Research Article A One-Dimensional Flow Analysis...

Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2013 Article ID 473512 19 pageshttpdxdoiorg1011552013473512

Research ArticleA One-Dimensional Flow Analysis for the Prediction ofCentrifugal Pump Performance Characteristics

Mohammed Ahmed El-Naggar

Department of Mechanical Power Engineering Mansoura University Mansoura 35516 Egypt

Correspondence should be addressed to Mohammed Ahmed El-Naggar naggarmmansedueg

Received 25 June 2013 Accepted 29 August 2013

Academic Editor Shigenao Maruyama

Copyright copy 2013 Mohammed Ahmed El-Naggar 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 one-dimensional flow procedure for analytical study of centrifugal pump performance is done applying the principle theories ofturbomachines Euler equation and energy equation are manipulated to find pump performance parameters at different dischargecoefficients Fluid slippage loss at impeller exit and volute loss are estimated The fluid slippage is modeled by the slip factorapproach using Wiesner empirical expression The volute loss model counts friction loss associated with the volute throw flowvelocity diffusion friction loss due to circulation associated with volute flow loss due to vanishing of radial flow at volute outletand loss inside pump volute throat Models for impeller hydraulic friction power loss disk friction power loss internal flow leakagepower loss and inlet shock circulation power loss are considered by suitable models Pump internal volumetric flow leakage andvolumetric efficiency are related to pump geometry and flow propertiesThe procedure adopted in this paper is capable of obtainingperformance characteristic curves of centrifugal pump in a dimensionless form Pump head coefficient manometric efficiencypower coefficient and required NPSH are characterizedThe predicted coefficients and obtained performance curves are consistentwith experimental characteristics of centrifugal pump

1 Introduction

Centrifugal pumps are used in various applications and areintegral to many industries Yet in spite of their prevalenceand relatively simple configurations compared to other turbo-machines designing an efficient and durable pump remainsa challenge

The design of centrifugal pumps is still determinedempirically because it relies on the use of a number ofexperimental and statistical rules However during the lastfew years the design and performance analysis of turbo-machinery have experienced great progress due to the jointevolution of computer power and the accuracy of numericalmethods

The one-dimensional performance analysis has provedto be an effective and important approach on pump design[1] Analytical calculations of pump characteristics dependon geometrical dimensions of pump and losses models indifferent parts of pump A series of formulae for calculatinglosses exist [2ndash5] but they lack accuracy when applied tocentrifugal pumps

In this work suggested models for calculating severallosses in pump are introduced to examine its validity inevaluating pump performanceThis paper is an effort towardstheoretically obtaining accurate centrifugal pump perfor-mance characteristics Pump characteristics and parametersare presented in dimensionless forms It presents a one-dimensional flow analysis procedure towards obtaining opti-mum centrifugal pump design parameters

2 Theoretical Analysis

Thepumpflow coefficient120595 and pump speed coefficient120593 aredefined as

120595 =1198811199032

radic2119892119867 120593 =

1199062

radic2119892119867 (1)

where119867 is the pump manometric head 1198811199032

is the flow radialvelocity at impeller outlet and 119906

2is the tangential velocity at

outlet of impeller

2 International Journal of Rotating Machinery

QQL

yc

Dey

eQ

D1

Vs

Veye

D2

b1

b2

D1

D2

t

N

d 2

d1L

u2

V2W2

u1

V1

W1

Q+QL

120587D2Z

120573b2

120573b1

tsin120573 b2

120573998400b2

Figure 1 Pump impeller notations

21 Pump Leakage Due to the difference between the outletpressure and the inlet pressure of the impeller a portion ofimpeller outlet flow rate119876

119871 returns to the impeller inlet from

the existing clearances between the impeller and the casingFigure 1 This internal discharge leakage 119876

119871 causes some

losses as the flow rate through the impeller (119876+119876119871) is greater

than the pump useful outlet discharge 119876The volumetric efficiency 120578vol of pump is defined as the

ratio of pump outlet discharge to the impeller discharge

120578vol =119876

119876119894

=119876

119876 + 119876119871

or 119876119871

119876=

1

120578volminus 1 (2)

in which119876119894= 119876 + 119876

119871= 1198811199032

sdot 120587119863211988721205762= 1198811199031

sdot 120587119863111988711205761 (3)

where 1198871is the bladewidth at inlet119863

1is the impeller diameter

at inlet 1205761is the blade thickness coefficient at impeller inlet

1205761= 1minus(119885120587)((119905 sin120573

1198871

)1198631) 1198872is the blade width at outlet

1198632is the impeller diameter at outlet 120576

2is the blade thickness

coefficient at impeller outlet 1205762= 1minus(119885120587)((119905 sin120573

1198872

)1198632) 119905

is the blade thickness 119885 is the number of blades 1205731198872

is theblade angle at impeller outlet and 120573

1198871

is the blade angleat impeller inlet The value of 120576

2is about 095 [6] The

relationship between 1205761and 1205762with constant blade thickness

is given as

1205761= 1 minus

(1 minus 1205762)

(11986311198632) (sin120573

1198871

sin1205731198872

) (4)

The impeller inlet flow velocity coefficient 1198621198811

=

1198811radic2119892119867 is calculated from (3) dividing both sides byradic2119892119867

and assuming that the flow enters the impeller without swirl(1198811= 1198811199031

)

1198621198811

=120595

(11986311198632) (11988711198872) (12057611205762) (5)

The pump discharge coefficient is 119862119876= 119876((11987360)119863

3

2)

and substituting for 119876 from (2) and (3)

119862119876= 120578vol1205762120587

21198872

1198632

sdot120595

120593 (6)

V2

W2

Vr2

1205722

Vu2act

W2e

q

u2

120573b2

V2eqΔWu2

ΔVu2

Vu2

Figure 2 Velocity diagram at impeller outlet

22 Outlet Velocity Diagram A fluid slippage occurs atimpeller exit due to the relative rotation of fluid in a directionopposite to that of impeller A slip factor 120590 defined as 120590 =

1198811199062 act1198811199062 could be estimated later in Section 23From the velocity diagram at the impeller outlet Figure 2

the tangential component of the outlet flow absolute velocityis given as

1198811199062

= 1198811199032

cot1205722= 1199062+ 1198811199032

cot1205731198872

(7)

The ratio of outlet swirl velocity to outlet tangential velocityis

1198811199062

1199062

= 1 +1198811199032

1199062

cot1205731198872

= 1 +120595

120593cot1205731198872

(8)

and thus1198811199062act

1199062

=120590 sdot 1198811199062

1199062

= 120590(1 +120595

120593cot1205731198872

) (9)

The outlet tangential velocity (from (7)) and the pumpspeed coefficient 120593 = 119906

2radic2119892119867 are given as

1199062= 1198811199032

(cot1205722minus cot120573

1198872

) (10)

120593 = 120595 (cot1205722minus cot120573

1198872

) (11)

International Journal of Rotating Machinery 3

d2

d1120596

minus120596

minus120596

120573b2

120587D2Z-tsin

120573b2

120573998400b2

Wav

Figure 3 Flow model for Stodola slip factor

From the outlet velocity diagram (Figure 2) and (7) the outletvelocity and the outlet velocity coefficient defined as 119862

1198812

=

1198812radic2119892119867 are

1198812

2= 1198812

1199032

+ 1198812

1199062

= 1198812

1199032

+ (1199062+ 1198811199032

cot1205731198872

)2

1198622

1198812

= 1205952

+ (120593 + 120595cot1205731198872

)2

(12)

23 Slip Factor

231 The Relative Eddy (Eddy Circulation) A simple expla-nation for the slip effect in an impeller is obtained fromthe idea of a relative eddy Suppose that an irrotational andfrictionless fluid flow is possible which passes through animpeller If the absolute flow enters the impeller without spinthen at outlet the spin of the absolute flow must still be zeroThe impeller itself has an angular velocity 120596 so that relativeto the impeller the fluid has an angular velocity of minus120596 thisis termed the relative eddy At outlet of impeller the relativeflow119882

2 can be regarded as a through flowonwhich a relative

eddy is superimposed The net effect of these two motionsis that the average relative flow emerging from the impellerpassages is at an angle to the vanes and in a direction oppositeto the blade motion

One of the earliest and simplest expressions for the slipfactor was obtained by Stodola cited in [7] Referring toFigure 3 the slip velocity due to relative eddyΔ119882

1199062

= Δ1198811199062

=

1198811199062

minus1198811199062 act is considered to be the product of the relative eddy

and the radius (11988922) of a circle which can be inscribedwithin

the channelThus

Δ1198821199062

= 1205961198892

2 (13)

An approximate expression for 1198892can be written if the

number of blades 119885 is not small

1198892=1205871198632

119885sin12057310158401198872

minus 119905 = 1205762

1205871198632

119885sin1205731198872

(14)

1198892

1198632

= 1205762

120587

119885sin1205731198872

(15)

Since the relative eddy angular velocity = 211990621198632 then

Δ1198821199062

= 1205762

1205871199062

119885sin1205731198872

orΔ1198821199062

1199062

= 1205762

120587

119885sin1205731198872

(16)

The slip factor is given by

120590 =1198811199062act

1198811199062

= 1 minusΔ1198811199062

1198811199062

= 1 minusΔ1198821199062

1199062

1198811199062

1199062

(17)

Substituting for Δ1198821199062

1199062from (16) and for 119881

1199062

1199062from

(8) the formulae proposed by Stodola cited in [7] for thecalculation of slip factor 120590 are obtained which are

120590 = 1 minus1205762(120587119885) sin120573

1198872

1 + (1198811199032

1199062) cot120573

1198872

= 1 minus1205762(120587119885) sin120573

1198872

1 + (120595120593) cot1205731198872

(18)

At 120595 = 0 (120590 = 1205900)

1 minus 1205900= 1205762

120587

119885sin1205731198872

(19)

Therefore

1 minus 120590 =1 minus 1205900

119910 (20)

where

119910 = 1 +120595

120593cot1205731198872

(21)

Wiesner [8] introduced an empirical expression whichextremely well fits the experimental results of slip factor forwide range of practical blade angles and number of blades Itis used in this work and given as

1 minus 1205900=

radicsin1205731198872

11988507for 119863

1

1198632

le 120576limit(22)

where 120576limit is the limiting diameter ratio

120576limit = 119890minus(816 sin120573

1198872119885)

(23)

It is assumed that the water entering the pump impelleris purely in the radial direction Relative to the impeller thefluid has an angular velocity of minus120596 relative eddy and thus therelative flow at blade inlet acquires an additional componentΔ1015840

1198821199061

opposite to rotational direction as seen in Figure 4which is

Δ1015840

1198821199061

= 1205961198891

2 (24)

where

1198891= 1205761

1205871198631

119885sin1205731198871

(25)

1198891

1198632

= 1205761

120587

119885

1198631

1198632

sin1205731198871

(26)

Hence

Δ1015840

1198821199061

= 1205761

120587

1198851199061sin 1205731198871

(27)


Vlowast1 = Vlowast

r1

120573lowastf1

120573b1

u1

Wlowast1b =

Vlowastr1

sin 120573b1

Δ998400Wu1

Wlowast1

(1 minus 1205761120587

Zsin 120573b1)u1

Figure 4 Velocity diagram at impeller inlet without shock

When the pump that operates at a discharge 119876 differsfrom that at designed condition 119876lowast the relative flow velocityat blade inlet tends to acquire an additional component incounter of the rotational directionΔ10158401015840119882

1199061

So the flow entersthe blade passage tangent to the blade surface and a shockeddy or a shock circulation exists prior to the blade leadingedge inside pump eye As could be noticed from Figure 5the relative velocity additional rotational speed at blade inletequals

Δ10158401015840

1198821199061

= 1199061(1 minus 120576

1

120587

119885sin 1205731198871

) minus 1198811199031

cot (180 minus 1205731198871

) (28)

24 Euler Equation of Turbomachines In the case that thefluid entering the pump impeller is purely in the radialdirection without swirl the pump Euler head is given as [6 7]

119867infin=11990621198811199062

119892 (29)

Due to the fluid slippage at impeller exit the actual head givento fluid by the impeller119867

0 is calculated from [6 7]

1198670=1199062sdot 1198811199062 act

119892=1199062sdot 1205901198811199062

119892= 120590 sdot 119867

infin (30)

And so the slippage head loss is

ℎ119897slp

= 119867infinminus 1198670= (1 minus 120590)119867

infin (31)

25 Pump Volute The flow that discharges from the impellerrequires careful handling in order to preserve the gains inenergy imparted to the fluid This requires the conversion ofvelocity head to pressure head by means of a diffuser andthis inevitably implies hydraulic losses The application ofmass conservation to a volute element [9] reveals that thedischarge flow from impeller is matched to the flow in thevolute if 119889119860

119881119889120579 = 119881

1199033

1198811199063

11990331198873 This requires a circumferen-

tially uniform rate of increase of the volute area 119860119881over the

entire development of the spiral (120579) Consequently for a givenimpeller there exist a specific volute angle 120572

119881and a specific

volute throat inlet area 119860119881th

for the volute geometryThe volute angle120572

119881 Figure 6 is chosen tomatch the angle

of flow entering the volute 1205723eq at a certain pump operating

Vlowast1 = Vlowast

r1

V1 = Vr1

Wlowast1b

Δ998400Wu1

W1

120573f1

120573b1

u1

Δ998400Wu1

Wlowast1

W1b =Vr1sin 120573b1

Δ998400998400Wu1

(1 minus 1205761120587

Zsin 120573b1)u1

Figure 5 Velocity diagram at blade inlet with shock eddy

point With a uniform rate of increase of the volute area thevolute angle is

tan120572119881=

119860119881th

12058711986331198873

=119860119881th

12058711986321198872(11986331198632) (11988731198872) (32)

The volute outlet flow velocity 1198814= 119876119860

119881thand the volute

outlet flow velocity coefficient 1198621198814

= 1198814radic2119892119867 are

1198814=120578vol sdot 119881119903

2

120587119863211988721205762

12058711986331198873sdot tan 120572

119881

=120578vol1205762tan 120572119881

1198632

1198633

1198872

1198873

sdot 1198811199032

1198621198814

=1205762120578vol

(11986331198632) (11988731198872) tan 120572

119881

sdot 120595

(33)

The throat is assumed to have an expanding angle 120572th andhence the throat diameter equals

119863th = 1205871198633tan 120572119881+ 119871 th tan120572th (34)

With the assumption that the throat height 119871 th = 1198633 thus

119863th1198633

= 120587 tan120572119881+ tan120572th (35)

The throat outlet flow velocity1198815= 119876119860 th and using (2) and

(3) then

1198815= 41205762120578vol

1198632

2

1198632th

1198872

1198632

sdot 1198811199032

(36)

The throat outlet flow velocity coefficient 1198621198815

(= 1198815radic2119892119867)

equals

1198621198815

=41205762120578vol (11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (37)

It is assumed that the throat diameter 119863th equals the eyediameter119863eye that is119863eye119863th = 1 which equals the suctionpipe diameter 119863

119904 Thus the throat outlet flow velocity 119881

5

equals the flow velocity at suction pipe 119881119904 Therefore

1198815= 119881119904= 120578vol sdot 119881eye (38)

And the eye velocity coefficient is

119862119881eye

=119881eye

radic2119892119867=1198621198815

120578vol=

41205762(11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (39)


120579

d120579 V3r

V4V3

V3

V3

V4 V5

V2

V3u

V3u

V5

Throat

Volute

Impeller

Lth

L th

y th

D3

D2

D1

xc

N

120572V

120572V

Vr2

V2eq

Vu2act

Case 1205723eq lt 120572V

Case 1205723eq gt 120572VV3eq

V3eq

120572V

120572V

V3p

V3p

V3p

V4 = V3pu

1205723eq lt 120572V1205723eq

V3d

V3d

V3d998400

Vr3 = 1205762120578volb2b3

D2

D3

middot Vr2

xc

120572V

b3

ThroatVolute

Dth

yth = 120587D3tan120572V

V3d + V3p equiv

V3eq equiv V3d + V3p

D3sin120572VZ

120587D2

120587D3

120587D3ta

n120572V

2lowast120587

Dth

120579th120572th

Vu2

ΔVu2

ΔVu3

Figure 6 Pump and volute notations

The eye diameter relative to the impeller inlet diameter istherefore

119863eye

1198631

=119863eye

119863th

119863th1198633

1198633

1198632

1198632

1198631

=(11986331198632) (120587 tan 120572

119881+ tan 120572th)

(11986311198632)

(40)

The ratio (119863eye1198631)must not exceed 1 and thus anupper limitis imposed on the volute angle

tan120572119881lt

(11986311198632)

120587 (11986331198632)minustan120572th120587

(41)

The throat has a cone angle 120579th where

tan(120579th2) =

(119863th minus 1198873) 2

119871 th (42)

Substituting from (35)

tan(120579th2) =

1

2[120587 tan120572

119881+ tan120572th minus

(11988731198872) (11988721198632)

(11986331198632)

]

(43)

26 Volute Loss Model Prediction models that account forthe main features of the swirling flow in volutes reviewed in[4] do not account for the circulatory flow initiated in voluteat off-design discharge operation of pump In this work amodel is proposed to account the volute head loss at off-design pump operation

The flow enters the volute with a through-velocity 1198813

at an angle 1205723(which may differ from that of the volute

angle 120572119881) on which an eddy of a tangential velocity Δ119881

1199063

=

Δ1198811199062

is superimposed opposite to impeller motionThis inlet

volute velocity 1198813is decomposed into a velocity parallel to

the direction of volute1198813119901 and another one in the tangential

direction of impeller11988110158403119889(Figure 6) The velocity component

1198811015840

3119889is the motive of a second circulatory motion given to

the volute flow in direction of impeller motion Thus the netcirculation velocity of flow in volute is 119881

3119889= 1198811015840

3119889minus Δ1198811199063

indirection of impeller motion The component of the velocity1198813119901

in the tangential direction denoted as 1198813119901119906

equals thevolute outlet velocity 119881

4

The volute loss model counts friction loss associated withthe volute throw flow velocity diffusion friction loss due tocirculation associated with volute flow loss due to vanishingof radial flow at volute outlet and loss inside pump volutethroat Consequently the volute head loss and the volute headloss relative to the pump head are written respectively as

ℎ119897119881

= 119862119891119881

1198812

3119901

2119892+ 119862119889119881

1198812

3119889

2119892+1198812

3119903

2119892+ ℎ119897th (44)

ℎ119897119881

119867= 119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

(45)

where 119862119891119881

is the volute friction loss coefficient

119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ119881

1198632

) (46)

where 119891119881is the volute friction coefficient 119863

ℎ119881

is the volutehydraulic diameter and 119871

119881is the average volute length

119871119881=1

2

1205871198633

cos120572119881

(47)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)


The average volute hydraulic diameter relative to impellerdiameter is

1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)

The volute friction coefficient119891119881which corresponds to a pipe

flow is function of the volute Reynolds number Re119881and the

roughness 120581 [10]

119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)

The volute Reynolds number is calculated as

Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)

In the first term of (44) and (45)1198813119901

is the volute throw flowvelocity (Figure 6) and given as

1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)

In the second term of (44) and (45) 119862119889119881

is the volutediffusion loss coefficient which could be assumed to have thevalue 119862

119889119881

= 08 and 1198813119889

is the volute circulatory velocitycomponent

1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)

The third term of (45) is

1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

In the fourth term of (44) and (45) the volute throat headloss ℎ119897thcould be calculated as

ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th

is the volute throat friction loss coefficientassumed to be [10]

119862119891th

= 05 + 26 lowast sin(120579th2) (56)

where 120579th is the throat cone angle

27 Pump Eye Head Loss ℎ119897eye Thehead loss in pump eye ℎ

119897eye

[2] and the eye head loss relative to the pump head are

ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)

where119881eye is the flow velocity in pump eye and119862eye is the eyeloss coefficient defined by (39)

28 PumpManometric Head119867 Thepumpmanometric headis the difference in static pressure heads between pump outletand pump eye

119867 =1199015minus 119901eye

120574 (58)

With the assumption that the throat diameter 119863th equals theeye diameter 119863eye the flow velocity is the same at pumpsuction pipe and pumpdelivery pipe (119881

119904= 1198815) and neglecting

the difference in elevation head across the pump then

119867 = 1198670minus ℎ119897119881

minus ℎ119897eye (59)

or

119867 = 119867infinminus ℎ119897slpminus ℎ119897119881


Define a parameter 119909 as

119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)

Using (45) as well as (57) and (61) then

1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)

The sum of volute and eye head losses from (59) is writtenas

ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)

Using (9) (29) and (59) the pump manometric head119867 is

119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)

Coefficients There are two additional groups of coefficientsnamely the pumphead coefficients and head loss coefficients


The pump head coefficients are the pump manometrichead coefficient 119862

119867 Euler head coefficient 119862

119867infin

and thehead coefficient at impeller outlet 119862

1198670

They are defined andgiven respectively as

119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)

According to (21) and (65b) 119910 = 119862119867infin

The head loss coefficients include the slippage head loss

coefficient 119862ℎ119897slp the volute-eye head loss coefficient 119862

ℎ119897119881+eye

the eye head loss coefficient 119862

ℎ119897eye and the volute head loss

coefficient 119862ℎ119897119881

They are given respectively as

119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)

The relationship between the pump speed coefficient 120593 andthe pump flow coefficient 120595 is derived as follows

Using (11) (64) becomes

119867 = 1199091205901199062

2

119892

cot1205722

cot1205722minus cot120573

1198872

(66)

Dividing both sides by 119867 noting that 11990622(2119892119867) = 120593

2 andusing (11) (64) becomes a quadric equation for cot120572

2

cot2 1205722minus cot120573

1198872

sdot cot1205722minus

1

21199091205901205952= 0 (67)

which has a solution (since cot1205722should be greater than

cot1205731198872

whence only the +ve sign is considered)

cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)

Multiplying (68) by 120595 and then subtracting 120595 cot1205731198872

fromboth sides and using (11) yield the following relation for 120593

120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)

29 Pump Shaft Power 119875sh and Pump Shaft Head 119867sh Thetotal shaft power required to drive the impeller is

119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0

= 120574(119876 + 119876119871)1198670is the impeller power given to

water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891

is the power lost in friction insideimpeller 1198751015840

119897cirin= 120574(119876+119876

119871)ℎ119897cirin

is the power needed for givencirculation to flow at impeller inlet and 119875

119897119863

is the power lostin friction on outside surface of impeller disks

The total shaft power and pump shaft head can besimplified respectively to

119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891

is the impeller skin friction head loss ℎ119897cirin

isthe inlet shock circulation head loss ℎ

119897volis the volumetric

(leakage) head loss and ℎ119897119863

is the disk friction head lossThe shaft head 119867sh = 119875sh (120574119876) and the shaft head

coefficient 119862119867sh

= 119867sh(1199062

2119892) are given respectively as

119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062

2119892) is the impeller skin friction head loss

coefficient119862ℎ119897cirin

= ℎ119897cirin

(1199062

2119892) is the inlet shock circulation

head loss coefficient 119862ℎ119897vol

= ℎ119897vol(1199062

2119892) is the volumetric

head loss coefficient and119862ℎ119897119863

= ℎ119897119863

(1199062

2119892) is the disk friction

head loss coefficient

210 Pump Efficiency (Manometric Efficiency) 120578 The pumpmanometric efficiency is the ratio of gained water power (119875

119908)

to the pump shaft power (119875sh) supplied to pump impellerAccording to its definition it takes the following forms

120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=

119867infinminus ℎ119897slpminus ℎ119897119881

minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin

minus 119862ℎ119897slpminus 119862ℎ119897119881+eye

1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)


211 Pump Shaft Power Coefficient 119862119875119904ℎ

The pump shaftpower coefficient and the pump water power coefficient aregiven respectively as

119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)

212 Impeller Skin Friction Power Head Loss ℎ119897119891

The impellerhydraulic friction head loss ℎ

119897119891

is estimated by the theory offlow through pipes and is given by [11]

ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894

is the dissipation coefficient 119871119887is the blade

length119863hyd is the hydraulic diameter and119882av is the averagerelative velocity Therefore the impeller skin friction headloss coefficient could be calculated from

119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)

The hydraulic diameter and the average relative velocity aregiven respectively as Gulich [11]

119863hyd =4 lowast AreaPerimeter

=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)

Substituting for 1198891 (26) 119889

2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)

The blade length is 119871119887= ((12)(119863

2minus 1198631)) sin120573bm and thus

119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)

The impeller dissipation coefficient is given as Gulich [11]

4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)

The impeller friction coefficient 119891119894is function of the average

impeller Reynolds number Re and the roughness 120581 [10]

119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)

Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)

where Re2is defined by (53)

213 Disk Friction Head Loss ℎ119897119863

Thedisk friction power loss119875119897119863

is the power loss in the fluid between external surfacesof the impeller disks and internal walls of the pump casingFigure 7 The 119875

119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)

where 120591 is the shear stress in circumferential direction and119891119863

is the disk skin friction coefficient and it is assumed constantalong the disk surface

Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)

Therefore the final expression after the integration for thedisk friction power loss and consequently the impeller diskfriction head loss are

119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)

The impeller disk friction head loss coefficient 119862ℎ119897119863

is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)

where119870119863is the impeller disk loss coefficient

119870119863=

1

40

119891119863

1205762(11988721198632) (92a)

It is derived by substituting for 119876 from (2) and (3) and usingthe definition of 120595 which yields 119876 = 120578vol1205762 sdot 120595radic2119892119867 sdot 120587119863

21198872


Correlations for 119891119863

were obtained by Kruyt cited in[12] Four different regimes were identified Figure 8 RegimeI (laminar flow boundary layers have merged) Regime II(laminar flow with two separate boundary layers) Regime III(turbulent flow boundary layers have merged) and RegimeIV (turbulent flow with two separate boundary layers) Theseregimes are characterized by the Reynolds number Re

2=

1199062sdot (11986322)] and a nondimensional gap parameter 119866 =

1199100(11986322) where 119910

0is the axial gap between impeller disk

and casing (Figure 7) The equations of curves 1 to 5 inFigure 8 are 119866 = 162Reminus511

2 119866 = 188Reminus910

2 119866 =

057 lowast 10minus6Re15162

Re = 158 lowast 105 and 119866 = 0402Reminus316

2

respectively The disk friction coefficient for each regime isgiven as

Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04

120587119866minus16Reminus14

2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)

214 Volumetric Head Loss ℎ119897Vol The volumetric (leakage)

head loss and the volumetric head loss coefficient are givenas follows after using (2)

ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1

120578volminus 1) sdot (119867

0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)

The leakage flow rate 119876119871can be estimated using orifice

formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)

where 119886119888is the clearance area of wearing ring (= 120587119863eye119910119888) 119910119888

is the clearance of wearing ring Figure 7 119862dL is the leakagedischarge coefficient (asymp06) and Δ119867

119888is the pressure head

drop across the clearance

Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)

where119901119888is the pressure before clearance and119901

119904is the pressure

after clearance at suction side of the impeller (Figure 7)The pressure head difference between volute and pump

suction side equals1199013minus 119901119904

120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps

Figure 7 Disk and wearing ring clearances

0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2

Figure 8 Disk friction flow regimes [12]

The pressure distribution along the radial direction ofimpeller shroud is parabolic [13] and thus the pressure headdifference between impeller outlet and before clearance isgiven by

1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)

Thus the leakage flow rate and the leakage flow rate coeffi-cient 119862

119876119871

are

119876119871= 119862dL sdot 120587119863eye119910119888 sdot 1199062radic2

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862

119876 the pump vol-

umetric efficiency can be deduced after some mathematicalmanipulations

120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)

215 Inlet Shock CirculationHead Loss ℎ119897cirin

At impeller inleta shock circulation exists inside pump eye when the flowdischarge differs from that of designed one As discussedbefore in Section 23 the tangential velocity responsible forthis shock eddy is Δ10158401015840119882

1199061

which could be estimated from(28)

Therefore the inlet shock circulation head loss equals

ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)

Thus the inlet shock circulation head loss coefficient is

119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2

minuscot (180 minus 120573

1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)

216 Required Net Positive Suction Head 119873119875119878119867R The pumpnet positive suction head NPSH is defined as the differencebetween the fluid inlet stagnation pressure head and vapourpressure head [13]

NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)

where 1199011 1198811are the absolute pressure and absolute velocity

of fluid at impeller inlet and119901V is the absolute vapour pressureof fluid at the corresponding fluid temperature

In the vicinity of the leading edge of the impeller bladesthe fluid has to accelerate in order to follow the rotatingmovement of the blades This acceleration leads to a drop ofthe static pressure which results in a local minimumpressureat blade inlet

119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)

Therefore the pump required net positive suction head is

NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)

The pump NPSH119877coefficient is defined as

119862NPSH119877

=NPSH

119877

11990622119892

(105)

Referring to Figure 51198821119887= (1198811199031

sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)

The term ((119901min minus119901V)120574)(1199062

2119892) is a design parameter for the

pump and could be assumed to be 002From (3) 119881

1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore

the NPSH119877coefficient is

119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)

217 Pump Specific Speed 119899119904 The pump specific speed is

defined with suitable units as

119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for

119876 = 1000120578vol1205762 sdot 12058711986321198872 sdot 120595radic2119892119867 (lits)

119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield

119899119904=60 lowast radic1000 lowast (2119892)

34

lowast radic1205762

radic120587radic(

1198872

1198632

) lowast radic120578vol sdot 120593radic120595

(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632

) sdot radic120578vol sdot 120593radic120595 (110)

3 Calculation Procedure and Results

An iterative procedure using Microsoft Office EXCEL pro-gram is performed to calculate the pump design parametersdescribed by the preceding derived equations The inputparameters are the blades number 119885 impeller inlet diam-eter to outlet diameter ratio (119863

11198632) volute diameter to

impeller outlet diameter ratio (11986331198632) ratio of blade width

at impeller outlet to impeller outlet diameter (11988721198632) blade

width at impeller inlet to blade width at impeller outlet ratio(11988711198872) volute width to blade width at impeller outlet ratio

(11988731198872) blade angle at impeller outlet 120573

1198872

blade angle at


Table 1 Pump input-parameters

Input parameter Value

Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10

impeller inlet 1205731198871

volute angle 120572119881 throat angle 120572th ratio of

wearing ring clearanceimpeller eye diameter (119910119888119863eye) leak-

age discharge coefficient119862dL volute diffusion loss coefficient119862119889119881

eye loss coefficient 119862eye blade thickness coefficient atimpeller outlet 120576

2 with constant-thickness blades ratio of

axial gap between impeller disk and casing to impeller outletradius 119910

0(11986322) Reynolds number at impeller outlet Re

2

and relative roughness (1205811198632) approximated values are used

by putting 1198632asymp 0209m 119873 asymp 1450 rpm 120581 asymp 00001m

] asymp 10minus6m2119904 Re

2= 116 lowast 10

6 and 1205811198632= 0000478

For a certain flow coefficient 120595 all pump performanceparameters and coefficients are calculated after iterationsCase Study For reason of comparison between results ofpresent analytical study and experimental pump perfor-mance obtained by Baun and Flack [14] calculations of cen-trifugal pump performance are performed with the followinginput parameters given in Table 1

Other pump constant-parameters are calculated accord-ing to the present procedure as shown in Table 2

The sequence of calculations by using the procedurederived equations of pump variable parameters is listed inTable 3

00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593

Experimental head coeff [14]Present procedure head coeff

Experimental [14]Present procedure

120578

120578

Efficiency

CHCH

Figure 9 Comparison between present procedure results andexperimental results by Baun and Flack [14]

The manometric head coefficient and pump efficiencyof both present procedure and experimental measurementsby Baun and Flack [14] are plotted in Figure 9 versus theratio (120595120593) (defined as flow coefficient in [14]) It shows agood similarity between the present procedure results and theexperimental ones Only in range of low ratios of 120595120593 thecalculated efficiency is relatively highThis is attributed to theuncounted mechanical power loss in the prediction of pumpshaft power and hence in efficiency

The pump performance characteristics are presentedby the calculated pump head coefficient efficiency powercoefficients and required NPSH as shown in Figure 10 Fromthe figure the pump best efficiency point (BEP) occurs atdischarge coefficient 119862

119876asymp 00675 giving an efficiency 120578 asymp

753 a manometric head coefficient 119862119867

asymp 0468 a shaftpower coefficient 119862

119875shasymp 0414 a water power coefficient

119862119875119908

asymp 0312 and a pump required net positive suction headcoefficient 119862NPSH

119877

asymp 0129Themaximumshaftpower coefficient119862

119875shasymp 0428 occurs

at 119862119876

asymp 0085 and the maximum water power coefficient119862119875119908

asymp 03165 occurs at 119862119876asymp 0075

The variations of pump flow coefficient 120595 and pumpspeed coefficient 120593 with the discharge coefficient are shownin Figures 11 and 12 respectively Figure 11 shows also thevariation of the ratio 120595120593 with the discharge coefficient 119862

119876

The linear relationship between 119862119876and 120595120593 is evident as

given by (6)The dotted lines in the figures correspond to thevalue of 119862

119876that gives the pump best efficiency At pump best

efficiency point (119862119876asymp 00675) 120595 asymp 0063 (Figure 11) and

120593 asymp 1034 (Figure 12) Also it is indicated from Figure 12 that


Table 2 Pump constant-parameters

Constant parameter equation Equation no

Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632

le 120598limit = 119890minus816 sin1205731198872 119885 (22)

1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)

the speed coefficient takes a minimum value of 120593 asymp 0932

at 119862119876asymp 0024 which corresponds to the maximum 119862

119867in

Figure 10Figure 13 shows the theoretical dimensionless head-

discharge curve (Euler head) which is a straight line and119862119867infin

decreases with the increase of 119862119876for the proposed outlet

blade angle 1205731198872

= 164∘

gt 90∘ The actual impeller outlet

dimensionless head-discharge curve 1198621198670

is obtained takinginto consideration the slip or eddy circulation of flow insideimpeller which is nearly constant Actually the effect of slipis not a loss but a discrepancy not accounted by the basicassumptions The second loss shown is the volute-and-eyeloss which is minimum at 119862

119876asymp 0051 Reduced or increased

119862119876 from the value 0051 increases the volute-and-eye lossFigure 14 gives the variation of different head loss coef-

ficients with 119862119876

in addition to the 119862119867

and 1198621198670

curvesGenerally these head loss coefficients decrease with theincrease of 119862

119876 At very low discharge coefficients both the

disk head loss coefficient and volumetric head loss coefficientbecome very big The inlet circulation head loss is big at low

119862119876and decreases until it reaches zero at 119862

119876asymp 006 and

then its direction is inverse (becomes negative) whichmeansthat the inlet circulation adds power to impeller and does notbleed power from impeller

The variation of the flow velocity coefficients and thepump volumetric efficiency 120578vol with the pump dischargecoefficient is shown in Figure 15The coefficients119862

1198811

119862119881Δ1198811199062

and 119862

1198814

have the trends of increasing with the increase of119862119876 whereas the two coefficients 119862

1198813119889

and 11986211988131198891015840have the

trends of decreasing with the increase of 119862119876 These two later

coefficients have maximum values at 119862119876= 0 At 119862

119876asymp 0085

the coefficient 1198621198813119889

becomes zero which indicates that at thispoint of operation the flow inside volute is without circulationand 119881

4= 1198811199062 act At increased 119862119876 the 1198621198813119889 becomes negative

which means that the flow circulation inside the volutechanges its direction of rotation opposite to impeller motionwhereas 119881

4gt 1198811199062 act (Figure 6)

The pump specific speed is presented in Figure 16 whichshows that 119899

119904asymp 870 at pump best efficiency point


Table 3 Pump variable-parameters

Order Variable parameter equation Eq no

120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593

120595lowast radic119862

1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632


Figure 17 shows the procedure results of centrifugal pumpperformance when the pump handles fluids with differentkinematic viscosities The head and efficiency for the pumpwhenhandling oils are lower than thosewhen handlingwaterBut the required power when handling oil is higher than thatwhen handling water

The pump head decreases slightly due to the increasein volute friction loss as fluid viscosity increases while theincrease in pump power is high due to the increase in bothhydraulic friction inside impeller and disk friction powerlosses Hence the drop in pump efficiency is very high asthe fluid viscosity increases This result is in accordance withexperimental results by Shojaee Fard and Boyaghchi [15]

4 ConclusionsA one-dimensional flow procedure for analytical study ofcentrifugal pump performance is accomplished applying the

principle theories of turbomachinesTheprocedure is capableof providing the performance characteristic of centrifugalpump in a dimensionless information form The predictedcoefficients and performance curves obtained have beenfound to be in a reasonable agreement with experimentalmeasurements The present procedure is also capable of pre-dicting the effects of handling viscous fluids on the centrifugalpump performance The input form for this procedure ofpump flow analysis makes it an effective tool analysis and canbe used in the pump conceptual design

Notations

119860 Impeller net area m2119886119888 Clearance area of wearing ring m2

119860119881 Volute area m2

119860 th Throat outlet area 1205871198632th4 m2

119860119881th Volute throat inlet area m2


000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh

Figure 10 Characteristics of centrifugal pump obtained by presentstudy

000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ

Figure 11 Variation of flow coefficient and ratio of flow coefficientto speed coefficient with discharge coefficient

119887 Impeller width m119862119889119894

Impeller dissipation coefficient119862dL Leakage discharge coefficient119862119889119881

Volute diffusion loss coefficient119862eye Eye loss coefficient119862119891th Volute throat friction loss coefficient

119862119891119881

Volute friction loss coefficient119862119867 Manometric head coefficient119867(119906

2

2119892)

1198621198670

Head coefficient at impeller outlet1198670(1199062

2119892)

119862119867infin

Euler head coefficient119867infin(1199062

2119892)

119862119867sh

Shaft head coefficient119867sh(1199062

2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593

Figure 12 Speed coefficient variation with pump discharge coeffi-cient

00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin

Figure 13 Variation of head coefficients and head loss coefficientswith discharge coefficient

119862ℎ119897cirin

Inlet shock circulation head loss coefficientℎ119897cirin

(1199062

2119892)

119862ℎ119897119891

Impeller skin friction head loss coefficient119862ℎ119897eye Eye head loss coefficient ℎ

119897eye(1199062

2119892)

119862ℎ119897119863

Disk friction head loss coefficient ℎ119897119863

(1199062

2119892)

119862ℎ119897slp Slip head loss coefficient ℎ

119897slp(1199062

2119892)

119862ℎ119897119881

Volute head loss coefficient ℎ119897119881

(1199062

2119892)

119862ℎ119897119881+eye

Volute-eye head loss coefficient(ℎ119897119881

+ ℎ119897eye)(1199062

2119892)

119862ℎ119897vol Volumetric head loss coefficient ℎ

119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

Impeller hydraulic friction loss

BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh

Figure 14 Head loss coefficients affecting pump power at differentdischarge coefficients

00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2

Figure 15 Values of volumetric efficiency and different pumpvelocity coefficients

119862NPSH119877

Coefficient of pump NPSH119877 NPSH

119877(1199062

2119892)

119862119875sh Shaft power coefficient 119875sh120588(11987360)

3

1198635

2

119862119875119908

Water power coefficient 119875119908120588(11987360)

3

1198635

2

119862119876 Pump discharge coefficient 119876(11987360)119863

3

2

119862119876119871

Leakage flow rate coefficient119862119881 Velocity coefficient 119881radic2119892119867

1198632 Impeller outer diameter m

0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns

Figure 16 Pump specific speed values at different discharge coeffi-cient

00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ

minus6 m2s = 100 lowast 10minus6 m2s = 200 lowast 10

minus6m2s = 1 lowast 10

CPsh

CPsh

Figure 17 Effect of pumped fluid viscosity on the performance ofcentrifugal pump

119863eye Pump eye diameter m119863ℎ119881

Average volute hydraulic diameter m119863th Throat diameter m119891 Hydraulic friction coefficient119891119863 Disk friction coefficient

119892 Acceleration of gravity ms2119866 Impeller disk gap parameter 119910

0(11986322)

ℎ119897cirin

Inlet shock circulation head loss mℎ119897119863

Disk friction head loss m


ℎ119897119891

Impeller skin friction head loss mℎ119897slp Slippage head loss m

ℎ119897119881

Volute head loss mℎ119897vol Volumetric head loss m

119867 Manometric head of pump m1198670 Water head at impeller outlet m

119867sh Pump shaft head m119867infin Euler pump head m

119870119863 Impeller disk loss coefficient

119871119887 Blade length m

119871 th Throat length m119871119881 Volute length m

119873 Pump rotational speed rpmNPSH Pump net positive suction head mNPSH

119877 Pump required net positive suction head m

119899119904 Pump specific speed

119901 Pressure Nm2119901119888 Pressure before wearing ring clearance Nm2

119901119904 Pressure at pump suction side Nm2

119875sh Pump shaft power W119875119908 Pumped water power W

119875sh0

Impeller Euler power W119875119897cirin

Inlet shock circulation power loss W119875119897119891

Impeller friction power loss W119875119897119863

Disk friction power loss W119875119897vol Volumetric power loss W

119876 Pump discharge m3s119876119894 Impeller discharge m3s

119876119871 Pump internal discharge leakage m3s

119903 Radius m119905 Blade thickness m119906 Tangential flow velocity ms119881 Absolute flow velocity ms1198813119901 Throughflow component of volute velocity

ms1198813119889 Circulation velocity of flow in volute ms

119882 Relative velocity of flow ms119909 Ratio119867119867

0

119910 Group parameter (119910 = 119862119867infin

)1199100 Axial gap between impeller disk and casing

m119910119888 Clearance of wearing ring m

119885 Number of blades

Greek Letters

120572 Theoretical absolute velocity angle degrees120572119881 Volute angle degrees

120573119887 Blade angle degrees

120579th Volute throat angle degrees120598 Blade thickness coefficient120593 Pump speed coefficient 119906

2radic2119892119867

120578 Manometric efficiency of pump120578vol Volumetric efficiency] Fluid kinematic viscosity m2s120588 Density of fluid kgm3120590 Slip factor 119881

1199062act1198811199062

120595 Pump flow coefficient 1198811199032

radic2119892119867

120581 Roughness height m

Subscripts

1 Impeller inlet2 Impeller outlet3 Volute inlet4 Volute outlet (throat inlet)5 Pump outlet (throat outlet)av Averageact Actualeye Pump eye119888 Clearancehl Head loss119894 Impeller119897 Loss119903 Radial direction119904 Suctionsh Shaftth Volute throat119906 Tangential directionvol Volumetric

References

[1] M Asuaje F Bakir S Kouidri F Kenyery and R Rey ldquoNumer-ical modelization of the flow in centrifugal pump voluteinfluence in velocity and pressure fieldsrdquo International Journalof Rotating Machinery vol 2005 no 3 pp 244ndash255 2005

[2] H W Oh and M K Chung ldquoOptimum values of design vari-ables versus specific speed for centrifugal pumpsrdquo Proceedingsof the Institution of Mechanical Engineers A vol 213 no 3 pp219ndash226 1999

[3] H W Oh and M K Chung ldquoDesign and performance analysisof centrifugal pumprdquo World Academy of Science Engineeringand Technology vol 36 pp 422ndash429 2008

[4] R A van den Braembussche ldquoFlow and loss mechanisms involutes of centrifugal pumpsrdquo in Design and Analysis of HighSpeed Pumps pp 121ndash12-26 Educational Notes RTO-EN-AVT-143 Paper 12 Neuilly-sur-Seine France 2006

[5] J Tuzson Centrifugal Pump Design John Wiley amp Sons NewYork NY USA 2000

[6] E Logan Jr and R Roy Handbook of Turbomachinery MarcelDekker New York NY USA 2003

[7] S L Dixon Fluid Mechanics andThermodynamics of Turboma-chinery Elsevier Butterworth-HeinemannNewYorkNYUSA5th edition 2005

[8] F J Wiesner ldquoA review of slip factors for centrifugal impellersrdquoJournal for Engineering for Power vol 89 no 4 pp 558ndash5721967

[9] C E Brennen Hydrodynamics of Pump Oxford UniversityPress Oxford UK 1994

[10] B K Hodge Analysis and Design of Energy Systems Prentice-Hall New York NY USA 2nd edition 1990

[11] J F GulichCentrifugal Pumps Springer Berlin Germany 2010[12] N P Kruyt Lecture Notes Fluid Mechanics of Turbomachines

II Turbomachinery Laboratory University of Twente Amster-dam The Netherlands 2009


[13] K M Srinivasan Rotodynamic Pumps (Centrifugal and Axial)New Age International New Delhi India 2008

[14] DO Baun andRD Flack ldquoEffects of volute design andnumberof impeller blades on lateral impeller forces and hydraulicperformancerdquo International Journal of Rotating Machinery vol9 no 2 pp 145ndash152 2003

[15] M H Shojaee Fard and F A Boyaghchi ldquoStudies on the influ-ence of various blade outlet angles in a centrifugal pump whenhandling viscous fluidsrdquo American Journal of Applied Sciencesvol 4 no 9 pp 718ndash724 2007

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DistributedSensor Networks



QQL

yc

Dey

eQ

D1

Vs

Veye

D2

b1

b2

D1

D2

t

N

d 2

d1L

u2

V2W2

u1

V1

W1

Q+QL

120587D2Z

120573b2

120573b1

tsin120573 b2

120573998400b2

Figure 1 Pump impeller notations

21 Pump Leakage Due to the difference between the outletpressure and the inlet pressure of the impeller a portion ofimpeller outlet flow rate119876

119871 returns to the impeller inlet from

the existing clearances between the impeller and the casingFigure 1 This internal discharge leakage 119876

119871 causes some

losses as the flow rate through the impeller (119876+119876119871) is greater

than the pump useful outlet discharge 119876The volumetric efficiency 120578vol of pump is defined as the

ratio of pump outlet discharge to the impeller discharge

120578vol =119876

119876119894

=119876

119876 + 119876119871

or 119876119871

119876=

1

120578volminus 1 (2)

in which119876119894= 119876 + 119876

119871= 1198811199032

sdot 120587119863211988721205762= 1198811199031

sdot 120587119863111988711205761 (3)

where 1198871is the bladewidth at inlet119863

1is the impeller diameter

at inlet 1205761is the blade thickness coefficient at impeller inlet

1205761= 1minus(119885120587)((119905 sin120573

1198871

)1198631) 1198872is the blade width at outlet

1198632is the impeller diameter at outlet 120576

2is the blade thickness

coefficient at impeller outlet 1205762= 1minus(119885120587)((119905 sin120573

1198872

)1198632) 119905

is the blade thickness 119885 is the number of blades 1205731198872

is theblade angle at impeller outlet and 120573

1198871

is the blade angleat impeller inlet The value of 120576

2is about 095 [6] The

relationship between 1205761and 1205762with constant blade thickness

is given as

1205761= 1 minus

(1 minus 1205762)

(11986311198632) (sin120573

1198871

sin1205731198872

) (4)

The impeller inlet flow velocity coefficient 1198621198811

=

1198811radic2119892119867 is calculated from (3) dividing both sides byradic2119892119867

and assuming that the flow enters the impeller without swirl(1198811= 1198811199031

)

1198621198811

=120595

(11986311198632) (11988711198872) (12057611205762) (5)

The pump discharge coefficient is 119862119876= 119876((11987360)119863

3

2)

and substituting for 119876 from (2) and (3)

119862119876= 120578vol1205762120587

21198872

1198632

sdot120595

120593 (6)

V2

W2

Vr2

1205722

Vu2act

W2e

q

u2

120573b2

V2eqΔWu2

ΔVu2

Vu2

Figure 2 Velocity diagram at impeller outlet

22 Outlet Velocity Diagram A fluid slippage occurs atimpeller exit due to the relative rotation of fluid in a directionopposite to that of impeller A slip factor 120590 defined as 120590 =

1198811199062 act1198811199062 could be estimated later in Section 23From the velocity diagram at the impeller outlet Figure 2

the tangential component of the outlet flow absolute velocityis given as

1198811199062

= 1198811199032

cot1205722= 1199062+ 1198811199032

cot1205731198872

(7)

The ratio of outlet swirl velocity to outlet tangential velocityis

1198811199062

1199062

= 1 +1198811199032

1199062

cot1205731198872

= 1 +120595

120593cot1205731198872

(8)

and thus1198811199062act

1199062

=120590 sdot 1198811199062

1199062

= 120590(1 +120595

120593cot1205731198872

) (9)

The outlet tangential velocity (from (7)) and the pumpspeed coefficient 120593 = 119906

2radic2119892119867 are given as

1199062= 1198811199032

(cot1205722minus cot120573

1198872

) (10)

120593 = 120595 (cot1205722minus cot120573

1198872

) (11)


d2

d1120596

minus120596

minus120596

120573b2

120587D2Z-tsin

120573b2

120573998400b2

Wav



1198812

=

1198812radic2119892119867 are

1198812

2= 1198812

1199032

+ 1198812

1199062

= 1198812

1199032

+ (1199062+ 1198811199032

cot1205731198872

)2

1198622

1198812

= 1205952

+ (120593 + 120595cot1205731198872

)2

(12)

23 Slip Factor





1199062

= Δ1198811199062

=

1198811199062



the channelThus

Δ1198821199062

= 1205961198892

2 (13)



1198892=1205871198632

119885sin12057310158401198872

minus 119905 = 1205762

1205871198632

119885sin1205731198872

(14)

1198892

1198632

= 1205762

120587

119885sin1205731198872

(15)


Δ1198821199062

= 1205762

1205871199062

119885sin1205731198872

orΔ1198821199062

1199062

= 1205762

120587

119885sin1205731198872

(16)


120590 =1198811199062act

1198811199062

= 1 minusΔ1198811199062

1198811199062

= 1 minusΔ1198821199062

1199062

1198811199062

1199062

(17)


1199062from (16) and for 119881

1199062

1199062from


120590 = 1 minus1205762(120587119885) sin120573

1198872

1 + (1198811199032

1199062) cot120573

1198872

= 1 minus1205762(120587119885) sin120573

1198872

1 + (120595120593) cot1205731198872

(18)

At 120595 = 0 (120590 = 1205900)

1 minus 1205900= 1205762

120587

119885sin1205731198872

(19)

Therefore

1 minus 120590 =1 minus 1205900

119910 (20)

where

119910 = 1 +120595

120593cot1205731198872

(21)


1 minus 1205900=

radicsin1205731198872

11988507for 119863

1

1198632

le 120576limit(22)


120576limit = 119890minus(816 sin120573

1198872119885)

(23)


1198821199061


Δ1015840

1198821199061

= 1205961198891

2 (24)

where

1198891= 1205761

1205871198631

119885sin1205731198871

(25)

1198891

1198632

= 1205761

120587

119885

1198631

1198632

sin1205731198871

(26)

Hence

Δ1015840

1198821199061

= 1205761

120587

1198851199061sin 1205731198871

(27)


Vlowast1 = Vlowast

r1

120573lowastf1

120573b1

u1

Wlowast1b =

Vlowastr1

sin 120573b1

Δ998400Wu1

Wlowast1

(1 minus 1205761120587

Zsin 120573b1)u1



1199061


Δ10158401015840

1198821199061

= 1199061(1 minus 120576

1

120587

119885sin 1205731198871

) minus 1198811199031

cot (180 minus 1205731198871

) (28)


119867infin=11990621198811199062

119892 (29)



1198670=1199062sdot 1198811199062 act

119892=1199062sdot 1205901198811199062

119892= 120590 sdot 119867

infin (30)


ℎ119897slp


infin (31)


119881119889120579 = 119881

1199033

1198811199063









Vlowast1 = Vlowast

r1

V1 = Vr1

Wlowast1b

Δ998400Wu1

W1

120573f1

120573b1

u1

Δ998400Wu1

Wlowast1


Δ998400998400Wu1

(1 minus 1205761120587

Zsin 120573b1)u1



tan120572119881=

119860119881th

12058711986331198873

=119860119881th

12058711986321198872(11986331198632) (11988731198872) (32)




= 1198814radic2119892119867 are

1198814=120578vol sdot 119881119903

2

120587119863211988721205762

12058711986331198873sdot tan 120572

119881

=120578vol1205762tan 120572119881

1198632

1198633

1198872

1198873

sdot 1198811199032

1198621198814

=1205762120578vol

(11986331198632) (11988731198872) tan 120572

119881

sdot 120595

(33)


119863th = 1205871198633tan 120572119881+ 119871 th tan120572th (34)


119863th1198633

= 120587 tan120572119881+ tan120572th (35)


(3) then

1198815= 41205762120578vol

1198632

2

1198632th

1198872

1198632

sdot 1198811199032

(36)


(= 1198815radic2119892119867)

equals

1198621198815

=41205762120578vol (11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (37)



5


1198815= 119881119904= 120578vol sdot 119881eye (38)


119862119881eye

=119881eye

radic2119892119867=1198621198815

120578vol=

41205762(11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (39)


120579

d120579 V3r

V4V3

V3

V3

V4 V5

V2

V3u

V3u

V5

Throat

Volute

Impeller

Lth

L th

y th

D3

D2

D1

xc

N

120572V

120572V

Vr2

V2eq

Vu2act

Case 1205723eq lt 120572V


V3eq

120572V

120572V

V3p

V3p

V3p

V4 = V3pu

1205723eq lt 120572V1205723eq

V3d

V3d

V3d998400

Vr3 = 1205762120578volb2b3

D2

D3

middot Vr2

xc

120572V

b3

ThroatVolute

Dth

yth = 120587D3tan120572V

V3d + V3p equiv


D3sin120572VZ

120587D2

120587D3

120587D3ta

n120572V

2lowast120587

Dth

120579th120572th

Vu2

ΔVu2

ΔVu3



119863eye

1198631

=119863eye

119863th

119863th1198633

1198633

1198632

1198632

1198631

=(11986331198632) (120587 tan 120572

119881+ tan 120572th)

(11986311198632)

(40)


tan120572119881lt

(11986311198632)

120587 (11986331198632)minustan120572th120587

(41)


tan(120579th2) =

(119863th minus 1198873) 2

119871 th (42)


tan(120579th2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

]

(43)





1199063

=

Δ1198811199062





1198811015840



3119889= 1198811015840

3119889minus Δ1198811199063




4


ℎ119897119881

= 119862119891119881

1198812

3119901

2119892+ 119862119889119881

1198812

3119889

2119892+1198812

3119903

2119892+ ℎ119897th (44)

ℎ119897119881

119867= 119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

(45)

where 119862119891119881


119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ119881

1198632

) (46)


ℎ119881



119871119881=1

2

1205871198633

cos120572119881

(47)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)



1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)




119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)


Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)



1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)



119889119881

= 08 and 1198813119889


1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)


1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)


ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th


119862119891th

= 05 + 26 lowast sin(120579th2) (56)



119897eye


ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)



119867 =1199015minus 119901eye

120574 (58)




119867 = 1198670minus ℎ119897119881


or




119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)


1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)


ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)


119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)





119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










d2

d1120596

minus120596

minus120596

120573b2

120587D2Z-tsin

120573b2

120573998400b2

Wav



1198812

=

1198812radic2119892119867 are

1198812

2= 1198812

1199032

+ 1198812

1199062

= 1198812

1199032

+ (1199062+ 1198811199032

cot1205731198872

)2

1198622

1198812

= 1205952

+ (120593 + 120595cot1205731198872

)2

(12)

23 Slip Factor





1199062

= Δ1198811199062

=

1198811199062



the channelThus

Δ1198821199062

= 1205961198892

2 (13)



1198892=1205871198632

119885sin12057310158401198872

minus 119905 = 1205762

1205871198632

119885sin1205731198872

(14)

1198892

1198632

= 1205762

120587

119885sin1205731198872

(15)


Δ1198821199062

= 1205762

1205871199062

119885sin1205731198872

orΔ1198821199062

1199062

= 1205762

120587

119885sin1205731198872

(16)


120590 =1198811199062act

1198811199062

= 1 minusΔ1198811199062

1198811199062

= 1 minusΔ1198821199062

1199062

1198811199062

1199062

(17)


1199062from (16) and for 119881

1199062

1199062from


120590 = 1 minus1205762(120587119885) sin120573

1198872

1 + (1198811199032

1199062) cot120573

1198872

= 1 minus1205762(120587119885) sin120573

1198872

1 + (120595120593) cot1205731198872

(18)

At 120595 = 0 (120590 = 1205900)

1 minus 1205900= 1205762

120587

119885sin1205731198872

(19)

Therefore

1 minus 120590 =1 minus 1205900

119910 (20)

where

119910 = 1 +120595

120593cot1205731198872

(21)


1 minus 1205900=

radicsin1205731198872

11988507for 119863

1

1198632

le 120576limit(22)


120576limit = 119890minus(816 sin120573

1198872119885)

(23)


1198821199061


Δ1015840

1198821199061

= 1205961198891

2 (24)

where

1198891= 1205761

1205871198631

119885sin1205731198871

(25)

1198891

1198632

= 1205761

120587

119885

1198631

1198632

sin1205731198871

(26)

Hence

Δ1015840

1198821199061

= 1205761

120587

1198851199061sin 1205731198871

(27)


Vlowast1 = Vlowast

r1

120573lowastf1

120573b1

u1

Wlowast1b =

Vlowastr1

sin 120573b1

Δ998400Wu1

Wlowast1

(1 minus 1205761120587

Zsin 120573b1)u1



1199061


Δ10158401015840

1198821199061

= 1199061(1 minus 120576

1

120587

119885sin 1205731198871

) minus 1198811199031

cot (180 minus 1205731198871

) (28)


119867infin=11990621198811199062

119892 (29)



1198670=1199062sdot 1198811199062 act

119892=1199062sdot 1205901198811199062

119892= 120590 sdot 119867

infin (30)


ℎ119897slp


infin (31)


119881119889120579 = 119881

1199033

1198811199063









Vlowast1 = Vlowast

r1

V1 = Vr1

Wlowast1b

Δ998400Wu1

W1

120573f1

120573b1

u1

Δ998400Wu1

Wlowast1


Δ998400998400Wu1

(1 minus 1205761120587

Zsin 120573b1)u1



tan120572119881=

119860119881th

12058711986331198873

=119860119881th

12058711986321198872(11986331198632) (11988731198872) (32)




= 1198814radic2119892119867 are

1198814=120578vol sdot 119881119903

2

120587119863211988721205762

12058711986331198873sdot tan 120572

119881

=120578vol1205762tan 120572119881

1198632

1198633

1198872

1198873

sdot 1198811199032

1198621198814

=1205762120578vol

(11986331198632) (11988731198872) tan 120572

119881

sdot 120595

(33)


119863th = 1205871198633tan 120572119881+ 119871 th tan120572th (34)


119863th1198633

= 120587 tan120572119881+ tan120572th (35)


(3) then

1198815= 41205762120578vol

1198632

2

1198632th

1198872

1198632

sdot 1198811199032

(36)


(= 1198815radic2119892119867)

equals

1198621198815

=41205762120578vol (11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (37)



5


1198815= 119881119904= 120578vol sdot 119881eye (38)


119862119881eye

=119881eye

radic2119892119867=1198621198815

120578vol=

41205762(11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (39)


120579

d120579 V3r

V4V3

V3

V3

V4 V5

V2

V3u

V3u

V5

Throat

Volute

Impeller

Lth

L th

y th

D3

D2

D1

xc

N

120572V

120572V

Vr2

V2eq

Vu2act

Case 1205723eq lt 120572V


V3eq

120572V

120572V

V3p

V3p

V3p

V4 = V3pu

1205723eq lt 120572V1205723eq

V3d

V3d

V3d998400

Vr3 = 1205762120578volb2b3

D2

D3

middot Vr2

xc

120572V

b3

ThroatVolute

Dth

yth = 120587D3tan120572V

V3d + V3p equiv


D3sin120572VZ

120587D2

120587D3

120587D3ta

n120572V

2lowast120587

Dth

120579th120572th

Vu2

ΔVu2

ΔVu3



119863eye

1198631

=119863eye

119863th

119863th1198633

1198633

1198632

1198632

1198631

=(11986331198632) (120587 tan 120572

119881+ tan 120572th)

(11986311198632)

(40)


tan120572119881lt

(11986311198632)

120587 (11986331198632)minustan120572th120587

(41)


tan(120579th2) =

(119863th minus 1198873) 2

119871 th (42)


tan(120579th2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

]

(43)





1199063

=

Δ1198811199062





1198811015840



3119889= 1198811015840

3119889minus Δ1198811199063




4


ℎ119897119881

= 119862119891119881

1198812

3119901

2119892+ 119862119889119881

1198812

3119889

2119892+1198812

3119903

2119892+ ℎ119897th (44)

ℎ119897119881

119867= 119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

(45)

where 119862119891119881


119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ119881

1198632

) (46)


ℎ119881



119871119881=1

2

1205871198633

cos120572119881

(47)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)



1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)




119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)


Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)



1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)



119889119881

= 08 and 1198813119889


1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)


1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)


ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th


119862119891th

= 05 + 26 lowast sin(120579th2) (56)



119897eye


ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)



119867 =1199015minus 119901eye

120574 (58)




119867 = 1198670minus ℎ119897119881


or




119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)


1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)


ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)


119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)





119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Vlowast1 = Vlowast

r1

120573lowastf1

120573b1

u1

Wlowast1b =

Vlowastr1

sin 120573b1

Δ998400Wu1

Wlowast1

(1 minus 1205761120587

Zsin 120573b1)u1



1199061


Δ10158401015840

1198821199061

= 1199061(1 minus 120576

1

120587

119885sin 1205731198871

) minus 1198811199031

cot (180 minus 1205731198871

) (28)


119867infin=11990621198811199062

119892 (29)



1198670=1199062sdot 1198811199062 act

119892=1199062sdot 1205901198811199062

119892= 120590 sdot 119867

infin (30)


ℎ119897slp


infin (31)


119881119889120579 = 119881

1199033

1198811199063









Vlowast1 = Vlowast

r1

V1 = Vr1

Wlowast1b

Δ998400Wu1

W1

120573f1

120573b1

u1

Δ998400Wu1

Wlowast1


Δ998400998400Wu1

(1 minus 1205761120587

Zsin 120573b1)u1



tan120572119881=

119860119881th

12058711986331198873

=119860119881th

12058711986321198872(11986331198632) (11988731198872) (32)




= 1198814radic2119892119867 are

1198814=120578vol sdot 119881119903

2

120587119863211988721205762

12058711986331198873sdot tan 120572

119881

=120578vol1205762tan 120572119881

1198632

1198633

1198872

1198873

sdot 1198811199032

1198621198814

=1205762120578vol

(11986331198632) (11988731198872) tan 120572

119881

sdot 120595

(33)


119863th = 1205871198633tan 120572119881+ 119871 th tan120572th (34)


119863th1198633

= 120587 tan120572119881+ tan120572th (35)


(3) then

1198815= 41205762120578vol

1198632

2

1198632th

1198872

1198632

sdot 1198811199032

(36)


(= 1198815radic2119892119867)

equals

1198621198815

=41205762120578vol (11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (37)



5


1198815= 119881119904= 120578vol sdot 119881eye (38)


119862119881eye

=119881eye

radic2119892119867=1198621198815

120578vol=

41205762(11988721198632)

(119863th1198633)2

(11986331198632)2sdot 120595 (39)


120579

d120579 V3r

V4V3

V3

V3

V4 V5

V2

V3u

V3u

V5

Throat

Volute

Impeller

Lth

L th

y th

D3

D2

D1

xc

N

120572V

120572V

Vr2

V2eq

Vu2act

Case 1205723eq lt 120572V


V3eq

120572V

120572V

V3p

V3p

V3p

V4 = V3pu

1205723eq lt 120572V1205723eq

V3d

V3d

V3d998400

Vr3 = 1205762120578volb2b3

D2

D3

middot Vr2

xc

120572V

b3

ThroatVolute

Dth

yth = 120587D3tan120572V

V3d + V3p equiv


D3sin120572VZ

120587D2

120587D3

120587D3ta

n120572V

2lowast120587

Dth

120579th120572th

Vu2

ΔVu2

ΔVu3



119863eye

1198631

=119863eye

119863th

119863th1198633

1198633

1198632

1198632

1198631

=(11986331198632) (120587 tan 120572

119881+ tan 120572th)

(11986311198632)

(40)


tan120572119881lt

(11986311198632)

120587 (11986331198632)minustan120572th120587

(41)


tan(120579th2) =

(119863th minus 1198873) 2

119871 th (42)


tan(120579th2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

]

(43)





1199063

=

Δ1198811199062





1198811015840



3119889= 1198811015840

3119889minus Δ1198811199063




4


ℎ119897119881

= 119862119891119881

1198812

3119901

2119892+ 119862119889119881

1198812

3119889

2119892+1198812

3119903

2119892+ ℎ119897th (44)

ℎ119897119881

119867= 119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

(45)

where 119862119891119881


119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ119881

1198632

) (46)


ℎ119881



119871119881=1

2

1205871198633

cos120572119881

(47)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)



1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)




119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)


Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)



1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)



119889119881

= 08 and 1198813119889


1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)


1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)


ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th


119862119891th

= 05 + 26 lowast sin(120579th2) (56)



119897eye


ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)



119867 =1199015minus 119901eye

120574 (58)




119867 = 1198670minus ℎ119897119881


or




119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)


1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)


ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)


119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)





119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120579

d120579 V3r

V4V3

V3

V3

V4 V5

V2

V3u

V3u

V5

Throat

Volute

Impeller

Lth

L th

y th

D3

D2

D1

xc

N

120572V

120572V

Vr2

V2eq

Vu2act

Case 1205723eq lt 120572V


V3eq

120572V

120572V

V3p

V3p

V3p

V4 = V3pu

1205723eq lt 120572V1205723eq

V3d

V3d

V3d998400

Vr3 = 1205762120578volb2b3

D2

D3

middot Vr2

xc

120572V

b3

ThroatVolute

Dth

yth = 120587D3tan120572V

V3d + V3p equiv


D3sin120572VZ

120587D2

120587D3

120587D3ta

n120572V

2lowast120587

Dth

120579th120572th

Vu2

ΔVu2

ΔVu3



119863eye

1198631

=119863eye

119863th

119863th1198633

1198633

1198632

1198632

1198631

=(11986331198632) (120587 tan 120572

119881+ tan 120572th)

(11986311198632)

(40)


tan120572119881lt

(11986311198632)

120587 (11986331198632)minustan120572th120587

(41)


tan(120579th2) =

(119863th minus 1198873) 2

119871 th (42)


tan(120579th2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

]

(43)





1199063

=

Δ1198811199062





1198811015840



3119889= 1198811015840

3119889minus Δ1198811199063




4


ℎ119897119881

= 119862119891119881

1198812

3119901

2119892+ 119862119889119881

1198812

3119889

2119892+1198812

3119903

2119892+ ℎ119897th (44)

ℎ119897119881

119867= 119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

(45)

where 119862119891119881


119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ119881

1198632

) (46)


ℎ119881



119871119881=1

2

1205871198633

cos120572119881

(47)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)



1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)




119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)


Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)



1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)



119889119881

= 08 and 1198813119889


1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)


1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)


ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th


119862119891th

= 05 + 26 lowast sin(120579th2) (56)



119897eye


ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)



119867 =1199015minus 119901eye

120574 (58)




119867 = 1198670minus ℎ119897119881


or




119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)


1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)


ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)


119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)





119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1

119863ℎ119881

1198632

=1

2 (11988731198872) (11988721198632)+

1

8 (120587119885) (11986331198632) sin120572

119881

(49)




119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881

) 37)111

]2 (50)

where

120581

119863ℎ119881

=(1205811198632)

(119863ℎ119881

1198632) (51)


Re119881=1198813119901sdot 119863ℎ119881

]= 2

1198621198813119901

120593(119863ℎ119881

1198632

) sdot Re2 (52)

where

Re2=1199062sdot 11986322

] (53)



1198813119901=

1198814

cos120572119881

or 1198621198813119901

=1198621198814

cos 120572119881

(54a)



119889119881

= 08 and 1198813119889


1198813119889= 1198811199062act

minus 1198814

1198621198813119889

=1198813119889

radic2119892119867= 120590 (120593 + 120595 cot120573

1198872

) minus 1198621198814

(54b)


1198621198813119903

=1198813119903

radic2119892119867=

1205762120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)


ℎ119897th= 119862119891th

1198812

4

2119892 (55)

where 119862119891th


119862119891th

= 05 + 26 lowast sin(120579th2) (56)



119897eye


ℎ119897eye

= 119862eye1198812

eye

2119892

ℎ119897eye

119867= 119862eye

1198812

eye

2119892119867= 119862eye sdot 119862

2

119881eye

(57)



119867 =1199015minus 119901eye

120574 (58)




119867 = 1198670minus ℎ119897119881


or




119909 =119867

1198670

=1

[1 + (ℎ119897119881

119867) + (ℎ119897eye119867)]

or 1

119909= 1 +

ℎ119897119881

119867+ℎ119897eye

119867

(61)


1

119909= 1 + 119862

119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

+ 119862eye sdot 1198622

119881eye

(62)


ℎ119897119881

+ ℎ119897eye

= (1 minus 119909)1198670= (1 minus 119909) 120590119867

infin (63)


119867 = 119909120590119867infin= 119909120590

1199062sdot 1198811199062

119892

119867 = 1199091205901199062

2

119892(1 +

120595

120593cot1205731198872

)

(64)





119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119867infin


1198670


119862119867=

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872

) (65a)

119862119867infin

=119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(65b)

1198621198670

=1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872

) (65c)




ℎ119897119881+eye



coefficient 119862ℎ119897119881


119862ℎ119897slp

=ℎ119897slp

11990622119892

= (1 minus 120590) 119862119867infin

(65d)

119862ℎ119897119881+eye

=ℎ119897119881

+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

119862ℎ119897eye

=ℎ119897eye

11990622119892

= 119862eye sdot 1198622

119881119890119910119890

sdot1

21205932 (65f)

119862ℎ119897119881

=ℎ119897119881

11990622119892

= (119862119891119881

sdot 1198622

1198813119901

+ 119862119889119881

sdot 1198622

1198813119889

+ 1198622

1198813119903

+ 119862119891thsdot 1198622

1198814

) sdot1

21205932

(65g)



119867 = 1199091205901199062

2

119892

cot1205722


1198872

(66)



2


1198872


1

21199091205901205952= 0 (67)


cot1205731198872


cot1205722=1

2cot1205731198872

+1

2radiccot2 120573

1198872

+2

1199091205901205952 (68)



120593 = minus1

2120595cot120573

1198872

+1

2radic1205952cot2 120573

1198872

+2

119909120590 (69)


119875sh = 1198751015840

sh0

+ 1198751015840

119897119891

+ 1198751015840

119897cirin+ 119875119897119863

(70)

where 1198751015840sh0


water 1198751015840119897119891

= 120574(119876 + 119876119871)ℎ119897119891


119897cirin= 120574(119876+119876

119871)ℎ119897cirin


119897119863



119875sh = 119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

(71)

in which

119875sh0

= 1205741198761198670 119875

119897119891

= 120574119876ℎ119897119891

119875119897cirin

= 120574119876ℎ119897cirin

119875119897vol

= 120574119876ℎ119897vol

= 120574119876119871(1198670+ ℎ119897119891

+ ℎ119897cirin

)

119875119897119863

= 120574119876ℎ119897119863

(72)

where ℎ119897119891







= 119867sh(1199062


119867sh = 1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

119862119867sh

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

where119862ℎ119897119891

= ℎ119897119891

(1199062



= ℎ119897cirin

(1199062



= ℎ119897vol(1199062



= ℎ119897119863

(1199062




119908)


120578 =119875119908

119875sh=

119875119908

119875sh0

+ 119875119897119891

+ 119875119897cirin

+ 119875119897vol+ 119875119897119863

120578 =119867

119867sh=


minus ℎ119897eye

1198670+ ℎ119897119891

+ ℎ119897cirin

+ ℎ119897vol+ ℎ119897119863

120578 =119862119867

119862119867119904ℎ

=

119862119867infin


1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(74)




119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119862119875sh

=119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh (75)

119862119875119908

=119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867 (76)



119897119891


ℎ119897119891

= 4119862119889119894

119871119887

119863hyd

1198822

av2119892

(77)

where 119862119889119894



119862ℎ119897119891

=ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av1199062

)

2

(78)



=2 (11988911198871+ 11988921198872)

1198891+ 1198871+ 1198892+ 1198872

119882av =119876119894119885

119860av=

119876119894119885

(11988911198871+ 11988921198872) 2

(79)


2 (15) and 119876

119894 (3) yield

119863hyd

1198632

=2 ((11988911198632) (11988711198872) (11988721198632) + (119889

21198632) (11988721198632))

(11988911198632) + (11988711198872) (11988721198632) + (119889

21198632) + (11988721198632)

(80)

119882av1199062

=1205762(120587119885) (119887

21198632)

12((11988911198632) (11988711198872) (11988721198632)+(11988921198632) (11988721198632))

sdot120595

120593

(81)



119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm (82a)

where

120573bm =(1205731198871

+ 1205731198872

)

2 (82b)


4119862119889119894

= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)



119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2 (84)

where

120581

119863hyd=

(1205811198632)

(119863hyd1198632) (85)


]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2 (86)





119897119863

can be estimated as

119875119897119863

= 2 lowast int

1199032

0

120596 sdot 119903 sdot 120591 sdot 2120587119903 119889119903 (87)

with

120591 =119891119863

4sdot1

21205881199062

=119891119863

4sdot1

2120588 (120596119903)

2

(88)



Thus

119875119897119863

=120587

21205881205963

int

1199032

0

1198911198631199034

119889119903 (89)


119875119897119863

=120587

211989111986312058812059631199035

2

5=

120587

401198911198631205881199063

21198632

2

ℎ119897119863

=119875119897119863

120588119892119876=

120587

40

119891119863

119892

1199063

21198632

2

119876

(90)


is givenas

119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595 (91)


119870119863=

1

40

119891119863

1205762(11988721198632) (92a)


21198872




2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












2=


1199100(11986322) where 119910



2 119866 = 188Reminus910

2 119866 =



2


Regime I 119891119863= 10119866

minus1Reminus12

Regime II 119891119863=185

120587119866110Reminus12

2

Regime III 119891119863=04


2

Regime IV 119891119863=051

120587119866110Reminus15

2

(92b)



ℎ119897vol

=119875119897vol

120574119876=119876119871

119876(1198670+ ℎ119897119891

+ ℎ119897cirin

)

= (1


0+ ℎ119897119891

+ ℎ119897cirin

)

119862ℎ119897vol

=ℎ119897vol

11990622119892

= (1


1198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

)

(93)


formula [13]

119876119871= 119862dL sdot 119886119888radic2119892Δ119867119888 (94)





Δ119867119888=119901119888minus 119901119904

120574=1199013minus 119901119904

120574minus1199013minus 119901119888

120574 (95)





120574=1199015minus 119901119904

120574+1199013minus 1199015

120574

= 119867 + [ℎ119897119881

minus (1198812

4

2119892minus1198812

5

2119892)]

= 1198670minus ℎ119897eye

minus (1198812

4

2119892minus1198812

5

2119892)

(96)

r 2=D22

w

Dey

e

yc

y0

p3

pc

ps


0000

0025

0050

0075

0100

1000 10000 100000 1000000

G

I

II

III

IV

1

2

3

4

5

Re2



1199013minus 119901119888

120574=1199062

2

8119892(1 minus

1198632

eye

11986322

) (97)


119876119871

are


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

119862119876119871

=119876119871

(11987360)11986332

=119910119888

119863eye

radic21205872

119862dL

(1198632119863eye)

2


lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8(1 minus

1198632

eye

11986322

))

12

(98)Using (2) 119876

119871119876 = 119862

119876119871

119862119876 and (8) for 119862



120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2 radic2119862dL(11988721198632) 1205762

120593

120595

lowast (1198621198670

minus

1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932

minus1

8[1 minus (

119863eye

1198631

)

2

(1198631

1198632

)

2

])

12

(99)



1199061



ℎ119897cirin

=Δ10158401015840

1198821199061

sdot 1199061

119892 (100)


119862ℎ119897cirin

=

ℎ119897cirin

11990622119892

= (1 minus 1205761

120587

119885sin1205731198871

)(1198631

1198632

)

2


1198871

)

(11988711198872) (12057611205762)sdot120595

120593

(101)


NPSH = (1199011

120574+1198812

1

2119892) minus

119901V

120574 (102)




119901min120574

=1199011

120574minus (

1198822

1119887

2119892minus1198812

1

2119892) (103)


NPSH119877=(119901min minus 119901V)

120574+1198822

1119887

2119892 (104)


119862NPSH119877

=NPSH

119877

11990622119892

(105)


sin1205731198871

) and thus

119862NPSH119877

=(119901min minus 119901V) 120574

11990622119892

+1

sin21205731198871

1198812

1199031

211990622

(106)




1199031

= 1198811199032

(11986321198631)(11988721198871)(12057621205761) and therefore


119862NPSH119877

= 002 +1

2sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12057611205762)2 (107)



119899119904=119873radic119876

11986734[119873 (rpm) 119867 (m) 119876 (lits)] (108)

Substitute for


119873 =60

1205871198632

120593radic2119892119867 and 1198872= 1198632(1198872

1198632

) yield


34

lowast radic1205762

radic120587radic(

1198872

1198632


(109)

Therefore

119899119904= 9977radic120576

2radic(

1198872

1198632









1198872

blade angle at




Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Impeller

1198631

1198632

051198872

1198632

01211198871

1198872

1

1205982

095119866 =

1199100

(11986322)

005

Re2=1199062sdot (11986322)

](1198632= 0209m)

(119873 = 1450 rpm)(] = 10

minus6m2s) water

116 lowast 106

120581

1198632

(120581 = 00001m)

0000478

Blades

119885 41205731198872

164∘

1205731198871

164∘1198633

1198632

1051198873

1198872

105

Volute120572119881

7∘

120572th 4∘

119862119889119881

08

Eye

119863eye

119863th

1

119910119888

119863eye0005

119862dL 06119862eye 10









2




2= 116 lowast 10

6 and 1205811198632= 0000478




00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

120595120593



120578

120578

Efficiency

CHCH








119862119875119908


119877



at 119862119876




119876









Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Impeller

1 minus 1205900=

radicsin1205731198872

11988507for 1198631

1198632


1205981= 1 minus

(1 minus 1205982)

(11986311198632) (sin120573

1198871 sin120573

1198872)

(4)

1198892

1198632

= 1205982

120587

119885sin1205731198872

(15)

1198891

1198632

= 1205981

120587

119885

1198631

1198632

sin1205731198871

(26)

119863hyd

1198632

=2 (((119889

11198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(11988911198632) + ((119887

11198872) (11988721198632)) + (119889

21198632) + (119887

21198632)

(80)

120581

119863hyd=

1205811198632

119863hyd1198632(85)

119870119863=

1

40

119891119863

1205982(11988721198632)

(92a)

119891119863= 119891119863(Re2 119866) (92b)

Blades119871119887

1198632

=1

2(1 minus

1198631

1198632

)1

sin120573bm(82a)

120573bm =1

2(1205731198871+ 1205731198872) (82b)

Volute

1

119863ℎ1198811198632

=1

2(11988731198872)(11988721198632)+

1

8(120587119885)(11986331198632) sin120572

119881

(49)

119871119881

1198632

=1

2

120587

cos120572119881

(1198633

1198632

) (48)

120581

119863ℎ119881

=(1205811198632)

(119863ℎ1198811198632)

(51)

tan(120579th

2) =

1

2[120587 tan120572


(11988731198872) (11988721198632)

(11986331198632)

] (43)

119862119891th

= 05 + 26 lowast sin(120579th

2) (56)

Eye119863eye

1198631

=(11986331198632)

(11986311198632)(120587 tan120572

119881+ tan120572th) (40)



119867in





= 164∘








and 1198621198670





119876asymp 006 and



1198811

119862119881Δ1198811199062

and 119862

1198814


1198813119889

and 11986211988131198891015840have the



119876asymp 0085





4gt 1198811199062 act (Figure 6)






120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120595 equiv1198811199032

radic2119892119867= 0002 to 012 step = 0001

Initial 120590 = 1205900 119909 = 1 120578vol = 1

Start of iteration

1 120593 = minus1

2120595 cot120573

1198872+1

2radic1205952cot2120573

1198872+

2

119909120590(69)

2 119910 = 119862119867infin

equiv119867infin

11990622119892

= 1 +120595

120593cot1205731198872

(21) (65b)

3 120590 = 1 minus(1 minus 120590

0)

119910(20)

4 119862ℎ119897slp

equivℎ119897slp

11990622119892

= (1 minus 120590)119862119867infin

(65d)

5 1198621198670

equiv1198670

11990622119892

= 120590(1 +120595

120593cot1205731198872) (65c)

6 1198621198815equiv

1198815

radic2119892119867=

41205982120578vol (11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (37)

7 1198621198814equiv

1198814

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872) tan120572

119881

sdot 120595 (33)

8 119862119881eye

equiv119881eye

radic2119892119867=

41205982(11988721198632)

(119863119905ℎ1198633)2

(11986331198632)2sdot 120595 (39)

9 1198621198813119901

=1198621198814

cos120572119881

(54a)

10 1198621198813119889

= 120590 (120593 + 120595cot1205731198872) minus 1198621198814

(54b)

11 1198621198813119903

equiv1198813119903

radic2119892119867=

1205982120578vol

(11986331198632) (11988731198872)sdot 120595 (54c)

12 120578vol = 1 minus (119910119888

119863eye)(

119863eye

1198631

)

2

(1198631

1198632

)

2

1

(11988721198632)

radic2119862dL

1205982

120593


1198670minus1198622

1198814

minus 1198622

1198815

+ 119862eye sdot 1198622

119881eye

21205932minus1

8[1 minus (

1198631

1198632

)

2

(119863eye

1198631

)

2

] (99)


Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 3 Continued


13 Re119881=1198813119901sdot119863ℎ119881

]= 2(

1198621198813119901

120593)(

119863ℎ119881

1198632

) sdot Re2

(52)

14 119891119881=

03086

log [(69Re119881) + ((120581119863

ℎ119881) 37)

111

]2

(50)

15 119862119891119881

= 119891119881

119871119881

119863ℎ119881

= 119891119881(119871119881

1198632

)(1

119863ℎ1198811198632

) (46)

16ℎ119897119881

119867= 1198621198911198811198622

1198813119901

+ 1198621198891198811198622

1198813119889

+ 1198622

1198813119903

+ 119862119891119905ℎ1198622

1198814

(45)

17ℎ119897eye

119867= 119862eye sdot 119862

2

119881eye(57)

18 1

119909= 1 +

ℎ119897119881

119867+

ℎ119897eye

119867(61)

End of iteration

19 119862ℎ119897119881+eye

equiv

ℎ119897119881+ ℎ119897eye

11990622119892

= (1 minus 119909) 120590119862119867infin

(65e)

20 119862119867equiv

119867

11990622119892

= 119909120590(1 +120595

120593cot1205731198872) (65a)

21 119862119876equiv

119876

(11987360)1198633

2

= 120578vol12059821205872

(1198872

1198632

) sdot (120595

120593) (6)

22119882av

1199062

=1205982(120587119885) (119887

21198632) sdot (120595120593)

12 (((11988911198632) (11988711198872) (11988721198632)) + ((119889

21198632) (11988721198632)))

(81)

23 Re =119882av sdot 119863hyd

]= 2(

119882av1199062

)(119863hyd

1198632

) sdot Re2

(86)

24 119891119894=

03086

log [(69Re) + ((120581119863hyd) 37)111

]2

(84)

25 4119862119889119894= (119891119894+ 0006) (11 + 4

1198872

1198632

) (83)


Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 3 Continued


26 119862ℎ119897119891

equiv

ℎ119897119891

11990622119892

= 4119862119889119894

(1198711198871198632)

(119863hyd1198632)

1

2(119882av

1199062

)

2

(78)

27 119862ℎ119897119863

equivℎ119897119863

11990622119892

= 119870119863sdot1

120578vol

120593

120595(91)

28 119862ℎ119897cirin

equiv

ℎ119897cirin

11990622119892

= (1 minus 1205981

120587

119885sin1205731198871)(

1198631

1198632

)

2


1198871)

(11988711198872) (12059811205982)sdot120595

120593(101)

29 119862ℎ119897vol

equivℎ119897vol

11990622119892

= (1


1198670+ 119862ℎ119897119891

+ 119862ℎ119897cirin

) (93)

30 119862119867sh

equiv119867sh

11990622119892

= 1198621198670

+ 119862ℎ119897119891

+ 119862ℎ119897cirin

+ 119862ℎ119897vol

+ 119862ℎ119897119863

(73)

31 120578 equiv119875119908

119875sh=

119862119867

119862119867sh

(74)

32 119862119875sh

equiv119875sh

120588(11987360)3

11986352

= 1205872

119862119876119862119867sh

(75)

33 119862119875119908

equiv119875119908

120588(11987360)3

11986352

= 1205872

119862119876119862119867

(76)

34 119862NPSH119877 = 002 +1

2 sin21205731198871

(120595120593)2

(11986311198632)2

(11988711198872)2

(12059811205982)2

(107)

35 119899119904= 9977radic1205982radic(

1198872

1198632






Notations






000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000 002 004 006 008 010 01200

01

02

03

04

05

06

07

08

120578

120578

CP119908CP119908

BEP

CQ

CH

CH

CNPSH119877

CNPSH 119877

CPshCPsh


000

002

004

006

008

010

012

014

016

018

020

000 002 004 006 008 010 012

BEP

120595

120595120595120601 120595120601

CQ


119887 Impeller width m119862119889119894



119862119891119881


2

2119892)

1198621198670


2119892)

119862119867infin


2119892)

119862119867sh


2119892)

07

08

09

10

11

12

13

14

15

16

000 002 004 006 008 010 012

BEP

CQ

120593


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents

BEP

Euler head

Volute and eye loss

Manometric head

CHCH0

Slip loss

CQ

CHinfin


119862ℎ119897cirin


(1199062

2119892)

119862ℎ119897119891


119897eye(1199062

2119892)

119862ℎ119897119863


(1199062

2119892)


119897slp(1199062

2119892)

119862ℎ119897119881


(1199062

2119892)

119862ℎ119897119881+eye


+ ℎ119897eye)(1199062

2119892)


119897vol(1199062

2119892)


00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

000 002 004 006 008 010 012

Hea

d an

d he

ad lo

ss co

effici

ents


BEP

CQ

Shaft head

Inlet cir loss

Volumetric loss

Disk loss

Manometric head

CH0

CH

CH

sh


00

01

02

03

04

05

06

07

08

09

10

000 002 004 006 008 010 012

BEP

CQ

CV

120578vol

120578vol

CV2

CV2act

C V4

CV1

CV3119889 998400

CV3119889

CVΔVu2


119862NPSH119877


119877(1199062

2119892)


3

1198635

2

119862119875119908


3

1198635

2


3

2

119862119876119871



0

200

400

600

800

1000

1200

1400

1600

1800

2000

000 002 004 006 008 010 012

BEP

CQ

ns


00

01

02

03

04

05

06

07

08

000 002 004 006 008 010 012

WaterOilOil

120578

120578

CH

CH

CQ



CPsh

CPsh





0(11986322)

ℎ119897cirin




ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ℎ119897119891


ℎ119897119881













119875sh0










0





Greek Letters




2radic2119892119867


1199062act1198811199062


radic2119892119867


Subscripts


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









Research Article A One-Dimensional Flow Analysis...

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Transcript of Research Article A One-Dimensional Flow Analysis...