performance improvement of a micro impulse water turbine based on orthogonal...

Research ArticlePerformance Improvement of a Micro Impulse Water TurbineBased on Orthogonal Array

Lingdi Tang Shouqi Yuan and Yue Tang

Research Center of Fluid Machinery Engineering and Technology Jiangsu University Zhenjiang 212013 China

Correspondence should be addressed to Lingdi Tang angelattld163com

Received 14 May 2017 Accepted 2 November 2017 Published 7 December 2017

Academic Editor Jian G Zhou

Copyright copy 2017 Lingdi Tang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The study on structural design and efficiency improvement of the micro impulse water turbine with the super-low specific speedhas rarely been reported in literature In this paper a micro impulse water turbine was optimized on the base of the orthogonalarray of L18(3

7) with six factors The range analysis and variance analysis were conducted to present the significance ranking offactors and the optimal combinations of factors aiming to improve the water turbine efficiency taken as the experimental indicatorin the orthogonal experiment And then the optimal parameter combination for the water turbine was calculated by orthogonalexperiment Moreover the internal flow field and hydraulic performance were simulated numerically to investigate the principle ofperformance improvement by comparing the optimized water turbine with the original Also the numerical method was verifiedby experimental result from performance tests of the original water turbine As a result the runner torque of the optimized waterturbine was 13 higher than that of the original and the water turbine efficiency was improved by 58 percentage points at the ratedoperating condition

1 Introduction

Water resource is only the clean and renewable energy andapplicable for the large-scale development by far Hydraulicturbine has been used to convert hydraulic energy intomechanical energy or electric energy for centuries in severalfields In industry the water turbine is applied to recyclethe energy from industrial water [1ndash4] or drive the coolingtower fan [5ndash11] In agriculture the water turbine in irrigationmachinery is used to rotate the nozzle andmove the sprinklertrolley [12ndash14] The water turbine is not only used in largehydropower station as large as a unit capacity of 700MWbut is also applied to power a sensor as small as a micropipeline turbine [15ndash17]The orthogonal experimental designis a method based on orthogonal array to obtain an optimalscheme with multiple factors and levels in a certain rangeThe orthogonal array can maximize the test coverage whileminimizing the number of test cases to consider and can beused in various fields An orthogonal experimental designconsidering the interactions was used to obtain an optimizedwind rotor to improve energy utilization of vertical axis

wind turbine Three factors including radius of curvatureinstallation angle and central angle of small arc were selectedin this orthogonal experimental design [18] The effectdegree of main geometry factors of splitter blade on theperformance of pump as turbine was obtained (in orderthe outlet deflection angle the outlet diameter the numberof blades and the blades circumferential biasing degrees)based on the L9(3

4) orthogonal design method [19] Thestructure of turbine blade was optimized by a seven-factororthogonal array with the Kriging model to maximize thefatigue life of turbine blade [20] The orthogonal experimentdesign method was applied to evaluate the performance ofASHP (air source heat pump) The optimized ASHP withoptimum parameter combination was proposed and themost significant parameter was identified [21] The controlparameter of the industrial controller was tuned through anorthogonal test considering the interactions to match thecharacteristics of the controlled system [22] Moreover anoptimal scheduling method of urban pumping stations wasproposed based on orthogonal array to cut down the energycost The total electric power consumption in a study case

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 5867101 15 pageshttpsdoiorg10115520175867101

2 Mathematical Problems in Engineering

with the optimized schedulingmethod reducedmore than 14compared with the traditional one [23]

Water turbine mainly has two types impulse turbine andreaction turbine the former works in nonpressure systemwhile the latter works in pressure system Moreover thepump reversal is also used as the turbine The water turbinefrom large scale to small scale has drawn enough attentionHowever researches on structural design and efficiencyimprovement of the micro water turbine with the powerless than 1 kW especially for the super-low specific speedone (less than 20msdotkW) are still inadequate In this paperthe orthogonal array was used to improve the performanceof the original water turbine by combining with numericalsimulations The result of orthogonal design was analyzedby the statistical method to rank the significance of factorsand the optimized water turbine with optimal combinationof factors was proposed

2 Materials and Methods

21 Geometry and Meshing for the Water Turbine The entityand three-dimensional model of the original water turbineare shown in Figure 1 The specific speed under the ratedworking condition is 1455 The rated rotational speed (119899119890) ofwater turbine is 700 rpm the rated flow (119876119890) is 175m3h andthe operation condition ranges from 15m3h to 21m3h

The polyhedral mesh was used in the entire compu-tational domain of water turbine as shown in Figure 2

The mesh was generated with a combination of surfaceremesh polyhedral mesher and prism layer mesherThe gridindependence was conducted and the result is presented inTable 1 It shows that the head and efficiency basically remainabout the same (the variation magnitude less than 2) whenthe grid number is greater than 106 Thus the mesh scheme 3was used in this work

22 Governing Equations The three-dimensional governingequation [24 25] of mass and momentum conservationfor steady turbulent incompressible flow can be written inCartesian tensor form as

120597120597119909119894 (120588119906119894) = 0120597120597119909119895 (120588119906119894119906119895) = minus

120597119901120597119909119894 +120597120597119909119895 [120583(

120597119906119894120597119909119895 +120597119906119895120597119909119894 )]

+ 120597120597119909119895 (minus12058811990610158401198941199061015840119895)(1)

where 120588 is liquid density and 120583 is dynamic viscosityThe SST 119896-120596 turbulence model was used for turbulence

closure The turbulence kinetic energy 119896 and the specificdissipation rate 120596 are obtained from the following transportequations

120597120597119905 (120588119896) + 120597120597119909119894 (120588119896119906119894) =120597120597119909119895 [(120583 +

120583119905120590119896)120597119896120597119909119895] + [min(minus12058811990610158401198941199061015840119895 120597119906119895120597119909119894 10120588120573lowast119896120596)] minus 120588120573lowast119896120596119896

120597120597119905 (120588120596) + 120597120597119909119895 (120588120596119906119895) =120597120597119909119895 [(120583 +

120583119905120590120596)120597120596120597119909119895] +

120596119896 (minus12058811990610158401198941199061015840119895120597119906119895120597119909119894 ) minus 1205881205731198941205962 + 2 (1 minus 1198651) 120588

11205961205901205962120597119896120597119909119895120597120596120597119909119895 120596

120583119905 = 120588119896120596 1max (1120572lowast 11987811986521205721120596)

120573119894 = 11986511205731198941 + (1 minus 1198651) 12057311989421198651 = tanh

min[max( radic119896009120596119910 5001205831205881199102120596) 41205881198961205901205962max (2120588 (11205901205962) (1120596) (120597119896120597119909119895) (120597120596120597119909119895) 10minus10) 1199102]

41198652 = tanh[max(2 radic119896009120596119910 5001205831205881199102120596)]

2

(2)

where 120583119905 is turbulent viscosity 119878 is strain rate and thecoefficients are as follows 120572lowast = 1 1205721 = 031 120573lowast = 0091205901198961 = 1176 1205901198962 = 10 1205901205961 = 20 1205901205962 = 1168 1205731198941 = 00751205731198942 = 0082823 Calculation Method and Boundary Conditions Three-dimensional steady flow field of water turbine was solvedby the commercial software Star-ccm+ The computational

domain includes inlet section (stationary domain) outletsection (stationary domain) and runner section (rotatingdomain) Data between neighboring domains was trans-mitted through interface The reference coordinate systemfor rotating domain is the rotating frame rotated aboutthe runner central axis and for stationary domain is thelab reference frame And the other setups for numericalcalculation are shown in Table 2

Mathematical Problems in Engineering 3

Figure 1 The original water turbine

(a) (b)

(c) (d)

Figure 2 Water turbine meshes (a) Three-dimensional meshes (b) Radial cross-section meshes (c) Local meshes of blade tip (d) Localmeshes of blade outlet


Table 1 Grid independence analysis

Mesh scheme 1 2 3 4 5Grid number 378861 569762 1141389 2386757 4696273Grid sizemm 6 4 2 1 05120578 3603 3501 3432 3402 3387119867m 1256 1232 1197 1195 1196

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Figure 3 Schematic diagram of experimental apparatus (1) Water supply pump (2) motor (3) electromagnetic flowmeter (4) regulatingvalve (5) pressure gauge (6) tested water turbine (7) torque-revolution speed transducer (8) magnetic powder brake (9) measuring system

Table 2 The primary setup for numerical calculation

Space Three-dimensionalTime SteadyMaterial LiquidFlow SegregatedEquation of state Constant densityViscous regime TurbulentReynolds-averaged turbulence 119870-omega turbulenceTransition Turbulence suppression119870-omega wall treatment All 119910+ wall treatmentGradient metrics Gradients

At the inlet boundary the uniform inlet velocity (Vin) isprescribed with the turbulence intensity (119868)

V119894119899 = 4119876in1205871198892in (3)

where 119876in is inlet flow rate and 119889in is inlet pipe diameter

119868 = 016Reminus18 = 016 (120588V119889120583 )minus18 (4)

where Re is Reynolds number V equals Vin and 119889 equals 119889inAt the outlet boundary the pressure outlet with the

relative static pressure of 03MPa according to the actualworking status and the turbulence intensity calculated by (4)

are fixed At thewall boundary a no-slip condition is imposedwith the roughness of 0025mm

3 Experimental Validation

The performance test of water turbine was conducted inan open test rig as shown in Figure 3 The test apparatusincludes a water turbine a water supply pump a regulatingvalve a magnetic powder brake a torque-revolution speedtransducer an electromagnetic flowmeter torque-rotationalspeed measurer and two pressure gauges

The water supply pump absorbs water from the reservoirthen the water is pressured by pump and flows to thewater turbine as the pressured water The water flow wasregulated by a valve and its magnitude is measured by aflowmeter installed between pump outlet and water turbineinlet The length of upstream pipeline of the flowmeteris about ten times the pipe diameter and the length ofdownstream pipeline of the flowmeter is about five timesthe pipe diameter Pressure measurements were conductedby pressure sensors installed at both the inlet and outletof the water turbine and measuring positions were bothplaced at the distance with double pipe diameter The shaftof water turbine torque sensor and magnetic powder brakewere connected by flexible coupling with the concentricityless than 20120583m

Comparisons between numerical simulation results andtest results are shown in Figure 4Themaximumrelative error


EXPCFD

n = 600 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)

15 25 35 455Q (Ｇ3middotＢminus1)

EXPCFD

n = 700 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 900 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 800 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


Figure 4 Comparisons between numerical simulation results and test results

is less than 5 which indicates a good agreement betweenmeasured and calculated results The efficiency curves ofthe original water turbine under different rotating speedconditions show that the performance in low rotation speedis lower than in high rotation speed with the increase ofrotation speed themaximum efficiency running pointmovestowards the direction of high flow and the high efficiency areagoes wider

4 Results and Discussions

41 Orthogonal Experimental Design The influential factorsof the water turbine runner were selected from all thegeometric parameters and each factor contained three levelsas shown in Table 3

The orthogonal experiment is supposed to present anew type water turbine runner with the high hydraulic


Table 3 Factors and levels of orthogonal experiment

Level

Factor

Number of blades119885 Blade tip clearance120575 Width of blade119887 Inlet blade angle120573119901Blade outletdiameter1198632

Inclination angle ofblade outlet1205742

1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70

Table 4 Orthogonal experiment schemes

Number A119885 B120575 C119887 D1205721 E1198632 F G1205742(1) A1 (10) B1 (7) C1 (26) D1 (155) E1 (65) F1 G1 (50)(2) A1 B2 (5) C2 (28) D2 (145) E2 (72) F2 G2 (60)(3) A1 B3 (2) C3 (31) D3 (135) E3 (80) F3 G3 (70)(4) A2 (12) B1 C1 D2 E2 F3 G3(5) A2 B2 C2 D3 E3 F1 G1(6) A2 B3 C3 D1 E1 F2 G2(7) A3 (14) B1 C2 D1 E3 F2 G3(8) A3 B2 C3 D2 E1 F3 G1(9) A3 B3 C1 D3 E2 F1 G2(10) A1 B1 C3 D3 E2 F2 G1(11) A1 B2 C1 D1 E3 F3 G2(12) A1 B3 C2 D2 E1 F1 G3(13) A2 B1 C2 D3 E1 F3 G2(14) A2 B2 C3 D1 E2 F1 G3(15) A2 B3 C1 D2 E3 F2 G1(16) A3 B1 C3 D2 E3 F1 G2(17) A3 B2 C1 D3 E1 F2 G3(18) A3 B3 C2 D1 E2 F3 G1Note A is number of blades B is blade tip clearance C is width of blade D is inlet blade angle E is blade outlet diameter F is vacancy and G is inclinationangle of blade outlet

performanceThe efficiency (120578)was regarded as the indicatorto evaluate the hydraulic performance of water turbine

120578 = 119875119904119875119908 times 100 (5)

where 119875119904 is the shaft power (output power) (W) and 119875119908 is thewater power (input power) (W)

119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)

where 120588 is the fluid density (kgm3) 119892 is the gravitationalacceleration (98Nkg) 119899 is the rotational speed (rpm)119872 isthe runner torque (Nsdotm) and Δ119901 is the pressure drop fromthe inlet of water turbine to the outlet of water turbine (Pa)

The orthogonal array without considering interactionsrequires six columns for six-factor analysis (three levelsfor each factor) and at least one vacant column for error

analysis The vacant column with no contributing factor canwell reflect the error induced by random chance and ittakes effect in the subsequent statistical data analysis Thevacant column (also called the error column) guarantees thereliability of the experimental results Thus the orthogonalarray of L18(3

7) was determined in this study The wholeexperimental arrangement including 18 times is shown inTable 4

The performance of the water turbine is expected to keepthe high efficiency in common used conditions Thus all thewater turbine efficiencies in the low-flow condition (08119876119890)the rated flow condition (10119876119890) and the high-flow condition(12119876119890) were observed as shown in Table 5

42Direct Analysis of theOrthogonal Experiment Results Thedirect analysis for the experimental results is to determinethe important order of all factors by means of the rangeanalysisThe range analysis in this study reveals the influenceextent for the efficiency of water turbine by comparisonsamong mean values and range values in different factorsRange analyses of orthogonal experiment results in threeoperating conditions are shown in Table 6 119870119898 is the sum of


Table 5Orthogonal experiment results at the rated rotational speed(119899119890 = 700 rmin)

Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158

efficiencies at the119898th level for a certain factor in one columnand 119896119898 is the mean value of 119870119898 119877 is the range for certainfactor in one column and can be calculated by (7) [26] Thevalue of 119896119898 indicates that the 119898th level for a certain factoris either a superior level or an inferior level and the valueof 119877 indicates the changing amplitude of the water turbineefficiency with the variation of levels for a certain factor Thehigher value of 119877 shows the greater effect on water turbineefficiency Therefore the descending order of the 119877 value isthe important order of factors as shown in Table 7

119877 = max 119896119898 minusmin 119896119898 (7)

The range of vacant column means the limit of error it isused to estimate whether a factor has an effect on the waterturbine efficiency The factor is an influential one if the rangeof this factor is larger than the range of vacant column whilethe factor is an effect-free one if the range of this factor is lessthan the range of vacant column and the range of this factoris considered to be caused by experimental error

As shown in Table 7 factor E (blade outlet diameter) isthe most important factor while factor B (blade tip clearance)is the least important factor for the observed operatingconditions (08119876119890 10119876119890 and 12119876119890) Factors G C andD always have the significance relation that G gt C gt DFactor A presents different significance in different operatingconditions It is a less important factor in low-flow operatingcondition and a secondary important factor in both rated andhigh-flow operating conditions

The relations between efficiencies and factors are shownin Figure 5 It clearly shows the effects of factor on theefficiency of water turbine and the variation trend of theefficiency with levels of factor

28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)

Figure 5 Relations between efficiencies and factors (a) 08119876119890 (b)10119876119890 (c) 12119876119890

The factors show the nearly same change rules in low-flow operating condition (08119876119890) rated operating condition(10119876119890) and high-flow operating condition (12119876119890)

For factor A the efficiency of level A2 is the highest andthe efficiency of level A3 is slightly less than that of level A2while the efficiency of level A1 is obviously lower than thatof level A2 and level A3 For factor B the efficiency of levelB1 is the highest followed by that of level B3 and level B2but the differences among them are quite modest For factorC the efficiency of level C3 is the highest and the efficiencyof level C1 is little higher than that of level A2 For factorD the efficiency of level D1 is the highest and efficiencies oflevels D1 D2 and D3 decrease in sequence For factor E theefficiency of level E1 is the highest and efficiencies of levelsE1 E2 and E3 descend in turn with the largest change rangeFactor F is vacant For factor G the calculated efficiency oflevel G3 is the highest and efficiencies of levelsG1 G2 andG3increase in sequence with the second largest change rangeThe level with the highest efficiency for each factor (A2 B1C3 D1 E1 and G3) can be combined to structure a newwaterturbine with the better performance


Table 6 Range analyses of orthogonal experiment results

08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325

Table 7 Important order of factors

Operating condition Majorrarrminor08119876119890 E G C D A B10119876119890 E A G C D B12119876119890 E A G C D B

In comparison to the change range of factor F (vacancy)factor B has basically the same change range all the factors AC andDhave slightly larger change ranges and both factors Eand G have significantly larger change ranges Therefore theeffect of factor B on water turbine efficiency can be neglectedwhile the effect of other factors on the efficiency should befurther investigated

43 Variance Analysis of the Orthogonal Experiment ResultsThe range analysis is simple and convenient but it does notconsider the influence of experimental error on experimentalindicator and not calculate the magnitude of experimen-tal error Thus it is unable to distinguish the change ofexperimental indicator resulting from factors various andexperimental error Moreover the range analysis is incapableof not only precisely calculating the effect degree of factors onexperimental indicator but also determining the significanceof each factor as a standard Therefore the variation analysisneeds to be added to determine the significance of each factor

The variation analysis for experimental results is toanalyze the significance of each factor by comparing valueof 119865 and critical value of 119865 119865 is calculated from the sum ofsquares and the degree of freedom The value of 119865 is used toshow that the effect of a factor on the efficiency is less than

that of experimental error if the value of 119865 is less than 1 Thevariation analysis process [26] is as follows

(1) Calculation for the Sum of Squares

(a) The Sum of Squares for Total The sum of squares for total(119878119879) is defined as

119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)

119878119879 resulting from variation of factor levels and experimentalerror shows the variation degree of the efficiency withdifferent parameter combinations

(b) The Sum of Squares for Factor The sum of squares forfactor is collectively named 119878119891 (119891 = A to G) then the sumof squares for factor A is defined as

119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2

minus 1n( 3sum119894=1

119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)

where 119909119894119895 is the experimental efficiency for factor A with the119894th level and the 119895th experiment 119878A shows the change of theefficiency with varying levels for factor A Sums of squares forother factors are calculated by the same way

(c) The Sum of Squares for ErrorThe sum of squares for error(119878119890) is defined as

119878119890 = 119878119879 minussum119878119891 (10)

where sum119878119891 is the sum of squares for all factors


Table 8 Variance analyses in the working condition of 08119876119890Source ofvariation

Sum ofsquares

Degree offreedom Mean square 119865 Critical value Significance

(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327

lowastB 693 2 347 242C 1254 2 627 437 lowastD 1225 2 613 427 lowastE 11801 2 5900 4116 lowast lowast lowastG 4138 2 2069 1443 lowast lowast lowastError 524 5 105Total 21041 17Note lowast lowast lowastmeans significant effect and lowastmeans slight effect


Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327

lowastlowastB 376 200 188 118C 1409 200 705 444 lowastD 1301 200 651 410 lowastE 9714 200 4857 3061 lowast lowast lowastG 3705 200 1852 1167 lowastlowastError 410 5 082Total 19176 17Note lowast lowast lowastmeans significant effect lowastlowastmeans visible effect and lowastmeans slight effect

(2) Calculation for the Degree of Freedom The total degree offreedom is 119891119879 defined as 119891119879 = 119899 minus 1 = 18 minus 1 = 17 where 119899 isthe total number of experiments The degree of freedom forfactor is collectively named 119891119891 (subscript 119891 = A to G) Takefor example the degree of freedom for factor A119891A = 119899Aminus1 =3 minus 1 = 2 where 119899A is the number of levels for factor A Thedegree of freedom for experimental error is 119891119890 calculated bythe total degree of freedom minus degrees of freedom for allfactors that is 119891119890 = 119891119879 minus 119891A minus 119891B minus 119891C minus 119891D minus 119891E minus 119891G =17 minus 2 minus 2 minus 2 minus 2 minus 2 minus 2 minus 2 = 5(3) Calculation for the Mean Square Value The sum ofsquares for factor (119878119891) affected by the number of additivedata does not accurately reflect the influence of each factorTherefore the mean value of the sum of squares is neededto be calculated The definitions for both the mean squarevalue of factor (119872119878119891) and the mean square value of error(119872119878119890) are given in the same way respectively and defined as119872119878119891 = 119878119891119891119891119872119878119890 = 119878119890119891119890(4) Calculation for Value of 119865 The value of 119865 shows the effectdegree of each factor on the efficiency which is defined as

119865 = 119872119878119891119872119878119890 (11)

(5) Significance Test for Factors Variation analyses in theworking condition of 08119876119890 10119876119890 and 12119876119890 using the above

method are shown in Tables 8ndash10 In the working conditionof 08119876119890 factors E and G have significant effects on the resultwhile factors A C and D have slight effects on the resultIn the working condition of 10119876119890 factor E has significanteffects on the result and factors A and G also have visibleeffects on the result while factors C and D have slight effectson the result In the working condition of 12119876119890 factor E hassignificant effects on the result and factor G also has visibleeffects on the result while factors A C and D have slighteffects on the result

The effect degree of each factor onwater turbine efficiencyis in order E G A C D B which has basically consistenttendency with the result of the range analysis Thereforethe water turbine with the optimal parameter combinationresulting from the range analysis is receivable

44 Flow Field Analysis The internal flow analysis was con-ducted to reveal the principle of performance improvementby comparing the optimized and original water turbinesTherunner of the optimized water turbine wasmodeled as shownin Figure 6

The velocity distribution in mid-axial section of waterturbine is shown in Figure 7 Both the velocities around bladeoutlet and draft tube wall of the optimized water turbineare lower than those of the original The optimized waterturbine just presents slight higher velocity in draft tube inletIt indicates that the optimized water turbine has the ability totransform more kinetic energy to pressure energy



Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327

lowastB 350 200 175 117C 1511 200 755 505 lowastD 1161 200 580 388 lowastE 9032 200 4516 3017 lowast lowast lowastG 3278 200 1639 1095 lowastlowastError 398 5 080Total 19048 17Note lowast lowast lowastmeans significant effect lowastlowastmeans visible effect and lowastmeans slight effect

Figure 6 Runner of optimized water turbine

1037933830726622518415311207104000

Velocity (ＧmiddotＭminus1)

(a) Original

1037933830726622518415311207104000


(b) Optimized

Figure 7 Velocity distribution in mid-axial section of water turbine

The pressure distribution in radial mid-span section ofwater turbine is shown in Figure 8 The pressure distributionin radial mid-span section of the optimized water turbineis more evenly than that of the original The high pressureregion around blade tip circumference of the optimizedwaterturbine is narrower than that of the original

The pressure distribution in mid-axial section of waterturbine is presented in Figure 9The pressure in inlet circle ofthe runner for the optimized water turbine is obviously lowerthan that for the original while the pressure in draft tubeinlet of the optimized water turbine is visibly higher than thatfor the original It indicates that the pressure energy rallies


Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized

Figure 8 Pressure distribution in radial mid-span section of water turbine

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized

Figure 9 Pressure distribution in mid-axial section of water turbine

(a) Original (b) Optimized

Figure 10 Vortex structure in draft tube

substantially in draft tube for the optimized water turbineand the low pressure region in the inlet of draft tube for theoriginal tends to induce the vortex

The method of 119876-Criterion is used to detect vorticesconsidering that the pressure distribution is unable to visually

display the vortex The 119876 is the second invariant of thevelocity gradient tensor The vortex structure in draft tubewith the level of 004119876 is shown in Figure 10The vortex zonefor the original almost occupies the whole bend section of thedraft tube and extends even longer in helicalmannerTheflow


Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized

Maximum force on working surface (FＧＲ)Force on working surface (FＭＯＬ)

0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)

Figure 11 Force on working surface of runner (a) Force on working surface of original (b) Force on working surface of optimized runner(c) Comparisons of force on working surface

structure with the large volume of the vortex likely inducesthe vibration of the draft tube However for the optimizedwater turbine the vortex just takes up half of the bend sectionof draft tube and has fully developed before going acrossthe bend section This behavior makes for improving flowcharacteristics in the draft tube and diminishes the vibrationinduced by the unsteady flows

The load distribution on the working surface of runner ispresented in Figure 11 Larger forces on the working surface

for both the optimized water turbine and the original con-centrate on the blade tip and blade lower-middle part but theforce magnitude of the optimized turbine is obviously largerthan that of the original In the rated flow the maximumforce on the working surface for the optimized turbine is 59higher than that for the original and the resultant force on theworking surface for the optimized turbine is 11 higher thanthat for the original The larger blade load generates greaterrotating torque for the runner Thus the optimized turbine is


Original Optimized0

11

22

33

44

0

1

2

3

4

Torque on runner (T)

TＭＯ

Ｌ(N

middotm)

T(N

middotm)

Torque on working surface (TＭＯＬ)

Figure 12 Torques on working surface and runner

capable of delivering greater torque than the original Torqueson theworking surface and the runner are shown in Figure 12In the rated flow the torque on the working surface of theoptimized turbine is 10 higher than that of the originaland the torque on the optimized runner is 13 higher thanthat of original The greater torque indicates higher output ofwater turbine So the optimized water turbine delivers greateroutput that is this turbine has the better performance

45 Hydraulic Performance Analysis The performance pre-dictions of the original and the optimized water turbine forthe rotational speed ranging from 500 rpm to 900 rpm areshown in Figure 13 The efficiency of the optimized waterturbine at the rated condition with the rotational speedof 700 rpm and the flow rate of 175m3h is 396 whichis 58 percentage points higher than that of the original(338) Thus the optimized water turbine has the obviousperformance improvement in the operating flow rate (from15m3h to 21m3h)

5 Conclusion

In this work the micro impulse water turbine with super-low specific speedwas optimized to improve the performancecharacteristics based on the orthogonal array together withthe numerical calculation The experiment for the originalwater turbine was conducted in an open test rig Its result hasa good agreement with the numerical simulations which wasused to verify the accuracy of numerical methods

Six main geometry parameters of the water turbinerunner including number of blades 119885 blade tip clearance 120575width of blade 119887 inlet blade angle 120573119901 blade outlet diameter1198632 and inclination angle of blade outlet 1205742 were selected asfactors in orthogonal array The efficiency of water turbinewas chosen as the experimental indicator in the process of

statistical analysis Specifically the range analysis presentsthe variation trend of the factor with different levels andthe optimal combination of factors (A2 B1 C3 D1 E1 G3)The variance analysis shows that the effect degree of eachfactor on water turbine efficiency is in order E(1198632) G(1205742)A(119885) C(119887) D(120573119901) B(120575) and factor B almost has no effecton the water turbine efficiency The variance analysis resulthas a basically consistent tendency with the range analysisresult so the optimal parameter combination calculated fromthe statistical analysis is receivable to structure a new waterturbine with the better performance

The internal flow characteristics and hydraulic perfor-mance of the optimized water turbine were investigated bycomparing with the original The pressure distribution inthe optimized water turbine is more evenly than that in theoriginal The vortex in draft tube of the optimized waterturbine generated from the lower velocity near the bladeoutlet and the higher pressure around the draft tube inlet issmaller than that in the original Moreover the blade inletof the optimized water turbine has a larger curvature so thatthe jet flow from the nozzle concentrates in the impacted areaof the blade As a result the runner torque of the optimizedwater turbine was 13 higher than that of the original andthe water turbine efficiency was improved by 58 percentagepoints at the rated operating condition

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Key Research andDevelopment Program (2016YFC0400202)


700 ＬＪＧ-original700 ＬＪＧ-optimized


10

15

20

25

30

35

40

45

()

(a)

500 ＬＪＧ-original600 ＬＪＧ-original

500 ＬＪＧ-optimized600 ＬＪＧ-optimized

10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)

Figure 13 Performance curves of the water turbine (a) Rated rotation speed condition (b) Low rotation speed condition (c) High rotationspeed condition

References

[1] P Bansal and N Marshall ldquoFeasibility of hydraulic powerrecovery from waste energy in bio-gas scrubbing processesrdquoApplied Energy vol 87 no 3 pp 1048ndash1053 2010

[2] S Mirza ldquoReduction of energy consumption in process plantsusing nanofiltration and reverse osmosisrdquo Desalination vol224 no 1-3 pp 132ndash142 2008

[3] H J Van Antwerpen and G P Greyvenstein ldquoUse of turbinesfor simultaneous pressure regulation and recovery in secondarycooling water systems in deep minesrdquo Energy Conversion andManagement vol 46 no 4 pp 563ndash575 2005

[4] Y Junhu Z Xuening Wang X et al ldquoProgress in energyrecovery hydraulic turbine researchrdquo Chemical Machinery vol38 no 6 pp 655ndash658 2011


[5] L Yanpin N Haipeng and C Dexin ldquoPerformance and typeselection of special hydraulic turbine in cooling towerrdquo Journalof Hydroelectric Engineering vol 30 no 1 pp 175ndash179 2011

[6] L Yanpin K Wang N Haipeng et al ldquoNan Haipeng Theapproximate calculation of the flow capacity of special turbinein cooling towerrdquo Journal of Xian University of Technology vol27 no 1 pp 79ndash82 2011

[7] Z Lanjin C Dexin R Yan et al ldquoHydraulic loss analysis of themicro francis turbine of cooling towersrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 33no 1 p 1 2012

[8] Z Lanjin L Yang C Dexin et al ldquoSaved-energy analysis offrancis turbine on hydraulic cooling towerrdquo Renewable EnergyResources vol 30 no 7 pp 123ndash128 2012

[9] Z Fei Y Zheng F Xiaojuan et al ldquoDesign and numericalsimulation of small francis turbine used in cooling towerrdquoWaterResources and Power vol 31 no 7 pp 165ndash168 2013 (Chinese)

[10] ZKaining L Li andXKaige ldquoHydraulic design andnumericalsimulation of super low specific speed francis turbine runnerrdquoEnergy Engineering vol 4 pp 6ndash9 2012

[11] L Zhang Y Zheng C Zhang Y Yin and J Liu ldquoStudy onFrancis turbine with super-low specific speed applied in coolingtowersrdquo Transactions of the Chinese Society of AgriculturalMachinery vol 41 no 1 pp 39ndash72 2010

[12] Y Tang Z Xiangyuan M Xingxin et al ldquoperformance test ofwater turbine of JP50 reel sprinklerrdquo China Rural Water andHydropower vol 2 pp 26ndash29 2014

[13] Y Shouqi N Guoping T Yue et al ldquoExperiment and numericalestimation of performance of hydraulic turbine of JP50 reelsprinklerdquo Journal of Drainage and Irrigation Machinery Engi-neering vol 32 no 7 pp 553ndash562 2014

[14] J Jun T Yue and L Tang ldquoEnergy consumption analysis ofhydraulic turbine of JP75 hose reel irrigatorrdquo Journal ofDrainageand Irrigation Machinery Engineering vol 34 no 11 pp 1008ndash1012 2016

[15] J Chen H X Yang C P Liu C H Lau and M Lo ldquoA novelvertical axis water turbine for power generation from waterpipelinesrdquo Energy vol 54 pp 184ndash193 2013

[16] S Wang C F Porres M Zuo et al ldquoStudy of impeller designfor pipe flow generator with CFD and RPrdquo in Proceedings ofthe 10th Asian International Conference on Fluid Machinery pp265ndash275 Kuala Lumpur Malaysia 2010

[17] S Wang and R Doblado ldquoComputational and experimentalstudy of a coaxial pipe flow generatorrdquo International Symposiumon Computer vol 132 no 42 pp 435ndash439

[18] L Song H-Z Liu and Z-X Yang ldquoOrthogonal analysisbased performance optimization for vertical axis wind turbinerdquoMathematical Problems in Engineering vol 2016 Article ID6241360 11 pages 2016

[19] J H Yang and S C Miao ldquoNumerical simulation and orthog-onal design method research effect of splitter bladersquos maingeometry factors on the performance of pump as turbinerdquoApplied Mechanics and Materials vol 456 pp 100ndash105 2014

[20] K-K Lee K-H Lee and S-HHan ldquoUse of an orthogonal arraybased on the Kriging model to maximize the fatigue life of aturbine bladerdquo International Journal of Structural Integrity vol2 no 3 pp 303ndash313 2011

[21] J Tang G Gong H Su F Wu and C Herman ldquoPerformanceevaluation of a novelmethod of frost prevention and retardationfor air source heat pumps using the orthogonal experimentdesign methodrdquo Applied Energy vol 169 pp 696ndash708 2016

[22] Z Li and O P Malik ldquoAn orthogonal test approach basedcontrol parameter optimization and its application to a hydro-turbine governorrdquo IEEE Transactions on Energy Conversion vol12 no 4 pp 388ndash393 1997

[23] J S Liu J L Cheng and Y Gong ldquoStudy on optimal schedulingmethods of urban drainage pumping stations based on orthog-onal testrdquo Applied Mechanics and Materials vol 373-375 pp2169ndash2174 2013

[24] F R Menter ldquoReview of the shear-stress transport turbulencemodel experience from an industrial perspectiverdquo InternationalJournal of Computational Fluid Dynamics vol 23 no 4 pp305ndash316 2009

[25] P Huang J Bardina and T Coakley ldquoTurbulencemodeling val-idationrdquo in Proceedings of the 28th Fluid Dynamics ConferenceSnowmass Village CO USA

[26] I Korkut and Y Kucuk ldquoExperimental analysis of the deviationfrom circularity of bored hole based on the taguchi methodrdquoStrojniski Vestnik Journal ofMechanical Engineering vol 56 no5 pp 340ndash346 2010

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with the optimized schedulingmethod reducedmore than 14compared with the traditional one [23]

Water turbine mainly has two types impulse turbine andreaction turbine the former works in nonpressure systemwhile the latter works in pressure system Moreover thepump reversal is also used as the turbine The water turbinefrom large scale to small scale has drawn enough attentionHowever researches on structural design and efficiencyimprovement of the micro water turbine with the powerless than 1 kW especially for the super-low specific speedone (less than 20msdotkW) are still inadequate In this paperthe orthogonal array was used to improve the performanceof the original water turbine by combining with numericalsimulations The result of orthogonal design was analyzedby the statistical method to rank the significance of factorsand the optimized water turbine with optimal combinationof factors was proposed

2 Materials and Methods

21 Geometry and Meshing for the Water Turbine The entityand three-dimensional model of the original water turbineare shown in Figure 1 The specific speed under the ratedworking condition is 1455 The rated rotational speed (119899119890) ofwater turbine is 700 rpm the rated flow (119876119890) is 175m3h andthe operation condition ranges from 15m3h to 21m3h

The polyhedral mesh was used in the entire compu-tational domain of water turbine as shown in Figure 2

The mesh was generated with a combination of surfaceremesh polyhedral mesher and prism layer mesherThe gridindependence was conducted and the result is presented inTable 1 It shows that the head and efficiency basically remainabout the same (the variation magnitude less than 2) whenthe grid number is greater than 106 Thus the mesh scheme 3was used in this work

22 Governing Equations The three-dimensional governingequation [24 25] of mass and momentum conservationfor steady turbulent incompressible flow can be written inCartesian tensor form as

120597120597119909119894 (120588119906119894) = 0120597120597119909119895 (120588119906119894119906119895) = minus

120597119901120597119909119894 +120597120597119909119895 [120583(

120597119906119894120597119909119895 +120597119906119895120597119909119894 )]

+ 120597120597119909119895 (minus12058811990610158401198941199061015840119895)(1)

where 120588 is liquid density and 120583 is dynamic viscosityThe SST 119896-120596 turbulence model was used for turbulence

closure The turbulence kinetic energy 119896 and the specificdissipation rate 120596 are obtained from the following transportequations

120597120597119905 (120588119896) + 120597120597119909119894 (120588119896119906119894) =120597120597119909119895 [(120583 +

120583119905120590119896)120597119896120597119909119895] + [min(minus12058811990610158401198941199061015840119895 120597119906119895120597119909119894 10120588120573lowast119896120596)] minus 120588120573lowast119896120596119896

120597120597119905 (120588120596) + 120597120597119909119895 (120588120596119906119895) =120597120597119909119895 [(120583 +

120583119905120590120596)120597120596120597119909119895] +

120596119896 (minus12058811990610158401198941199061015840119895120597119906119895120597119909119894 ) minus 1205881205731198941205962 + 2 (1 minus 1198651) 120588

11205961205901205962120597119896120597119909119895120597120596120597119909119895 120596

120583119905 = 120588119896120596 1max (1120572lowast 11987811986521205721120596)

120573119894 = 11986511205731198941 + (1 minus 1198651) 12057311989421198651 = tanh

min[max( radic119896009120596119910 5001205831205881199102120596) 41205881198961205901205962max (2120588 (11205901205962) (1120596) (120597119896120597119909119895) (120597120596120597119909119895) 10minus10) 1199102]

41198652 = tanh[max(2 radic119896009120596119910 5001205831205881199102120596)]

2

(2)

where 120583119905 is turbulent viscosity 119878 is strain rate and thecoefficients are as follows 120572lowast = 1 1205721 = 031 120573lowast = 0091205901198961 = 1176 1205901198962 = 10 1205901205961 = 20 1205901205962 = 1168 1205731198941 = 00751205731198942 = 0082823 Calculation Method and Boundary Conditions Three-dimensional steady flow field of water turbine was solvedby the commercial software Star-ccm+ The computational

domain includes inlet section (stationary domain) outletsection (stationary domain) and runner section (rotatingdomain) Data between neighboring domains was trans-mitted through interface The reference coordinate systemfor rotating domain is the rotating frame rotated aboutthe runner central axis and for stationary domain is thelab reference frame And the other setups for numericalcalculation are shown in Table 2



(a) (b)

(c) (d)





(1) (2) (3) (4) (5) (6) (7) (8) (9)





V119894119899 = 4119876in1205871198892in (3)


119868 = 016Reminus18 = 016 (120588V119889120583 )minus18 (4)









EXPCFD

n = 600 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 700 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 900 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 800 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)









Level

Factor



1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70




120578 = 119875119904119875119908 times 100 (5)


119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)









Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






(a) (b)

(c) (d)





(1) (2) (3) (4) (5) (6) (7) (8) (9)





V119894119899 = 4119876in1205871198892in (3)


119868 = 016Reminus18 = 016 (120588V119889120583 )minus18 (4)









EXPCFD

n = 600 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 700 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 900 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 800 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)









Level

Factor



1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70




120578 = 119875119904119875119908 times 100 (5)


119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)









Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(1) (2) (3) (4) (5) (6) (7) (8) (9)





V119894119899 = 4119876in1205871198892in (3)


119868 = 016Reminus18 = 016 (120588V119889120583 )minus18 (4)









EXPCFD

n = 600 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 700 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 900 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 800 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)









Level

Factor



1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70




120578 = 119875119904119875119908 times 100 (5)


119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)









Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





EXPCFD

n = 600 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 700 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 900 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)


EXPCFD

n = 800 ＬＪＧ

500

1000

1500

2000

2500

3000

3500

4000

4500

(

)









Level

Factor



1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70




120578 = 119875119904119875119908 times 100 (5)


119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)









Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Level

Factor



1 10 7 26 155 65 502 12 5 28 145 72 603 14 2 31 135 80 70




120578 = 119875119904119875119908 times 100 (5)


119875119908 = 120588119892119876119867119875119904 = 212058711989911987260119867 = Δ119901120588119892

(6)









Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Number 08119876119890 10119876119890 12119876119890(1) 3452 3492 3545(2) 3089 3146 3113(3) 2963 3039 3022(4) 3580 3632 3726(5) 2530 2691 2777(6) 3787 3910 3898(7) 3169 3274 3294(8) 3458 3546 3536(9) 3461 3530 3522(10) 2996 3046 3046(11) 2823 2948 2947(12) 3568 3585 3519(13) 3556 3649 3656(14) 3781 3927 3968(15) 2813 3036 3132(16) 3290 3355 3338(17) 3464 3542 3618(18) 3139 3177 3158


119877 = max 119896119898 minusmin 119896119898 (7)




28

31

34

37

(

)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A1

(a)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

(

)(b)

G3

G2

G1F3F2F1E3E2E1D3

D2

D1C3C2C1B3B2B1A3

A2

A129

32

35

38

()

(c)






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






08119876119890

1198701 18890 20042 19593 20151 21284 20082 183881198702 20045 19144 19050 19797 20046 19317 200061198703 19982 19731 20274 18969 17587 19518 205231198961 3148 3340 3265 3358 3547 3347 30651198962 3341 3191 3175 3299 3341 3219 33341198963 3330 3288 3379 3162 2931 3253 3421119877 193 150 204 197 616 128 356

10119876119890

1198701 19256 20448 20180 20728 21723 20579 189871198702 20846 19800 19522 20300 20457 19954 205381198703 20423 20276 20822 19497 18344 19991 210001198961 3209 3408 3363 3455 3621 3430 31651198962 3474 3300 3254 3383 3409 3326 34231198963 3404 3379 3470 3249 3057 3332 3500119877 265 108 217 205 563 104 335

12119876119890

1198701 19191 20605 20489 20809 21771 20669 191941198702 21158 19958 19516 20364 20533 20100 204731198703 20465 20250 20808 19640 18510 20045 211461198961 3198 3434 3415 3468 3628 3445 31991198962 3526 3326 3253 3394 3422 3350 34121198963 3411 3375 3468 3273 3085 3341 3524119877 328 108 215 195 543 104 325









119878119879 = 119899sum119896=1

1199092119896 minus 1119899 (119899sum119896=1

119909119896)2 (8)



119878A = 133sum119894=1

( 119886sum119895=1

x119894119895)2


119886sum119895=1

x119894119895)2

(119894 = 1 2 3 119895 = 1 sdot sdot sdot 119886) (9)



119878119890 = 119878119879 minussum119878119891 (10)




Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Sum ofsquares


(relative)A 1406 2 703 490

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



Sum ofsquares


(relative)A 2261 200 1130 712

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



119865 = 119872119878119891119872119878119890 (11)








Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Sum ofsquares


(relative)A 3319 200 1659 1108

119865010(2 5) = 378119865005(2 5) = 579119865001(2 5) = 1327



1037933830726622518415311207104000


(a) Original

1037933830726622518415311207104000


(b) Optimized





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(a) Original

Pressure (Pa)

269838

284702

299567

314431

329296

344160

359024

373889

388753

403618

418482

(b) Optimized


Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(a) Original

Pressure (Pa)

269930277634285339293043300747308452316156323860331564339269346973

(b) Optimized








Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Force (N)

00

05

09

14

19

(a)

Force (N)

00

05

09

14

19

(b)

Original Optimized


0

200

400

600

800

FＭＯ

Ｌ(N

)

0

08

16

24

FＧＲ

(N)

(c)






Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Original Optimized0

11

22

33

44

0

1

2

3

4


TＭＯ

Ｌ(N

middotm)

T(N

middotm)





5 Conclusion







Acknowledgments





10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







10

15

20

25

30

35

40

45

()

(a)



10

15

20

25

30

35

40

45

(

)


(b)



10

15

20

25

30

35

40

45

(

)


(c)


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




performance improvement of a micro impulse water turbine based on orthogonal...

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