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8/10/2019 CENTRIFUGAL PUMP IMPELLER VANE PROFILE
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CHAPTER 5
CENTRIFUGAL PUMP IMPELLER VANE PROFILE
The concept of impeller design and the application of inverse design
for the vane profile construction are discussed in this chapter. The vane
profile plays a vital role to develop the streamlined flow. In conventional
design, the designer uses vane arc method to develop the profile. Due to this
approach, the eddy and flow reversal may occur in the flow path. The main
focus on inverse design concept is explained here in detail for the vane profile
construction. Subsequently, the different vane profile geometry is constructed
based on this approach.
The design of the centrifugal pump impeller is not a universally
standardized one. Every firm depends on its designers experience, expertise
and technical intuition to design a good impeller. The fact that the impeller
flow physics has not been understood fully has led the designers to fall back
on tried and tested old design methodologies.
5.1 CONVENTIONAL DESIGN
Impeller dimensions have always been a direct fall down of the head
it has to develop and the discharge it has to supply. Previously used empirical
formulae and thumb rules have always been the design aid for designers. The
different methods developed by highly experienced and accomplished
hydraulic engineers like Lebonoff, Kurowzski, Anderson and Lazarkiewicz
also have elements of empirical design.
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5.2 DESIGN METHODOLOGY
The impeller dimensions are designed based on the head and
discharge. The following are the steps involved in designing a centrifugal
impeller (Figure 5.1):
From the head (H) and discharge (Q), the kinematic specific
speed (nsQ) is calculated
4/3sQ H
Qnn = (5.1)
From the head and discharge, the shaft power (Psh) required is
calculated.
=
75
QHPsh unit in hp (5.2)
Before finding the hub diameter, the shaft diameter (dsh) is
found using the formula
nP360000d
3sh
3
sh
= - Torsional Stress, (kP/cm2) (5.3)
Figure 5.1 Pump Impeller
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In order to trace the profile with four radii of curvature, four
more circles, that is, point 1 to 6, are drawn at equal intervals on
the axis. The curve is drawn through A, B, C, D and E based on
the positions G, H, I, J and K.
Figure 5.2 Vane profile construction From the point where inlet circle meets the horizontal axis, a line
at an angle of inlet vane angle (16) is drawn to the length of the
radius of curvature of the first arc (47 mm).
An arc is drawn with the end point of this line as the centre and
with the corresponding radius, till the arc meets the next circle.
From the point where the arc meets the next circle, a line is
drawn to the length equal to the next radius of curvature and
passing through the previous centre.
An arc is drawn with the end point of this line as the centre and
with the corresponding radius till the arc meets the next circle.
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based gradient with absolute velocity formulation are followed. The fluid is
assigned as water from the database at standard operating condition
(properties at standard atmospheric condition) and the flow is considered as
steady flow. For turbulence, the well agreed standard k- two-equation
turbulence model with a standard wall function is adopted. Among the
available various convection schemes, the Upwind Differencing is used for
the ease of convergence. Relaxation factor is applied for pressure, momentum
and turbulence parameters. The solution is initialized with atmospheric
operating condition and solved till it reaches the convergence. The
convergence is achieved up to 1 e-4
and the mass balance is checked till 1 e-5
of the mass flux. The static, dynamic and the total pressure values are
important in finding the new vane profile. The contours of static pressure
distribution and velocity distribution are useful in making inferences and are
shown in Figures 5.4 and 5.5.
Figure 5.3 Conventional designed model of the impeller
Inflow
Outflow
Rotational
Direction
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Figure 5.4 Static pressure distribution of conventionally designed model
Figure 5.5 Velocity distribution of conventionally designed model
Pascal
m/s
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In this case, the pressure increases gradually towards the outlet and
also the low pressure zone is extended till the outlet section. The peripheral
velocity (u2) is greater at outer diameters and the flow is oriented or guided
gradually towards the outlet. The low pressure zone present in the flow path
causes the flow separation, due to which the flow losses are more in the
conventional impeller. The redesign process reduces the losses and also
increases the static pressure at the outlet.
The area weighted average of the static pressure given below is
taken from Fluent software results:
The area weighted average static pressure value at the inlet = -35697.32 Pa.
The area weighted average static pressure value at the outlet = 266906.5 Pa.
5.5 VANE PROFILE OPTIMIZATION BY INVERSE DESIGN
METHOD
The real flow through an impeller is three dimensional, that is to say
the velocity of the fluid is the function of three positional coordinates, say, in
the cylindrical system, r, and z. Thus there is a variation of velocity not only
along the radius but also across the blade passage in any plane parallel to the
impeller rotation, say from upper side of one blade to the underside of the
adjacent blade, which constitutes an abrupt change - a discontinuity. Also
there is a variation of velocity in the meridional plane, i.e. along the axis of
the impeller. The velocity distribution is, therefore, very complex and
dependent upon the number of blades, their shapes and thickness as well as
the width of the impeller and its variation with radius.
The one-dimensional theory simplifies the problem considerably by
making the following assumptions
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The blades are infinitely thin and the pressure difference across
them is replaced by imaginary body forces acting on the fluid
producing torque.
The number of blades is infinitely large, so that the variation of
velocity across the blade passages is reduced and tends to zero.
This assumption is equivalent to stimulating axisymmetrical
flow, in which there is perfect symmetry with regard to the axis
of impeller rotation. Thus,
0v
=
Over that part of the impeller where transfer of energy takes
place (blade passages) there is no variation of velocity in the
meridional plane, i.e. across the width of the impeller.
0z
v=
The result of these assumptions is for the one-dimensional flow
= f (r) only, whereas in reality the flow is given as = f (r, , z). Note that
the suffix stipulates the assumption of an infinite number of blades and
hence, it is axisymmetry.
Furthermore, the assumption implies that the fluid stream lines are
confined to infinitely narrow inter blade passages and hence their paths are
congruent with the shape of the inter blade centerline. Thus the flow of fluid
through an impeller passage may be regarded as a flow of fluid particles along
the centerline of the inter blade passage.
The assumptions of the theory enable us to limit our analysis to
changes of conditions, which occur between impeller inlet and impeller outlet
without reference to the space in between where the real transfer of energy
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takes place. This space is treated as a black box having an input in the form
of an inlet velocity triangle and an output in the form of outlet velocity
triangle.
At inlet, the fluid moving with an absolute velocity 1 enters the
impeller through a cylindrical surface of radius r1 and makes an angle of 1
with the tangent at that radius as shown in Figure 5.6. At outlet, the fluid
leaves the impeller through a cylindrical surface of radius r2, with absolute
velocity 2inclined to the tangent at the outlet by the angle 2.
Figure 5.6 Velocity diagram of impeller
The inlet velocity triangle is constructed by first drawing the vector
representing the absolute velocity 1 at an angle 1. The tangential velocity ofthe impeller, u1, is then subtracted from it vectorially in order to obtain vr1, the
relative velocity of the fluid with respect to the impeller blade at the radius r1.
In this basic velocity triangle, the absolute velocity v1 is resolved into two
components: one is the radial direction, called velocity of flow vf1, and the
other, perpendicular to it and hence, in the tangential direction, vw1, sometimes
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called velocity of whirl. These two components are useful in the analysis and,
therefore, they are always shown as part of the velocity triangles.
Similarly, the outlet velocity triangle consists of the absolute fluid
velocity 2 making an angle 2 with the tangent at the outlet, subtracted from
which, vectorailly, is the tangential blade velocity u2 to give the relative
velocity vr2. Here again, the absolute fluid velocity is resolved into radial (vf2)
and tangential (vw2) components.
The general expression for the energy transfer between the impeller
and the fluid, based on the one dimensional theory and usually referred to as
Eulers turbine equation, is derived as follows .
From Newtons second law applied to angular motion,
Torque = Rate of change of angular momentum.
Now, Angular momentum = (Mass)(Tangential velocity)(Radius).
Therefore,
Angular momentum entering the impeller per second = m vw1r1
Angular momentum leaving the impeller per second = m vw2r2
in which mis the mass of fluid flowing per second. Therefore,
Rate of change of angular momentum = Avw2r2- Avw1r1
So that torque transmitted = A (vw2r2- vw1r1)
Since the work done in unit time is given by the product of torque and angular
velocity,
Work done per second = (Torque) = A (vw2r2- vw1r1)
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But = u/r, so that r2= u2and r1= u1. Hence, on substitution,
Work done per second,
Et= A (u2vw2- u1vw1) (5.13)
Since the work done per second by the impeller on the fluid, such as
in this case, is the rate of energy transfer, then:
Rate of energy transfer/Unit mass of fluid flowing, Y= gE=Et/m
The product gE = Y,known as specific energy, is of significance in
the case of pumps and fans.
From the specific energy, Eulers headEis given by
E= (1/g) (u2vw2 - u1vw1) (5.14)
From its mode of derivation it is apparent that Eulers equation
applies to pump (as derived) and to turbine. In the latter case, however,
u1vw1 > u2vw2,Ewould be negative, indicating the reversed direction of energy
transfer. It is, therefore, common to use reversed order of terms in the
brackets to yield positiveE. since the units ofEreduced to meters of the fluid
handled, is often referred to as Eulers head, and in the case of pumps or fans
it represents the ideal theoretical head developedHth.
It is useful to express Eulers head in terms of the absolute fluid
velocities rather than their components. From the velocity triangles
vw1 = 1 cos1, vw2 = 2 cos2
so that
E = (1/g) (u22 cos2 - u11 cos1) (5.15)
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But, using cosine rule
v2
r1= u2
1+ v21 2u1v1cos1
So that
u1v1 cos1 = (u2
1-v2r1+v
21)
Similarly
u2v2 cos2 = (u22-v
2r2+v2
2)
Substituting into
E = (1/2g) (u22
-u12
+v22
-v12
+v
2
r1-v
2
r2)and E = (v
22-v
21)/2g + (u
22u
21)/2g + (v
2r1-v
2r2)/2g (5.16)
In this expression, the first term denotes the increase in kinetic
energy of the fluid in the impeller. The second term represents the energy
used in the setting the fluid in a circular motion about the impeller axis
(forced vortex). The third term is the region of static head due to a reduction
in the relative velocity in the fluid passing through the impeller.
Theoretical pressure values along the vane profile are obtained by
drawing the velocity triangles at the desired points. The velocity triangles are
drawn by assuming that the fluid leaves the impeller with a relative velocity
tangential to the blade at outlet, and in order to draw the outlet velocity
triangles, must be known. The direction of vris then drawn, as well as the vf
vector, which is radial and whose magnitude is calculated from the continuity
equation. It is, thus, possible to draw the u vector perpendicular to vf andstarting from the intersection with the direction of vr. The absolute velocity v
is then obtained by completing the triangle. The pressure values are found at
each point by substituting the velocity values obtained at the points.
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5.5.1 Lagrange Interpolation Polynomial
The actual and theoretical pressure distribution data obtained on the
vane of the impeller were used to develop the equations using Lagranges
method of a polynomial of n degree in the following form.
PN(x) passing through (N+1) points {x0, f(x0)},{x1, f(x1)},
{xN, f(xN)} is given by
=
=
=
ji,N,0i ij
ij
jjN
xx
xx
)x(L
)x(L)x(fP
(5.17)
The interpolation polynomialis used to find the actual and the target
pressure equation at four segments as shown in Figure 5.7.
Figure 5.7 Interpolation segment of impeller
The interpolation formulation is simplified by taking the four
segments instead of eighteen segments to reduce the order of polynomial for
design variable calculation. The corresponding pressure values are taken for
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the calculation of target and existing pressure interpolation function using the
equation (5.17).
Theoretical Pressure Equation Pi(RD)is given by
Pi (RD) =A (RD)3+ B (RD)
2+C (RD) +D (5.18)
where the constant values are tabulated in Table 5.1.
Table 5.1 Constants for equation (5.18)
A B C D
-38388046591 68616050976 - 40851961118 8101566489
Likewise the Conventional Pressure Equation Pe(RD) is
Pe(RD) =E (RD)3+F (RD)
2+G (RD) + H
(5.19)
where the constant values are tabulated in Table 5.2.
Table 5.2 Constants for equation (5.19)
E F G H
-11294428469 20169715455 11997691876 2377262682
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5.5.2 Minimization of Objective Function
The difference between theoretical static pressure distribution
Pi (RD) function and conventional static pressure distribution function Pe (RD)
is framed as the objective function f (RD) which should be minimized. TheRD
is the design variable called radius factor, which is the ratio of radius of
curvature to diametrical distance. The objective function is minimized using
the first derivative method.
Step 1: f (RD) = Pi (RD) Pe (RD)
Step 2: f(RD) = 0
Step 3: (RD)1and (RD)2are found out
Step 4: f(RD) is found out
Step 5: Substitute (RD)1and (RD)2 in f(RD)
Step 6: One of the (RD) is chosen which satisfies the condition f(R/D) is
positive
The value for the RD ratio is achieved as 0.5798 for this particular
impeller, which is used for constructing the vane profile at the intervals
5.5.3 Flow Passages
The vane profile can be a single arc with a centre and a uniform
radius of curvature. The profile can also be a composite one wherein we have
more than one arc with each having a different centre and different radius of
curvature.
In the redesigning procedure five different vane profiles have been
generated as shown in Figures 5.8 to 5.12.
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The first profile is a single arc with one centre and the radius of
curvature calculated from the obtained RD with diameter being
the mean diameter of the inlet and outlet diameters.
The second profile is a composite curve with two arcs, each
having a dedicated centre of its own. The radii of curvature are
calculated with two different diameters, the first one being the
average of inlet and mean diameter and the second being the
average of mean and the outlet diameters.
The third profile is generated in the same way by taking three
zones and their mean diameters.
The fourth profile is an extension of the previous profiles. The
fifth profile is in concept an extension of previous profiles, but it
has been generated with 17 different radii of curvature capturing
the effect of the optimized RDto the utmost.
Figure 5.8 Single radius Figure 5.9 Double radii
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Figure 5.10 Triple radii Figure 5.11 Quadruple radii
Figure 5.12 Seventeen radii
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5.5.4 Pressure and Velocity for Different Vane Profiles
Figures 5.13 to 5.22 show the changes in the pressure and velocity
distribution from single radius model to seventeen radii model. The uniform
pressure distribution over the entire flow field is achieved by increasing the
number of segments for creating the vane profile.
5.5.4.1 Single Radius Model
The pressure and velocity distribution (Figures 5.13 and 5.14) show
that the low-pressure and high velocity zones are observed in the flow path.
The flow distortion is observed across the flow direction. The large area of
passage extending form the pressure side to passage center is traversed by a
uniform flow. On the contrary, the remaining passage is dominated by an
important velocity gradient and an accumulation of low momentum fluid in
the suction side. The velocity value at the suction side is observed as
minimum. The low pressure area causes the recirculation in the flow path.
Due to this phenomenon, the transfer of kinetic energy is less efficient, which
results in low static pressure rise.
The area weighted average static pressure value at the inlet = -35527.7 Pa
The area weighted average static pressure value at the outlet = 251648.2 Pa.
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Figure 5.13 Static pressure distribution of single radius model
Figure 5.14 Velocity distribution of single radius model
Pascal
m/s
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Figure 5.16 Velocity distribution of double radii model
5.5.4.3 Triple Radii Model
The static pressure value is further improved in the triple radii
model as the flow losses are reduced, which is evident (Figures 5.17 and
5.18). A pressure jump near the exit is visible in the flow path, which will
drop the pressure. At the pressure side of the vane, a concentrated pressure
zone near the exit section can be observed. The pressure variation from the
mid of the passage to the suction side is less compared to the earlier models.
The pressure values for the triple radii model are given below:
The area weighted average static pressure value at the inlet = -34666.28 Pa
The area weighted average static pressure value at the outlet = 256091.9 Pa.
m/s
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Figure 5.17 Static pressure distribution of triple radii model
Figure 5.18 Velocity distribution of triple radii model
Pascal
m/s
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5.5.4.4 Quadruple Radii Model
The pressure and velocity plot (Figures 5.19 5.20) shows the
uniform distribution of pressure and velocity from inlet to outlet section. The
pressure jump location is moved further towards exit compared to the triple
radii model. The momentum gained by the fluid is diffused except at the exit
section and the velocity value at the suction side is improved. A considerable
improvement in the static pressure value at the outlet is observed as given
below:
The area weighted average static pressure value at the inlet = -35765.03 Pa
The area weighted average static pressure value at the outlet = 261602.1 Pa.
Figure 5.19 Static pressure distribution of quadruple radii model
Pascal
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Figure 5.20 Velocity distribution of quadruple radii model
5.5.4.5 Seventeen Radii Model
The seventeen segment arc model was tried and the pressure and
velocity (Figures 5.21 - 5.22) plot reveals the uniform distribution of the flow
throughout its passage. This shows that the pressure distribution is controlled
by the flow path developed by the inverse method.
The area weighted average static pressure value at the inlet = -35531.6 Pa
The area weighted average static pressure value at the outlet = 327352.6 Pa.
The results obtained from the analysis of all the five models, reveal
that as the number of radii of curvature increases, the static pressure value at
the outlet also increases. This is due to the essence of the optimized
m/s
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Figure 5.21 Static pressure distribution of seventeen radii model
Figure 5.22 Velocity distribution of seventeen radii model
Pascal
m/s
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radius factor (RD) captured in more number of points. Here the model is
limited to 17 radii of curvature with 5mm interval because of modeling
difficulties.
5.6 CONVENTIONAL AND INVERSE DESIGN COMPARISON
The redesigned single arc vane profile produces the flow separation
similar to the conventional impeller. This is due to the fact that it does not
guide the flow uniformly towards the exit. The static pressure improvement is
further tried by increasing the number of arcs up to seventeen segments. The
optimized design variable does not improve the flow pattern by single arc due
to complex flow behavior, which is not captured by the equation. The number
of segments is incremented up to seventeen as shown in Figure 5.23 and
further increase in the segment is restricted due to modeling difficulty. The
improvement in outlet total pressure is achieved around 38000 Pa from the
original to modified impeller. It shows around 10% of improvement in its
performance.
Figure 5.23 Comparison of vane profile
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Figures 5.24 - 5.26 and Table 5.3 compare the conventional and
redesigned impeller performance. The pressure distribution in the
conventional impeller has some pressure jumps in the flow path compared to
the redesigned one. The uniform flow path in the redesigned impeller
improves the pressure head.
Figure 5.24 Conventionally designed impeller static pressure distribution
Figure 5.25 Redesigned impeller static pressure distribution
Pascal
Pascal
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Table 5.3 Static Pressure (Pascal) of conventionally designed and
seventeen radii model
Model Conventionally designed(Pa)
Seventeen radius model(Pa)
Inlet -35687.32 -35531.6
Outlet 266906.5 327352.6
Figure 5.26 compares the pressure variation with the reference
points taken for the pressure calculation. The trend curve shows the
improvement achieved from the existing model. It shows that the scope ofpressure recovery is more at the exit section of the impeller where the leakage
occurs.
Figure 5.26 Comparison of static pressure distribution
The impeller design calculation using the conventional and the
redesign methodology can be made with a computer program. This makes the
process simple to the designer to get the impeller dimensions and vane
profile.
Comparison of Static Pressure Distribution in
Pascal
0
100000
200000
300000
400000
500000
600000
1 3 5 7 9 11 13 15 17 19
Reference Points
Pressur Ideal
Existing
Redesigned
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By knowing the required head, discharge and the speed of the
motor, the conventional vane profile can be obtained. This curve can
be optimized by entering the pressure values obtained from Fluent.
The redesigned vane profile can be derived by entering the optimized
RD. This computer code (Figure 5.27) helps the designer to minimize
the time taken for design and drafting.
Figure 5.27 Flow chart for the process of computer program
Start
Get Input
(Discharge, head,speed)
Find the Design parameters
Check for the
suitability
Get the Pressure Distribution by
CFD simulation
Develop the theoretical maximum
pressure
Compare the pressure and develop the
optimum parameter
End
NO
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To improve the efficiency of the impeller, the vane profile is taken as a
parameter to redesign. The static pressure gain is increased because more and
more kinetic energy of the impeller are transferred to the fluid. The increase
in transfer of kinetic energy is due to the minimum of loss in the flow
passage. The usual losses like eddy formation and flow separation are reduced
to a great extent. The increase in efficiency is also due to subtle changes in the
velocity profile all across the flow passage. The computer program is
developed based on this methodology, which will serve as useful tool in the
designing process, thus bypassing the time consuming processes of design
and drafting. Further, the efficiency can be increased by optimizing other
parameters independently and collectively.
In this chapter, the design procedure follows the conventional approach
to develop the impeller. Then the model is simulated using CFD to calculate
the pressure and velocity distribution. The head developed by the
conventional model is around 266906.5 Pa. To improve the performance, the
inverse design approach is followed to develop the vane profile. The pressure
and velocity plots (Figure 5.13 - 5.22) show the incremental improvement in
the flow performance. The pressure developed by the seventeen radii arc
model is around 327352.6 Pa. The approach to design the impeller is made
simpler by introducing the computer program.