Performance and Flow Field of a Gravitation Vortex Type...

Research ArticlePerformance and Flow Field of a GravitationVortex Type Water Turbine

Yasuyuki Nishi and Terumi Inagaki

Department of Mechanical Engineering Ibaraki University 4-12-1 Nakanarusawa-cho Hitachi-shi Ibaraki 316-8511 Japan

Correspondence should be addressed to Yasuyuki Nishi yasuyukinishifevcibarakiacjp

Received 28 November 2016 Revised 15 February 2017 Accepted 5 March 2017 Published 16 March 2017

Academic Editor Ryoichi Samuel Amano

Copyright copy 2017 Yasuyuki Nishi and Terumi Inagaki This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A gravitation vortex type water turbine which mainly comprises a runner and a tank generates electricity by introducing a flowof water into the tank and using the gravitation vortex generated when the water drains from the bottom of the tank This waterturbine is capable of generating electricity using a low head and a low flow rate with relatively simple structure However because itsflow field has a free surface this water turbine is extremely complicated and thus its relevance to performance for the generation ofelectricity has not been clarifiedThis study aims to clarify the performance and flow field of a gravitation vortex type water turbineWe conducted experiments and numerical analysis taking the free surface into consideration As a result the experimental andcomputational values of the torque turbine output turbine efficiency and effective head agreed with one anotherThe performanceof this water turbine can be predicted by this analysis It has been shown that when the rotational speed increases at the runnerinlet the forward flow area expands However when the air area decreases the backward flow area also expands

1 Introduction

Many large-scale conventional hydraulic power generationsmainly use medium- or high-heads and water turbines [1 2]for conduits such as the Francis water turbine Recentlyhowever as public consciousness about renewable energieshas risen the demand for small-scale hydraulic power gener-ation with a water turbine [3ndash7] for open channels has beenincreasing with the use of so-far unused common rivers orwaterways that have low heads and low flow rates

Therefore we focused on a water turbine used in theGravitation Water Vortex Power Plant (GWVPP) [8] whichgenerates electricity with a low head and a low flow rate Thisgravitation vortex typewater turbinemainly comprises a run-ner and a tank On introducing a flow of water into the tankthe turbine generates electricity from the gravitation vortexthat occurs while draining the water from the bottom of thetank In addition it is thought that this water turbine has anaeration function to raise the dissolved oxygen concentrationof the downstream water by rolling up the air above the freesurface around the runner Despite the fact that this waterturbine has a relatively simple structure the flow field is

extremely complicated because of its free surface Howeveralthough some studies on other types of runners related tothis kind of water turbine have been presented [9ndash12] theirflowfields have not been investigated in detail To improve theperformance of the water turbine it is important to study theflow field in detail in order to determine its relevance to theperformance characteristic Although a numerical analysis iseffective for this because this water turbine operates by usinga gravitation vortex it is necessary to conduct a numericalanalysis with consideration to the free surface Because anumerical analysis with consideration to the free surfacerequires a large computational load there are few examplesof where it has been applied to a water turbine Recentlyhowever it has started to be applied to a spiral water turbine[13] an undershot cross-flow water turbine [14 15] and apropeller water turbine [16 17]

In light of this background this study aims to clarify theperformance of a gravitation vortex type water turbine andelucidate its flow field We performed numerical analysis byconsidering the free surface conducted a performance testand a visualization experiment and verified the validity ofour analysis Furthermore we examined the flowfield around

HindawiInternational Journal of Rotating MachineryVolume 2017 Article ID 2610508 11 pageshttpsdoiorg10115520172610508

2 International Journal of Rotating Machinery

Section A-A

A A

Tank inlet

100Flow

Runner

Tank

Rotationx

y

490z

Figure 1 Test water turbine

Rotation

Section B-B

BB

b2 D2D1 x

y

b12b2

b1

b1

Figure 2 Runner

the runner at the center of the blade width in detail using anumerical analysis

2 Experimental Apparatus and Methods

21 Gravitation Vortex Type Water Turbine An overview ofthe gravitation vortex type water turbine is shown in Figure 1This water turbine mainly comprises a runner and a tank andgenerates electricity from the gravitation vortex that occursin the tank when the water is drained An overview of therunner is illustrated in Figure 2 and its specifications aregiven in Table 1 This runner has a centrifugal form whichis different from the form of the paddle-type runner that hasbeen used in previous studies [9ndash12]The blade inlet diameter(outer diameter) isD1 = 140mm blade outlet diameter (innerdiameter) is D2 = 90mm blade inlet width is b1 = 91mmblade outlet width is b2 = 91mm and number of blades isz = 20 Section B-B in Figure 2 is a section at the center ofthe blade width The inner diameter of the cylindrical tankis 490mm and the diameter of the hole at the bottom oftank is 100mm In addition the coordinate system is definedas shown in Figure 1 The circumferential angle 120579 is definedas 120579 = 0∘ on the positive y axis and its positive direction iscounterclockwise

Table 1 Specifications of runner

Outer diameter1198631 014mInner diameter1198632 009mInlet width 1198871 0091mOutlet width 1198872 0091mInlet angle 1205731198871 719∘

Outlet angle 1205731198872 190∘

Tip clearance 120575 05mmNumber of blades 119885 20

Motor

Torque meter

Flow meter

Tachometer

Flow control valve

TankRunner

Reserve tank

Pump

Figure 3 Experimental apparatus

22 Experimental Apparatus and Methods An overview ofthe experimental apparatus is shown in Figure 3 The flowrate Q of water supplied by the pump was measured with anelectromagnetic flow meter (Toshiba Corporation LF620)The experiment was conducted under the condition ofconstant flow rate Q = 000285m3s The load to the runnerwas controlled by a motor and an inverter and the rotationalspeed was arbitrarily set The rotational speed n and torque Tweremeasured with amagnetic rotation detector (Ono SokkiCo LtdMP-981) and a torque detector (Ono Sokki Co LtdSS-005) respectively The turbine output P was obtained by

119875 = 212058711989911987960 (1)

Here the torque T was corrected by measuring the idlingtorque without the runnerThe effective headH is defined bythe following as shown in Figure 4

119867 = ℎ1015840 + ℎ3 + V232119892 minus ℎ4 minus

V242119892 (2)

Here the upstream water depth h3 was measured at thetank inlet in the vicinity of thewall surface on the +y-axiswitha ruler The downstream water depth h4 was measured by apoint gauge (Kenek Corporation PH-102) at five points fromthe vicinity of the wall surface on the +y-axis to the centerat the position of 6D1 downstream from the downstreamatmospheric opening from which the average water depth

International Journal of Rotating Machinery 3

Tank inlet322g 2D1 6D1

H

422g

3ℎ3

4 ℎ4

Reference positionReference position

ℎ

Figure 4 Definition of performance evaluation

was obtained The upstream velocity v3 and downstreamvelocity v4 were calculated by

V3 = 1198761198613ℎ3 (3)

V4 = 1198761198614ℎ4 (4)

Here B3 and B4 are the waterway widths of the tankinlet and downstream respectively In addition the turbineefficiency 120578 was calculated by

120578 = 119875120588119892119876119867 (5)

A digital camera (Casio Computer Co Ltd EXILIMEX-F1) was used to visualize the flow field at a frame rate of 30frames per second (fps)

3 Numerical Analysis Method and Conditions

In this study three-dimensional unsteady flow analysis wasperformed by considering the free surface The general-purpose thermal fluid analysis software ANSYS CFX150(ANSYS Inc) was used for the calculations Moreover thevolume of fluid (VOF) method [18] which is suitable for aflow field that has a clear interface between two phases andexpresses the actual performance [13ndash17] in a free surface flowanalysis of a water turbine was also used The working fluidswere water and air The governing equations were the massconservation equation momentum conservation equationand volume conservation equation A guideline for using aturbulence model that is suitable for the VOF method hasnot been clarified Therefore the shear stress transport (SST)model [19] which can model the actual performance [13 1617] of a free surface flow analysis of a water turbine using theVOF method has been used as the turbulence model

The entire area of calculation is shown in Figure 5This isdivided into four main areas runner tank upstream water-way and downstream waterway The downstream waterwayis 10D1 in length from the atmospheric opening to the outletboundary The reference position of the upstream side wasset to the tank inlet which was the same as that in theexperiment The reference position of the downstream sidewas set to 6D1 downstream from the atmospheric opening

Outlet

Inlet

FlowOpening

Runner

Tank

z

yx

Figure 5 Calculating area

of the downstream waterway At these reference positionsthe distribution of each water depth was obtained in thewidth direction assuming that the water surface is equivalentto that of a water volume fraction of 05 The upstreamwater depth h3 and downstream water depth h4 are theaverages obtained from the distribution of water depth in thewidth direction As an example the grids used in the runnerand tank calculations are illustrated in Figures 6(a) and6(b) respectively The runner tank upstream waterway anddownstreamwaterway have approximately 446 000 541 000434 000 and 780 000 computational elements respectivelywhich totals to 2 201000 Computational grids [14 15] usingan undershot cross-flow water turbine were prepared asthese are able to verify the free surface flow analysis and theexperiments of the flow field relatively well As boundaryconditions the mass flow rate (2838 kgs) was given to theinlet boundary an open boundary (total pressure of 0 Pafor the inflow or relative static pressure of 0 Pa for theoutflow) to the outlet boundary and an arbitrary rotationalspeed to the runner In addition the upper surfaces of thetank and downstream waterway were set to open boundaries(relative static pressure of 0 Pa) so that air could enter andexit freely The wall surface was set to the no-slip conditionWith reference to the calculations a steady flow analysiswas first conducted followed by an unsteady flow analysisusing the steady flow results as the initial conditions In theunsteady flow analysis the boundary between a rotationaland a stationary area was connected using the transientrotor-stator methodThe calculation continued until the flowbecame almost stable as determined by its fluctuations Atotal of 180 time steps were used during which the runnercompletes one rotation

4 Results and Discussion

41 Comparison of Water Turbine Performance A compar-ison between the experimental and calculation results forthis water turbine in relation to its performance is shown inFigures 7(a) and 7(b) It is observed that the experimental andcalculation results are in good agreement in terms of torque119879 turbine output119875 turbine efficiency 120578 and effective headHAs the rotational speed n is increased the torque T decreasesand the effective headH increasesmarginallyThe reasons forthis appear to be that the increase of effective head H derivesmainly from the increase of upstream water depth h3 andthat the resistance of the runner increases with the increase


(a) Runner (b) Tank

Figure 6 Computational grids

T (Exp)T (Cal)

P (Exp)P (Cal)

000

005

010

015

020

025

T(N

middotm)

30 60 90 120 150 180000

05

10

15

20

25P

(W)

n (minminus1)

(a) Torque and turbine output

H (Exp)H (Cal)

(Exp) (Cal)

00

01

02

03

04

05

000

005

010

015

020

025

H(m

)

30 60 90 120 150 1800

(mdash

)

n (minminus1)

(b) Turbine efficiency and effective head

Figure 7 Turbine performance

of rotational speed n Both turbine output P and turbineefficiency 120578 show maximum values at the rotational speedn = 122minminus1 The maximum experimentally determinedefficiency is approximately 0354 at a specific speed 119899119904 ofapproximately 47 [minminus1 kW m]

42 Comparison of Free Surface Flow Field The experimentalresults for the free surface flow field of this water turbine areshown in Figures 8(a)ndash8(c) and the calculation results areshown in Figures 9(a)ndash9(c) In these calculation results thewaterair boundary is defined as VF = 05 and this isosurfaceis illustrated The water depth around the runner varies inthe circumferential direction in both the experimental andcalculation results becoming lower in the +119909 direction In

addition the water depth varies greatly according to therotational speedThe experimentally determined free surfaceshape with change in the rotational speed can be seen toagree qualitatively with the calculation results Because theflow speed depends on the water depth in the circumferentialdirection if it is combined with a change of operationalconditions such as the rotational speed it is presumed thatthe flow field around the runner will become extremelycomplicatedTherefore in this study we next discuss the flowfield at the center of the blade width (section B-B in Figure 1)

43 Flow Field at the Center of the BladeWidth First in orderto identify the waterair interface the circumferential distri-bution of water volume fraction VF1 obtained numerically


Runner

Flow

Flow

(a) n = 81minminus1 (b) n = 122minminus1 (c) n = 162minminus1

Figure 8 Flow field by experiment

Free surfaceFlow Runner

Flow


Figure 9 Flow field by calculation

n = 81

n = 122

n = 162

00

02

04

06

08

10

12

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

VF 1

Figure 10 Volume fraction of water at runner inlet (Cal)

for a runner inlet at the center of the blade width is shownin Figure 10 Here VF1 is the time average value during onerotation of the runner Air is represented by 0 le VF lt 05water is represented by 05ltVFle 1 and the interface betweenthem is represented by VF = 05 These notations are the

Rotation

① Inlet

② Outlet①

y

x

w

u

w

u

②

Figure 11 Velocity triangles

same as those of a runner outlet which will be describedsubsequently According to Figure 10 at the maximum-efficiency rotational speed n = 122minminus1 the water area ofVF gt 05 is at 120579 = 0∘ndash93∘ and 120579 = 168∘ndash360∘ and thus itstotal effective angle is approximately 285∘ This water areareduces with decrease in the rotational speed and increaseswith increase of it This appears to be because the resistanceof the runner increases with increase in rotational speed asdescribed before

Here velocity triangles of the water turbine are illustratedin Figure 11 The velocity vr is defined in the radially inwarddirection vu in the rotation direction and va in the +z


minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(a) Radial component

minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

Theoretical value (n = 81)Theoretical value (n = 122)Theoretical value (n = 162)

(c) Circumferential component

Figure 12 Absolute velocity at runner inlet (Cal)

direction Figures 12(a)ndash12(c) show circumferential distri-butions of the radial component vr1 axial component va1and circumferential component vu1 of the absolute velocityof a runner inlet at the center of the blade width in thenumerical analysis Here each component is the time averagevalue during one rotation of the runner and displays a waterarea identified only from the water volume fraction Thesenotations are the same as those of a runner outlet which willbe described subsequentlyThe theoretical vu1 value obtainedfrom the following expression that is based on the assumption

that the tank has a free vortex type flow is also shown inFigure 12(c)

V1199061 = 1199033V31199031 (6)

where r1 is the outer radius of the runner inlet and r3 is therepresentative radius of the tank inlet The radius (705m)at the measuring point on the periphery of the runner inletand the radius (245mm) in the tank were used as r1 and r3respectively The value obtained from (3) was used as v3


From Figure 12(a) it can be observed that vr1 is notuniform in the circumferential direction at any rotationalspeed For 119899 = 122minminus1 the forward flow area in whichvr1 is positive appears at 120579 = 0∘ndash24∘ and 120579 = 168∘ndash360∘its total effective angle is approximately 216∘ In additionthe backward flow area appears at 120579 = 27∘ndash93∘ its effectiveangle is approximately 66∘ In this backward flow area va1 isnegatively large as is shown in Figure 12(b) With increasein the rotational speed the forward flow area increases butbecause of the reduction in the air area the backward flowarea also increases Moreover with increase in the rotationalspeed the value of 120579 at which vr1 changes from positive tonegative shifts to the large 120579 side In Figure 12(c) although thecomputational and theoretical vu1 values are relatively similarat 120579 = 0∘ near the tank inlet they differ in other 120579 regionsAlthough the theoretical vu1 value decreases as the rotationalspeed increases the computational vu1 value increases as therotational speed increases and it becomes large at both endsof the air area at any rotational speed Because the waterarea for low values of 120579 of the air area is a backward flowarea the water area for large values of 120579 of the air area isconsidered to greatly contribute to the conversion of energyFor the circumferential velocity u1 at the runner inlet u1 =0594ms at n = 81minminus1 u1 = 0894ms at n = 122minminus1and u1 = 1188ms at n = 162minminus1 At n = 122 and 162minminus1vu1 at both ends of the air area is nearly the same as u1Therefore a flow in the tank of this water turbine is not aperfect free vortex and it is greatly influenced by the rotationof the runner near the runner inlet Because thiswater turbinedoes not have guide vane upstream of the runner a uniformand strong circumferential spiral flow can be produced bydesigning the tank shape that improves the turbine output

The numerically determined circumferential distributionof the relative flow angle 1205731 of a runner inlet at the centerof the blade width is shown in Figure 13 At n = 81minminus1the relative flow angle 1205731 for 120579 = 240∘ndash300∘ shows a relativelyclose value at the blade inlet angle 120573b1 = 719∘ However at n= 122 and n = 162minminus1 the relative flow angle 1205731 dissociatesgreatly from the blade inlet angle and the shock loss appearsto increase Therefore it is necessary to control the flow inthe tank and homogenize the relative flow angle 1205731 in thecircumferential direction in order to decrease the shock lossat the blade inlet

The numerically determined circumferential distributionof water volume fraction VF2 of a runner outlet at the centerof the blade width is shown in Figure 14 At n = 122minminus1the largest water area is for 120579 = 0∘ndash15∘ and 120579 = 228∘ndash360∘ thetotal effective angle is approximately 147∘ This water area isconsiderably smaller than that of a runner inlet and does notchange with the rotational speed

Figures 15(a)ndash15(c) show the numerically determinedtime average values of the radial component vr2 axial compo-nent va2 and circumferential component vu2 of the absolutevelocity of a runner outlet at the center of the blade widthAccording to Figure 15(a) vr2 distribution at each rotationalspeed is almost the same and does not show a backward flowat any rotational speed From Figure 15(b) va2 componentof the water area at the large 120579 side of the air area decreases

minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

Figure 13 Relative flow angle at runner inlet (Cal)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2

Figure 14 Volume fraction of water at runner outlet (Cal)

rapidly as 120579 increases From Figure 15(c) it can be observedthat at n = 122minminus1 vu2 is the largest around 120579 = 15∘which shows the remains of the rotational component but isrelatively small for 120579 = 228∘ndash285∘ However for this range of120579 a negative rotation remains at n = 81minminus1 and a positiverotation remains at n = 162minminus1

The flow rate and the angular momentum per unit timethat flow in and out at the runner inlet and outlet relates tothe torque of a water turbine studied


minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


Figure 15 Absolute velocity at runner outlet (Cal)

The flow rates 1199021 and 1199022 per unit blade width (1mm) atthe runner inlet and outlet are expressed as the followingequations

1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)

V1199031 and V1199032 were obtained from the following equationsby using the time average values of vr1 and vr2 during onerotation of the runner at each measuring point

V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)

Here 120579119908 is the circumferential angle of the water area


0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2

Figure 16 Flow rate per unit blade width

Therefore the angular momentums L1 and L2 per unitblade width (1mm) and unit time at the runner inlet andoutlet can be expressed as the following equations

1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)

V1199061 and V1199062 were obtained from the following equationsby using the time average values of vr1vu1 and vr2vu2 duringone rotation of the runner at each measuring point

V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)

The relationship between the rotational speed n at thecenter of the blade width and the flow rates q1 and q2 perunit blade width are shown in Figure 16 The relationshipsbetween the rotational speedn and the angularmomentumsL1 and L2 per unit blade width and unit time are shown inFigure 17

In Figure 16 although q1 is rather small at n = 122minminus1it is nearly constant when the rotational speed changesTherefore when the rotational speed increases the forwardflow area expandsHowever as previously stated the flow ratethat flows in from the center of blade width barely changesConversely q2 is relatively similar to q1 at n = 81minminus1Whenthe rotational speed increases q2 at n = 81minminus1 is approx-imately 206 lower than q1 Since va2 is a negative value asstated above the flow through the runner comes close to thetip side (bottom of the tank) Therefore in order to designa high-performance runner it is necessary to study the

00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2

Figure 17 Angular momentum per unit blade width and unit time

three-dimensional flow including the direction of the bladewidth

In Figure 17 although L1 is rather large at n = 122minminus1it is nearly constant when the rotational speed changesHowever L2 has a large negative value at n = 81minminus1 and alarge positive value at n = 162minminus1 When only the center ofthe blade width is considered because the difference betweenL1 and L2 is the theoretical torque of the water turbinechange in torque when the rotational speed changes showsthat difference of the angular momentum that remains at therunner outlet has a large influence Since both the positiveand negative angular momentums at the runner outlet causean increase in the loss of waste they are considered to be oneof the factors related to the decrease in efficiency at low orhigh rotational speeds

Figures 18(a)ndash18(c) illustrate the relative velocity vectorsand water volume fraction of the runner The cross sectionrepresents the center of the blade width (section B-B) Itcan be seen that with increase in the rotational speed ata runner inlet as described above the air area reducesbut the backward flow area increases In addition at n =81minminus1 the relative water flow is relatively smooth alongthe blade but at n = 122 and 162minminus1 it flows in at smallangles

5 Conclusions

The following matters were determined by our research ofthe performance of a gravitation vortex type water turbineand the flow field at the center of blade width throughexperiments and free surface flow analysis

(1) The experimental and computational values of thetorque turbine output turbine efficiency and effec-tive head agree well with one another Thus the


Rotation

00

01

02

03

04

05

06

07

08

09

10

VF

(a) n = 81minminus100

01

02

03

04

05

06

07

08

09

10

VF

(b) n = 122minminus1

00

01

02

03

04

05

06

07

08

09

10

VF

(c) n = 162minminus1

Figure 18 Relative velocity vectors and volume fraction of water (Cal)

performance of this water turbine can be predicted bythis analysis

(2) With increase in the rotational speed at a runner inletthe forward flow area increases as does the backwardflow area because of the reduction in the air areaHowever the flow rate that flows in from the centerof the blade width barely changes

(3) The flow in the tank of this water turbine is not aperfect free vortex and it is greatly influenced by therotation of the runner near the runner inlet

(4) The water area of a runner outlet is considerablysmaller than that of a runner inlet and does notchange with the rotational speed In addition back-ward flow does not occur at a runner outlet

(5) When the rotational speed changes the angularmomentum per unit time that flows from the runnerinlet is nearly constant The angular momentum perunit time that flows from the runner outlet showsa large negative value at low-speed rotations and alarge positive value at high-speed rotations It also hasa large influence on the torque when the rotationalspeed changes

Nomenclature

119887 Blade width m119861 Waterway width m119863 Runner diameter m119892 Gravitational acceleration ms2ℎ Water depth mℎ1015840 The difference in height between the

bottom surface of tank and the bottomsurface of downstream waterway m119867 Effective head m(= ℎ1015840 + ℎ3 + V232119892 minus ℎ4 minus V242119892)119871 Angular momentum per unit blade widthand unit time Nsdotmmmsdots119899 Rotational speed minminus1119899119904 Specific speed minminus1 kW m(= 119899(1198751000)1211986754)119875 Turbine output W (= 212058711989911987960)119902 Flow rate per unit blade width m3mmsdots119876 Flow rate m3s119879 Torque Nsdotm119906 Circumferential velocity msV Absolute velocity ms


VF Volume fraction of water119908 Relative velocity ms

Greek Letters

120573 Relative flow angle ∘120573119887 Blade angle ∘120578 Turbine efficiency (= 119875120588119892119876119867)120579 Circumferential angle ∘120588 Density of water kgm3119908 Water area

Subscripts

1 Runner inlet2 Runner outlet3 Upstream4 Downstream119886 Axial componentℎ Hub119903 Radial component119905 Tip119906 Circumferential component

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors acknowledge the support of Shinoda Co Ltdin the design and production of the experimental apparatusThey are also grateful to Tomoaki Tanemura and KentaroHatano graduate students of Ibaraki University at the timewho supported us with the experiments and numericalanalysis Here we express our sincere gratitude for theircooperation

References

[1] C Nicolet A Zobeiri P Maruzewski and F Avellan ldquoExperi-mental investigations on upper part load vortex rope pressurefluctuations in francis turbine draft tuberdquo International Journalof Fluid Machinery and Systems vol 4 no 1 pp 179ndash190 2011

[2] T Vu M Koller M Gauthier and C Deschenes ldquoFlowsimulation and efficiency hill chart prediction for a Propellerturbinerdquo International Journal of Fluid Machinery and Systemsvol 4 no 2 pp 243ndash254 2011

[3] Y Nishi T Inagaki K Okubo and N Kikuchi ldquoStudy onan axial flow hydraulic turbine with collection devicerdquo Inter-national Journal of Rotating Machinery vol 2014 Article ID308058 11 pages 2014

[4] A Georgescu S Georgescu C I Cosoiu and N AlboiuldquoEfficiency of marine hydropower farms consisting of multi-plevertical axis cross-flow turbinesrdquo International Journal ofFluid Machinery and Systems vol 4 no 1 pp 150ndash160 2011

[5] M Anyi and B Kirke ldquoEvaluation of small axial flow hydroki-netic turbines for remote communitiesrdquo Energy for SustainableDevelopment vol 14 no 2 pp 110ndash116 2010

[6] A Inagaki and T Kanemoto ldquoPerformance of gyro-typehydraulic turbine suitable for shallow streamrdquo Turbomachineryvol 33 no 10 pp 614ndash621 2005

[7] M Nakajima S Iio and T Ikeda ldquoPerformance of savoniusrotor for environmentally friendly hydraulic turbinerdquo Journalof Fluid Science and Technology vol 3 no 3 pp 420ndash429 2008

[8] F Zotloterer ldquoHydroelectric power plantrdquo Patent WO 2004061295 A3 2004

[9] S Wanchat R Suntivarakorn S Wanchat K Tonmit and PKayanyiem ldquoA parametric study of a gravitation vortex powerplantrdquo Advanced Materials Research vol 805-806 pp 811ndash8172013

[10] S Dhakal S Nakarmi P Pun A B Thapa and T RBajracharya ldquoDevelopment and testing of runner and conicalbasin for gravitational water vortex power plantrdquo Journal of theInstitute of Engineering vol 10 no 1 pp 140ndash148 2014

[11] H M Shabara O B Yaakob Y M Ahmed A H Elbatranand M S M Faddir ldquoCFD validation for efficient gravitationalvortex pool systemrdquo Jurnal Teknologi vol 74 no 5 pp 97ndash1002015

[12] C Power A McNabola and P Coughlan ldquoA parametricexperimental investigation of the operating conditions of grav-itational vortex hydropower (GVHP)rdquo Journal of Clean EnergyTechnologies vol 4 no 2 pp 112ndash119 2015

[13] J Matsui ldquoInternal flow and performance of the spiral waterturbinerdquo Turbomachinery vol 38 no 6 pp 358ndash364 2010

[14] Y Nishi T Inagaki Y Li R Omiya and K Hatano ldquoTheflow field of undershot cross-flow water turbines based on PIVmeasurements and numerical analysisrdquo International Journal ofFluid Machinery and Systems vol 7 no 4 pp 174ndash182 2014

[15] YNishi T Inagaki Y Li andKHatano ldquoStudy on anundershotcross-flow water turbine with straight bladesrdquo InternationalJournal of Rotating Machinery vol 2015 Article ID 817926 10pages 2015

[16] N Kolekar and A Banerjee ldquoPerformance characterization andplacement of a marine hydrokinetic turbine in a tidal chan-nel under boundary proximity and blockage effectsrdquo AppliedEnergy vol 148 pp 121ndash133 2015

[17] J Riglin W Chris Schleicher I-H Liu and A OztekinldquoCharacterization of a micro-hydrokinetic turbine in closeproximity to the free surfacerdquo Ocean Engineering vol 110 pp270ndash280 2015

[18] C W Hirt and B D Nichols ldquoVolume of fluid (VOF) methodfor the dynamics of free boundariesrdquo Journal of ComputationalPhysics vol 39 no 1 pp 201ndash225 1981

[19] F R Menter ldquoTwo-equation eddy-viscosity turbulence modelsfor engineering applicationsrdquo AIAA journal vol 32 no 8 pp1598ndash1605 1994

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Section A-A

A A

Tank inlet

100Flow

Runner

Tank

Rotationx

y

490z

Figure 1 Test water turbine

Rotation

Section B-B

BB

b2 D2D1 x

y

b12b2

b1

b1

Figure 2 Runner

the runner at the center of the blade width in detail using anumerical analysis

2 Experimental Apparatus and Methods

21 Gravitation Vortex Type Water Turbine An overview ofthe gravitation vortex type water turbine is shown in Figure 1This water turbine mainly comprises a runner and a tank andgenerates electricity from the gravitation vortex that occursin the tank when the water is drained An overview of therunner is illustrated in Figure 2 and its specifications aregiven in Table 1 This runner has a centrifugal form whichis different from the form of the paddle-type runner that hasbeen used in previous studies [9ndash12]The blade inlet diameter(outer diameter) isD1 = 140mm blade outlet diameter (innerdiameter) is D2 = 90mm blade inlet width is b1 = 91mmblade outlet width is b2 = 91mm and number of blades isz = 20 Section B-B in Figure 2 is a section at the center ofthe blade width The inner diameter of the cylindrical tankis 490mm and the diameter of the hole at the bottom oftank is 100mm In addition the coordinate system is definedas shown in Figure 1 The circumferential angle 120579 is definedas 120579 = 0∘ on the positive y axis and its positive direction iscounterclockwise

Table 1 Specifications of runner

Outer diameter1198631 014mInner diameter1198632 009mInlet width 1198871 0091mOutlet width 1198872 0091mInlet angle 1205731198871 719∘

Outlet angle 1205731198872 190∘

Tip clearance 120575 05mmNumber of blades 119885 20

Motor

Torque meter

Flow meter

Tachometer

Flow control valve

TankRunner

Reserve tank

Pump

Figure 3 Experimental apparatus

22 Experimental Apparatus and Methods An overview ofthe experimental apparatus is shown in Figure 3 The flowrate Q of water supplied by the pump was measured with anelectromagnetic flow meter (Toshiba Corporation LF620)The experiment was conducted under the condition ofconstant flow rate Q = 000285m3s The load to the runnerwas controlled by a motor and an inverter and the rotationalspeed was arbitrarily set The rotational speed n and torque Tweremeasured with amagnetic rotation detector (Ono SokkiCo LtdMP-981) and a torque detector (Ono Sokki Co LtdSS-005) respectively The turbine output P was obtained by

119875 = 212058711989911987960 (1)

Here the torque T was corrected by measuring the idlingtorque without the runnerThe effective headH is defined bythe following as shown in Figure 4

119867 = ℎ1015840 + ℎ3 + V232119892 minus ℎ4 minus

V242119892 (2)

Here the upstream water depth h3 was measured at thetank inlet in the vicinity of thewall surface on the +y-axiswitha ruler The downstream water depth h4 was measured by apoint gauge (Kenek Corporation PH-102) at five points fromthe vicinity of the wall surface on the +y-axis to the centerat the position of 6D1 downstream from the downstreamatmospheric opening from which the average water depth



H

422g

3ℎ3

4 ℎ4


ℎ



V3 = 1198761198613ℎ3 (3)

V4 = 1198761198614ℎ4 (4)


120578 = 119875120588119892119876119867 (5)





Outlet

Inlet

FlowOpening

Runner

Tank

z

yx






(a) Runner (b) Tank


T (Exp)T (Cal)

P (Exp)P (Cal)

000

005

010

015

020

025

T(N

middotm)

30 60 90 120 150 180000

05

10

15

20

25P

(W)

n (minminus1)


H (Exp)H (Cal)

(Exp) (Cal)

00

01

02

03

04

05

000

005

010

015

020

025

H(m

)

30 60 90 120 150 1800

(mdash

)

n (minminus1)








Runner

Flow

Flow




Flow



n = 81

n = 122

n = 162

00

02

04

06

08

10

12

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

VF 1



Rotation

① Inlet

② Outlet①

y

x

w

u

w

u

②





minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)


minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)






V1199061 = 1199033V31199031 (6)







minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











H

422g

3ℎ3

4 ℎ4


ℎ



V3 = 1198761198613ℎ3 (3)

V4 = 1198761198614ℎ4 (4)


120578 = 119875120588119892119876119867 (5)





Outlet

Inlet

FlowOpening

Runner

Tank

z

yx






(a) Runner (b) Tank


T (Exp)T (Cal)

P (Exp)P (Cal)

000

005

010

015

020

025

T(N

middotm)

30 60 90 120 150 180000

05

10

15

20

25P

(W)

n (minminus1)


H (Exp)H (Cal)

(Exp) (Cal)

00

01

02

03

04

05

000

005

010

015

020

025

H(m

)

30 60 90 120 150 1800

(mdash

)

n (minminus1)








Runner

Flow

Flow




Flow



n = 81

n = 122

n = 162

00

02

04

06

08

10

12

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

VF 1



Rotation

① Inlet

② Outlet①

y

x

w

u

w

u

②





minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)


minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)






V1199061 = 1199033V31199031 (6)







minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(a) Runner (b) Tank


T (Exp)T (Cal)

P (Exp)P (Cal)

000

005

010

015

020

025

T(N

middotm)

30 60 90 120 150 180000

05

10

15

20

25P

(W)

n (minminus1)


H (Exp)H (Cal)

(Exp) (Cal)

00

01

02

03

04

05

000

005

010

015

020

025

H(m

)

30 60 90 120 150 1800

(mdash

)

n (minminus1)








Runner

Flow

Flow




Flow



n = 81

n = 122

n = 162

00

02

04

06

08

10

12

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

VF 1



Rotation

① Inlet

② Outlet①

y

x

w

u

w

u

②





minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)


minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)






V1199061 = 1199033V31199031 (6)







minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Runner

Flow

Flow




Flow



n = 81

n = 122

n = 162

00

02

04

06

08

10

12

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

VF 1



Rotation

① Inlet

② Outlet①

y

x

w

u

w

u

②





minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)


minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)






V1199061 = 1199033V31199031 (6)







minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus02

minus01

00

01

02

03

04 r

1(m

s)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)


minus04

minus03

minus02

minus01

00

01

a1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

(b) Axial component

00

02

04

06

08

10

12

14

u1

(ms

)

n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)






V1199061 = 1199033V31199031 (6)







minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














minus90

minus60

minus30

0

30

60

90

120

150

180

210

1

(∘)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


n = 81

n = 122

n = 162

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

00

02

04

06

08

10

12

VF 2





minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus02

minus01

00

01

02

03

04 r

2(m

s)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162


minus06

minus04

minus05

minus03

minus02

minus01

00

01

a2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162

(b) Axial component

minus04

minus02

00

02

04

06

08

10

u2

(ms

)

30 60 90 120 150 180 210 240 270 300 330 3600

(∘)

n = 81

n = 122

n = 162




1199021 = 21205871199031V11990311000

1199022 = 21205871199032V11990321000 (7)


V1199031 = 1120579119908 int120579119908 V1199031119889120579119908

V1199032 = 1120579119908 int120579119908 V1199032119889120579119908(8)



0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0000000

0000005

0000010

0000015

0000020

0000025

q(m

3m

mmiddots)

30 60 90 120 150 1800n (min

minus1)

q1q2



1198711 = 12058811990211199031V11990611198712 = 12058811990221199032V1199062

(9)


V1199061 = 1V1199031120579119908 int120579119908 V1199031V1199061119889120579119908

V1199062 = 1V1199032120579119908 int120579119908 V1199032V1199062119889120579119908(10)



00000

00002

00004

00006

00008

00010

00012

L(N

middotmm

mmiddots)

30 60 90 120 150 1800minus00002

n (minminus1)

L1

L2





5 Conclusions




Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Rotation

00

01

02

03

04

05

06

07

08

09

10

VF


01

02

03

04

05

06

07

08

09

10

VF


00

01

02

03

04

05

06

07

08

09

10

VF








Nomenclature





Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Greek Letters


Subscripts




Acknowledgments


References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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