Hindawi Publishing Corporation
International Journal of Rotating Machinery
Volume 2013, Article ID 473512, 19 pages
http://dx.doi.org/10.1155/2013/473512
Research Article
A One-Dimensional Flow Analysis for the Prediction of
Centrifugal 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; naggarm@mans.edu.eg
Received 25 June 2013; Accepted 29 August 2013
Academic Editor: Shigenao Maruyama
Copyright © 2013 Mohammed Ahmed El-Naggar. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
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.
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 onedimensional flow analysis procedure towards obtaining optimum centrifugal pump design parameters.
2. Theoretical Analysis
The pump flow coefficient 𝜓 and pump speed coefficient 𝜑 are
defined as
𝜓=
𝑉𝑟2
√2𝑔𝐻
,
𝜑=
𝑢2
,
√2𝑔𝐻
(1)
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.
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International Journal of Rotating Machinery
Q
in
Vs
D2
D1
Q + QL
𝛽b2
𝛽 b2
t
u2
W1
L
V1
d1
𝛽 b1
Deye
yc
𝜋D2 /Z
t/ s
Veye
V2
W2
𝛽󳰀b2
d2
QL
D1
u1
b1
D2
N
Q
b2
Figure 1: Pump impeller notations.
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.
The volumetric efficiency 𝜂vol of pump is defined as the
ratio of pump outlet discharge to the impeller discharge:
𝜂vol =
𝑄
𝑄
=
𝑄𝑖 𝑄 + 𝑄𝐿
or
𝑄𝐿
1
− 1,
=
𝑄
𝜂vol
ΔWu2
W
W2
𝛼2
V2
2
eq
V2eq
Vr2
𝛽b2
u2
Vu2act
(2)
ΔVu2
Vu2
in which
𝑄𝑖 = 𝑄 + 𝑄𝐿 = 𝑉𝑟2 ⋅ 𝜋𝐷2 𝑏2 𝜀2 = 𝑉𝑟1 ⋅ 𝜋𝐷1 𝑏1 𝜀1 ,
(3)
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/ sin 𝛽𝑏1 )/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/ sin 𝛽𝑏2 )/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
𝜀1 = 1 −
(1 − 𝜀2 )
(𝐷1 /𝐷2 ) (sin 𝛽𝑏1 / sin 𝛽𝑏2 )
.
(4)
The impeller inlet flow velocity coefficient 𝐶𝑉1 =
𝑉1 /√2𝑔𝐻 is calculated from (3) dividing both sides by √2𝑔𝐻
and assuming that the flow enters the impeller without swirl
(𝑉1 = 𝑉𝑟1 ):
𝐶𝑉1
𝜓
=
.
(𝐷1 /𝐷2 ) (𝑏1 /𝑏2 ) (𝜀1 /𝜀2 )
The pump discharge coefficient is 𝐶𝑄 =
and substituting for 𝑄 from (2) and (3);
𝑏 𝜓
𝐶𝑄 = 𝜂vol 𝜀2 𝜋 2 ⋅ .
𝐷2 𝜑
2
(5)
𝑄/((𝑁/60)𝐷23 ),
(6)
Figure 2: Velocity diagram at impeller outlet.
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 factor 𝜎 defined as 𝜎 =
𝑉𝑢2 act /𝑉𝑢2 could be estimated later in Section 2.3.
From the velocity diagram at the impeller outlet, Figure 2,
the tangential component of the outlet flow absolute velocity
is given as
𝑉𝑢2 = 𝑉𝑟2 cot 𝛼2 = 𝑢2 + 𝑉𝑟2 cot 𝛽𝑏2 .
(7)
The ratio of outlet swirl velocity to outlet tangential velocity
is
𝑉𝑟
𝑉𝑢2
𝜓
(8)
= 1 + 2 cot 𝛽𝑏2 = 1 + cot 𝛽𝑏2 ,
𝑢2
𝑢2
𝜑
and thus,
𝑉𝑢2 act
𝑢2
=
𝜎 ⋅ 𝑉𝑢2
𝑢2
= 𝜎 (1 +
𝜓
cot 𝛽 𝑏2 ) .
𝜑
(9)
The outlet tangential velocity (from (7)) and the pump
speed coefficient 𝜑 = 𝑢2 /√2𝑔𝐻 are given as
𝑢2 = 𝑉𝑟2 (cot 𝛼2 − cot 𝛽𝑏2 ) ,
(10)
𝜑 = 𝜓 (cot 𝛼2 − cot 𝛽𝑏2 ) .
(11)
International Journal of Rotating Machinery
3
Since the relative eddy angular velocity = 2𝑢2 /𝐷2 , then
𝛽
/sin b2
/Z-t
𝜋D 2
−𝜔
𝛽󳰀b2
Δ𝑊𝑢2 = 𝜀2
𝛽b2
d2
𝜋𝑢2
sin 𝛽𝑏2 ,
𝑍
𝜎=
−𝜔
d1
= 𝜀2
𝜋
sin 𝛽𝑏2 .
𝑍
(16)
𝜎=1−
From the outlet velocity diagram (Figure 2) and (7), the outlet
velocity and the outlet velocity coefficient defined as 𝐶𝑉2 =
𝑉2 /√2𝑔𝐻 are
2
𝑉22 = 𝑉𝑟22 + 𝑉𝑢22 = 𝑉𝑟22 + (𝑢2 + 𝑉𝑟2 cot 𝛽𝑏2 ) ,
2
𝑉𝑢2 act
=1−
𝑉𝑢2
Δ𝑉𝑢2
𝑉𝑢2
=1−
Δ𝑊𝑢2 /𝑢2
𝑉𝑢2 /𝑢2
.
(17)
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
𝜔
Figure 3: Flow model for Stodola slip factor.
2
𝑢2
The slip factor is given by
Wav
2
𝐶𝑉
2
Δ𝑊𝑢2
or
𝜀2 (𝜋/𝑍) sin 𝛽𝑏2
=1−
1 + (𝑉𝑟2 /𝑢2 ) cot 𝛽𝑏2
At 𝜓 = 0 (𝜎 = 𝜎0 )
(12)
1−𝜎=
= 𝜓 + (𝜑 + 𝜓cot 𝛽𝑏2 ) .
1 + (𝜓/𝜑) cot 𝛽𝑏2
. (18)
𝜋
sin 𝛽𝑏2 .
𝑍
(19)
1 − 𝜎0
,
𝑦
(20)
𝜓
cot 𝛽𝑏2 .
𝜑
(21)
1 − 𝜎0 = 𝜀2
Therefore,
𝜀2 (𝜋/𝑍) sin 𝛽𝑏2
where
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 flow must 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 −𝑉𝑢2 act , 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.
Thus,
Δ𝑊𝑢2 = 𝜔
𝑑2
.
2
(13)
An approximate expression for 𝑑2 can be written if the
number of blades 𝑍 is not small:
𝜋𝐷2
𝜋𝐷2
(14)
sin 𝛽𝑏󸀠2 − 𝑡 = 𝜀2
sin 𝛽𝑏2 ,
𝑑2 =
𝑍
𝑍
𝑑2
𝜋
= 𝜀2 sin 𝛽𝑏2 .
𝐷2
𝑍
(15)
𝑦=1+
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
1 − 𝜎0 =
√sin 𝛽𝑏2
𝑍0.7
for
𝐷1
≤ 𝜀limit ,
𝐷2
(22)
where 𝜀limit is the limiting diameter ratio,
𝜀limit = 𝑒−(8.16 sin 𝛽𝑏2 /𝑍) .
(23)
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
Δ󸀠 𝑊𝑢1 = 𝜔
where
𝑑1 = 𝜀1
𝑑1
,
2
𝜋𝐷1
sin 𝛽𝑏1 ,
𝑍
(24)
(25)
𝑑1
𝜋 𝐷1
= 𝜀1
sin 𝛽𝑏1 .
𝐷2
𝑍 𝐷2
(26)
𝜋
Δ󸀠 𝑊𝑢1 = 𝜀1 𝑢1 sin 𝛽𝑏1 .
𝑍
(27)
Hence,
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International Journal of Rotating Machinery
Δ󳰀 Wu1
V1∗
=
∗
Vr1
∗
V1∗ = Vr1
∗
Vr1
∗
W1b
=
sin 𝛽b1
W1∗
V1 = Vr1
𝛽b1
Δ󳰀 Wu1
𝜋
sin 𝛽b1 )u1
Z
Δ Wu1 Δ󳰀󳰀 Wu1
W1b =
W1
𝛽∗f1
(1 − 𝜀1
∗
W1b
W1∗
󳰀
Vr1
sin 𝛽b1
𝛽b1
𝛽f1
u1
(1 − 𝜀1
u1
𝜋
sin 𝛽b1 )u1
Z
Figure 4: Velocity diagram at impeller inlet without shock.
Figure 5: Velocity diagram at blade inlet with shock eddy.
When the pump that operates at a discharge 𝑄 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
point. With a uniform rate of increase of the volute area, the
volute angle is
Δ󸀠󸀠 𝑊𝑢1 = 𝑢1 (1 − 𝜀1
𝜋
sin 𝛽𝑏1 ) − 𝑉𝑟1 cot (180 − 𝛽𝑏1 ) . (28)
𝑍
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]
𝐻∞ =
𝑢2 𝑉𝑢2
𝑔
tan 𝛼𝑉 =
𝑢2 ⋅ 𝑉𝑢2 act
𝑔
=
𝑔
𝐴 𝑉th
𝜋𝐷2 𝑏2 (𝐷3 /𝐷2 ) (𝑏3 /𝑏2 )
.
𝑉4 =
𝜂vol ⋅ 𝑉𝑟2 𝜋𝐷2 𝑏2 𝜀2
𝜋𝐷3 𝑏3 ⋅ tan 𝛼𝑉
𝐶𝑉4
=
𝜂vol 𝜀2 𝐷2 𝑏2
⋅𝑉 ,
tan 𝛼𝑉 𝐷3 𝑏3 𝑟2
𝜀2 𝜂vol
=
⋅ 𝜓.
(𝐷3 /𝐷2 ) (𝑏3 /𝑏2 ) tan 𝛼𝑉
(29)
= 𝜎 ⋅ 𝐻∞ .
(30)
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 /𝑉𝑢3 𝑟3 𝑏3 . This requires a circumferentially uniform rate of increase of the volute area 𝐴 𝑉 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 𝐴 𝑉th for the volute geometry.
The volute angle 𝛼𝑉, Figure 6, is chosen to match the angle
of flow entering the volute 𝛼3eq at a certain pump operating
(34)
𝐷th
= 𝜋 tan 𝛼𝑉 + tan 𝛼th .
𝐷3
(35)
The throat outlet flow velocity 𝑉5 = 𝑄/𝐴 th , and using (2) and
(3), then
𝑉5 = 4𝜀2 𝜂vol
(31)
(33)
With the assumption that the throat height 𝐿 th = 𝐷3 , thus
And so, the slippage head loss is
ℎ𝑙slp = 𝐻∞ − 𝐻0 = (1 − 𝜎) 𝐻∞ .
(32)
The throat is assumed to have an expanding angle 𝛼th , and
hence the throat diameter equals
𝐷th = 𝜋𝐷3 tan 𝛼𝑉 + 𝐿 th tan 𝛼th .
.
𝑢2 ⋅ 𝜎𝑉𝑢2
𝜋𝐷3 𝑏3
=
The volute outlet flow velocity 𝑉4 = 𝑄/𝐴 𝑉th and the volute
outlet flow velocity coefficient 𝐶𝑉4 = 𝑉4 /√2𝑔𝐻 are
Due to the fluid slippage at impeller exit, the actual head given
to fluid by the impeller, 𝐻0 , is calculated from [6, 7]:
𝐻0 =
𝐴 𝑉th
𝐷22 𝑏2
⋅ 𝑉𝑟2 .
2 𝐷
𝐷th
2
(36)
The throat outlet flow velocity coefficient 𝐶𝑉5 (= 𝑉5 /√2𝑔𝐻)
equals
𝐶𝑉5 =
4𝜀2 𝜂vol (𝑏2 /𝐷2 )
2
2
(𝐷th /𝐷3 ) (𝐷3 /𝐷2 )
⋅ 𝜓.
(37)
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,
𝑉5 = 𝑉𝑠 = 𝜂vol ⋅ 𝑉eye .
(38)
And the eye velocity coefficient is
𝐶𝑉eye =
𝑉eye
√2𝑔𝐻
=
𝐶𝑉5
𝜂vol
=
4𝜀2 (𝑏2 /𝐷2 )
2
2
(𝐷th /𝐷3 ) (𝐷3 /𝐷2 )
⋅ 𝜓.
(39)
International Journal of Rotating Machinery
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Case 𝛼3eq > 𝛼V :
V3d
V3eq
V3d + V3p ≡
V 3p
Dth V5
D2
𝜃th
V3
V4
yth = 𝜋D3 tan 𝛼V
D1
D3
Impeller
N
𝛼V
V 3p
V3eq
V3u
𝜃
xc
d𝜃
V3u
2∗𝜋D3 sin 𝛼V /Z
b3
Volute
𝛼V
xc
Volute
V 3p
V3
𝜋D3
Vr2
V3d󳰀
V3
Vr3 = 𝜀2 𝜂vol
Vu2act
V4 = V3pu
V3r
V2
𝛼3eq < 𝛼V
𝛼3eq 𝛼V
𝛼V
V2eq ΔVu
2
ΔVu3
V3d
b2 D2
·V
b3 D3 r2
Vu2
V4
Dth
Throat
𝛼th
Case 𝛼3eq < 𝛼V :
V3eq ≡ V3d + V3p
𝜋D3tan 𝛼V
L th
𝛼V
V5
y th
Throat
L th
𝜋D2
Figure 6: Pump and volute notations.
The eye diameter relative to the impeller inlet diameter is
therefore
𝐷eye
𝐷1
=
𝐷eye 𝐷th 𝐷3 𝐷2
𝐷th 𝐷3 𝐷2 𝐷1
=
(𝐷3 /𝐷2 ) (𝜋 tan 𝛼𝑉 + tan 𝛼th )
.
(𝐷1 /𝐷2 )
(40)
The ratio (𝐷eye /𝐷1 ) must not exceed 1, and thus an upper limit
is imposed on the volute angle:
tan 𝛼𝑉 <
(𝐷1 /𝐷2 )
tan 𝛼th
−
.
𝜋
𝜋 (𝐷3 /𝐷2 )
(41)
The throat has a cone angle 𝜃th , where
tan (
(𝐷th − 𝑏3 ) /2
𝜃th
.
)=
2
𝐿 th
(42)
volute velocity 𝑉3 is decomposed into a velocity parallel to
the direction of volute, 𝑉3𝑝 , and another one in the tangential
󸀠
(Figure 6). The velocity component
direction of impeller, 𝑉3𝑑
󸀠
𝑉3𝑑 is the motive of a second circulatory motion given to
the volute flow in direction of impeller motion. Thus, the net
󸀠
− Δ𝑉𝑢3 in
circulation velocity of flow in volute is 𝑉3𝑑 = 𝑉3𝑑
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
ℎ𝑙𝑉 = 𝐶𝑓𝑉
Substituting from (35),
tan (
(𝑏 /𝑏 ) (𝑏 /𝐷 )
𝜃th
1
) = [𝜋 tan 𝛼𝑉 + tan 𝛼th − 3 2 2 2 ] .
2
2
(𝐷3 /𝐷2 )
(43)
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 offdesign pump operation.
The flow enters the volute with a through-velocity 𝑉3
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
2
𝑉3𝑝
2𝑔
+ 𝐶𝑑𝑉
2
𝑉3𝑑
𝑉2
+ 3𝑟 + ℎ𝑙th ,
2𝑔
2𝑔
ℎ𝑙𝑉
2
2
2
2
+ 𝐶𝑑𝑉 ⋅ 𝐶𝑉
+ 𝐶𝑉
+ 𝐶𝑓th ⋅ 𝐶𝑉
,
= 𝐶𝑓𝑉 ⋅ 𝐶𝑉
3𝑝
3𝑑
3𝑟
4
𝐻
where 𝐶𝑓𝑉 is the volute friction loss coefficient:
𝐶𝑓𝑉 = 𝑓𝑉
𝐿𝑉
𝐿
1
= 𝑓𝑉 ( 𝑉 ) (
),
𝐷ℎ𝑉
𝐷2
𝐷ℎ𝑉 /𝐷2
(44)
(45)
(46)
where 𝑓𝑉 is the volute friction coefficient, 𝐷ℎ𝑉 is the volute
hydraulic diameter, and 𝐿 𝑉 is the average volute length:
𝐿𝑉 =
1 𝜋𝐷3
,
2 cos 𝛼𝑉
𝐿𝑉 1 𝜋
𝐷
=
( 3).
𝐷2 2 cos 𝛼𝑉 𝐷2
(47)
(48)
6
International Journal of Rotating Machinery
The average volute hydraulic diameter relative to impeller
diameter is
2.7. Pump Eye Head Loss ℎ𝑙eye . The head loss in pump eye ℎ𝑙eye ,
[2], and the eye head loss relative to the pump head are
1
1
1
=
.
+
𝐷ℎ𝑉 /𝐷2 2 (𝑏3 /𝑏2 ) (𝑏2 /𝐷2 ) 8 (𝜋/𝑍) (𝐷3 /𝐷2 ) sin 𝛼𝑉
(49)
The volute friction coefficient 𝑓𝑉 which corresponds to a pipe
flow is function of the volute Reynolds number Re𝑉 and the
roughness 𝜅, [10]:
𝑓𝑉 =
0.3086
,
1.11 2
{log [(6.9/Re𝑉) + ((𝜅/𝐷ℎ𝑉 ) /3.7) ]}
(50)
(𝜅/𝐷2 )
𝜅
.
=
𝐷ℎ𝑉 (𝐷ℎ /𝐷2 )
𝑉
(51)
Re𝑉 =
]
=2
𝜑
(
𝐷ℎ𝑉
𝐷2
) ⋅ Re2 ,
(52)
where
Re2 =
𝑢2 ⋅ 𝐷2 /2
.
]
𝑉4
,
cos 𝛼𝑉
or 𝐶𝑉3𝑝 =
𝐶𝑉4
cos 𝛼𝑉
.
(54a)
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:
𝑉3𝑑 = 𝑉𝑢2 act − 𝑉4
𝐶𝑉3𝑑
(54b)
𝜀2 𝜂vol
𝑉3𝑟
=
⋅ 𝜓.
(𝐷3 /𝐷2 ) (𝑏3 /𝑏2 )
√2𝑔𝐻
(54c)
In the fourth term of (44) and (45), the volute throat head
loss ℎ𝑙th could be calculated as
𝑝5 − 𝑝eye
𝛾
.
(58)
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
𝐻 = 𝐻0 − ℎ𝑙𝑉 − ℎ𝑙eye
(59)
𝐻 = 𝐻∞ − ℎ𝑙slp − ℎ𝑙𝑉 − ℎ𝑙eye .
(60)
or
Define a parameter 𝑥 as
𝑥=
or
𝐻
1
=
,
𝐻0
[1 + (ℎ𝑙𝑉 /𝐻) + (ℎ𝑙eye /𝐻)]
ℎ𝑙𝑉
ℎ𝑙eye
(61)
1
=1+
+
.
𝑥
𝐻
𝐻
1
2
2
2
2
+ 𝐶𝑑𝑉 ⋅ 𝐶𝑉
+ 𝐶𝑉
+ 𝐶𝑓th ⋅ 𝐶𝑉
= 1 + 𝐶𝑓𝑉 ⋅ 𝐶𝑉
3𝑝
3𝑑
3𝑟
4
𝑥
(62)
The sum of volute and eye head losses, from (59), is written
as
ℎ𝑙𝑉 + ℎ𝑙eye = (1 − 𝑥) 𝐻0 = (1 − 𝑥) 𝜎𝐻∞ .
(63)
Using (9), (29), and (59), the pump manometric head 𝐻 is
𝑉2
ℎ𝑙th = 𝐶𝑓th 4 ,
2𝑔
(55)
where 𝐶𝑓th is the volute throat friction loss coefficient
assumed to be, [10],
where 𝜃th is the throat cone angle.
2
,
= 𝐶eye ⋅ 𝐶𝑉
eye
2
+ 𝐶eye ⋅ 𝐶𝑉
.
eye
The third term of (45) is
𝐶𝑓th = 0.5 + 2.6 ∗ sin (
2𝑔𝐻
Using (45) as well as (57) and (61) then
𝑉3𝑑
=
= 𝜎 (𝜑 + 𝜓 cot 𝛽𝑏2 ) − 𝐶𝑉4 .
√2𝑔𝐻
𝐶𝑉3𝑟 =
= 𝐶eye
(57)
where 𝑉eye is the flow velocity in pump eye and 𝐶eye is the eye
loss coefficient defined by (39).
(53)
In the first term of (44) and (45), 𝑉3𝑝 is the volute throw flow
velocity (Figure 6) and given as
𝑉3𝑝 =
𝐻
,
2
𝑉eye
𝐻=
The volute Reynolds number is calculated as
𝐶𝑉3𝑝
ℎ𝑙eye
2𝑔
2.8. Pump Manometric Head 𝐻. The pump manometric head
is the difference in static pressure heads between pump outlet
and pump eye:
where
𝑉3𝑝 ⋅ 𝐷ℎ𝑉
ℎ𝑙eye = 𝐶eye
2
𝑉eye
𝜃th
),
2
(56)
𝐻 = 𝑥𝜎𝐻∞ = 𝑥𝜎
𝑢2 ⋅ 𝑉𝑢2
𝑔
,
𝑢2
𝜓
𝐻 = 𝑥𝜎 2 (1 + cot 𝛽𝑏2 ) .
𝑔
𝜑
(64)
Coefficients. There are two additional groups of coefficients,
namely, the pump head coefficients and head loss coefficients.
International Journal of Rotating Machinery
7
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
2.9. Pump Shaft Power 𝑃sh , and Pump Shaft Head 𝐻sh . The
total shaft power required to drive the impeller is
󸀠
+ 𝑃𝑙󸀠𝑓 + 𝑃𝑙󸀠cir + 𝑃𝑙𝐷 ,
𝑃sh = 𝑃sh
0
(70)
in
𝐶𝐻 =
𝜓
𝐻
= 𝑥𝜎 (1 + cot 𝛽𝑏2 ) ,
2
𝜑
𝑢2 /𝑔
(65a)
𝜓
𝐻∞
= 1 + cot 𝛽𝑏2 ,
2
𝜑
𝑢2 /𝑔
(65b)
𝜓
𝐻0
= 𝜎 (1 + cot 𝛽𝑏2 ) .
2
𝜑
𝑢2 /𝑔
(65c)
𝐶𝐻∞ =
𝐶𝐻0 =
According to (21) and (65b), 𝑦 = 𝐶𝐻∞ .
The head loss coefficients include the slippage head loss
coefficient, 𝐶ℎ𝑙 , the volute-eye head loss coefficient, 𝐶ℎ𝑙 ,
𝑉
ℎ𝑙slp
𝐶ℎ𝑙 =
𝑢22 /𝑔
slp
𝑉+eye
𝐶ℎ𝑙
eye
𝐶ℎ𝑙 =
𝑉
= (1 − 𝜎) 𝐶𝐻∞ ,
ℎ𝑙𝑉 + ℎ𝑙eye
=
𝑢22 /𝑔
ℎ𝑙eye
=
ℎ𝑙𝑉
impeller, 𝑃𝑙󸀠cir = 𝛾(𝑄 + 𝑄𝐿 )ℎ𝑙cir is the power needed for given
in
in
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
𝑃sh = 𝑃sh0 + 𝑃𝑙𝑓 + 𝑃𝑙cir + 𝑃𝑙vol + 𝑃𝑙𝐷 ,
(71)
in
𝑉+eye
slp
the eye head loss coefficient, 𝐶ℎ𝑙 eye , and the volute head loss
coefficient, 𝐶ℎ𝑙 . They are given, respectively, as
𝐶ℎ𝑙
󸀠
where 𝑃sh
= 𝛾(𝑄 + 𝑄𝐿 )𝐻0 is the impeller power given to
0
󸀠
water, 𝑃𝑙𝑓 = 𝛾(𝑄 + 𝑄𝐿 )ℎ𝑙𝑓 is the power lost in friction inside
𝑢22 /𝑔
𝑢22 /𝑔
= (1 − 𝑥) 𝜎𝐶𝐻∞ ,
2
= 𝐶eye ⋅ 𝐶𝑉
⋅
𝑒𝑦𝑒
1
,
2𝜑2
+ 𝐶𝑓th ⋅
1
⋅ 2.
2𝜑
𝑃sh0 = 𝛾𝑄𝐻0 ,
𝑃𝑙𝑓 = 𝛾𝑄ℎ𝑙𝑓 ,
𝑃𝑙cir = 𝛾𝑄ℎ𝑙cir ,
in
in
𝑃𝑙vol = 𝛾𝑄ℎ𝑙vol = 𝛾𝑄𝐿 (𝐻0 + ℎ𝑙𝑓 + ℎ𝑙cir ) ,
(65d)
in
(72)
𝑃𝑙𝐷 = 𝛾𝑄ℎ𝑙𝐷 ,
(65e)
(65f)
where ℎ𝑙𝑓 is the impeller skin friction head loss, ℎ𝑙cir is
in
the inlet shock circulation head loss, ℎ𝑙vol is the volumetric
(leakage) head loss, and ℎ𝑙𝐷 is the disk friction head loss.
The shaft head 𝐻sh = 𝑃sh /(𝛾𝑄) and the shaft head
coefficient 𝐶𝐻sh = 𝐻sh /(𝑢22 /𝑔) are given, respectively, as
(65g)
𝐻sh = 𝐻0 + ℎ𝑙𝑓 + ℎ𝑙cir + ℎ𝑙vol + ℎ𝑙𝐷 ,
2
2
2
= (𝐶𝑓𝑉 ⋅ 𝐶𝑉
+ 𝐶𝑑𝑉 ⋅ 𝐶𝑉
+ 𝐶𝑉
3𝑝
3𝑑
3𝑟
2
𝐶𝑉
)
4
in which
in
𝐶𝐻sh = 𝐶𝐻0 + 𝐶ℎ𝑙 + 𝐶ℎ𝑙 cir + 𝐶ℎ𝑙 vol + 𝐶ℎ𝑙 𝐷 ,
𝑓
(73)
in
The relationship between the pump speed coefficient 𝜑 and
the pump flow coefficient 𝜓 is derived as follows.
Using (11), (64) becomes
where 𝐶ℎ𝑙 = ℎ𝑙𝑓 /(𝑢22 /𝑔) is the impeller skin friction head loss
𝑢2
cot 𝛼2
.
𝐻 = 𝑥𝜎 2
𝑔 cot𝛼2 − cot 𝛽𝑏2
head loss coefficient, 𝐶ℎ𝑙 vol = ℎ𝑙vol /(𝑢22 /𝑔) is the volumetric
head loss coefficient, and 𝐶ℎ𝑙 𝐷 = ℎ𝑙𝐷 /(𝑢22 /𝑔) is the disk friction
head loss coefficient.
in
(66)
Dividing both sides by 𝐻, noting that 𝑢22 /(2𝑔𝐻) = 𝜑2 , and
using (11), (64) becomes a quadric equation for cot𝛼2 :
cot2 𝛼2 − cot 𝛽𝑏2 ⋅ cot 𝛼2 −
1
= 0,
2𝑥𝜎𝜓2
𝑓
coefficient, 𝐶ℎ𝑙 cir = ℎ𝑙cir /(𝑢22 /𝑔) is the inlet shock circulation
(67)
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:
which has a solution (since cot 𝛼2 should be greater than
cot 𝛽𝑏2 , whence only the +ve sign is considered)
1
1
2
cot 𝛼2 = cot 𝛽𝑏2 + √cot2 𝛽𝑏2 +
.
2
2
𝑥𝜎𝜓2
𝜂=
𝑃𝑤
𝑃𝑤
=
,
𝑃sh 𝑃sh0 + 𝑃𝑙𝑓 + 𝑃𝑙cir + 𝑃𝑙vol + 𝑃𝑙𝐷
in
(68)
Multiplying (68) by 𝜓 and then subtracting 𝜓 cot 𝛽𝑏2 from
both sides and using (11) yield the following relation for 𝜑:
1
2
1
.
𝜑 = − 𝜓cot 𝛽𝑏2 + √ 𝜓2 cot2 𝛽𝑏2 +
2
2
𝑥𝜎
in
(69)
𝜂=
𝐻∞ − ℎ𝑙slp − ℎ𝑙𝑉 − ℎ𝑙eye
𝐻
=
,
𝐻sh 𝐻0 + ℎ𝑙𝑓 + ℎ𝑙cir + ℎ𝑙vol + ℎ𝑙𝐷
in
𝜂=
𝐶𝐻∞ − 𝐶ℎ𝑙 slp − 𝐶ℎ𝑙 𝑉+eye
𝐶𝐻
=
.
𝐶𝐻𝑠ℎ 𝐶𝐻0 + 𝐶ℎ𝑙 + 𝐶ℎ𝑙 cir + 𝐶ℎ𝑙 vol + 𝐶ℎ𝑙 𝐷
𝑓
in
(74)
8
International Journal of Rotating Machinery
2.11. Pump Shaft Power Coefficient 𝐶𝑃𝑠ℎ . The pump shaft
power coefficient and the pump water power coefficient are
given, respectively, as
𝑃sh
𝐶𝑃sh =
𝜌(𝑁/60)3 𝐷25
𝐶𝑃𝑤 =
𝑃𝑤
3
𝜌(𝑁/60) 𝐷25
2
The impeller friction coefficient 𝑓𝑖 is function of the average
impeller Reynolds number Re and the roughness 𝜅 [10]:
𝑓𝑖 =
= 𝜋 𝐶𝑄𝐶𝐻sh ,
(75)
= 𝜋2 𝐶𝑄𝐶𝐻.
(76)
𝐿 𝑏 𝑊av2
,
𝐷hyd 2𝑔
(77)
where 𝐶𝑑𝑖 is the dissipation coefficient, 𝐿 𝑏 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
𝐶ℎ𝑙 𝑓
ℎ𝑙𝑓
(𝐿 𝑏 /𝐷2 ) 1 𝑊av 2
= 2 = 4𝐶𝑑𝑖
).
(
𝑢2 /𝑔
(𝐷hyd /𝐷2 ) 2 𝑢2
2 (𝑑1 𝑏1 + 𝑑2 𝑏2 )
4 ∗ Area
,
=
Perimeter 𝑑1 + 𝑏1 + 𝑑2 + 𝑏2
𝑄 /𝑍
𝑄𝑖 /𝑍
=
𝑊av = 𝑖
.
𝐴 av
(𝑑1 𝑏1 + 𝑑2 𝑏2 ) /2
𝐷2
=
(79)
Re =
𝑊av ⋅ 𝐷hyd
𝜀2 (𝜋/𝑍) (𝑏2 /𝐷2 )
𝜓
𝑊av
=
⋅ .
𝑢2
1/2((𝑑1 /𝐷2 ) (𝑏1 /𝑏2 ) (𝑏2 /𝐷2 )+(𝑑2 /𝐷2 ) (𝑏2 /𝐷2 )) 𝜑
(81)
The blade length is 𝐿 𝑏 = ((1/2)(𝐷2 − 𝐷1 ))/ sin 𝛽bm , and thus
(82a)
2
(82b)
.
The impeller dissipation coefficient is given as, Gülich [11]
4𝐶𝑑𝑖 = (𝑓𝑖 + 0.006) (1.1 + 4
𝑏2
).
𝐷2
(84)
𝐷hyd
𝑊av
)(
) ⋅ Re2 ,
𝑢2
𝐷2
(85)
(86)
2.13. Disk Friction Head Loss ℎ𝑙𝐷 . The disk friction power loss
𝑃𝑙𝐷 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
𝑟2
𝑃𝑙𝐷 = 2 ∗ ∫ 𝜔 ⋅ 𝑟 ⋅ 𝜏 ⋅ 2𝜋𝑟 𝑑𝑟
0
𝜏=
𝑓𝐷 1 2 𝑓𝐷 1
⋅ 𝜌𝑢 =
⋅ 𝜌 (𝜔𝑟)2 ,
4 2
4 2
(87)
(83)
(88)
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.
Thus,
𝜋 3 𝑟2
𝜌𝜔 ∫ 𝑓𝐷𝑟4 𝑑𝑟.
2
0
(89)
Therefore, the final expression after the integration for the
disk friction power loss and, consequently, the impeller disk
friction head loss are
𝑃𝑙𝐷 =
𝑟5
𝜋
𝜋
𝑓𝐷 𝜌𝜔3 2 = 𝑓𝐷𝜌𝑢23 𝐷22 ,
2
5
40
3 2
𝜋 𝑓𝐷 𝑢2 𝐷2
ℎ𝑙𝐷 =
=
.
𝜌𝑔𝑄 40 𝑔 𝑄
𝑃𝑙𝐷
(90)
The impeller disk friction head loss coefficient 𝐶ℎ𝑙 is given
𝐷
as
𝐷
(𝛽𝑏1 + 𝛽𝑏2 )
,
where Re2 is defined by (53).
𝐶ℎ𝑙 ≡
where
𝛽bm =
= 2(
]
𝑃𝑙𝐷 =
2 ((𝑑1 /𝐷2 ) (𝑏1 /𝑏2 ) (𝑏2 /𝐷2 ) + (𝑑2 /𝐷2 ) (𝑏2 /𝐷2 ))
,
(𝑑1 /𝐷2 ) + (𝑏1 /𝑏2 ) (𝑏2 /𝐷2 ) + (𝑑2 /𝐷2 ) + (𝑏2 /𝐷2 )
(80)
𝐿𝑏 1
𝐷
1
= (1 − 1 )
,
𝐷2 2
𝐷2 sin 𝛽bm
2
]}
with
Substituting for 𝑑1 , (26), 𝑑2 , (15), and 𝑄𝑖 , (3), yield:
𝐷hyd
1.11
(𝜅/𝐷2 )
𝜅
.
=
𝐷hyd (𝐷hyd /𝐷2 )
(78)
The hydraulic diameter and the average relative velocity are
given, respectively, as, Gülich [11]
𝐷hyd =
{log [(6.9/ Re) + ((𝜅/𝐷hyd ) /3.7)
where
2.12. Impeller Skin Friction Power Head Loss ℎ𝑙𝑓 . The impeller
hydraulic friction head loss ℎ𝑙𝑓 is estimated by the theory of
flow through pipes and is given by [11]:
ℎ𝑙𝑓 = 4𝐶𝑑𝑖
0.3086
ℎ𝑙𝐷
𝑢22 /𝑔
= 𝐾𝐷 ⋅
1 𝜑
,
𝜂vol 𝜓
(91)
where 𝐾𝐷 is the impeller disk loss coefficient,
𝐾𝐷 =
𝑓𝐷
1
.
40 𝜀2 (𝑏2 /𝐷2 )
(92a)
It is derived by substituting for 𝑄 from (2) and (3) and using
the definition of 𝜓 which yields 𝑄 = 𝜂vol 𝜀2 ⋅ 𝜓√2𝑔𝐻 ⋅ 𝜋𝐷2 𝑏2 .
International Journal of Rotating Machinery
9
Correlations for 𝑓𝐷 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)/], 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
, 𝐺 = 188Re−9/10
, 𝐺 =
Figure 8 are 𝐺 = 1.62Re−5/11
2
2
−3/16
5
,
Re
=
1.58
∗
10
,
and
𝐺
=
0.402Re
,
0.57 ∗ 10−6 Re15/16
2
2
respectively. The disk friction coefficient for each regime is
given as
Regime I : 𝑓𝐷 = 10𝐺−1 Re−1
2
18.5 1/10 −1/2
Regime II : 𝑓𝐷 =
𝐺 Re2
𝜋
0.4 −1/6 −1/4
Regime III : 𝑓𝐷 =
𝐺 Re2
𝜋
0.51 1/10 −1/5
Regime IV : 𝑓𝐷 =
𝐺 Re2 .
𝜋
ℎ𝑙vol
𝐶ℎ𝑙
vol
=
ℎ𝑙vol
𝑢22 /𝑔
(92b)
Deye
r2 = D2 /2
Figure 7: Disk and wearing ring clearances.
IV
4
0.075
5
1
3
0.025
III
I
2
10000
100000
1000000
Figure 8: Disk friction flow regimes [12].
(93)
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
(94)
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
𝑝3 − 𝑝𝑠 𝑝5 − 𝑝𝑠 𝑝3 − 𝑝5
=
+
𝛾
𝛾
𝛾
𝑉42 𝑉52
−
).
2𝑔 2𝑔
y0
Re2
where 𝑎𝑐 is the clearance area of wearing ring (= 𝜋𝐷eye 𝑦𝑐 ), 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:
𝑝 − 𝑝𝑠 𝑝3 − 𝑝𝑠 𝑝3 − 𝑝𝑐
=
−
,
Δ𝐻𝑐 = 𝑐
(95)
𝛾
𝛾
𝛾
= 𝐻0 − ℎ𝑙eye − (
w
0.000
1000
The leakage flow rate 𝑄𝐿 can be estimated using orifice
formula [13]:
𝑉2 𝑉2
= 𝐻 + [ℎ𝑙𝑉 − ( 4 − 5 )]
2𝑔 2𝑔
ps
G 0.050
1
− 1) ⋅ (𝐶𝐻0 + 𝐶ℎ𝑙 𝑓 + 𝐶ℎ𝑙 cir ) .
in
𝜂vol
𝑄𝐿 = 𝐶dL ⋅ 𝑎𝑐 √2𝑔Δ𝐻𝑐 ,
yc
II
𝑃𝑙vol
=(
pc
0.100
2.14. Volumetric Head Loss ℎ𝑙Vol . The volumetric (leakage)
head loss and the volumetric head loss coefficient are given
as follows after using (2):
𝑄
=
= 𝐿 (𝐻0 + ℎ𝑙𝑓 + ℎ𝑙cir )
in
𝛾𝑄
𝑄
1
− 1) ⋅ (𝐻0 + ℎ𝑙𝑓 + ℎ𝑙cir )
=(
in
𝜂vol
p3
2
𝐷eye
𝑝3 − 𝑝𝑐 𝑢22
=
(1 − 2 ) .
𝛾
8𝑔
𝐷2
(97)
Thus, the leakage flow rate and the leakage flow rate coefficient, 𝐶𝑄𝐿 , are
𝑄𝐿 = 𝐶dL ⋅ 𝜋𝐷eye 𝑦𝑐 ⋅ 𝑢2 √2
∗ (𝐶𝐻0 −
2
2
2
𝐶𝑉
− 𝐶𝑉
+ 𝐶eye ⋅ 𝐶𝑉
4
5
eye
2𝜑2
1/2
2
𝐷eye
1
− (1 − 2 ))
8
𝐷2
(96)
𝐶𝑄𝐿 =
,
𝑦𝑐 √2𝜋2 𝐶dL
𝑄𝐿
=
(𝑁/60) 𝐷23 𝐷eye (𝐷 /𝐷 )2
2
eye
10
International Journal of Rotating Machinery
∗ (𝐶𝐻0 −
−
2
2
2
𝐶𝑉
− 𝐶𝑉
+ 𝐶eye ⋅ 𝐶𝑉
4
5
eye
Therefore, the pump required net positive suction head is
2𝜑2
1/2
2
𝐷eye
1
(1 − 2 )) .
8
𝐷2
Using (2), 𝑄𝐿 /𝑄 = 𝐶𝑄𝐿 /𝐶𝑄, and (8) for 𝐶𝑄, the pump volumetric efficiency can be deduced after some mathematical
manipulations:
𝐶NPSH𝑅 =
2
2
2
𝐶𝑉
− 𝐶𝑉
+ 𝐶eye ⋅ 𝐶𝑉
4
5
eye
2
𝐷eye
𝐷 2
1
− [1 − (
) ( 1 ) ])
8
𝐷1
𝐷2
𝐶NPSH𝑅
(99)
2𝜑2
1/2
.
in
Δ󸀠󸀠 𝑊𝑢1 ⋅ 𝑢1
𝑔
.
(100)
𝐶NPSH𝑅 = 0.02 +
𝑛𝑠 =
𝑁√𝑄
[𝑁 (rpm) , 𝐻 (m) , 𝑄 (lit/s)] .
𝐻3/4
(108)
Substitute for
𝑏
60
𝜑√2𝑔𝐻, and 𝑏2 = 𝐷2 ( 2 ) yield :
𝜋𝐷2
𝐷2
3/4
𝑛𝑠 =
60 ∗ √1000 ∗ (2𝑔)
√𝜋
∗ √𝜀2
√(
𝑏2
) ∗ √𝜂vol ⋅ 𝜑√𝜓.
𝐷2
(109)
Therefore,
𝑛𝑠 = 9977√𝜀2 √ (
𝑏2
) ⋅ √𝜂vol ⋅ 𝜑√𝜓.
𝐷2
(110)
(102)
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:
𝑊2
𝑉2
𝑝min 𝑝1
=
− ( 1𝑏 − 1 ) .
𝛾
𝛾
2𝑔
2𝑔
(𝜓/𝜑)
1
. (107)
2sin2 𝛽𝑏1 (𝐷1 /𝐷2 )2 (𝑏1 /𝑏2 )2 (𝜀1 /𝜀2 )2
2.17. Pump Specific Speed 𝑛𝑠 . The pump specific speed is
defined with suitable units as
𝑁=
2.16. Required Net Positive Suction Head 𝑁𝑃𝑆𝐻R . The pump
net positive suction head NPSH is defined as the difference
between the fluid inlet stagnation pressure head and vapour
pressure head [13]:
𝑝
𝑝1 𝑉12
+
)− V,
𝛾
2𝑔
𝛾
(106)
𝑄 = 1000𝜂vol 𝜀2 ⋅ 𝜋𝐷2 𝑏2 ⋅ 𝜓√2𝑔𝐻 (lit/s) ,
𝐷 2 cot (180 − 𝛽𝑏1 ) 𝜓
𝜋
⋅ .
sin 𝛽𝑏1 ) ( 1 ) −
𝑍
𝐷2
(𝑏1 /𝑏2 ) (𝜀1 /𝜀2 ) 𝜑
(101)
NPSH = (
2
(𝑝min − 𝑝V ) /𝛾
1 𝑉𝑟1
+
=
.
sin2 𝛽𝑏1 2𝑢22
𝑢22 /𝑔
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
Thus, the inlet shock circulation head loss coefficient is
ℎ𝑙cir
𝐶ℎ𝑙 = 2 in
cirin
𝑢2 /𝑔
= (1 − 𝜀1
(105)
2
2.15. Inlet Shock Circulation Head Loss ℎ𝑙cir . At impeller inlet,
in
a shock circulation exists inside pump eye when the flow
discharge differs from that of designed one. As discussed
before in Section 2.3, 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
ℎ𝑙cir =
NPSH𝑅
.
𝑢22 /𝑔
Referring to Figure 5, 𝑊1𝑏 = (𝑉𝑟1 / sin 𝛽𝑏1 ), and thus
𝐷eye 2 𝐷1 2 √2𝐶dL 𝜑
𝑦𝑐
)(
)( )
𝐷eye
𝐷1
𝐷2 (𝑏2 /𝐷2 ) 𝜀2 𝜓
∗ (𝐶𝐻0 −
(104)
The pump NPSH𝑅 coefficient is defined as
(98)
𝜂vol = 1 − (
2
(𝑝min − 𝑝V ) 𝑊1𝑏
+
.
𝛾
2𝑔
NPSH𝑅 =
(103)
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
International Journal of Rotating Machinery
11
Table 1: Pump input-parameters.
Impeller
Blades
Volute
Input parameter
𝐷1
𝐷2
𝑏2
𝐷2
𝑏1
𝑏2
𝜖2
𝑦0
𝐺=
(𝐷2 /2)
𝑢 ⋅ (𝐷2 /2)
Re2 = 2
]
(𝐷2 = 0.209 m)
(𝑁 = 1450 rpm)
(] = 10−6 m2 /s) water
𝜅
𝐷2
(𝜅 = 0.0001 m)
𝑍
𝛽𝑏2
𝛽𝑏1
𝐷3
𝐷2
𝑏3
𝑏2
𝛼𝑉
𝛼th
𝐶𝑑𝑉
𝐷eye
Eye
𝐷th
𝑦𝑐
𝐷eye
𝐶dL
𝐶eye
0.8
𝜂
Value
0.5
0.121
1
0.95
0.7
𝜂
0.6
CH
CH
0.5
0.4
0.05
0.3
1.16 ∗ 106
0.2
0.1
0.000478
4
164∘
164∘
1.05
1.05
7∘
4∘
0.8
1
0.005
0.6
1.0
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 ≈ 0.209 m, 𝑁 ≈ 1450 rpm, 𝜅 ≈ 0.0001 m,
] ≈ 10−6 m2 /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.
Other pump constant-parameters are calculated according to the present procedure as shown in Table 2.
The sequence of calculations by using the procedure
derived equations of pump variable parameters is listed in
Table 3.
0.0
0.00
0.02
0.04
0.06
𝜓/𝜑
0.08
0.10
0.12
Experimental head coeff. [14]
Present procedure head coeff.
Efficiency
Experimental [14]
Present procedure
Figure 9: Comparison between present procedure results and
experimental results by Baun and Flack [14].
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.
The pump performance characteristics are presented
by the calculated pump head coefficient, efficiency, power
coefficients, and required NPSH as shown in Figure 10. 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.
The maximum shaft power coefficient 𝐶𝑃sh ≈ 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 coefficient 𝜓 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
12
International Journal of Rotating Machinery
Table 2: Pump constant-parameters.
Constant parameter equation
1 − 𝜎0 =
√sin 𝛽𝑏2
𝑍0.7
𝜖1 = 1 −
Impeller
𝐷hyd
𝐷2
Blades
Volute
Eye
for
𝐷1
≤ 𝜖limit = 𝑒−8.16 sin 𝛽𝑏2 /𝑍
𝐷2
(1 − 𝜖2 )
(𝐷1 /𝐷2 ) (sin 𝛽𝑏1 / sin 𝛽𝑏2 )
𝑑2
𝜋
= 𝜖2 sin 𝛽𝑏2
𝐷2
𝑍
𝑑1
𝜋 𝐷1
= 𝜖1
sin 𝛽𝑏1
𝐷2
𝑍 𝐷2
2 (((𝑑1 /𝐷2 ) (𝑏1 /𝑏2 ) (𝑏2 /𝐷2 )) + ((𝑑2 /𝐷2 ) (𝑏2 /𝐷2 )))
=
(𝑑1 /𝐷2 ) + ((𝑏1 /𝑏2 ) (𝑏2 /𝐷2 )) + (𝑑2 /𝐷2 ) + (𝑏2 /𝐷2 )
𝜅/𝐷2
𝜅
=
𝐷hyd 𝐷hyd /𝐷2
𝑓𝐷
1
𝐾𝐷 =
40 𝜖2 (𝑏2 /𝐷2 )
𝑓𝐷 = 𝑓𝐷 (Re2 , 𝐺)
𝐿𝑏
𝐷
1
1
= (1 − 1 )
𝐷2 2
𝐷2 sin 𝛽bm
1
𝛽bm = (𝛽𝑏1 + 𝛽𝑏2 )
2
1
1
1
=
+
𝐷ℎ𝑉 /𝐷2 2(𝑏3 /𝑏2 )(𝑏2 /𝐷2 ) 8(𝜋/𝑍)(𝐷3 /𝐷2 ) sin 𝛼𝑉
𝐿𝑉 1 𝜋
𝐷
=
( 3)
𝐷2 2 cos 𝛼𝑉 𝐷2
(𝜅/𝐷2 )
𝜅
=
𝐷ℎ𝑉
(𝐷ℎ𝑉 /𝐷2 )
(𝑏 /𝑏 ) (𝑏 /𝐷 )
𝜃
1
tan ( th ) = [𝜋 tan 𝛼𝑉 + tan 𝛼th − 3 2 2 2 ]
2
2
(𝐷3 /𝐷2 )
𝜃th
𝐶𝑓th = 0.5 + 2.6 ∗ sin ( )
2
𝐷eye (𝐷3 /𝐷2 )
=
(𝜋 tan 𝛼𝑉 + tan 𝛼th )
𝐷1
(𝐷1 /𝐷2 )
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 13 shows the theoretical dimensionless headdischarge 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 14 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
Equation no.
(22)
(4)
(15)
(26)
(80)
(85)
(92a)
(92b)
(82a)
(82b)
(49)
(48)
(51)
(43)
(56)
(40)
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.
The variation of the flow velocity coefficients and the
pump volumetric efficiency 𝜂vol with the pump discharge
coefficient is shown in Figure 15. The coefficients 𝐶𝑉1 , 𝐶𝑉Δ𝑉𝑢 ,
2
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 = 𝑉𝑢2 act . 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 > 𝑉𝑢2 act (Figure 6).
The pump specific speed is presented in Figure 16, which
shows that 𝑛𝑠 ≈ 870 at pump best efficiency point.
International Journal of Rotating Machinery
13
Table 3: Pump variable-parameters.
Order
Variable parameter equation
𝜓≡
𝑉𝑟2
√2𝑔𝐻
step = 0.001
𝜎 = 𝜎0 , 𝑥 = 1, 𝜂vol = 1
Start of iteration
1
1
2
𝜑 = − 𝜓 cot 𝛽𝑏2 + √ 𝜓2 cot2 𝛽𝑏2 +
2
2
𝑥𝜎
Initial
1
𝑦 = 𝐶𝐻∞ ≡
2
𝜓
𝐻∞
= 1 + cot𝛽𝑏2
𝑢22 /𝑔
𝜑
𝜎=1−
3
4
𝐶ℎ𝑙
5
𝐶𝐻0 ≡
slp
𝐶𝑉4 ≡
7
ℎ𝑙slp
𝑢22 /𝑔
(21), (65b)
(1 − 𝜎0 )
𝑦
(20)
= (1 − 𝜎) 𝐶𝐻∞
(65d)
(65c)
4𝜖2 𝜂vol (𝑏2 /𝐷2 )
𝑉5
⋅𝜓
=
√2𝑔𝐻 (𝐷𝑡ℎ /𝐷3 )2 (𝐷3 /𝐷2 )2
(37)
𝜖2 𝜂vol
𝑉4
⋅𝜓
=
√2𝑔𝐻 (𝐷3 /𝐷2 ) (𝑏3 /𝑏2 ) tan 𝛼𝑉
(33)
𝐶𝑉eye ≡
8
≡
𝑉eye
√2𝑔𝐻
=
4𝜖2 (𝑏2 /𝐷2 )
2
2
(𝐷𝑡ℎ /𝐷3 ) (𝐷3 /𝐷2 )
𝐶𝑉3𝑝 =
9
⋅𝜓
𝐶𝑉4
cos 𝛼𝑉
𝐶𝑉3𝑑 = 𝜎 (𝜑 + 𝜓cot𝛽𝑏2 ) − 𝐶𝑉4
10
𝐶𝑉3𝑟 ≡
11
𝜂vol = 1 − (
(69)
𝜓
𝐻0
= 𝜎 (1 + cot𝛽𝑏2 )
𝑢22 /𝑔
𝜑
𝐶𝑉5 ≡
6
12
= 0.002 to 0.12
Eq. no.
𝜖2 𝜂vol
𝑉3𝑟
=
⋅𝜓
(𝐷
/𝐷
√2𝑔𝐻
3
2 ) (𝑏3 /𝑏2 )
2
2
𝐶𝑉2 − 𝐶𝑉2 5 + 𝐶eye ⋅ 𝐶𝑉eye
𝐷eye 2 𝐷1 2 1
√2𝐶dL 𝜑
𝑦𝑐
𝐷1 2 𝐷eye
1
)(
)( )
−
)
(
)
]
∗ √ 𝐶𝐻0 − 4
[1
−
(
𝐷eye
𝐷1
𝐷2 (𝑏2 /𝐷2 ) 𝜖2 𝜓
2𝜑2
8
𝐷2
𝐷1
(39)
(54a)
(54b)
(54c)
(99)
14
International Journal of Rotating Machinery
Table 3: Continued.
Order
13
14
15
16
Variable parameter equation
Re𝑉 =
𝑓𝑉 =
𝑉3𝑝 ⋅𝐷ℎ𝑉
]
𝐻
22
23
24
25
𝐷2
) ⋅ Re2
(52)
1.11
]}
2
𝑉+eye
(57)
ℎ𝑙eye
ℎ𝑙
1
=1+ 𝑉 +
𝑥
𝐻
𝐻
(61)
End of iteration
ℎ𝑙𝑉 + ℎ𝑙eye
≡
= (1 − 𝑥) 𝜎𝐶𝐻∞
𝑢22 /𝑔
(65e)
𝜓
𝐻
= 𝑥𝜎 (1 + cot 𝛽𝑏2 )
𝑢22 /𝑔
𝜑
(65a)
𝜓
𝑏
𝑄
= 𝜂vol 𝜖2 𝜋2 ( 2 ) ⋅ ( )
3
𝐷
𝜑
(𝑁/60) 𝐷2
2
(6)
𝐶𝐻 ≡
𝐶𝑄 ≡
(45)
= 𝐶eye ⋅ 𝐶𝑉2 eye
𝐻
𝐶ℎ𝑙
𝜖2 (𝜋/𝑍) (𝑏2 /𝐷2 ) ⋅ (𝜓/𝜑)
𝑊av
=
𝑢2
1/2 (((𝑑1 /𝐷2 ) (𝑏1 /𝑏2 ) (𝑏2 /𝐷2 )) + ((𝑑2 /𝐷2 ) (𝑏2 /𝐷2 )))
Re =
𝑓𝑖 =
(50)
(46)
= 𝐶𝑓𝑉 𝐶𝑉2 3𝑝 + 𝐶𝑑𝑉 𝐶𝑉2 3𝑑 + 𝐶𝑉2 3𝑟 + 𝐶𝑓𝑡ℎ 𝐶𝑉2 4
18
21
𝐷ℎ𝑉
𝐿𝑉
𝐿
1
= 𝑓𝑉 ( 𝑉 ) (
)
𝐷ℎ𝑉
𝐷2
𝐷ℎ𝑉 /𝐷2
ℎ𝑙eye
20
𝜑
)(
{log [(6.9/Re𝑉 ) + ((𝜅/𝐷ℎ𝑉 ) /3.7)
17
19
𝐶𝑉3𝑝
0.3086
𝐶𝑓𝑉 = 𝑓𝑉
ℎ𝑙𝑉
= 2(
Eq. no.
𝑊av ⋅ 𝐷hyd
]
= 2(
𝐷hyd
𝑊av
)(
) ⋅ Re2
𝑢2
𝐷2
(86)
0.3086
{log [(6.9/ Re) + ((𝜅/𝐷hyd ) /3.7)
4𝐶𝑑𝑖 = (𝑓𝑖 + 0.006) (1.1 + 4
1.11
𝑏2
)
𝐷2
(81)
2
(84)
]}
(83)
International Journal of Rotating Machinery
15
Table 3: Continued.
Order
Variable parameter equation
𝐶ℎ𝑙𝑓 ≡
26
= 4𝐶𝑑𝑖
𝑢22 /𝑔
𝐶ℎ𝑙𝐷 ≡
27
28
ℎ𝑙𝑓
𝐶ℎ𝑙cir
in
≡
ℎ𝑙cir
in
𝑢22 /𝑔
29
𝐶ℎ𝑙vol ≡
30
𝐶𝐻sh ≡
= (1 − 𝜖1
ℎ𝑙vol
𝑢22 /𝑔
ℎ𝑙𝐷
𝑢22 /𝑔
Eq. no.
(𝐿 𝑏 /𝐷2 ) 1 𝑊av 2
)
(
(𝐷hyd /𝐷2 ) 2 𝑢2
(78)
1 𝜑
𝜂vol 𝜓
(91)
= 𝐾𝐷 ⋅
𝐷 2 cot (180 − 𝛽𝑏1 ) 𝜓
𝜋
⋅
sin 𝛽𝑏1 ) ( 1 ) −
𝑍
𝐷2
(𝑏1 /𝑏2 ) (𝜖1 /𝜖2 ) 𝜑
(101)
1
− 1) ⋅ (𝐶𝐻0 + 𝐶ℎ𝑙𝑓 + 𝐶ℎ𝑙cir )
in
𝜂vol
(93)
=(
𝐻sh
= 𝐶𝐻0 + 𝐶ℎ𝑙𝑓 + 𝐶ℎ𝑙cir + 𝐶ℎ𝑙vol + 𝐶ℎ𝑙𝐷
in
𝑢22 /𝑔
𝜂≡
31
𝑃𝑤
𝐶
= 𝐻
𝑃sh 𝐶𝐻sh
(73)
(74)
32
𝐶𝑃sh ≡
𝑃sh
= 𝜋2 𝐶𝑄𝐶𝐻sh
𝜌(𝑁/60)3 𝐷25
(75)
33
𝐶𝑃𝑤 ≡
𝑃𝑤
= 𝜋2 𝐶𝑄𝐶𝐻
𝜌(𝑁/60)3 𝐷25
(76)
34
𝐶NPSH𝑅 = 0.02 +
2
35
(𝜓/𝜑)
1
2 sin2 𝛽𝑏1 (𝐷1 /𝐷2 )2 (𝑏1 /𝑏2 )2 (𝜖1 /𝜖2 )2
(107)
𝑏2
) ⋅ √𝜂vol ⋅ 𝜑√𝜓
𝐷2
(110)
𝑛𝑠 = 9977√𝜖2 √(
Figure 17 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.
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.
Notations
𝐴:
𝑎𝑐 :
𝐴 𝑉:
𝐴 th :
𝐴 𝑉th :
Impeller net area, m2
Clearance area of wearing ring, m2
Volute area, m2
2
Throat outlet area, 𝜋𝐷th
/4, m2
Volute throat inlet area, m2
16
International Journal of Rotating Machinery
0.8
1.6
BEP
1.5
𝜂
0.7
1.4
𝜂
0.6
𝜑
CH
CH
1.3
1.2
0.5
1.1
0.4
1.0
CPsh
0.3
0.9
CP𝑤
CP𝑤
BEP
CPsh
0.8
0.2
CNPSH 𝑅
0.1
0.7
0.00
𝑅
PSH
CN
0.0
0.00
0.02
0.04
0.02
0.04
0.06
0.08
0.10
0.12
CQ
0.06
0.08
0.10
Figure 12: Speed coefficient variation with pump discharge coefficient.
0.12
CQ
Figure 10: Characteristics of centrifugal pump obtained by present
study.
1.4
1.3
1.2
1.1
Head and head loss coefficients
0.20
0.18
𝜓/𝜙
0.16
0.14
0.12
0.10
𝜓/𝜙
𝜓
0.08
0.06
0.02
0.00
0.00
0.02
0.04
0.06
Slip
loss
0.7
0.6
CH∞
CH0
Volu
te an
d
CH
0.5
0.4
0.2
0.08
0.10
0.1
0.12
0.0
0.00
CQ
Figure 11: Variation of flow coefficient and ratio of flow coefficient
to speed coefficient with discharge coefficient.
𝑏:
𝐶𝑑𝑖 :
𝐶dL :
𝐶𝑑𝑉 :
𝐶eye :
𝐶𝑓th :
𝐶𝑓𝑉 :
𝐶𝐻:
𝐶𝐻0 :
0.8
head
0.3
BEP
0.04
Eule
r
0.9
Impeller width, m
Impeller dissipation coefficient
Leakage discharge coefficient
Volute diffusion loss coefficient
Eye loss coefficient
Volute throat friction loss coefficient
Volute friction loss coefficient
Manometric head coefficient, 𝐻/(𝑢22 /𝑔)
Head coefficient at impeller outlet,
𝐻0 /(𝑢22 /𝑔)
𝐶𝐻∞ : Euler head coefficient, 𝐻∞ /(𝑢22 /𝑔)
𝐶𝐻sh : Shaft head coefficient, 𝐻sh /(𝑢22 /𝑔)
BEP
𝜓
1.0
0.02
0.04
0.06
eye l
oss
M
an
om
etr
ic
he
ad
0.08
0.10
0.12
CQ
Figure 13: Variation of head coefficients and head loss coefficients
with discharge coefficient.
𝐶ℎ𝑙
: Inlet shock circulation head loss coefficient,
ℎ𝑙cir /(𝑢22 /𝑔)
in
𝐶ℎ𝑙 𝑓 : Impeller skin friction head loss coefficient
𝐶ℎ𝑙 : Eye head loss coefficient, ℎ𝑙eye /(𝑢22 /𝑔)
cirin
eye
𝐶ℎ𝑙 :
𝐷
𝐶ℎ𝑙 slp :
𝐶ℎ𝑙 :
𝑉
𝐶ℎ𝑙 :
𝑉+eye
𝐶ℎ𝑙 :
vol
Disk friction head loss coefficient, ℎ𝑙𝐷 /(𝑢22 /𝑔)
Slip head loss coefficient, ℎ𝑙slp /(𝑢22 /𝑔)
Volute head loss coefficient, ℎ𝑙𝑉 /(𝑢22 /𝑔)
Volute-eye head loss coefficient,
(ℎ𝑙𝑉 + ℎ𝑙eye )/(𝑢22 /𝑔)
Volumetric head loss coefficient, ℎ𝑙vol /(𝑢22 /𝑔)
International Journal of Rotating Machinery
17
1.4
2000
1.3
1800
1.2
0.9
0.8
1400
1200
C
H
1000
sh
CH0
Impeller hydraulic
friction loss
0.7
CH
0.6
1600
800
600
400
0.5
200
0.4
0.3
BEP
0.2
0.1
0.0
0.00
0.02
0.04
0.06
0
0.00
M
an
om
etr
ic
he
ad
0.08
0.10
BEP
Head and head loss coefficients
1.0
ns
ad
he loss ss
o
.
aft
cir c l
Sh
tr i
let
In me
s
lu
os
Vo isk l
D
1.1
0.02
0.04
0.06
0.08
0.10
0.12
CQ
Figure 16: Pump specific speed values at different discharge coefficient.
0.12
CQ
0.8
Figure 14: Head loss coefficients affecting pump power at different
discharge coefficients.
𝜂
0.7
𝜂
1.0
CH
𝜂vol
0.9
0.5
CV2
𝜂vol 0.8
0.4
C
V
CV
CPsh
3𝑑 󳰀
0.7
CV2
0.6
0.3
act
CH
0.2
C
0.5
V
3𝑑
0.1
0.4
0.3
4
V
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
CQ
Figure 15: Values of volumetric efficiency and different pump
velocity coefficients.
Coefficient of pump NPSH𝑅 , NPSH𝑅 /(𝑢22 /𝑔)
Shaft power coefficient, 𝑃sh /𝜌(𝑁/60)3 𝐷25
Water power coefficient, 𝑃𝑤 /𝜌(𝑁/60)3 𝐷25
Pump discharge coefficient, 𝑄/(𝑁/60)𝐷23
Leakage flow rate coefficient
Velocity coefficient, 𝑉/√2𝑔𝐻
Impeller outer diameter, m
0.02
0.04
0.06
0.08 0.10
CQ
0.12
Water = 1 ∗ 10−6 m2 /s
Oil
= 100 ∗ 10−6 m2 /s
= 200 ∗ 10−6 m2 /s
Oil
BEP
CV1
0.1
0.0
0.00
C
CVΔVu2
0.2
𝐶NPSH𝑅 :
𝐶𝑃sh :
𝐶𝑃𝑤 :
𝐶𝑄:
𝐶𝑄𝐿 :
𝐶𝑉 :
𝐷2 :
CPsh
0.6
Figure 17: Effect of pumped fluid viscosity on the performance of
centrifugal pump.
𝐷eye :
𝐷ℎ𝑉 :
𝐷th :
𝑓:
𝑓𝐷:
𝑔:
𝐺:
ℎ𝑙cir :
in
ℎ𝑙𝐷 :
Pump eye diameter, m
Average volute hydraulic diameter, m
Throat diameter, m
Hydraulic friction coefficient
Disk friction coefficient
Acceleration of gravity, m/s2
Impeller disk gap parameter, 𝑦0 /(𝐷2 /2)
Inlet shock circulation head loss, m
Disk friction head loss, m
18
International Journal of Rotating Machinery
ℎ𝑙𝑓 :
ℎ𝑙slp :
ℎ𝑙𝑉 :
ℎ𝑙vol :
𝐻:
𝐻0 :
𝐻sh :
𝐻∞ :
𝐾𝐷:
𝐿 𝑏:
𝐿 th :
𝐿 𝑉:
𝑁:
NPSH:
NPSH𝑅 :
𝑛𝑠 :
𝑝:
𝑝𝑐 :
𝑝𝑠 :
𝑃sh :
𝑃𝑤 :
𝑃sh0 :
𝑃𝑙cir :
in
𝑃𝑙𝑓 :
𝑃𝑙𝐷 :
𝑃𝑙vol :
𝑄:
𝑄𝑖 :
𝑄𝐿 :
𝑟:
𝑡:
𝑢:
𝑉:
𝑉3𝑝 :
𝑉3𝑑 :
𝑊:
𝑥:
𝑦:
𝑦0 :
𝑦𝑐 :
𝑍:
Impeller skin friction head loss, m
Slippage head loss, m
Volute head loss, m
Volumetric head loss, m
Manometric head of pump, m
Water head at impeller outlet, m
Pump shaft head, m
Euler pump head, m
Impeller disk loss coefficient
Blade length, m
Throat length, m
Volute length, m
Pump rotational speed, rpm
Pump net positive suction head, m
Pump required net positive suction head, m
Pump specific speed
Pressure, N/m2
Pressure before wearing ring clearance, N/m2
Pressure at pump suction side, N/m2
Pump shaft power, W
Pumped water power, W
Impeller Euler power, W
Inlet shock circulation power loss, W
Impeller friction power loss, W
Disk friction power loss, W
Volumetric power loss, W
Pump discharge, m3 /s
Impeller discharge, m3 /s
Pump internal discharge leakage, m3 /s
Radius, m
Blade thickness, m
Tangential flow velocity, m/s
Absolute flow velocity, m/s
Throughflow component of volute velocity,
m/s
Circulation velocity of flow in volute, m/s
Relative velocity of flow, m/s
Ratio, 𝐻/𝐻0
Group parameter, (𝑦 = 𝐶𝐻∞ )
Axial gap between impeller disk and casing,
m
Clearance of wearing ring, m
Number of blades.
Greek Letters
𝛼:
𝛼𝑉 :
𝛽𝑏 :
𝜃th :
𝜖:
𝜑:
𝜂:
𝜂vol :
]:
𝜌:
𝜎:
Theoretical absolute velocity angle, degrees
Volute angle, degrees
Blade angle, degrees
Volute throat angle, degrees
Blade thickness coefficient
Pump speed coefficient, 𝑢2 /√2𝑔𝐻
Manometric efficiency of pump
Volumetric efficiency
Fluid kinematic viscosity, m2 /s
Density of fluid, kg/m3
Slip factor, 𝑉𝑢2act /𝑉𝑢2
𝜓: Pump flow coefficient, 𝑉𝑟2 /√2𝑔𝐻
𝜅: Roughness height, m.
Subscripts
1:
2:
3:
4:
5:
av:
act:
eye:
𝑐:
hl:
𝑖:
𝑙:
𝑟:
𝑠:
sh:
th:
𝑢:
vol:
Impeller inlet
Impeller outlet
Volute inlet
Volute outlet (throat inlet)
Pump outlet (throat outlet)
Average
Actual
Pump eye
Clearance
Head loss
Impeller
Loss
Radial direction
Suction
Shaft
Volute throat
Tangential direction
Volumetric.
References
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Hindawi Publishing Corporation
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Volume 2014
Volume 2014
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Journal of
Journal of
Electrical and Computer
Engineering
Robotics
Hindawi Publishing Corporation
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Hindawi Publishing Corporation
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Volume 2014
Volume 2014
VLSI Design
Advances in
OptoElectronics
International Journal of
Navigation and
Observation
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Volume 2014
Hindawi Publishing Corporation
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Chemical Engineering
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Volume 2014
Volume 2014
Active and Passive
Electronic Components
Antennas and
Propagation
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Aerospace
Engineering
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Volume 2014
International Journal of
International Journal of
International Journal of
Modelling &
Simulation
in Engineering
Volume 2014
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Volume 2014
Shock and Vibration
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Volume 2014
Advances in
Acoustics and Vibration
Hindawi Publishing Corporation
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Volume 2014