage of mixed-flow pump isdetermined by the lines of hub and shroud. Because ofits significant influence on impeller flow capacity,hydraulic performance and cavitation performance,the meridional flow passage should be smooth.While drawing it, the influence of specific speedshould be fully considered, and mixed-flow pumpmodels with similar specific speeds and favorable per-formance should be drawn on. Hub and shroud areconnected smoothly by arcs and straight lines,respectively. To adjust the blade rotational angle con-veniently, spherical flow passage was adopted for huband shroud in the blade zone.

Direct calculation

During the mixed-flow pump design, the governingequations of the direct calculation are equal to thecontinuity equation and the motion equation of thefluid

r � �Cð Þ ¼ 0

W� ðr � CÞ ¼ �Fþ rEr

�ð1Þ

where � is the blade expelling coefficient, W is therelative velocity vector of fluid motion, C is the abso-lute velocity vector of fluid motion, F is the fluid massforce per mass unit and Er is the mechanical energy offluid relative motion per mass unit.

Figure 1 shows the orthogonal curvilinear coordin-ate system. Three coordinates, q1, q2 and q3, each

stands for meridional streamline, flow cross-sectionline and central angle.

The continuity equation in the orthogonal curvilin-ear coordinate system is as follows

@ ð�C1H2H3Þ

@q1þ@ ð�C2H1H3Þ

@q2þ@ ð�C3H1H2Þ

@q3¼ 0

ð2Þ

where C1, C2 and C3 are components of the absolutevelocity C in three directions, respectively. H1, H2 andH3 are Lame coefficients as follows

H1 ¼ H1 q1, q2ð Þ

H2 ¼ H2 q1, q2ð Þ

H3 ¼ r

8><>: ð3Þ

where r is the meridional plane projection radius.The motion equation in the orthogonal curvilinear

coordinate system can be written as

W2�3 �W3�2 ¼ �F1 þ@Er

H1@q1

W3�1 �W1�3 ¼ �F2 þ@Er

H2@q2

W1�2 �W2�1 ¼ �F3 þ@Er

H3@q3

8>>>>>>>><>>>>>>>>:

ð4Þ

where W1, W2 and W3 indicate components ofthe relative velocity W in three directions, respect-ively. F1, F2 and F3 are components of the massforce F in three directions, respectively. �1, �2 and�3 indicate

�1 ¼1

H2H3

@ ðC3H3Þ

@q2�@ ðC2H2Þ

@q3

� �

�2 ¼1

H1H3

@ ðC1H1Þ

@q3�@ ðC3H3Þ

@q1

� �

�3 ¼1

H1H2

@ ðC2H2Þ

@q1�@ ðC1H1Þ

@q2

� �

8>>>>>>>>><>>>>>>>>>:

ð5Þ

Figure 1. Orthogonal curvilinear coordinate system.

568 Proc IMechE Part A: J Power and Energy 227(5)



Calculation of S1 stream surface. Assuming that the flowon S1 stream surface is irrotational, the continuityequation and the motion equation on S1 stream sur-face can be written as

@

@q3ðH1C1Þ �

@

@q1ðH3C3Þ ¼ 0

@

@q1ð�H2H3C1Þ þ

@

@q3ð�H2H3C3Þ ¼ 0

8>><>>: ð6Þ

After conformal transformation of the above equa-tion set, the velocity potential function equation onthe conformal transformation plane (x–y plane inFigure 2) can be obtained as follows

@

@xh@�

@x

� �þ@

@yh@�

@y

� �¼ 0 ð7Þ

where h stands for flow layer thickness on S1 streamsurface, h¼ h(y), and � is the velocity potential func-tion on the conformal transformation plane.

After setting boundary conditions, finite elementmethod was applied to obtain numerical results ofequation (7). The boundary conditions were set asfollows: velocities of the inlet and outlet are uniformdistribution in circumferential direction; the relativevelocity of the solid wall surface is 0 in normal direc-tion; periodic conditions were given on the boundarieswith periodic variations.

Calculation of S2 stream surface. Assuming that the flowon S2 stream surface meets axisymmetric conditions,then the continuity equation, motion equation,streamline equation and orthogonal equation on S2

stream surface can be written as

@

@q1ð�C1H2H3Þ ¼ 0

W� ðr � CÞ ¼ �Fþ rEr

W � n ¼ 0

F� n ¼ 0

8>>>>><>>>>>:

ð8Þ

where n denotes normal vector.

In orthogonal curvilinear coordinate system, it isdifficult to directly solve equation (8). So this articleused streamline curvature method instead, convertingthe above equation set to a meridional velocity gradi-ent equation along any quasi-orthogonal line l(Figure 3).

Basic geometrical relationships in Figure 3 are asfollows

sin � ¼H1dq1dl

cos � ¼H2dq2dl

sin � ¼@r

H1@q1¼ �

@z

H2@q2

cos � ¼@r

H2@q2¼ �

@z

H1@q1

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð9Þ

where � stands for the angle between normal vector ofquasi-orthogonal line and meridional streamline, and� denotes the angle between plumb line and merid-ional streamline.

Any physical quantity f on S2 stream surface can bewritten as follows

df

dl¼

@f

H1@q1

H1dq1dlþ

@f

H2@q2

H2dq2dl

ð10Þ

With equations (9) and (10), equation (8) canbe converted to a meridional velocity gradientequation along any quasi-orthogonal line asfollows

dCm

dl¼ ACm þ Bþ

D

Cmð11Þ

where Cm denotes meridional velocity, and A, B, Dare equation coefficients which vary in and outside theblade zone.

Figure 3. Geometrical relationships on S2 stream surface.

Figure 2. S1 stream surface before (left) and after (right)

conformal transformation.

Bing and Cao 569



In the blade zone

A¼1

1þ ðr @�@mÞ2

1

cos �

@�

@m1þ r2 sin �

d�

dl

@�

@m

� ��

�1

�

@�

@mþsin�

r

�þ

1

cos �

d�

dl

�sin �þ r2

d�

dl

@�

@m

� �

þd�

dl

@

@mr2@�

@m

� ��@�

@m

d

dlr2@�

@m

� ��

B¼2!r

1þ r @�@m� �2 d�

dlsin� �

@�

@mcosð� � �Þ

� �

D¼1

1þ r @�@m� �2 dEr

dl

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ð12Þ

Outside the blade zone

A ¼1

cos �

@�

@m�d�

dlsin �

� ��sin �

rsin �

B ¼ 0

D ¼!r2 � Cur

r2dðCurÞ

dlþdEr

dl

8>>>>><>>>>>:

ð13Þ

where m stands for meridional streamline length, l forquasi-orthogonal line length, � for central angle, ! forrotation angle velocity and Cu for circumferentialcomponent of the absolute velocity.

Iteration of S1 and S2 stream surfaces. To calculate theimpeller internal flow field more accurately and pre-cisely, iteration needs to be conducted in S1 and S2

stream surfaces. Steps are as follows.

1. Use conventional design method to get the merid-ional streamline as the baseline. Rotate the base-line around the rotation axis to generate initial S1

stream surface.2. Solve equation (7) on the conformal transform-

ation plane of S1 stream surface.3. According to the principle of equal flow rate,

select central streamlines of each S1 stream surfaceto constitute S2 stream surface.

4. By iteration, solve equation (11) on S2 stream sur-face. According to the principle that ‘any sub-flowpassage formed by two adjacent streamlines hasthe same flow rate’, new meridional streamlinecan be determined.

5. Revolve the new meridional streamline around therotation axis to generate new S1 stream surface.

6. Repeat steps (2)–(5). When position deviation ofthe two meridional streamlines obtained by twoconsecutive rounds of calculation satisfies the seterror range, the iteration converges and the directcalculation is finished.

Inverse design

The inverse design of mixed-flow pump impellersinvolves blade camber line drawing, thickening andsmoothing.

Blade camber line drawing. Under the assumption thatblades are of infinite number and are infinitively thin,blade camber lines completely overlap with spatialstreamlines in the blade zone. Hence, blade camberline drawing can be converted to the solution of theposition and shape of the spatial streamline. Aftercalculating meridional flow field, both meridionaland radial coordinates of the spatial streamline aredetermined. When coordinate in circumferential dir-ection is determined, the position and shape of spatialstreamline can be completely determined, thus obtain-ing the shape of blade camber line.

As is shown in Figure 4, the direction in whichmeridional streamline increases along radius is set aspositive, and the direction opposite to circumferentialrotation direction as positive. The distance that fluidmicelle covers along spatial streamline in time perioddt is dM. dM has its meridional projection of dm

dm ¼ Cmdt ð14Þ

Within the same period of time, the distance that fluidmicelle motion covers in circumferential direction isdMu. Hence

dMu ¼Wu dt ¼ r d’ ð15Þ

where Wu denotes the circumferential component ofrelative velocity, d’ denotes the rotation angle ofmeridional plane, on which the fluid micelle stayswithin the time period of dt.

From the velocity triangle shown in Figure 5, thefollowing relationship can be obtained

Wu ¼ U� Cu ð16Þ

Figure 4. Fluid micelle motion on the stream surface: (a)

meridional plane projection and (b) section projection vertical

to rotation axis.




Combining equations (14) to (16), then

d’ ¼U� Cu

Cmrdm ð17Þ

Put U¼!r into equation (17). Hence

d’ ¼!r2 � Cur

Cmr2dm ð18Þ

The velocity triangle holds such a relationship asfollows:

tg� ¼Cm

U� Cuð19Þ

where � is the flow angle. Under the assumption thatblades are of infinite number and are infinitively thin,� is equal to the blade angle �b.

Combining equations (17) and (19), thus

d’ ¼1

tan� � rdm ð20Þ

Likewise, the velocity triangle also holds such arelationship

U� Cu ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2 � C2

m

qð21Þ

Combining equations (17) and (21), thus

d’ ¼1

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW2

C2m

� 1

sdm ð22Þ

Equations (18), (20) and (22) are three forms of dif-ferential equation of blade camber line. They eachadopt different parameters to represent position vari-ation d’ in circumferential direction caused by pos-ition variation dm on meridional streamline in theblade zone. Based on any one of the three equations,by giving Cur, or �, or W, the unique correspondingrelationship between dm and d’ can be fully deter-mined. Thus, the blade wrap angle can be obtainedby integration, and blade camber line shape couldtherefore be obtained.

The velocity moment distribution pattern alongmeridional streamline, while closely related to the

loading distribution on mixed-flow pump blade sur-face, also influences the hydraulic and cavitation per-formance of the impeller. The following equationdemonstrates the relationship among blade pressuresurface pressure, suction surface pressure and velocitymoment variation gradient along meridionalstreamline

pp � ps ¼2�

Z� �Wm

@ Curð Þ

@mð23Þ

where pp and ps denote pressure on blade pressure sur-face and suction surface, respectively, and Wm is themeridional component of the relative velocity.

To better control the loading distribution onmixed-flow pump blade surface, blade camber linedifferential equation (18), which includes Cur, wasselected to be the basic equation for blade camberline drawing.

Equation (18) was point-by-point integratedalong meridional streamline, thus obtaining theblade wrap angle of each node on the meridionalflow net

’ ¼

Z mi

0

!r2 � Cur

Cmr2dm ð24Þ

where mi denotes the length of meridional streamlinein the blade zone. In equation (24), ! is the givendesign parameter, r and Cm are calculated based onmeridional flow field. If the variation pattern of Curalong meridional streamline is given, the variationpattern of blade wrap angle ’ along meridionalstreamline can be fully determined.

Blade thickening. To ensure that the blade meets struc-tural strength requirement, the blade has to be thick-ened after blade camber line drawing. This articleadopted conformal transformation thickeningmethod to conduct the thickening. Through confor-mal transformation, spatial camber line was trans-formed into a conformal transformation plane onwhich blade thickening was completed. The bladecamber line obtained previously is the one for bladepressure surface. Based on the given thickness distri-bution pattern, blade suction surface can be calcu-lated, and therefore thickening is finished.

The steps are as follows: select the conformal trans-formation plane radius R0. The variation pattern ofblade wrap angle ’ with streamline length m is inte-grated based on blade camber line integration equa-tion (24). By transforming spatial camber line ontothe conformal transformation plane, the correspond-ing relation between (’, m) and coordinates (x, y) onthe conformal transformation plane is obtained. Onthe plane, blade thickening is achieved by calculationof coordinates x and y.

In the coordinate system on the conformal trans-formation plane (Figure 6), abscissa and ordinate are

Figure 5. Velocity triangle.

Bing and Cao 571



defined as follows

x ¼ R0’

y ¼

Z m

0

R0

rdm

8<: ð25Þ

Blade thickness on the spatial stream surface is

S ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

cos2 �btan2

rð26Þ

where � denotes the given actual thickness and denotes the angle between meridional transversaland meridional streamline.

After conformal transformation, blade thicknessobtained on the conformal transformation plane is

S0 ¼R0

rS ð27Þ

Based on the geometrical relationship shown inFigure 6, coordinates of the blade suction surface onthe conformal transformation plane are as follows

xs ¼ xp þ S0 sin�b

ys ¼ yp � S0 cos�b

�ð28Þ

where xp and yp are coordinates of blade pressuresurface on the conformal transformation planeobtained by transformation of ’ and m on bladecamber line through equation (25). When all thedata about the blade suction surface were calculatedaccording to equation (28), coordinates x and y wereconverted back to the meridional plane and bladethickening was therefore completed.

Blade smoothing. After completing the blade thicken-ing, the leading and trailing edges of the blade are

not smooth enough. This will bring relatively largeflow loss. To improve flow performance near bladeleading and trailing edges, these two edges must besmoothed.

This article selected conformal transformationsmoothing method to perform the smoothing.On the conformal transformation plane, by adoptingthe pressure and suction surface coordinates x and yafter blade thickening, a pair of streamlines on pres-sure and suction surface was selected first. Then, twoinscribed circles with radius of �l and �t were drawnon blade leading and trailing edges, respectively.By repeating it for many times between streamlineson the blade pressure and suction surfaces, blade lead-ing and trailing shapes after smoothing were obtained.Figure 7 demonstrates groups of streamlines afterblade leading edge smoothing on the conformal trans-formation plane.

Three-dimensional design platform

Based on direct calculation and inverse design, thisarticle introduced a DIIDM for mixed-flow pumpimpeller design. The steps are as follows: (1) designinitial impeller based on the conventional method, (2)through iteration calculation of two kinds of streamsurfaces, obtain meridional flow field of the mixed-flow pump and (3) based on the meridional flowfield, finish inverse design and obtain the three-dimen-sional blade. Repeat steps (2) and (3), until positiondeviation of blade meridional transversals obtainedby two consecutive rounds of designs satisfies devi-ation range requirement.

Based on this method, a TDP for mixed-flow pumpwas programmed with FORTRAN language.Figure 8 shows the comparison between this platform(Figure 8(a)) and the conventional method(Figure 8(b)) in design process. Compared to the

Figure 7. Streamlines of the blade leading edge after

smoothing on the conformal transformation plane.Figure 6. Blade thickening on the conformal transformation

plane.




conventional method, the TDP possesses advantagesas follows.

1. The meridional flow field calculation (shown inFigure 8, dotted box (1)) for TDP does not useone-dimensional or two-dimensional flow assump-tion. Instead, the calculation is done by iterationcalculation between S1 stream surface and S2

stream surface with solution to the continuityequation and motion equation of the fluid at thesame time. Thus, the method ensures that themeridional flow field calculation results can reflectthe actual flow inside the impeller.

2. The TDP adopts DIIDM (shown in Figure 8,dotted box (2)) to improve impeller design. Afterthe blade camber line drawing, thickening andsmoothing, position deviation of the two blademeridional transversals obtained in two consecu-tive rounds of design needs to be examined to seeif it is within set error range. If not, then flow fieldcalculation and blade drawing should be redone.This platform applies meridional flow field infor-mation obtained by direct calculation to inversedesign, and applies the impeller model obtainedby inverse design to direct calculation.

Thus, flow field distribution variation can feedback to blade shape variation in time and viceversa. This effectively ensures that the finaldesign of the blade shape well fits the actual flowinside the impeller and improves calculation anddesign.

Case study on the TDP

To examine the effect of the TDP, a group of designparameters was selected. They were used by two-dimensional design method based on two-dimensionalflow assumption and TDP, respectively, to designmixed-flow pump impellers. The design parametersof the mixed-flow pump impeller are shown inTable 1.

Figure 9 shows the meridional velocity distribu-tions on the trailing edge under different meridionalflow net. It can be observed that from inlet to outlet ofthe calculation domain, when the number of gridsequals to or is larger than 90� 40, the calculationresults of the meridional velocity on the trailingedge no longer varies with the number of grids.Therefore, 90 calculation nodes were selected alongmeridional flow direction and 40 calculation nodesfrom hub to shroud. Together, a grid of 90� 40 waschosen to conduct on the meridional flow field calcu-lation (shown in Figure 10).

Figure 8. Impeller design process: (a) TDP and (b) conven-

tional design method.

TDP: three-dimensional design platform.

Figure 9. Meridional velocity distributions on the trailing

edge under different meridional flow net.

Table 1. Design parameters of the mixed-flow pump impeller.

Parameters

Design volume flow rate Q (m3/h) 1944

Head H (m) 17

Rotational speed n (r/min) 1450

Specific speed ns 465

Blade number Z1 5

Diffuser vane number Z2 6

Maximum diameter of impeller (mm) 420

Bing and Cao 573



By integrating equation (24), blade camber surfacewas obtained and taken as blade pressure surface.According to the blade relative thickness �r distribu-tion pattern obtained from Figure 11, blade thicken-ing was finished and the blade suction surface wasobtained. �r denotes the ratio of blade actual thicknessto maximum thickness. Afterward, blade leading andtrailing edges smoothing was carried out. Figure 12shows blade leading edge shape before and aftersmoothing. Then, the three-dimensional design wasfinished with a three-dimensional model for themixed-flow pump impeller (shown in Figure 13).

Expelling coefficient distribution in the blade zone. Whencalculating meridional flow fields, conventional meth-ods usually give expelling coefficient distributionalong meridional streamline in blade zone based onexperience, and then calculate meridional flow fieldunder certain assumptions. Figure 14(a) shows theexpelling coefficient distribution in the blade zonegiven in impeller design based on two-dimensionalflow assumption. In the figure, mr denotes relativestreamline length (with shroud streamline length ms

as the benchmark). From hub to shroud, meridionalstreamlines are numbered as J¼ 1, 2, . . . , 40. Whenthe impeller was designed with TDP, the initial

expelling coefficient distribution in the blade zonealso adopted what is shown in Figure 14(a).However, after iteration design, the actual expellingcoefficient distribution witnesses a significant change

Figure 14. Expelling coefficient distribution pattern in

the blade zone: (a) distribution pattern given initially and

(b) distribution pattern after three-dimensional iteration

design.

Figure 13. Three-dimensional model for the mixed-flow

pump.

Figure 12. Blade leading edge shape before and after

smoothing: (a) before smoothing and (b) after smoothing.

Figure 10. Meridional flow net.

Figure 11. Blade relative thickness distribution pattern.




(shown in Figure 14(b)). The expelling coefficient onthe thickest position of the blade significantlydecreases, suggesting a marked difference betweenthe coefficient distribution given based on experienceand the actual blade thickness distribution, which canseriously compromise the accuracy of meridional flowfield calculation.

Meridional velocity distribution. Figure 15(a) and (b)shows the distribution of meridional velocity Cm

obtained by direct calculation based on the impellerdesigned with two-dimensional flow assumption andTDP, respectively. The impellers designed by the twomethods have similar meridional velocities in the inletzone (shown in solid lines), which correspond witheven inflow condition. Meridional velocities in theblade zone of both designs (shown in dotted lines)decrease monotonously along the streamline withlarge gradient near the shroud, while they firstincrease and then decrease along streamline near thehub in quasi-parabolas. In the outlet zone, bothmeridional velocities (shown in solid lines) are in non-uniform distribution with smaller velocities near huband larger velocities near shroud. The difference isthat the deviation between meridional velocity nearhub and the one near shroud in the outlet zone ofthe impeller designed based on two-dimensional flowassumption is approximately 2m/s, while the devi-ation with TDP is 4m/s. This significant variation of

velocity nonuniform in the outlet zone also reflectsthat the TDP, during the course of direct and inverseiteration, significantly changed the blade shape.

Relative circulation distribution in the blade zone.

Figure 16(a) shows the relative circulation �r (circu-lation �s of blade trailing edge on the shroud as thebenchmark) distribution in the blade zone obtainedby direct calculation of the impeller design based ontwo-dimensional flow assumption. It can be noticedthat relative circulation in the blade zone graduallyincreases in flow direction, but increase rate is highlydifferent among streamlines. The nonuniform circula-tion distribution on the blade trailing edge is in an ‘S’form, with the circulation on streamlines near huband central part of the flow passage significantlylower than the one on other streamlines. Thishollow circulation distribution on the blade trailingedge suggests that blade designed based on two-dimensional flow assumption fails to satisfy therequirement for uniform circulation distribution onthe blade trailing edge, which means the blade doesnot have sufficient energy conversion capacity.

Figure 16(b) shows the relative circulation distribu-tion of the impeller designed with TDP in the bladezone. Circulation variation along each meridionalstreamline is even, suggesting an even loading distri-bution on the blade surface, which is beneficial toimproving impeller internal flow and reducing flow

Figure 16. Relative circulation distribution in the blade zone

obtained by direct calculation: (a) impeller designed based on

two-dimensional flow assumption and (b) impeller designed

with TDP.


Figure 15. Meridional velocity distribution obtained by direct

calculation: (a) impeller designed based on two-dimensional

flow assumption and (b) impeller designed with TDP.


Bing and Cao 575



loss. It is worth noticing that on blade trailing edge,circulation on each meridional streamline is basicallythe same, satisfying requirement for uniform circula-tion distribution on the blade trailing edge. This dem-onstrates that the TDP has obvious advantages overthe design method based on two-dimensional flowassumption.

Blade wrap angle control strategy

Blade wrap angle size directly influences the work cap-acity of the mixed-flow pump impeller and flow lossinside flow passage. The blade wrap angle, if toosmall, could lead to insufficient work capacity andperformance to meet design requirements. So aproper enlargement of the blade wrap angle willhelp increase energy conversion efficiency of theimpeller. However, if the blade wrap angle is toolarge, hydraulic friction loss inside the flow passagewill increase rapidly, which will also compromisehydraulic performance of the impeller. As a result, aproperly controlled blade wrap angle after consider-ing mixed-flow pump characteristics will be vital toimpeller performance improvement. With the TDPfor mixed-flow pump established before, the followingpart will inspect the blade wrap angle control strategy.

Parametrization of velocity moment distribution

From blade camber line integration equation (24), itcan be noticed that blade velocity moment distribu-tion pattern directly influences blade wrap angle size.Velocity moment distribution pattern along anymeridional streamline is given by the followingequation

Cur ¼ Cu1r1 þ f xð Þ � Cu2r2 � Cu1r1ð Þ ð29Þ

where x denotes relative length of the meridionalstreamline in the blade zone. According to theequation, x on the blade leading and trailingedges is 0 and 1, respectively. The subscripts 1and 2 represent parameters of pump impeller lead-ing and trailing edges, respectively. f(x) is thedimensionless velocity moment distribution func-tion, and 04 f(x)4 1. The function can be writtenas a polynomial as follows

f xð Þ ¼ ax4 þ bx3 þ cx2 þ dxþ e ð30Þ

a, b, c, d and e are coefficients, whose values are deter-mined by following conditions: flow on the blade lead-ing edge is without circulation, i.e. f(0)¼ 0; flow onblade trailing edge satisfies given velocity moment, i.e.f(1)¼ 1; on the blade leading edge, to reduce bladeloading and prevent cavitation, df(0)/dx¼ 0 is given;on blade trailing edge, df(1)/dx¼P is given (P is aconstant) and considering the characteristic of bladeworking on fluid, df(x)/dxjxe[0,1]5 0 in the blade zone.

Based on the above five conditions, five coefficientscan be solved. The results are as follows

x� 1ð Þ 2x� 1ð Þa5 3� 1:5Pð Þxþ P� 3ð Þ, x 2 ½0, 1�

b¼�2aþP� 2

c¼ a�Pþ 3

d¼ 0

e¼ 0

8>>>>>><>>>>>>:

ð31Þ

Both d and e are 0. b and c can be represented by Pand a, while the value ranges of P and a are deter-mined by the first inequality in equation set (31).Therefore, by controlling the values of P and a, vel-ocity moment distribution pattern can be completelycontrolled. Thus, P and a were selected as dimension-less velocity moment distribution parameters.Following is the discussion of value range for P anda on four kinds of conditions

a53� 1:5Pð Þxþ P� 3ð Þ

x� 1ð Þ 2x� 1ð Þ, x 2 ½0, 0:5Þ

P46, x ¼ 0:5

a43� 1:5Pð Þxþ P� 3ð Þ

x� 1ð Þ 2x� 1ð Þ, x 2 0:5, 1ð Þ

P50, x ¼ 1

8>>>>>>><>>>>>>>:

ð32Þ

Hence, the constant P has a value range of [0,6].Assign

g xð Þ ¼3� 1:5Pð Þxþ P� 3ð Þ

x� 1ð Þ 2x� 1ð Þð33Þ

By solving function g(x), a maximum value a1 within(0, 0.5), and a minimum value a2 within (0.5, 1) wereobtained. Thus, the value range of a is [a1, a2]. Becausethis range varies with the value of P, a0, the relativevalue of a was defined as follows to facilitatediscussion

a0 ¼a� a1a2 � a1

ð34Þ

From the definition of equation (34), it can be con-cluded that with different values of P, the value rangeof a0 remains [0,1].

Based on the analysis above, if the value of P isdetermined within [0,6], and the value of a0 is deter-mined within [0,1], then the velocity moment distribu-tion pattern can be determined and is unique. P and a0are therefore velocity moment distribution parameters.

Influence of velocity moment distributionon blade wrap angle

Six values of velocity moment distribution parameterP were assigned: 0.58, 1.43, 2.14, 3.27, 4.33 and 5.67.




For each P, five values of a0 were assigned: 0.2, 0.35,0.5, 0.65 and 0.8. Thus, 30 different kinds of velocitymoment distribution patterns were formed. Figure 17shows dimensionless velocity moment distributionfunction curves with different values of P and a0.

Figure 18 shows the blade wrap angles of themixed-flow pump designed with TDP with differentvalues of P and a0. The blade wrap angle increaseswith P increasing, and decreases with a0 increasing.When P is fixed, blade wrap angle varies with a0 in alinear relationship. When P is 0.58, 1.43, 2.14, 3.27 or4.33, five corresponding curves share similar slopes.However, when P is 5.67, the curve slope has a smallerabsolute value than other curves.

Influence of velocity moment distributionon internal flow

Figure 19 shows the relative velocity distribution onS1 stream surface obtained by direct calculation when

P is 3.27 and a0 is 0.2, 0.5 or 0.8. It can be noticed thatwith a0 increasing, relative velocity distribution on S1

stream surface becomes less uniform, especially on thepressure surface near blade leading edge, relative vel-ocity witnesses a significant decrease. This suggeststhat when P is given, with a0 increasing, blade wrapangle decreases, blade control over fluid motion weak-ens and therefore relative velocity distributionbecomes less even. When setting velocity moment dis-tribution pattern, a moderately small value of a0 willhelp to obtain a more uniform flow field distribution.

Figure 20 shows the relative velocity distributionon S1 stream surface obtained by direct calculationwhen a0 is 0.2 and P is 0.58, 3.27 or 5.67. It can benoticed that with P increasing, relative velocity distri-bution on S1 stream surface becomes more uniform.The increase in relative velocity on the blade pressuresurface and in the blade outlet zone is especially sig-nificant. Meanwhile, with P increasing, on S1 streamsurface near the hub, streamline direction in the bladeoutlet zone changes from deviating to correspondingwith the streamline direction in the blade zone, sug-gesting a stronger control from blade over fluidmotion. In summary, after the value of a0 is given,with value of P increasing, blade wrap angle increases,blade has a stronger control over fluid motion, andrelative velocity distribution becomes more uniform.This is especially significant near the hub. Whilegiving velocity moment distribution pattern, a moder-ately large value of P will be available to obtain arelatively uniform distribution of flow field.

Parametrization of blade leading and trailingedges positions

Velocity moment distribution directly influences bladewrap angle size. Blade leading and trailing edges

Figure 17. Dimensionless velocity moment distribution function curves with different values of P and a0. (a) P¼ 0.58, (b) P¼ 1.43,

(c) P¼ 2.14, (d) P¼ 3.27, (e), P¼ 4.33 and (f) P¼ 5.67.

Figure 18. Blade wrap angle with different velocity moment

distribution.

Bing and Cao 577



positions also influence blade wrap angle size.Figure 21 shows meridional plane projection of theblade zone. During design process, to convenientlyadjust blade rotational angle, hub and shroud in theblade zone both adopted spherical flow passage withO as the center of sphere. OA is the rotation axis forthe impeller and OB is the rotation axis for the blades.H and T1 are intersections of blade leading edge withhub and shroud, respectively. T2 is the intersection ofblade trailing edge with shroud. �O is the anglebetween blade rotation axis and pump impeller rota-tion axis. �h is the angle between OH and blade rota-tion axis. �t1 and �t2 are angles between OT1, OT2 andblade rotation axis, respectively.

While drawing the blade camber line of a mixed-flow pump, the positions of H, T1 and T2 were usuallydetermined first. By connecting H and T1 with a curvewhich corresponds with a certain variation pattern, theposition of the leading edge was then completely deter-mined. After giving the velocity moment distributionpattern of each meridional streamline, the blade wrapangle and spatial camber line on the shroud can bedetermined firstly based on equation (24). Afterward,

based on the principle that each meridional streamlinehas the same blade wrap angle, blade camber line wasdrawn one by one from the shroud to hub. After draw-ing all the blade camber lines, end points of each bladecamber line were connected, and therefore the bladetrailing edge was obtained. Thus, during the impellerdesign process, after giving positions of H, T1 and T2,and velocity moment distribution pattern, the bladeleading and trailing edge positions can be fullydetermined.

To spherical flow passage, there is

OHj j ¼ rh

OT1j j ¼ OT2j j ¼ rt

�ð35Þ

where rh and rt are the spherical radius on hub andshroud, respectively.

Since OB is the blade rotation axis, to ensure rota-tion of blade, OT1 and OT2 were usually set to besymmetrical around blade rotation axis. Hence

�t1 ¼ �t2 ð36Þ

Figure 19. Relative velocity distribution on S1 stream surface

(P¼ 3.27. Unit: m/s): (a) near hub, (b) mid-span and (c) near

shroud.


(a0¼ 0.2. Unit: m/s): (a) near hub, (b) mid-span and (c) near

shroud.




Assign �t¼ �t1¼ �t2, then the r and Z coordinates ofH, T1 and T2 can be solved as follows

rH ¼ rh � sin �O � �hð Þ

ZH ¼ ZO � rh � cos �O � �hð Þ

rT1 ¼ rt � sin �O � �tð Þ

ZT1 ¼ ZO � rt � cos �O � �tð Þ

rT2 ¼ rt � sin �O þ �tð Þ

ZT2 ¼ ZO � rt � cos �O þ �tð Þ

8>>>>>>>>>>><>>>>>>>>>>>:

ð37Þ

From the above equation set, it can be concluded thatafter determining the blade rotation axis position, rhand rt, positions of H, T1 and T2 were completelydetermined by the parameters �h and �t. By control-ling the last two parameters, blade leading and trailingedges positions could thus be controlled, i.e. �h and �tare parameters of the blade leading and trailing edgespositions.

Influence of blade leading and trailing edgespositions on blade wrap angle

The position parameter �h of the blade leading edgeon the hub was assigned with five values: 18�, 19�, 20�,21� and 22�. For each �h, five values were assigned to�t, the position parameter of blade leading and trailingedge positions on the shroud: 7�, 8�, 9�, 10� and 11�,thus forming 25 types of combination of leading andtrailing edge positions.

The design results suggested that �h, the positionparameter of the blade leading edge on the hub, did

not influence blade wrap angle size, while �t, the pos-ition parameter of the blade leading and trailing edgeson the shroud significantly influenced blade wrapangle size. Based on this, �h was assigned as 20� toanalyze the influence of different values of �t on bladewrap angle.

Figure 22 demonstrates the mixed-flow pump bladewrap angle with different values of �t designed withTDP. The blade wrap angle size increases with �tincreasing, showing a linear relationship.

Influence of blade leading and trailing edgespositions on impeller internal flow

Figure 23 shows the relative velocity distribution onS1 stream surface obtained by direct calculation when�h is 20�, and �t is 7�, 9� or 11�. It can noticed thatwith �t increasing, relative velocity distribution on S1

stream surface becomes more uniform. Especially onblade pressure surface, relative velocity significantlyincreases. When designing mixed-flow pump impeller,a proper increase of �t will improve flow in the bladezone. However, if �t is too large, the hydraulic frictionloss will also increase. So the value of �t should bedetermined after considering flow improvement, lossincrease and other factors.

Analysis of blade wrap angle control strategy

When designing a mixed-flow pump impeller, the vel-ocity moment distribution pattern and blade leadingand trailing edges positions, i.e. velocity moment dis-tribution parameters P and a0, and blade leading andtrailing edge position parameters �h and �t, need to begiven based on specific speed of the pump and previ-ous design experience. Previous analyses have sug-gested that except �h, other three parameters alldirectly influence blade wrap angle size. Hence, thesequence and basis of assigning values to these par-ameters are of crucial importance to blade wrap anglecontrol. The following part will introduce blade wrapangle control strategy with specific examples.

Figure 21. Meridional plane projection of the blade zone.

Figure 22. Blade wrap angle with different blade leading and

trailing edges positions (�h¼ 20�).

Bing and Cao 579



Figure 24 shows blade wrap angle curves varyingwith �t with different values of P and a0 when �h is 20

�.By linear fitting these curves, the following equationwas obtained

’ ¼ k�t þ l ð38Þ

Table 2 shows values for k and l according to differentvalues of P and a0. When P is fixed, and a0 increasesarithmetically, k and l decrease arithmetically, pre-senting a linear relationship.

By linear fitting coefficients k and l, equation (38)can be converted to as follows

’ ¼ k1 þ k2a0ð Þ�t þ l1 þ l2a0 ð39Þ

Table 3 shows values of k1, k2, l1 and l2 with differentvalues of P.

By conducting quadratic polynomial fits for coeffi-cients k1, k2, l1 and l2, these coefficients can be writtenas follows

k1 ¼ �0:015P2 þ 0:285Pþ 6:934

k2 ¼ 0:062P2 � 0:275P� 1:046

l1 ¼ 0:033Pþ 1:882

l2 ¼ 0:003P2 � 0:012P� 0:058

8>>>>><>>>>>:

ð40Þ

In fit result of l1, the quadratic coefficient is 0, sug-gesting that the fit is linear. Figure 25 demonstrates acomparison between quadratic polynomial fit curvesof k1, k2, l1 and l2 and initial data, showing that thetwo share similar variation trend.

Figure 24. Blade wrap angle curves varying with �t under different velocity moment distribution. (a) P¼ 0.58, (b) P¼ 1.43,

(c) P¼ 2.14, (d) P¼ 3.27, (e) P¼ 4.33, (f) P¼ 5.67.


(�h¼ 20�. Unit: m/s): (a) near hub, (b) mid-span and (c) near

shroud.




Equations (39) and (40) show blade wrap angle ’variation pattern varying with velocity moment distri-bution parameters P and a0, and parameter �t forblade leading and trailing edge positions on theshroud. While designing the mixed-flow pump impel-ler, these equations can be used as the control equa-tion for blade wrap angle.

The specific control strategy for mixed-flow pumpblade wrap angle is as follows.

1. Since the value of �h significantly influencesthe internal flow and cavitation performanceon the blade leading edge, it should be deter-mined after considering how to reduce inci-dence loss and prevent cavitation on bladeleading edge.

2. After determining the value for �h, the velocitymoment distribution pattern, i.e. the values for Pand a0 should be given according to the energyconversion requirement for the mixed-flow pumpimpeller.

3. After determining velocity moment distributionpattern, the linear variation pattern of bladewrap angle ’ varying with �t can be obtainedfrom equation (38). Based on the given size ofthe blade wrap angle, the value of �t can be calcu-lated and therefore realizing accurate control ofblade wrap angle.

Test validation

Based on the mixed-flow pump design parameters(Table 1), by drawing on other mixed-flow pumpswith similar specific speeds, blade wrap angle wasassigned to 70�, and parameter �h for blade leadingedge position on the hub to 20�. According to impellerenergy conversion requirement, velocity moment dis-tribution parameters P was assigned to 4.33 and a0 as0.3. Thus, based on equations (39) and (40), param-eter �t for blade leading and trailing edges positionson the shroud can be determined as 9�.

After determining the above parameters, hydraulicdesign of the mixed-flow pump impeller was com-pleted based on TDP and blade wrap angle was70.3�. By giving leading and trailing edge positionsof the diffuser vane, and diffuser vane angles distribu-tion pattern along meridional streamlines, the hydrau-lic design of mixed-flow pump diffuser was completed.Based on hydraulic design, the structural design of themixed-flow pump device was finished, as shown inFigure 26.

Based on the structural design, by using thehydraulic machinery test rig from BeifangInvestigation, Design & Research Co. Ltd, the per-formance test was carried out. This test rig has arandom error within� 0.1%, and a composite errorwithin� 0.3%. The mixed-flow pump performance

Table 2. Linear fit coefficients of curves.

k P¼ 0.58 P¼ 1.43 P¼ 2.14 P¼ 3.27 P¼ 4.33 P¼ 5.67

a0¼ 0.20 6.873 7.036 7.186 7.439 7.690 7.911

a0¼ 0.35 6.693 6.840 6.988 7.250 7.523 7.826

a0¼ 0.50 6.512 6.644 6.791 7.061 7.356 7.741

a0¼ 0.65 6.332 6.448 6.593 6.873 7.189 7.656

a0¼ 0.80 6.151 6.253 6.395 6.684 7.022 7.571

l P¼ 0.58 P¼ 1.43 P¼ 2.14 P¼ 3.27 P¼ 4.33 P¼ 5.67

a0¼ 0.20 1.887 1.915 1.940 1.979 2.016 2.059

a0¼ 0.35 1.877 1.905 1.929 1.968 2.007 2.054

a0¼ 0.50 1.867 1.894 1.918 1.958 1.998 2.049

a0¼ 0.65 1.858 1.883 1.907 1.948 1.989 2.045

a0¼ 0.80 1.848 1.873 1.897 1.938 1.980 2.040

Table 3. Linear fit coefficients for k and l.

P¼ 0.58 P¼ 1.43 P¼ 2.14 P¼ 3.27 P¼ 4.33 P¼ 5.67

k1 7.111 7.298 7.451 7.692 7.913 8.024

k2 �1.200 �1.307 �1.320 �1.260 �1.113 �0.567

l1 1.901 1.926 1.956 1.991 2.028 2.067

l2 �0.067 �0.067 �0.073 �0.067 �0.060 �0.033

Bing and Cao 581



test was conducted according to Hydraulic turbines,storage pumps and pump-turbines — Model acceptancetests (IEC60193-1999) and Code for model pumpacceptance tests (SL140-2006). Figure 27 shows themixed-flow pump test rig and key components suchas the impeller, diffuser, inlet and outlet piezometrictubes, motor and inlet tank.

When mixed-flow pump was designed, the bladerotational angle at this time was defined as 0�.When the blade rotates from 0� to another side, if

the impeller has a higher flow capacity, then theblade rotational angle is positive. Otherwise, it isnegative.

Eight blade rotational angles, �10�, �8�, �6�, �4�,�2�, 0�, þ2� and þ4�, were selected to complete thisperformance test. Figure 28 shows curves that dem-onstrate the efficiency, head and shaft power of themixed-flow pump varying with flow rates with differ-ent blade rotational angles. In high efficiency oper-ation range, data collection nodes were increased toensure an accurate performance curve of the range.

It can be observed from these eight efficiencycurves that the mixed-flow pump designed with TDPachieves its maximum efficiency of 87.2% when bladerotational angle is �6�. With different blade rota-tional angles, the maximum efficiency varies within1%, suggesting a wide adjustment range for theblade to operate efficiently. Also, with differentblade rotational angles, efficiency curves of themixed-flow pump all have a flat and wide range ofhigh efficiency operation. For example, when bladerotational angle is 0�, the flow rate range is 0.07m3/s, with the efficiency 1% smaller than maximum effi-ciency, while the design flow rate for the mixed-flowpump is 0.54m3/s. This means, while design flow ratevaries within 13%, the pump efficiency decreases lessthan 1%. It suggests that the pump designed withTDP could achieve a high efficiency operation witha wide flow rate range, significantly improving themixed-flow pump operation performance when flowrate varies frequently.

Figure 25. Curve fitting results: (a) relationship between k1

and P and its fitting curve, (b) relationship between k2 and P and

its fitting curve, (c) relationship between l1 and P and its fitting

curve and (d) relationship between l2 and P and its fitting curve.

Figure 27. Mixed-flow pump performance test rig.

Figure 26. Structural design of mixed-flow pump device.




In mixed-flow pump cavitation performance test,the critical cavitation allowance NPSHc was definedas the available cavitation allowance NPSHa when thepump efficiency decreases 1%. By measuring NPSHc

with different flow rates, then the variation curve wasobtained.

Figure 29 demonstrates NPSHc varying with flowrates when blade rotational angle is �6�, �2� andþ4�. It is shown that with the blade rotational angleof �6�, the maximum efficiency is achieved, when thepump cavitation performance is guaranteed as well,obtaining a combination of optimal efficiency andfavorable cavitation performance. With the bladerotating positively, cavitation performance weakens,

suggesting that the pump moderately rotating bladeto negative side will improve cavitation performanceof the impeller.

Performance test results showed that the mixed-flow pump designed with TDP had a maximum effi-ciency of 87.2%. Within 14� of blade rotational anglevariation, the maximum efficiency with each anglevaried less than 1%, suggesting a wide adjustmentrange for the blade. Furthermore, the mixed-flowpump had a flat and wide range of high efficiencyoperation, effectively improving the pump operationperformance when flow rate changes frequently. Also,the cavitation performance of the pump was favor-able. During the test, the test equipment remainedstable, demonstrating favorable stability.

Conclusion

Based on quasi-three-dimensional flow calculation, aDIIDM for mixed-flow pump impeller has been intro-duced, and a TDP for mixed-flow pump impeller hasbeen established. By comparing and analyzing merid-ional velocity distribution, relative circulation distri-bution and expelling coefficient distribution in bladezone, it has been proved that the impeller designedwith TDP, compared with the one designed basedon two-dimensional flow assumption, has a betterhydraulic performance, especially a significantimprovement of blade energy conversion capacity.

Figure 28. Mixed-flow pump performance curves with different blade rotational angles: (a) �10�, (b) �8�, (c) �6�, (d) �4�, (e) �2�,

(f) 0�, (g) þ2� and (h) þ4�.

Figure 29. Critical cavitation allowance curves for the mixed-

flow pump.

Bing and Cao 583



With TDP for mixed-flow pump impellers, param-etrization is adopted to velocity moment distributionand blade leading and trailing edges positions. Theinfluence of velocity moment distribution parametersP and a0, and parameters �h and �t for blade leadingand trailing edges positions on blade wrap angle andimpeller internal flow is analyzed. By curves fitting,the functional relationship ’(�t,P, a0) is obtainedamong blade wrap angle ’ and parameters �t, P anda0. The blade wrap angle control strategy in mixed-flow pump impeller design is proposed, thus realizingthree-dimensional design of mixed-flow pump bladeswith controllable blade wrap angles. The performancetest for the mixed-flow pump model suggests that thethree-dimensional design method with controllableblade wrap angle, by controlling blade wrap anglesize and adopting DIIDM, ensures that the mixed-flow pump impeller designed with TDP has a betterhydraulic and cavitation performance.

Funding

This work was supported by the National ScienceFoundation of China (grant number 51176088).

Conflict of interest

The authors declare that they have no conflict of interest.

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