# A One-Dimensional Flow Analysis for the Prediction of Centrifugal Pump Performance Characteristics

Authors: Mohammed Ahmed El-Naggar
Journal: International Journal of Rotating Machinery
 (2013)

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## Abstract

A one-dimensional flow procedure for analytical study of centrifugal pump performance is done applying the principle theories of turbomachines. Euler equation and energy equation are manipulated to find pump performance parameters at different discharge coefficients. Fluid slippage loss at impeller exit and volute loss are estimated. The fluid slippage is modeled by the slip factor approach using Wiesner empirical expression. The volute loss model counts friction loss associated with the volute throw flow velocity, diffusion friction loss due to circulation associated with volute flow, loss due to vanishing of radial flow at volute outlet, and loss inside pump volute throat. Models for impeller hydraulic friction power loss, disk friction power loss, internal flow leakage power loss, and inlet shock circulation power loss are considered by suitable models. Pump internal volumetric flow leakage and volumetric efficiency are related to pump geometry and flow properties. The procedure adopted in this paper is capable of obtaining performance characteristic curves of centrifugal pump in a dimensionless form. Pump head coefficient, manometric efficiency, power coefficient, and required NPSH are characterized. The predicted coefficients and obtained performance curves are consistent with experimental characteristics of centrifugal pump.


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## Body

## 1. Introduction

Centrifugal pumps are used in various applications and are integral to many industries. Yet, in spite of their prevalence and relatively simple configurations compared to other turbomachines, designing an efficient and durable pump remains a challenge.The design of centrifugal pumps is still determined empirically because it relies on the use of a number of experimental and statistical rules. However, during the last few years, the design and performance analysis of turbomachinery have experienced great progress due to the joint evolution of computer power and the accuracy of numerical methods.The one-dimensional performance analysis has proved to be an effective and important approach on pump design [1]. Analytical calculations of pump characteristics depend on geometrical dimensions of pump and losses models in different parts of pump. A series of formulae for calculating losses exist [2–5], but they lack accuracy when applied to centrifugal pumps.In this work, suggested models for calculating several losses in pump are introduced to examine its validity in evaluating pump performance. This paper is an effort towards theoretically obtaining accurate centrifugal pump performance characteristics. Pump characteristics and parameters are presented in dimensionless forms. It presents a one-dimensional flow analysis procedure towards obtaining optimum centrifugal pump design parameters.


## 2. Theoretical Analysis

The pump flow coefficientψ and pump speed coefficient φ are defined as(1)ψ=Vr22gH,φ=u22gH,
where H is the pump manometric head, Vr2 is the flow radial velocity at impeller outlet, and u2 is the tangential velocity at outlet of impeller.


### 2.1. Pump Leakage

Due to the difference between the outlet pressure and the inlet pressure of the impeller, a portion of impeller outlet flow rate,QL, returns to the impeller inlet from the existing clearances between the impeller and the casing, Figure 1. This internal discharge leakage, QL, causes some losses as the flow rate through the impeller (Q+QL) is greater than the pump useful outlet discharge Q.Figure 1
Pump impeller notations.The volumetric efficiencyηvol of pump is defined as the ratio of pump outlet discharge to the impeller discharge:
(2)ηvol=QQi=QQ+QLorQLQ=1ηvol-1,
in which
(3)Qi=Q+QL=Vr2·πD2b2ε2=Vr1·πD1b1ε1,
where b1 is the blade width at inlet, D1 is the impeller diameter at inlet, ε1 is the blade thickness coefficient at impeller inlet, ε1=1-(Z/π)((t/sinβb1)/D1),  b2 is the blade width at outlet, D2 is the impeller diameter at outlet, ε2 is the blade thickness coefficient at impeller outlet, ε2=1-(Z/π)((t/sinβb2)/D2),  t is the blade thickness, Z is the number of blades, βb2 is the blade angle at impeller outlet, and βb1 is the blade angle at impeller inlet. The value of ε2 is about 0.95 [6]. The relationship between ε1 and ε2 with constant blade thickness is given as
(4)ε1=1-(1-ε2)(D1/D2)(sinβb1/sinβb2).The impeller inlet flow velocity coefficientCV1=V1/2gH is calculated from (3) dividing both sides by 2gH and assuming that the flow enters the impeller without swirl (V1=Vr1):
(5)CV1=ψ(D1/D2)(b1/b2)(ε1/ε2).The pump discharge coefficient isCQ=Q/((N/60)D23), and substituting for Q from (2) and (3);
(6)CQ=ηvolε2π2b2D2·ψφ.


### 2.2. Outlet Velocity Diagram

A fluid slippage occurs at impeller exit due to the relative rotation of fluid in a direction opposite to that of impeller. A slip factorσ defined as σ=Vu2act/Vu2 could be estimated later in Section 2.3.From the velocity diagram at the impeller outlet, Figure2, the tangential component of the outlet flow absolute velocity is given as
(7)Vu2=Vr2cotα2=u2+Vr2cotβb2.Figure 2
Velocity diagram at impeller outlet.The ratio of outlet swirl velocity to outlet tangential velocity is(8)Vu2u2=1+Vr2u2cotβb2=1+ψφcotβb2,
and thus,
(9)Vu2actu2=σ·Vu2u2=σ(1+ψφcotβb2).The outlet tangential velocity (from (7)) and the pump speed coefficient φ=u2/2gH are given as
(10)u2=Vr2(cotα2-cotβb2),(11)φ=ψ(cotα2-cotβb2).From the outlet velocity diagram (Figure2) and (7), the outlet velocity and the outlet velocity coefficient defined as CV2=V2/2gH are
(12)V22=Vr22+Vu22=Vr22+(u2+Vr2cotβb2)2,CV22=ψ2+(φ+ψcotβb2)2.


### 2.3. Slip Factor

#### 2.3.1. The Relative Eddy (Eddy Circulation)

A simple explanation for the slip effect in an impeller is obtained from the idea of a relative eddy. Suppose that an irrotational and frictionless fluid flow is possible which passes through an impeller. If the absolute flow enters the impeller without spin, then at outlet the spin of the absolute flowmust still be zero. The impeller itself has an angular velocity ω so that, relative to the impeller, the fluid has an angular velocity of -ω; this is termed the relative eddy. At outlet of impeller, the relative flow, W2, can be regarded as a through flow on which a relative eddy is superimposed. The net effect of these two motions is that the average relative flow emerging from the impeller passages is at an angle to the vanes and in a direction opposite to the blade motion.One of the earliest and simplest expressions for the slip factor was obtained by Stodola, cited in [7]. Referring to Figure 3, the slip velocity due to relative eddy, ΔWu2=ΔVu2=Vu2-Vu2act, is considered to be the product of the relative eddy and the radius (d2/2) of a circle which can be inscribed within the channel.Figure 3
Flow model for Stodola slip factor.Thus,(13)ΔWu2=ωd22.
An approximate expression for d2 can be written if the number of blades Z is not small:
(14)d2=πD2Zsinβb2′-t=ε2πD2Zsinβb2,(15)d2D2=ε2πZsinβb2.
Since the relative eddy angular velocity  =2u2/D2, then
(16)ΔWu2=ε2πu2Zsinβb2,orΔWu2u2=ε2πZsinβb2.
The slip factor is given by
(17)σ=Vu2actVu2=1-ΔVu2Vu2=1-ΔWu2/u2Vu2/u2.
Substituting for ΔWu2/u2 from (16) and for Vu2/u2 from (8), the formulae proposed by Stodola, cited in [7] for the calculation of slip factor, σ, are obtained which are
(18)σ=1-ε2(π/Z)sinβb21+(Vr2/u2)cotβb2=1-ε2(π/Z)sinβb21+(ψ/φ)cotβb2.
At ψ=0 (σ=σ0)
(19)1-σ0=ε2πZsinβb2.
Therefore,
(20)1-σ=1-σ0y,
where
(21)y=1+ψφcotβb2.Wiesner [8] introduced an empirical expression which extremely well fits the experimental results of slip factor for wide range of practical blade angles and number of blades. It is used in this work and given as
(22)1-σ0=sinβb2Z0.7forD1D2≤εlimit,
where εlimit is the limiting diameter ratio,
(23)εlimit=e-(8.16sinβb2/Z).It is assumed that the water entering the pump impeller is purely in the radial direction. Relative to the impeller, the fluid has an angular velocity of-ω, relative eddy, and thus the relative flow at blade inlet acquires an additional component Δ′Wu1 opposite to rotational direction, as seen in Figure 4, which is
(24)Δ′Wu1=ωd12,
where
(25)d1=ε1πD1Zsinβb1,(26)d1D2=ε1πZD1D2sinβb1.
Hence,
(27)Δ′Wu1=ε1πZu1sinβb1.Figure 4
Velocity diagram at impeller inlet without shock.When the pump that operates at a dischargeQ differs from that at designed condition Q*, the relative flow velocity at blade inlet tends to acquire an additional component in counter of the rotational direction, Δ′′Wu1. So, the flow enters the blade passage tangent to the blade surface, and a shock eddy or a shock circulation exists prior to the blade leading edge inside pump eye. As could be noticed from Figure 5, the relative velocity additional rotational speed at blade inlet equals
(28)Δ′′Wu1=u1(1-ε1πZsinβb1)-Vr1cot(180-βb1).Figure 5
Velocity diagram at blade inlet with shock eddy.


### 2.4. Euler Equation of Turbomachines

In the case that the fluid entering the pump impeller is purely in the radial direction without swirl, the pump Euler head is given as [6, 7]
(29)H∞=u2Vu2g.
Due to the fluid slippage at impeller exit, the actual head given to fluid by the impeller, H0, is calculated from [6, 7]:
(30)H0=u2·Vu2actg=u2·σVu2g=σ·H∞.
And so, the slippage head loss is
(31)hlslp=H∞-H0=(1-σ)H∞.


### 2.5. Pump Volute

The flow that discharges from the impeller requires careful handling in order to preserve the gains in energy imparted to the fluid. This requires the conversion of velocity head to pressure head by means of a diffuser, and this inevitably implies hydraulic losses. The application of mass conservation to a volute element, [9], reveals that the discharge flow from impeller is matched to the flow in the volute if dAV/dθ=Vr3/Vu3r3b3. This requires a circumferentially uniform rate of increase of the volute area AV over the entire development of the spiral (θ). Consequently, for a given impeller, there exist a specific volute angle αV and a specific volute throat inlet area AVth for the volute geometry.The volute angleαV, Figure 6, is chosen to match the angle of flow entering the volute α3eq at a certain pump operating point. With a uniform rate of increase of the volute area, the volute angle is
(32)tanαV=AVthπD3b3=AVthπD2b2(D3/D2)(b3/b2).
The volute outlet flow velocity V4=Q/AVth and the volute outlet flow velocity coefficient CV4=V4/2gH are
(33)V4=ηvol·Vr2πD2b2ε2πD3b3·tanαV=ηvolε2tanαVD2D3b2b3·Vr2,CV4=ε2ηvol(D3/D2)(b3/b2)tanαV·ψ.
The throat is assumed to have an expanding angle αth, and hence the throat diameter equals
(34)Dth=πD3tanαV+Lthtanαth.
With the assumption that the throat height Lth=D3, thus
(35)DthD3=πtanαV+tanαth.
The throat outlet flow velocity V5=Q/Ath, and using (2) and (3), then
(36)V5=4ε2ηvolD22Dth2b2D2·Vr2.
The throat outlet flow velocity coefficient CV5(=V5/2gH) equals
(37)CV5=4ε2ηvol(b2/D2)(Dth/D3)2(D3/D2)2·ψ.
It is assumed that the throat diameter Dth equals the eye diameter Deye; that is, Deye/Dth=1, which equals the suction pipe diameter Ds. Thus, the throat outlet flow velocity, V5, equals the flow velocity at suction pipe, Vs. Therefore,
(38)V5=Vs=ηvol·Veye.
And the eye velocity coefficient is
(39)CVeye=Veye2gH=CV5ηvol=4ε2(b2/D2)(Dth/D3)2(D3/D2)2·ψ.
The eye diameter relative to the impeller inlet diameter is therefore
(40)DeyeD1=DeyeDthDthD3D3D2D2D1=(D3/D2)(πtanαV+tanαth)(D1/D2).
The ratio (Deye/D1) must not exceed 1, and thus an upper limit is imposed on the volute angle:
(41)tanαV<(D1/D2)π(D3/D2)-tanαthπ.
The throat has a cone angle θth, where
(42)tan(θth2)=(Dth-b3)/2Lth.
Substituting from (35),
(43)tan(θth2)=12[πtanαV+tanαth-(b3/b2)(b2/D2)(D3/D2)].Figure 6
Pump and volute notations.


### 2.6. Volute Loss Model

Prediction models that account for the main features of the swirling flow in volutes, reviewed in [4], do not account for the circulatory flow initiated in volute at off-design discharge operation of pump. In this work, a model is proposed to account the volute head loss at off-design pump operation.The flow enters the volute with a through-velocityV3 at an angle α3 (which may differ from that of the volute angle αV) on which an eddy of a tangential velocity ΔVu3=ΔVu2 is superimposed opposite to impeller motion. This inlet volute velocity V3 is decomposed into a velocity parallel to the direction of volute, V3p, and another one in the tangential direction of impeller, V3d′ (Figure 6). The velocity component V3d′ is the motive of a second circulatory motion given to the volute flow in direction of impeller motion. Thus, the net circulation velocity of flow in volute is V3d=V3d′-ΔVu3 in direction of impeller motion. The component of the velocity V3p in the tangential direction denoted as V3pu equals the volute outlet velocity V4.The volute loss model counts friction loss associated with the volute throw flow velocity, diffusion friction loss due to circulation associated with volute flow, loss due to vanishing of radial flow at volute outlet, and loss inside pump volute throat. Consequently, the volute head loss and the volute head loss relative to the pump head are written, respectively, as(44)hlV=CfVV3p22g+CdVV3d22g+V3r22g+hlth,(45)hlVH=CfV·CV3p2+CdV·CV3d2+CV3r2+Cfth·CV42,
where CfV is the volute friction loss coefficient:
(46)CfV=fVLVDhV=fV(LVD2)(1DhV/D2),
where fV is the volute friction coefficient, DhV is the volute hydraulic diameter, and LV is the average volute length:
(47)LV=12πD3cosαV,(48)LVD2=12πcosαV(D3D2).
The average volute hydraulic diameter relative to impeller diameter is
(49)1DhV/D2=12(b3/b2)(b2/D2)+18(π/Z)(D3/D2)sinαV.
The volute friction coefficient fV which corresponds to a pipe flow is function of the volute Reynolds number ReV and the roughness κ, [10]:
(50)fV=0.3086{log[(6.9/ReV)+((κ/DhV)/3.7)1.11]}2,
where
(51)κDhV=(κ/D2)(DhV/D2).
The volute Reynolds number is calculated as
(52)ReV=V3p·DhVν=2CV3pφ(DhVD2)·Re2,
where
(53)Re2=u2·D2/2ν.
In the first term of (44) and (45), V3p is the volute throw flow velocity (Figure 6) and given as(54a)V3p=V4cosαV,orCV3p=CV4cosαV.
In the second term of (44) and (45), CdV is the volute diffusion loss coefficient which could be assumed to have the value CdV=0.8, and V3d is the volute circulatory velocity component:
(54b)V3d=Vu2act-V4CV3d=V3d2gH=σ(φ+ψcotβb2)-CV4.
The third term of (45) is
(54c)CV3r=V3r2gH=ε2ηvol(D3/D2)(b3/b2)·ψ.In the fourth term of (44) and (45), the volute throat head loss hlth could be calculated as
(55)hlth=CfthV422g,
where Cfth is the volute throat friction loss coefficient assumed to be, [10],
(56)Cfth=0.5+2.6*sin(θth2),
where θth is the throat cone angle.


### 2.7. Pump Eye Head Losshleye

The head loss in pump eyehleye, [2], and the eye head loss relative to the pump head are
(57)hleye=CeyeVeye22g,hleyeH=CeyeVeye22gH=Ceye·CVeye2,
where Veye is the flow velocity in pump eye and Ceye is the eye loss coefficient defined by (39).


### 2.8. Pump Manometric HeadH

The pump manometric head is the difference in static pressure heads between pump outlet and pump eye:(58)H=p5-peyeγ.
With the assumption that the throat diameter Dth equals the eye diameter Deye, the flow velocity is the same at pump suction pipe and pump delivery pipe (Vs=V5) and neglecting the difference in elevation head across the pump, then
(59)H=H0-hlV-hleye
or
(60)H=H∞-hlslp-hlV-hleye.
Define a parameter x as
(61)x=HH0=1[1+(hlV/H)+(hleye/H)],or1x=1+hlVH+hleyeH.
Using (45)  as well as (57) and (61) then
(62)1x=1+CfV·CV3p2+CdV·CV3d2+CV3r2+Cfth·CV42+Ceye·CVeye2.
The sum of volute and eye head losses, from (59), is written as
(63)hlV+hleye=(1-x)H0=(1-x)σH∞.
Using (9), (29), and (59), the pump manometric head H is
(64)H=xσH∞=xσu2·Vu2g,H=xσu22g(1+ψφcotβb2).Coefficients. There are two additional groups of coefficients, namely, the pump head coefficients and head loss coefficients.The pump head coefficients are the pump manometric head coefficient,CH, Euler head coefficient, CH∞, and the head coefficient at impeller outlet, CH0. They are defined and given, respectively, as (65a)CH=Hu22/g=xσ(1+ψφcotβb2),(65b)CH∞=H∞u22/g=1+ψφcotβb2,(65c)CH0=H0u22/g=σ(1+ψφcotβb2).
According to (21) and (65b), y=CH∞.The head loss coefficients include the slippage head loss coefficient,Chlslp, the volute-eye head loss coefficient, ChlV+eye, the eye head loss coefficient, Chleye, and the volute head loss coefficient, ChlV. They are given, respectively, as
(65d)Chlslp=hlslpu22/g=(1-σ)CH∞,(65e)ChlV+eye=hlV+hleyeu22/g=(1-x)σCH∞,(65f)Chleye=hleyeu22/g=Ceye·CVeye2·12φ2,(65g)ChlV=hlVu22/g=(CfV·CV3p2+CdV·CV3d2+CV3r2CV3p2+Cfth·CV42)·12φ2.The relationship between the pump speed coefficient φ and the pump flow coefficient ψ is derived as follows.Using (11), (64) becomes
(66)H=xσu22gcotα2cotα2-cotβb2.
Dividing both sides by H, noting that u22/(2gH)=φ2, and using (11), (64) becomes a quadric equation for cotα2:
(67)cot2α2-cotβb2·cotα2-12xσψ2=0,
which has a solution (since cotα2 should be greater than cotβb2, whence only the +ve sign is considered)
(68)cotα2=12cotβb2+12cot2βb2+2xσψ2.
Multiplying (68) by ψ and then subtracting ψcotβb2 from both sides and using (11) yield the following relation for φ:
(69)φ=-12ψcotβb2+12ψ2cot2βb2+2xσ.


### 2.9. Pump Shaft PowerPsh, and Pump Shaft Head Hsh

The total shaft power required to drive the impeller is(70)Psh=Psh0′+Plf′+Plcirin′+PlD,
where Psh0′=γ(Q+QL)H0 is the impeller power given to water, Plf′=γ(Q+QL)hlf is the power lost in friction inside impeller, Plcirin′=γ(Q+QL)hlcirin is the power needed for given circulation to flow at impeller inlet, and PlD is the power lost in friction on outside surface of impeller disks.The total shaft power and pump shaft head can be simplified, respectively, to(71)Psh=Psh0+Plf+Plcirin+Plvol+PlD,
in which
(72)Psh0=γQH0,Plf=γQhlf,Plcirin=γQhlcirin,Plvol=γQhlvol=γQL(H0+hlf+hlcirin),PlD=γQhlD,
where hlf is the impeller skin friction head loss, hlcirin is the inlet shock circulation head loss, hlvol is the volumetric (leakage) head loss, and hlD is the disk friction head loss.The shaft headHsh=Psh/(γQ) and the shaft head coefficient CHsh=Hsh/(u22/g) are given, respectively, as
(73)Hsh=H0+hlf+hlcirin+hlvol+hlD,CHsh=CH0+Chlf+Chlcirin+Chlvol+ChlD,
where Chlf=hlf/(u22/g) is the impeller skin friction head loss coefficient, Chlcirin=hlcirin/(u22/g) is the inlet shock circulation head loss coefficient, Chlvol=hlvol/(u22/g) is the volumetric head loss coefficient, and ChlD=hlD/(u22/g) is the disk friction head loss coefficient.


### 2.10. Pump Efficiency (Manometric Efficiency)η

The pump manometric efficiency is the ratio of gained water power (Pw) to the pump shaft power (Psh) supplied to pump impeller. According to its definition, it takes the following forms:
(74)η=PwPsh=PwPsh0+Plf+Plcirin+Plvol+PlD,η=HHsh=H∞-hlslp-hlV-hleyeH0+hlf+hlcirin+hlvol+hlD,η=CHCHsh=CH∞-Chlslp-ChlV+eyeCH0+Chlf+Chlcirin+Chlvol+ChlD.


### 2.11. Pump Shaft Power CoefficientCPsh

The pump shaft power coefficient and the pump water power coefficient are given, respectively, as(75)CPsh=Pshρ(N/60)3D25=π2CQCHsh,(76)CPw=Pwρ(N/60)3D25=π2CQCH.


### 2.12. Impeller Skin Friction Power Head Losshlf

The impeller hydraulic friction head losshlf is estimated by the theory of flow through pipes and is given by [11]:
(77)hlf=4CdiLbDhydWav22g,
where Cdi is the dissipation coefficient, Lb is the blade length, Dhyd is the hydraulic diameter, and Wav is the average relative velocity. Therefore, the impeller skin friction head loss coefficient could be calculated from
(78)Chlf=hlfu22/g=4Cdi(Lb/D2)(Dhyd/D2)12(Wavu2)2.
The hydraulic diameter and the average relative velocity are given, respectively, as, Gülich [11]
(79)Dhyd=4*AreaPerimeter=2(d1b1+d2b2)d1+b1+d2+b2,Wav=Qi/ZAav=Qi/Z(d1b1+d2b2)/2.
Substituting for d1, (26), d2, (15), and Qi, (3), yield:
(80)DhydD2=2((d1/D2)(b1/b2)(b2/D2)+(d2/D2)(b2/D2))(d1/D2)+(b1/b2)(b2/D2)+(d2/D2)+(b2/D2),(81)Wavu2=ε2(π/Z)(b2/D2)1/2((d1/D2)(b1/b2)(b2/D2)+(d2/D2)(b2/D2))·ψφ.
The blade length is Lb=((1/2)(D2-D1))/sinβbm, and thus (82a)LbD2=12(1-D1D2)1sinβbm,
where
(82b)βbm=(βb1+βb2)2.The impeller dissipation coefficient is given as, Gülich [11]
(83)4Cdi=(fi+0.006)(1.1+4b2D2).
The impeller friction coefficient fi is function of the average impeller Reynolds number Re and the roughness κ [10]:
(84)fi=0.3086{log[(6.9/Re)+((κ/Dhyd)/3.7)1.11]}2,
where
(85)κDhyd=(κ/D2)(Dhyd/D2).(86)Re=Wav·Dhydν=2(Wavu2)(DhydD2)·Re2,
where Re2 is defined by (53).


### 2.13. Disk Friction Head LosshlD

The disk friction power lossPlD is the power loss in the fluid between external surfaces of the impeller disks and internal walls of the pump casing, Figure 7. The PlD can be estimated as
(87)PlD=2*∫0r2ω·r·τ·2πrdr
with
(88)τ=fD4·12ρu2=fD4·12ρ(ωr)2,
where τ is the shear stress in circumferential direction and fD is the disk skin friction coefficient, and it is assumed constant along the disk surface.Figure 7
Disk and wearing ring clearances.Thus,(89)PlD=π2ρω3∫0r2fDr4dr.
Therefore, the final expression after the integration for the disk friction power loss and, consequently, the impeller disk friction head loss are
(90)PlD=π2fDρω3r255=π40fDρu23D22,hlD=PlDρgQ=π40fDgu23D22Q.The impeller disk friction head loss coefficientChlD is given as
(91)ChlD≡hlDu22/g=KD·1ηvolφψ,
where KD is the impeller disk loss coefficient,(92a)KD=140fDε2(b2/D2).
It is derived by substituting for Q from (2) and (3) and using the definition of ψ which yields Q=ηvolε2·ψ2gH·πD2b2.Correlations forfD were obtained by Kruyt, cited in [12]. Four different regimes were identified, Figure 8: Regime I (laminar flow, boundary layers have merged), Regime II (laminar flow with two separate boundary layers), Regime III (turbulent flow, boundary layers have merged), and Regime IV (turbulent flow with two separate boundary layers). These regimes are characterized by the Reynolds number, Re2=u2·(D2/2)/ν, and a nondimensional gap parameter, G=y0/(D2/2), where y0 is the axial gap between impeller disk and casing (Figure 7). The equations of curves 1 to 5 in Figure 8 are G=1.62Re2-5/11, G=188Re2-9/10, G=0.57*10-6Re215/16, Re=1.58*105, and G=0.402Re2-3/16, respectively. The disk friction coefficient for each regime is given as
(92b)RegimeI:fD=10G-1Re2-1RegimeII:fD=18.5πG1/10Re2-1/2RegimeIII:fD=0.4πG-1/6Re2-1/4RegimeIV:fD=0.51πG1/10Re2-1/5.Figure 8
Disk friction flow regimes [12].


### 2.14. Volumetric Head Losshlvol

The volumetric (leakage) head loss and the volumetric head loss coefficient are given as follows after using (2):
(93)hlvol=PlvolγQ=QLQ(H0+hlf+hlcirin)=(1ηvol-1)·(H0+hlf+hlcirin)Chlvol=hlvolu22/g=(1ηvol-1)·(CH0+Chlf+Chlcirin).
The leakage flow rate QL can be estimated using orifice formula [13]:
(94)QL=CdL·ac2gΔHc,
where ac is the clearance area of wearing ring (=πDeyeyc), yc is the clearance of wearing ring, Figure 7, CdL is the leakage discharge coefficient (≈0.6), and ΔHc is the pressure head drop across the clearance:
(95)ΔHc=pc-psγ=p3-psγ-p3-pcγ,
where pc is the pressure before clearance and ps is the pressure after clearance at suction side of the impeller (Figure 7).The pressure head difference between volute and pump suction side equals(96)p3-psγ=p5-psγ+p3-p5γ=H+[hlV-(V422g-V522g)]=H0-hleye-(V422g-V522g).
The pressure distribution along the radial direction of impeller shroud is parabolic [13], and thus the pressure head difference between impeller outlet and before clearance is given by
(97)p3-pcγ=u228g(1-Deye2D22).Thus, the leakage flow rate and the leakage flow rate coefficient,CQL, are
(98)QL=CdL·πDeyeyc·u22*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18(1-Deye2D22))1/2,CQL=QL(N/60)D23=ycDeye2π2CdL(D2/Deye)2*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18(1-Deye2D22)).1/2
Using (2), QL/Q=CQL/CQ, and (8) for CQ, the pump volumetric efficiency can be deduced after some mathematical manipulations:
(99)ηvol=1-(ycDeye)(DeyeD1)2(D1D2)22CdL(b2/D2)ε2φψ*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18[1-(DeyeD1)2(D1D2)2])1/2.


### 2.15. Inlet Shock Circulation Head Losshlcirin

At impeller inlet, a shock circulation exists inside pump eye when the flow discharge differs from that of designed one. As discussed before in Section2.3, the tangential velocity responsible for this shock eddy is Δ′′Wu1, which could be estimated from (28).Therefore, the inlet shock circulation head loss equals(100)hlcirin=Δ′′Wu1·u1g.
Thus, the inlet shock circulation head loss coefficient is
(101)Chlcirin=hlcirinu22/g=(1-ε1πZsinβb1)(D1D2)2-cot(180-βb1)(b1/b2)(ε1/ε2)·ψφ.


### 2.16. Required Net Positive Suction HeadNPSHR

The pump net positive suction headNPSH is defined as the difference between the fluid inlet stagnation pressure head and vapour pressure head [13]:
(102)NPSH=(p1γ+V122g)-pvγ,
where p1, V1 are the absolute pressure and absolute velocity of fluid at impeller inlet and pv is the absolute vapour pressure of fluid at the corresponding fluid temperature.In the vicinity of the leading edge of the impeller blades, the fluid has to accelerate in order to follow the rotating movement of the blades. This acceleration leads to a drop of the static pressure, which results in a local minimum pressure at blade inlet:(103)pminγ=p1γ-(W1b22g-V122g).
Therefore, the pump required net positive suction head is
(104)NPSHR=(pmin-pv)γ+W1b22g.
The pump NPSHR coefficient is defined as
(105)CNPSHR=NPSHRu22/g.
Referring to Figure 5, W1b=(Vr1/sinβb1), and thus
(106)CNPSHR=(pmin-pv)/γu22/g+1sin2βb1Vr122u22.
The term ((pmin-pv)/γ)/(u22/g) is a design parameter for the pump and could be assumed to be 0.02.From (3) Vr1=Vr2(D2/D1)(b2/b1)(ε2/ε1), and therefore the NPSHR coefficient is
(107)CNPSHR=0.02+12sin2βb1(ψ/φ)2(D1/D2)2(b1/b2)2(ε1/ε2)2.


### 2.17. Pump Specific Speedns

The pump specific speed is defined with suitable units as(108)ns=NQH3/4[N(rpm),H(m),Q(lit/s)].
Substitute for
(109)Q=1000ηvolε2·πD2b2·ψ2gH(lit/s),N=60πD2φ2gH,andb2=D2(b2D2)yield:ns=60*1000*(2g)3/4*ε2π(b2D2)*ηvol·φψ.
Therefore,
(110)ns=9977ε2(b2D2)·ηvol·φψ.


## 2.1. Pump Leakage

Due to the difference between the outlet pressure and the inlet pressure of the impeller, a portion of impeller outlet flow rate,QL, returns to the impeller inlet from the existing clearances between the impeller and the casing, Figure 1. This internal discharge leakage, QL, causes some losses as the flow rate through the impeller (Q+QL) is greater than the pump useful outlet discharge Q.Figure 1
Pump impeller notations.The volumetric efficiencyηvol of pump is defined as the ratio of pump outlet discharge to the impeller discharge:
(2)ηvol=QQi=QQ+QLorQLQ=1ηvol-1,
in which
(3)Qi=Q+QL=Vr2·πD2b2ε2=Vr1·πD1b1ε1,
where b1 is the blade width at inlet, D1 is the impeller diameter at inlet, ε1 is the blade thickness coefficient at impeller inlet, ε1=1-(Z/π)((t/sinβb1)/D1),  b2 is the blade width at outlet, D2 is the impeller diameter at outlet, ε2 is the blade thickness coefficient at impeller outlet, ε2=1-(Z/π)((t/sinβb2)/D2),  t is the blade thickness, Z is the number of blades, βb2 is the blade angle at impeller outlet, and βb1 is the blade angle at impeller inlet. The value of ε2 is about 0.95 [6]. The relationship between ε1 and ε2 with constant blade thickness is given as
(4)ε1=1-(1-ε2)(D1/D2)(sinβb1/sinβb2).The impeller inlet flow velocity coefficientCV1=V1/2gH is calculated from (3) dividing both sides by 2gH and assuming that the flow enters the impeller without swirl (V1=Vr1):
(5)CV1=ψ(D1/D2)(b1/b2)(ε1/ε2).The pump discharge coefficient isCQ=Q/((N/60)D23), and substituting for Q from (2) and (3);
(6)CQ=ηvolε2π2b2D2·ψφ.


## 2.2. Outlet Velocity Diagram

A fluid slippage occurs at impeller exit due to the relative rotation of fluid in a direction opposite to that of impeller. A slip factorσ defined as σ=Vu2act/Vu2 could be estimated later in Section 2.3.From the velocity diagram at the impeller outlet, Figure2, the tangential component of the outlet flow absolute velocity is given as
(7)Vu2=Vr2cotα2=u2+Vr2cotβb2.Figure 2
Velocity diagram at impeller outlet.The ratio of outlet swirl velocity to outlet tangential velocity is(8)Vu2u2=1+Vr2u2cotβb2=1+ψφcotβb2,
and thus,
(9)Vu2actu2=σ·Vu2u2=σ(1+ψφcotβb2).The outlet tangential velocity (from (7)) and the pump speed coefficient φ=u2/2gH are given as
(10)u2=Vr2(cotα2-cotβb2),(11)φ=ψ(cotα2-cotβb2).From the outlet velocity diagram (Figure2) and (7), the outlet velocity and the outlet velocity coefficient defined as CV2=V2/2gH are
(12)V22=Vr22+Vu22=Vr22+(u2+Vr2cotβb2)2,CV22=ψ2+(φ+ψcotβb2)2.


## 2.3. Slip Factor

### 2.3.1. The Relative Eddy (Eddy Circulation)

A simple explanation for the slip effect in an impeller is obtained from the idea of a relative eddy. Suppose that an irrotational and frictionless fluid flow is possible which passes through an impeller. If the absolute flow enters the impeller without spin, then at outlet the spin of the absolute flowmust still be zero. The impeller itself has an angular velocity ω so that, relative to the impeller, the fluid has an angular velocity of -ω; this is termed the relative eddy. At outlet of impeller, the relative flow, W2, can be regarded as a through flow on which a relative eddy is superimposed. The net effect of these two motions is that the average relative flow emerging from the impeller passages is at an angle to the vanes and in a direction opposite to the blade motion.One of the earliest and simplest expressions for the slip factor was obtained by Stodola, cited in [7]. Referring to Figure 3, the slip velocity due to relative eddy, ΔWu2=ΔVu2=Vu2-Vu2act, is considered to be the product of the relative eddy and the radius (d2/2) of a circle which can be inscribed within the channel.Figure 3
Flow model for Stodola slip factor.Thus,(13)ΔWu2=ωd22.
An approximate expression for d2 can be written if the number of blades Z is not small:
(14)d2=πD2Zsinβb2′-t=ε2πD2Zsinβb2,(15)d2D2=ε2πZsinβb2.
Since the relative eddy angular velocity  =2u2/D2, then
(16)ΔWu2=ε2πu2Zsinβb2,orΔWu2u2=ε2πZsinβb2.
The slip factor is given by
(17)σ=Vu2actVu2=1-ΔVu2Vu2=1-ΔWu2/u2Vu2/u2.
Substituting for ΔWu2/u2 from (16) and for Vu2/u2 from (8), the formulae proposed by Stodola, cited in [7] for the calculation of slip factor, σ, are obtained which are
(18)σ=1-ε2(π/Z)sinβb21+(Vr2/u2)cotβb2=1-ε2(π/Z)sinβb21+(ψ/φ)cotβb2.
At ψ=0 (σ=σ0)
(19)1-σ0=ε2πZsinβb2.
Therefore,
(20)1-σ=1-σ0y,
where
(21)y=1+ψφcotβb2.Wiesner [8] introduced an empirical expression which extremely well fits the experimental results of slip factor for wide range of practical blade angles and number of blades. It is used in this work and given as
(22)1-σ0=sinβb2Z0.7forD1D2≤εlimit,
where εlimit is the limiting diameter ratio,
(23)εlimit=e-(8.16sinβb2/Z).It is assumed that the water entering the pump impeller is purely in the radial direction. Relative to the impeller, the fluid has an angular velocity of-ω, relative eddy, and thus the relative flow at blade inlet acquires an additional component Δ′Wu1 opposite to rotational direction, as seen in Figure 4, which is
(24)Δ′Wu1=ωd12,
where
(25)d1=ε1πD1Zsinβb1,(26)d1D2=ε1πZD1D2sinβb1.
Hence,
(27)Δ′Wu1=ε1πZu1sinβb1.Figure 4
Velocity diagram at impeller inlet without shock.When the pump that operates at a dischargeQ differs from that at designed condition Q*, the relative flow velocity at blade inlet tends to acquire an additional component in counter of the rotational direction, Δ′′Wu1. So, the flow enters the blade passage tangent to the blade surface, and a shock eddy or a shock circulation exists prior to the blade leading edge inside pump eye. As could be noticed from Figure 5, the relative velocity additional rotational speed at blade inlet equals
(28)Δ′′Wu1=u1(1-ε1πZsinβb1)-Vr1cot(180-βb1).Figure 5
Velocity diagram at blade inlet with shock eddy.


## 2.3.1. The Relative Eddy (Eddy Circulation)

A simple explanation for the slip effect in an impeller is obtained from the idea of a relative eddy. Suppose that an irrotational and frictionless fluid flow is possible which passes through an impeller. If the absolute flow enters the impeller without spin, then at outlet the spin of the absolute flowmust still be zero. The impeller itself has an angular velocity ω so that, relative to the impeller, the fluid has an angular velocity of -ω; this is termed the relative eddy. At outlet of impeller, the relative flow, W2, can be regarded as a through flow on which a relative eddy is superimposed. The net effect of these two motions is that the average relative flow emerging from the impeller passages is at an angle to the vanes and in a direction opposite to the blade motion.One of the earliest and simplest expressions for the slip factor was obtained by Stodola, cited in [7]. Referring to Figure 3, the slip velocity due to relative eddy, ΔWu2=ΔVu2=Vu2-Vu2act, is considered to be the product of the relative eddy and the radius (d2/2) of a circle which can be inscribed within the channel.Figure 3
Flow model for Stodola slip factor.Thus,(13)ΔWu2=ωd22.
An approximate expression for d2 can be written if the number of blades Z is not small:
(14)d2=πD2Zsinβb2′-t=ε2πD2Zsinβb2,(15)d2D2=ε2πZsinβb2.
Since the relative eddy angular velocity  =2u2/D2, then
(16)ΔWu2=ε2πu2Zsinβb2,orΔWu2u2=ε2πZsinβb2.
The slip factor is given by
(17)σ=Vu2actVu2=1-ΔVu2Vu2=1-ΔWu2/u2Vu2/u2.
Substituting for ΔWu2/u2 from (16) and for Vu2/u2 from (8), the formulae proposed by Stodola, cited in [7] for the calculation of slip factor, σ, are obtained which are
(18)σ=1-ε2(π/Z)sinβb21+(Vr2/u2)cotβb2=1-ε2(π/Z)sinβb21+(ψ/φ)cotβb2.
At ψ=0 (σ=σ0)
(19)1-σ0=ε2πZsinβb2.
Therefore,
(20)1-σ=1-σ0y,
where
(21)y=1+ψφcotβb2.Wiesner [8] introduced an empirical expression which extremely well fits the experimental results of slip factor for wide range of practical blade angles and number of blades. It is used in this work and given as
(22)1-σ0=sinβb2Z0.7forD1D2≤εlimit,
where εlimit is the limiting diameter ratio,
(23)εlimit=e-(8.16sinβb2/Z).It is assumed that the water entering the pump impeller is purely in the radial direction. Relative to the impeller, the fluid has an angular velocity of-ω, relative eddy, and thus the relative flow at blade inlet acquires an additional component Δ′Wu1 opposite to rotational direction, as seen in Figure 4, which is
(24)Δ′Wu1=ωd12,
where
(25)d1=ε1πD1Zsinβb1,(26)d1D2=ε1πZD1D2sinβb1.
Hence,
(27)Δ′Wu1=ε1πZu1sinβb1.Figure 4
Velocity diagram at impeller inlet without shock.When the pump that operates at a dischargeQ differs from that at designed condition Q*, the relative flow velocity at blade inlet tends to acquire an additional component in counter of the rotational direction, Δ′′Wu1. So, the flow enters the blade passage tangent to the blade surface, and a shock eddy or a shock circulation exists prior to the blade leading edge inside pump eye. As could be noticed from Figure 5, the relative velocity additional rotational speed at blade inlet equals
(28)Δ′′Wu1=u1(1-ε1πZsinβb1)-Vr1cot(180-βb1).Figure 5
Velocity diagram at blade inlet with shock eddy.


## 2.4. Euler Equation of Turbomachines

In the case that the fluid entering the pump impeller is purely in the radial direction without swirl, the pump Euler head is given as [6, 7]
(29)H∞=u2Vu2g.
Due to the fluid slippage at impeller exit, the actual head given to fluid by the impeller, H0, is calculated from [6, 7]:
(30)H0=u2·Vu2actg=u2·σVu2g=σ·H∞.
And so, the slippage head loss is
(31)hlslp=H∞-H0=(1-σ)H∞.


## 2.5. Pump Volute

The flow that discharges from the impeller requires careful handling in order to preserve the gains in energy imparted to the fluid. This requires the conversion of velocity head to pressure head by means of a diffuser, and this inevitably implies hydraulic losses. The application of mass conservation to a volute element, [9], reveals that the discharge flow from impeller is matched to the flow in the volute if dAV/dθ=Vr3/Vu3r3b3. This requires a circumferentially uniform rate of increase of the volute area AV over the entire development of the spiral (θ). Consequently, for a given impeller, there exist a specific volute angle αV and a specific volute throat inlet area AVth for the volute geometry.The volute angleαV, Figure 6, is chosen to match the angle of flow entering the volute α3eq at a certain pump operating point. With a uniform rate of increase of the volute area, the volute angle is
(32)tanαV=AVthπD3b3=AVthπD2b2(D3/D2)(b3/b2).
The volute outlet flow velocity V4=Q/AVth and the volute outlet flow velocity coefficient CV4=V4/2gH are
(33)V4=ηvol·Vr2πD2b2ε2πD3b3·tanαV=ηvolε2tanαVD2D3b2b3·Vr2,CV4=ε2ηvol(D3/D2)(b3/b2)tanαV·ψ.
The throat is assumed to have an expanding angle αth, and hence the throat diameter equals
(34)Dth=πD3tanαV+Lthtanαth.
With the assumption that the throat height Lth=D3, thus
(35)DthD3=πtanαV+tanαth.
The throat outlet flow velocity V5=Q/Ath, and using (2) and (3), then
(36)V5=4ε2ηvolD22Dth2b2D2·Vr2.
The throat outlet flow velocity coefficient CV5(=V5/2gH) equals
(37)CV5=4ε2ηvol(b2/D2)(Dth/D3)2(D3/D2)2·ψ.
It is assumed that the throat diameter Dth equals the eye diameter Deye; that is, Deye/Dth=1, which equals the suction pipe diameter Ds. Thus, the throat outlet flow velocity, V5, equals the flow velocity at suction pipe, Vs. Therefore,
(38)V5=Vs=ηvol·Veye.
And the eye velocity coefficient is
(39)CVeye=Veye2gH=CV5ηvol=4ε2(b2/D2)(Dth/D3)2(D3/D2)2·ψ.
The eye diameter relative to the impeller inlet diameter is therefore
(40)DeyeD1=DeyeDthDthD3D3D2D2D1=(D3/D2)(πtanαV+tanαth)(D1/D2).
The ratio (Deye/D1) must not exceed 1, and thus an upper limit is imposed on the volute angle:
(41)tanαV<(D1/D2)π(D3/D2)-tanαthπ.
The throat has a cone angle θth, where
(42)tan(θth2)=(Dth-b3)/2Lth.
Substituting from (35),
(43)tan(θth2)=12[πtanαV+tanαth-(b3/b2)(b2/D2)(D3/D2)].Figure 6
Pump and volute notations.


## 2.6. Volute Loss Model

Prediction models that account for the main features of the swirling flow in volutes, reviewed in [4], do not account for the circulatory flow initiated in volute at off-design discharge operation of pump. In this work, a model is proposed to account the volute head loss at off-design pump operation.The flow enters the volute with a through-velocityV3 at an angle α3 (which may differ from that of the volute angle αV) on which an eddy of a tangential velocity ΔVu3=ΔVu2 is superimposed opposite to impeller motion. This inlet volute velocity V3 is decomposed into a velocity parallel to the direction of volute, V3p, and another one in the tangential direction of impeller, V3d′ (Figure 6). The velocity component V3d′ is the motive of a second circulatory motion given to the volute flow in direction of impeller motion. Thus, the net circulation velocity of flow in volute is V3d=V3d′-ΔVu3 in direction of impeller motion. The component of the velocity V3p in the tangential direction denoted as V3pu equals the volute outlet velocity V4.The volute loss model counts friction loss associated with the volute throw flow velocity, diffusion friction loss due to circulation associated with volute flow, loss due to vanishing of radial flow at volute outlet, and loss inside pump volute throat. Consequently, the volute head loss and the volute head loss relative to the pump head are written, respectively, as(44)hlV=CfVV3p22g+CdVV3d22g+V3r22g+hlth,(45)hlVH=CfV·CV3p2+CdV·CV3d2+CV3r2+Cfth·CV42,
where CfV is the volute friction loss coefficient:
(46)CfV=fVLVDhV=fV(LVD2)(1DhV/D2),
where fV is the volute friction coefficient, DhV is the volute hydraulic diameter, and LV is the average volute length:
(47)LV=12πD3cosαV,(48)LVD2=12πcosαV(D3D2).
The average volute hydraulic diameter relative to impeller diameter is
(49)1DhV/D2=12(b3/b2)(b2/D2)+18(π/Z)(D3/D2)sinαV.
The volute friction coefficient fV which corresponds to a pipe flow is function of the volute Reynolds number ReV and the roughness κ, [10]:
(50)fV=0.3086{log[(6.9/ReV)+((κ/DhV)/3.7)1.11]}2,
where
(51)κDhV=(κ/D2)(DhV/D2).
The volute Reynolds number is calculated as
(52)ReV=V3p·DhVν=2CV3pφ(DhVD2)·Re2,
where
(53)Re2=u2·D2/2ν.
In the first term of (44) and (45), V3p is the volute throw flow velocity (Figure 6) and given as(54a)V3p=V4cosαV,orCV3p=CV4cosαV.
In the second term of (44) and (45), CdV is the volute diffusion loss coefficient which could be assumed to have the value CdV=0.8, and V3d is the volute circulatory velocity component:
(54b)V3d=Vu2act-V4CV3d=V3d2gH=σ(φ+ψcotβb2)-CV4.
The third term of (45) is
(54c)CV3r=V3r2gH=ε2ηvol(D3/D2)(b3/b2)·ψ.In the fourth term of (44) and (45), the volute throat head loss hlth could be calculated as
(55)hlth=CfthV422g,
where Cfth is the volute throat friction loss coefficient assumed to be, [10],
(56)Cfth=0.5+2.6*sin(θth2),
where θth is the throat cone angle.


## 2.7. Pump Eye Head Losshleye

The head loss in pump eyehleye, [2], and the eye head loss relative to the pump head are
(57)hleye=CeyeVeye22g,hleyeH=CeyeVeye22gH=Ceye·CVeye2,
where Veye is the flow velocity in pump eye and Ceye is the eye loss coefficient defined by (39).


## 2.8. Pump Manometric HeadH

The pump manometric head is the difference in static pressure heads between pump outlet and pump eye:(58)H=p5-peyeγ.
With the assumption that the throat diameter Dth equals the eye diameter Deye, the flow velocity is the same at pump suction pipe and pump delivery pipe (Vs=V5) and neglecting the difference in elevation head across the pump, then
(59)H=H0-hlV-hleye
or
(60)H=H∞-hlslp-hlV-hleye.
Define a parameter x as
(61)x=HH0=1[1+(hlV/H)+(hleye/H)],or1x=1+hlVH+hleyeH.
Using (45)  as well as (57) and (61) then
(62)1x=1+CfV·CV3p2+CdV·CV3d2+CV3r2+Cfth·CV42+Ceye·CVeye2.
The sum of volute and eye head losses, from (59), is written as
(63)hlV+hleye=(1-x)H0=(1-x)σH∞.
Using (9), (29), and (59), the pump manometric head H is
(64)H=xσH∞=xσu2·Vu2g,H=xσu22g(1+ψφcotβb2).Coefficients. There are two additional groups of coefficients, namely, the pump head coefficients and head loss coefficients.The pump head coefficients are the pump manometric head coefficient,CH, Euler head coefficient, CH∞, and the head coefficient at impeller outlet, CH0. They are defined and given, respectively, as (65a)CH=Hu22/g=xσ(1+ψφcotβb2),(65b)CH∞=H∞u22/g=1+ψφcotβb2,(65c)CH0=H0u22/g=σ(1+ψφcotβb2).
According to (21) and (65b), y=CH∞.The head loss coefficients include the slippage head loss coefficient,Chlslp, the volute-eye head loss coefficient, ChlV+eye, the eye head loss coefficient, Chleye, and the volute head loss coefficient, ChlV. They are given, respectively, as
(65d)Chlslp=hlslpu22/g=(1-σ)CH∞,(65e)ChlV+eye=hlV+hleyeu22/g=(1-x)σCH∞,(65f)Chleye=hleyeu22/g=Ceye·CVeye2·12φ2,(65g)ChlV=hlVu22/g=(CfV·CV3p2+CdV·CV3d2+CV3r2CV3p2+Cfth·CV42)·12φ2.The relationship between the pump speed coefficient φ and the pump flow coefficient ψ is derived as follows.Using (11), (64) becomes
(66)H=xσu22gcotα2cotα2-cotβb2.
Dividing both sides by H, noting that u22/(2gH)=φ2, and using (11), (64) becomes a quadric equation for cotα2:
(67)cot2α2-cotβb2·cotα2-12xσψ2=0,
which has a solution (since cotα2 should be greater than cotβb2, whence only the +ve sign is considered)
(68)cotα2=12cotβb2+12cot2βb2+2xσψ2.
Multiplying (68) by ψ and then subtracting ψcotβb2 from both sides and using (11) yield the following relation for φ:
(69)φ=-12ψcotβb2+12ψ2cot2βb2+2xσ.


## 2.9. Pump Shaft PowerPsh, and Pump Shaft Head Hsh

The total shaft power required to drive the impeller is(70)Psh=Psh0′+Plf′+Plcirin′+PlD,
where Psh0′=γ(Q+QL)H0 is the impeller power given to water, Plf′=γ(Q+QL)hlf is the power lost in friction inside impeller, Plcirin′=γ(Q+QL)hlcirin is the power needed for given circulation to flow at impeller inlet, and PlD is the power lost in friction on outside surface of impeller disks.The total shaft power and pump shaft head can be simplified, respectively, to(71)Psh=Psh0+Plf+Plcirin+Plvol+PlD,
in which
(72)Psh0=γQH0,Plf=γQhlf,Plcirin=γQhlcirin,Plvol=γQhlvol=γQL(H0+hlf+hlcirin),PlD=γQhlD,
where hlf is the impeller skin friction head loss, hlcirin is the inlet shock circulation head loss, hlvol is the volumetric (leakage) head loss, and hlD is the disk friction head loss.The shaft headHsh=Psh/(γQ) and the shaft head coefficient CHsh=Hsh/(u22/g) are given, respectively, as
(73)Hsh=H0+hlf+hlcirin+hlvol+hlD,CHsh=CH0+Chlf+Chlcirin+Chlvol+ChlD,
where Chlf=hlf/(u22/g) is the impeller skin friction head loss coefficient, Chlcirin=hlcirin/(u22/g) is the inlet shock circulation head loss coefficient, Chlvol=hlvol/(u22/g) is the volumetric head loss coefficient, and ChlD=hlD/(u22/g) is the disk friction head loss coefficient.


## 2.10. Pump Efficiency (Manometric Efficiency)η

The pump manometric efficiency is the ratio of gained water power (Pw) to the pump shaft power (Psh) supplied to pump impeller. According to its definition, it takes the following forms:
(74)η=PwPsh=PwPsh0+Plf+Plcirin+Plvol+PlD,η=HHsh=H∞-hlslp-hlV-hleyeH0+hlf+hlcirin+hlvol+hlD,η=CHCHsh=CH∞-Chlslp-ChlV+eyeCH0+Chlf+Chlcirin+Chlvol+ChlD.


## 2.11. Pump Shaft Power CoefficientCPsh

The pump shaft power coefficient and the pump water power coefficient are given, respectively, as(75)CPsh=Pshρ(N/60)3D25=π2CQCHsh,(76)CPw=Pwρ(N/60)3D25=π2CQCH.


## 2.12. Impeller Skin Friction Power Head Losshlf

The impeller hydraulic friction head losshlf is estimated by the theory of flow through pipes and is given by [11]:
(77)hlf=4CdiLbDhydWav22g,
where Cdi is the dissipation coefficient, Lb is the blade length, Dhyd is the hydraulic diameter, and Wav is the average relative velocity. Therefore, the impeller skin friction head loss coefficient could be calculated from
(78)Chlf=hlfu22/g=4Cdi(Lb/D2)(Dhyd/D2)12(Wavu2)2.
The hydraulic diameter and the average relative velocity are given, respectively, as, Gülich [11]
(79)Dhyd=4*AreaPerimeter=2(d1b1+d2b2)d1+b1+d2+b2,Wav=Qi/ZAav=Qi/Z(d1b1+d2b2)/2.
Substituting for d1, (26), d2, (15), and Qi, (3), yield:
(80)DhydD2=2((d1/D2)(b1/b2)(b2/D2)+(d2/D2)(b2/D2))(d1/D2)+(b1/b2)(b2/D2)+(d2/D2)+(b2/D2),(81)Wavu2=ε2(π/Z)(b2/D2)1/2((d1/D2)(b1/b2)(b2/D2)+(d2/D2)(b2/D2))·ψφ.
The blade length is Lb=((1/2)(D2-D1))/sinβbm, and thus (82a)LbD2=12(1-D1D2)1sinβbm,
where
(82b)βbm=(βb1+βb2)2.The impeller dissipation coefficient is given as, Gülich [11]
(83)4Cdi=(fi+0.006)(1.1+4b2D2).
The impeller friction coefficient fi is function of the average impeller Reynolds number Re and the roughness κ [10]:
(84)fi=0.3086{log[(6.9/Re)+((κ/Dhyd)/3.7)1.11]}2,
where
(85)κDhyd=(κ/D2)(Dhyd/D2).(86)Re=Wav·Dhydν=2(Wavu2)(DhydD2)·Re2,
where Re2 is defined by (53).


## 2.13. Disk Friction Head LosshlD

The disk friction power lossPlD is the power loss in the fluid between external surfaces of the impeller disks and internal walls of the pump casing, Figure 7. The PlD can be estimated as
(87)PlD=2*∫0r2ω·r·τ·2πrdr
with
(88)τ=fD4·12ρu2=fD4·12ρ(ωr)2,
where τ is the shear stress in circumferential direction and fD is the disk skin friction coefficient, and it is assumed constant along the disk surface.Figure 7
Disk and wearing ring clearances.Thus,(89)PlD=π2ρω3∫0r2fDr4dr.
Therefore, the final expression after the integration for the disk friction power loss and, consequently, the impeller disk friction head loss are
(90)PlD=π2fDρω3r255=π40fDρu23D22,hlD=PlDρgQ=π40fDgu23D22Q.The impeller disk friction head loss coefficientChlD is given as
(91)ChlD≡hlDu22/g=KD·1ηvolφψ,
where KD is the impeller disk loss coefficient,(92a)KD=140fDε2(b2/D2).
It is derived by substituting for Q from (2) and (3) and using the definition of ψ which yields Q=ηvolε2·ψ2gH·πD2b2.Correlations forfD were obtained by Kruyt, cited in [12]. Four different regimes were identified, Figure 8: Regime I (laminar flow, boundary layers have merged), Regime II (laminar flow with two separate boundary layers), Regime III (turbulent flow, boundary layers have merged), and Regime IV (turbulent flow with two separate boundary layers). These regimes are characterized by the Reynolds number, Re2=u2·(D2/2)/ν, and a nondimensional gap parameter, G=y0/(D2/2), where y0 is the axial gap between impeller disk and casing (Figure 7). The equations of curves 1 to 5 in Figure 8 are G=1.62Re2-5/11, G=188Re2-9/10, G=0.57*10-6Re215/16, Re=1.58*105, and G=0.402Re2-3/16, respectively. The disk friction coefficient for each regime is given as
(92b)RegimeI:fD=10G-1Re2-1RegimeII:fD=18.5πG1/10Re2-1/2RegimeIII:fD=0.4πG-1/6Re2-1/4RegimeIV:fD=0.51πG1/10Re2-1/5.Figure 8
Disk friction flow regimes [12].


## 2.14. Volumetric Head Losshlvol

The volumetric (leakage) head loss and the volumetric head loss coefficient are given as follows after using (2):
(93)hlvol=PlvolγQ=QLQ(H0+hlf+hlcirin)=(1ηvol-1)·(H0+hlf+hlcirin)Chlvol=hlvolu22/g=(1ηvol-1)·(CH0+Chlf+Chlcirin).
The leakage flow rate QL can be estimated using orifice formula [13]:
(94)QL=CdL·ac2gΔHc,
where ac is the clearance area of wearing ring (=πDeyeyc), yc is the clearance of wearing ring, Figure 7, CdL is the leakage discharge coefficient (≈0.6), and ΔHc is the pressure head drop across the clearance:
(95)ΔHc=pc-psγ=p3-psγ-p3-pcγ,
where pc is the pressure before clearance and ps is the pressure after clearance at suction side of the impeller (Figure 7).The pressure head difference between volute and pump suction side equals(96)p3-psγ=p5-psγ+p3-p5γ=H+[hlV-(V422g-V522g)]=H0-hleye-(V422g-V522g).
The pressure distribution along the radial direction of impeller shroud is parabolic [13], and thus the pressure head difference between impeller outlet and before clearance is given by
(97)p3-pcγ=u228g(1-Deye2D22).Thus, the leakage flow rate and the leakage flow rate coefficient,CQL, are
(98)QL=CdL·πDeyeyc·u22*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18(1-Deye2D22))1/2,CQL=QL(N/60)D23=ycDeye2π2CdL(D2/Deye)2*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18(1-Deye2D22)).1/2
Using (2), QL/Q=CQL/CQ, and (8) for CQ, the pump volumetric efficiency can be deduced after some mathematical manipulations:
(99)ηvol=1-(ycDeye)(DeyeD1)2(D1D2)22CdL(b2/D2)ε2φψ*(CH0-CV42-CV52+Ceye·CVeye22φ2cccc-18[1-(DeyeD1)2(D1D2)2])1/2.


## 2.15. Inlet Shock Circulation Head Losshlcirin

At impeller inlet, a shock circulation exists inside pump eye when the flow discharge differs from that of designed one. As discussed before in Section2.3, the tangential velocity responsible for this shock eddy is Δ′′Wu1, which could be estimated from (28).Therefore, the inlet shock circulation head loss equals(100)hlcirin=Δ′′Wu1·u1g.
Thus, the inlet shock circulation head loss coefficient is
(101)Chlcirin=hlcirinu22/g=(1-ε1πZsinβb1)(D1D2)2-cot(180-βb1)(b1/b2)(ε1/ε2)·ψφ.


## 2.16. Required Net Positive Suction HeadNPSHR

The pump net positive suction headNPSH is defined as the difference between the fluid inlet stagnation pressure head and vapour pressure head [13]:
(102)NPSH=(p1γ+V122g)-pvγ,
where p1, V1 are the absolute pressure and absolute velocity of fluid at impeller inlet and pv is the absolute vapour pressure of fluid at the corresponding fluid temperature.In the vicinity of the leading edge of the impeller blades, the fluid has to accelerate in order to follow the rotating movement of the blades. This acceleration leads to a drop of the static pressure, which results in a local minimum pressure at blade inlet:(103)pminγ=p1γ-(W1b22g-V122g).
Therefore, the pump required net positive suction head is
(104)NPSHR=(pmin-pv)γ+W1b22g.
The pump NPSHR coefficient is defined as
(105)CNPSHR=NPSHRu22/g.
Referring to Figure 5, W1b=(Vr1/sinβb1), and thus
(106)CNPSHR=(pmin-pv)/γu22/g+1sin2βb1Vr122u22.
The term ((pmin-pv)/γ)/(u22/g) is a design parameter for the pump and could be assumed to be 0.02.From (3) Vr1=Vr2(D2/D1)(b2/b1)(ε2/ε1), and therefore the NPSHR coefficient is
(107)CNPSHR=0.02+12sin2βb1(ψ/φ)2(D1/D2)2(b1/b2)2(ε1/ε2)2.


## 2.17. Pump Specific Speedns

The pump specific speed is defined with suitable units as(108)ns=NQH3/4[N(rpm),H(m),Q(lit/s)].
Substitute for
(109)Q=1000ηvolε2·πD2b2·ψ2gH(lit/s),N=60πD2φ2gH,andb2=D2(b2D2)yield:ns=60*1000*(2g)3/4*ε2π(b2D2)*ηvol·φψ.
Therefore,
(110)ns=9977ε2(b2D2)·ηvol·φψ.


## 3. Calculation Procedure and Results

An iterative procedure using Microsoft Office EXCEL program is performed to calculate the pump design parameters described by the preceding derived equations. The input parameters are the blades number,Z, impeller inlet diameter to outlet diameter ratio, (D1/D2), volute diameter to impeller outlet diameter ratio, (D3/D2), ratio of blade width at impeller outlet to impeller outlet diameter, (b2/D2), blade width at impeller inlet to blade width at impeller outlet ratio, (b1/b2), volute width to blade width at impeller outlet ratio, (b3/b2), blade angle at impeller outlet, βb2, blade angle at impeller inlet, βb1, volute angle, αV, throat angle, αth, ratio of wearing ring clearance/impeller eye diameter, (yc/Deye), leakage discharge coefficient, CdL, volute diffusion loss coefficient, CdV, eye loss coefficient, Ceye, blade thickness coefficient at impeller outlet, ε2, with constant-thickness blades, ratio of axial gap between impeller disk and casing to impeller outlet radius, y0/(D2/2), Reynolds number at impeller outlet, Re2, and relative roughness, (κ/D2) approximated values are used by putting D2≈0.209m, N≈1450rpm, κ≈0.0001m, ν≈10-6m2/s: Re2=1.16*106, and κ/D2=0.000478.For a certain flow coefficient,ψ, all pump performance parameters and coefficients are calculated after iterations.Case Study. For reason of comparison between results of present analytical study and experimental pump performance obtained by Baun and Flack [14], calculations of centrifugal pump performance are performed with the following input parameters given in Table 1.Table 1
Pump input-parameters.
Input parameter
Value
Impeller
D
1
D
2
0.5
b
2
D
2
0.121
b
1
b
2
1
ϵ
2
0.95
G
=
y
0
(
D
2
/
2
)
0.05
Re
2
=
u
2
·
(
D
2
/
2
)
ν (D2=0.209m) (N=1450rpm) (ν=10-6m2/s) water
1.16
*
10
6
κ
D
2 (κ=0.0001m)
0.000478
Blades
Z
4
β
b
2
164°
β
b
1
164°
D
3
D
2
1.05
b
3
b
2
1.05
Volute
α
V
7°
α
th
4°
C
d
V
0.8
Eye
D
eye
D
th
1
y
c
D
eye
0.005
C
dL
0.6
C
eye
1.0Other pump constant-parameters are calculated according to the present procedure as shown in Table2.Table 2
Pump constant-parameters.
Constant parameter equation
Equation no.
Impeller
1
-
σ
0
=
sin
β
b
2
Z
0.7 for D1D2≤ϵlimit=e-(8.16sinβb2)/(Z)
(22)
ϵ
1
=
1
-
(
1
-
ϵ
2
)
(
D
1
/
D
2
)
(
sin
β
b
1
/
sin
β
b
2
)
(4)
d
2
D
2
=
ϵ
2
π
Z
sin
β
b
2
(15)
d
1
D
2
=
ϵ
1
π
Z
D
1
D
2
sin
β
b
1
(26)
D
hyd
D
2
=
2
(
(
(
d
1
/
D
2
)
(
b
1
/
b
2
)
(
b
2
/
D
2
)
)
+
(
(
d
2
/
D
2
)
(
b
2
/
D
2
)
)
)
(
d
1
/
D
2
)
+
(
(
b
1
/
b
2
)
(
b
2
/
D
2
)
)
+
(
d
2
/
D
2
)
+
(
b
2
/
D
2
)
(80)
κ
D
hyd
=
κ
/
D
2
D
hyd
/
D
2
(85)
K
D
=
1
40
f
D
ϵ
2
(
b
2
/
D
2
)
(92a)
f
D
=
f
D
(
Re
2
,
G
)
(92b)
Blades
L
b
D
2
=
1
2
(
1
-
D
1
D
2
)
1
sin
β
bm
(82a)
β
bm
=
1
2
(
β
b
1
+
β
b
2
)
(82b)
Volute
1
D
h
V
/
D
2
=
1
2
(
b
3
/
b
2
)
(
b
2
/
D
2
)
+
1
8
(
π
/
Z
)
(
D
3
/
D
2
)
sin
α
V
(49)
L
V
D
2
=
1
2
π
cos
α
V
(
D
3
D
2
)
(48)
κ
D
h
V
=
(
κ
/
D
2
)
(
D
h
V
/
D
2
)
(51)
tan
(
θ
th
2
)
=
1
2
[
π
tan
α
V
+
tan
α
th
-
(
b
3
/
b
2
)
(
b
2
/
D
2
)
(
D
3
/
D
2
)
]
(43)
C
f
th
=
0.5
+
2.6
*
sin
(
θ
th
2
)
(56)
Eye
D
eye
D
1
=
(
D
3
/
D
2
)
(
D
1
/
D
2
)
(
π
tan
α
V
+
tan
α
th
)
(40)The sequence of calculations by using the procedure derived equations of pump variable parameters is listed in Table3.Table 3
Pump variable-parameters.
Order
Variable parameter equation
Eq. no.
ψ
≡
V
r
2
2
g
H
=
0.002 to 0.12 step=0.001
Initial
σ
=
σ
0, x=1, ηvol=1
Start of iteration
1
φ
=
-
1
2
ψ
cot
β
b
2
+
1
2
ψ
2
cot
2
β
b
2
+
2
x
σ
(69)
2
y
=
C
H
∞
≡
H
∞
u
2
2
/
g
=
1
+
ψ
φ
cot
β
b
2
(21), (65b)
3
σ
=
1
-
(
1
-
σ
0
)
y
(20)
4
C
h
l
slp
≡ hlslpu22/g=(1-σ)CH∞
(65d)
5
C
H
0
≡
H
0
u
2
2
/
g
=
σ
(
1
+
ψ
φ
cot
β
b
2
)
(65c)
6
C
V
5
≡
V
5
2
g
H
=
4
ϵ
2
η
vol
(
b
2
/
D
2
)
(
D
t
h
/
D
3
)
2
(
D
3
/
D
2
)
2
·
ψ
(37)
7
C
V
4
≡
V
4
2
g
H
=
ϵ
2
η
vol
(
D
3
/
D
2
)
(
b
3
/
b
2
)
tan
α
V
·
ψ
(33)
8
C
V
eye
≡
V
eye
2
g
H
=
4
ϵ
2
(
b
2
/
D
2
)
(
D
t
h
/
D
3
)
2
(
D
3
/
D
2
)
2
·
ψ
(39)
9
C
V
3
p
=
C
V
4
cos
α
V
(54a)
10
C
V
3
d
=
σ
(
φ
+
ψ
cot
β
b
2
)
-
C
V
4
(54b)
11
C
V
3
r
≡
V
3
r
2
g
H
=
ϵ
2
η
vol
(
D
3
/
D
2
)
(
b
3
/
b
2
)
·
ψ
(54c)
12
η
vol
=
1
-
(
y
c
D
eye
)
(
D
eye
D
1
)
2
(
D
1
D
2
)
2
1
(
b
2
/
D
2
)
2
C
dL
ϵ
2
φ
ψ
*
C
H
0
-
C
V
4
2
-
C
V
5
2
+
C
eye
·
C
V
2
eye
2
φ
2
-
1
8
[
1
-
(
D
1
D
2
)
2
(
D
eye
D
1
)
2
]
(99)
13
Re
V
=
V
3
p
·
D
h
V
ν
=
2
(
C
V
3
p
φ
)
(
D
h
V
D
2
)
·
Re
2
(52)
14
f
V
=
0.3086
{
log
[
(
6.9
/
Re
V
)
+
(
(
κ
/
D
h
V
)
/
3.7
)
1.11
]
}
2
(50)
15
C
f
V
=
f
V
L
V
D
h
V
=
f
V
(
L
V
D
2
)
(
1
D
h
V
/
D
2
)
(46)
16
h
l
V
H
=
C
f
V
C
V
3
p
2
+
C
d
V
C
V
3
d
2
+
C
V
3
r
2
+
C
f
t
h
C
V
4
2
(45)
17
h
l
eye
H
=
C
eye
·
C
V
eye
2
(57)
18
1
x
=
1
+
h
l
V
H
+
h
l
eye
H
(61)
End of iteration
19
C
h
l
V
+eye
≡
h
l
V
+
h
l
eye
u
2
2
/
g
=
(
1
-
x
)
σ
C
H
∞
(65e)
20
C
H
≡
H
u
2
2
/
g
=
x
σ
(
1
+
ψ
φ
cot
β
b
2
)
(65a)
21
C
Q
≡
Q
(
N
/
60
)
D
2
3
=
η
vol
ϵ
2
π
2
(
b
2
D
2
)
·
(
ψ
φ
)
(6)
22
W
av
u
2
=
ϵ
2
(
π
/
Z
)
(
b
2
/
D
2
)
·
(
ψ
/
φ
)
1
/
2
(
(
(
d
1
/
D
2
)
(
b
1
/
b
2
)
(
b
2
/
D
2
)
)
+
(
(
d
2
/
D
2
)
(
b
2
/
D
2
)
)
)
(81)
23
Re
=
W
av
·
D
hyd
ν
=
2
(
W
av
u
2
)
(
D
hyd
D
2
)
·
Re
2
(86)
24
f
i
=
0.3086
{
log
[
(
6.9
/
Re
)
+
(
(
κ
/
D
hyd
)
/
3.7
)
1.11
]
}
2
(84)
25
4
C
d
i
=
(
f
i
+
0.006
)
(
1.1
+
4
b
2
D
2
)
(83)
26
C
h
l
f
≡ hlfu22/g=4Cdi(Lb/D2)(Dhyd/D2)12(Wavu2)2
(78)
27
C
h
l
D
≡ hlDu22/g=KD·1ηvolφψ
(91)
28
C
h
l
cir
in
≡ hlcirinu22/g=(1-ϵ1πZsinβb1)(D1D2)2-cot(180-βb1)(b1/b2)(ϵ1/ϵ2)·ψφ
(101)
29
C
h
l
vol
≡ hlvolu22/g=(1ηvol-1)·(CH0+Chlf+Chlcirin)
(93)
30
C
H
sh
≡
H
sh
u
2
2
/
g
=
C
H
0
+
C
h
l
f
+
C
h
l
cir
in
+
C
h
l
vol
+
C
h
l
D
(73)
31
η
≡
P
w
P
sh
=
C
H
C
H
sh
(74)
32
C
P
sh
≡
P
sh
ρ
(
N
/
60
)
3
D
2
5
=
π
2
C
Q
C
H
sh
(75)
33
C
P
w
≡
P
w
ρ
(
N
/
60
)
3
D
2
5
=
π
2
C
Q
C
H
(76)
34
C
NPSH
R
=
0.02
+
1
2
sin
2
β
b
1
(
ψ
/
φ
)
2
(
D
1
/
D
2
)
2
(
b
1
/
b
2
)
2
(
ϵ
1
/
ϵ
2
)
2
(107)
35
n
s
=
9977
ϵ
2
(
b
2
D
2
)
·
η
vol
·
φ
ψ
(110)The manometric head coefficient and pump efficiency of both present procedure and experimental measurements by Baun and Flack [14] are plotted in Figure 9 versus the ratio (ψ/φ) (defined as flow coefficient in [14]). It shows a good similarity between the present procedure results and the experimental ones. Only in range of low ratios of ψ/φ, the calculated efficiency is relatively high. This is attributed to the uncounted mechanical power loss in the prediction of pump shaft power and hence in efficiency.Figure 9
Comparison between present procedure results and experimental results by Baun and Flack [14].The pump performance characteristics are presented by the calculated pump head coefficient, efficiency, power coefficients, and required NPSH as shown in Figure10. From the figure, the pump best efficiency point (BEP) occurs at discharge coefficient CQ≈0.0675 giving an efficiency η≈75.3%, a manometric head coefficient CH≈0.468, a shaft power coefficient CPsh≈0.414, a water power coefficient CPw≈0.312, and a pump required net positive suction head coefficient CNPSHR≈0.129.Figure 10
Characteristics of centrifugal pump obtained by present study.The maximum shaft power coefficientCPsh≈0.428 occurs at CQ≈0.085, and the maximum water power coefficient CPw≈0.3165 occurs at CQ≈0.075.The variations of pump flow coefficientψ and pump speed coefficient φ with the discharge coefficient are shown in Figures 11 and 12, respectively. Figure 11 shows also the variation of the ratio ψ/φ with the discharge coefficient CQ. The linear relationship between CQ and ψ/φ is evident as given by (6). The dotted lines in the figures correspond to the value of CQ that gives the pump best efficiency. At pump best efficiency point (CQ≈0.0675),ψ≈0.063 (Figure 11) and φ≈1.034 (Figure 12). Also, it is indicated from Figure 12 that the speed coefficient takes a minimum value of φ≈0.932 at CQ≈0.024, which corresponds to the maximum CH in Figure 10.Figure 11
Variation of flow coefficient and ratio of flow coefficient to speed coefficient with discharge coefficient.Figure 12
Speed coefficient variation with pump discharge coefficient.Figure13 shows the theoretical dimensionless head-discharge curve (Euler head) which is a straight line, and CH∞ decreases with the increase of CQ for the proposed outlet blade angle βb2=164°>90°. The actual impeller outlet dimensionless head-discharge curve CH0 is obtained taking into consideration the slip or eddy circulation of flow inside impeller which is nearly constant. Actually, the effect of slip is not a loss but a discrepancy not accounted by the basic assumptions. The second loss shown is the volute-and-eye loss, which is minimum at CQ≈0.051. Reduced or increased CQ, from the value 0.051, increases the volute-and-eye loss.Figure 13
Variation of head coefficients and head loss coefficients with discharge coefficient.Figure14 gives the variation of different head loss coefficients with CQ in addition to the CH and CH0 curves. Generally, these head loss coefficients decrease with the increase of CQ. At very low discharge coefficients, both the disk head loss coefficient and volumetric head loss coefficient become very big. The inlet circulation head loss is big at low CQ and decreases until it reaches zero at CQ≈0.06, and then, its direction is inverse (becomes negative) which means that the inlet circulation adds power to impeller and does not bleed power from impeller.Figure 14
Head loss coefficients affecting pump power at different discharge coefficients.The variation of the flow velocity coefficients and the pump volumetric efficiencyηvol with the pump discharge coefficient is shown in Figure 15. The coefficients CV1, CVΔVu2, and CV4 have the trends of increasing with the increase of CQ, whereas the two coefficients CV3d and CV3d′ have the trends of decreasing with the increase of CQ. These two later coefficients have maximum values at CQ=0. At CQ≈0.085, the coefficient CV3d becomes zero, which indicates that at this point of operation the flow inside volute is without circulation and V4=Vu2act. At increased CQ, the CV3d becomes negative which means that the flow circulation inside the volute changes its direction of rotation opposite to impeller motion whereas V4>Vu2act (Figure 6).Figure 15
Values of volumetric efficiency and different pump velocity coefficients.The pump specific speed is presented in Figure16, which shows that ns≈870 at pump best efficiency point.Figure 16
Pump specific speed values at different discharge coefficient.Figure17 shows the procedure results of centrifugal pump performance when the pump handles fluids with different kinematic viscosities. The head and efficiency for the pump when handling oils are lower than those when handling water. But the required power when handling oil is higher than that when handling water.Figure 17
Effect of pumped fluid viscosity on the performance of centrifugal pump.The pump head decreases slightly due to the increase in volute friction loss as fluid viscosity increases, while the increase in pump power is high due to the increase in both hydraulic friction inside impeller and disk friction power losses. Hence, the drop in pump efficiency is very high as the fluid viscosity increases. This result is in accordance with experimental results by Shojaee Fard and Boyaghchi [15].


## 4. Conclusions

A one-dimensional flow procedure for analytical study of centrifugal pump performance is accomplished applying the principle theories of turbomachines. The procedure is capable of providing the performance characteristic of centrifugal pump in a dimensionless information form. The predicted coefficients and performance curves obtained have been found to be in a reasonable agreement with experimental measurements. The present procedure is also capable of predicting the effects of handling viscous fluids on the centrifugal pump performance. The input form for this procedure of pump flow analysis makes it an effective tool analysis and can be used in the pump conceptual design.



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*Source: 473512-2013-12-07.xml*