research article a one-dimensional flow analysis...
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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)
4 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 5
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)
6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
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)
4 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 5
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)
6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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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)
4 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 5
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)
6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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4 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 5
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)
6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of Rotating Machinery 5
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)
6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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6 International Journal of Rotating Machinery
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
minus ℎ119897eye (60)
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
International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of Rotating Machinery 7
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)
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
8 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
International Journal of Rotating Machinery 9
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
120578volminus 1) sdot (119862
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
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
10 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
International Journal of Rotating Machinery 11
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
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
12 International Journal of Rotating Machinery
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
119881+ tan120572th minus
(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
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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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International Journal of
International Journal of Rotating Machinery 13
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)
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 International Journal of Rotating Machinery
Table 3 Continued
Order Variable parameter equation Eq no
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)
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Rotating Machinery 15
Table 3 Continued
Order Variable parameter equation Eq no
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
minuscot (180 minus 120573
1198871)
(11988711198872) (12059811205982)sdot120595
120593(101)
29 119862ℎ119897vol
equivℎ119897vol
11990622119892
= (1
120578volminus 1) sdot (119862
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
) sdot radic120578vol sdot 120593radic120595 (110)
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
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 International Journal of Rotating Machinery
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)
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Rotating Machinery 17
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
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 International Journal of Rotating Machinery
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
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Rotating Machinery 19
[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
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of