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



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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:



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:



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 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.

CENTRIFUGAL PUMP IMPELLER VANE PROFILE

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