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**Authors :**J. C. Statharas , E. S. Valamontes , P. Filiousis , N. W. Vlachakis -
**Paper ID :**IJERTV8IS060319 -
**Volume & Issue :**Volume 08, Issue 06 (June 2019) -
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#### Strategies for Predicting Centrifugal Pump Performance Characteristics by Validating Blade Shape Configurations. Introducing the Trojan Horse Method

J. C. Statharasa*, E. S. Valamontesb, P. Filiousisa, N. W. Vlachakisa

aNational and Kapodistrian University of Athens, Sciences General Department, Psachna, Evias, 34400, Greece.

bE.S. Valamontes, Professor, Department of Electrical & Electronics Engineering, University of West Attica, Greece.

a Professor, Corresponding Author a Ingenieur, External Research Fellow a Professor, Research Fellow

AbstractA simple and easy to apply numerical method – called Trojan Horse method-for pump head and efficiency estimation is presented. This is a contribution on centrifugal pump characteristics prediction using empirical relations based on performance maps of known pumps developed by some researchers. The present method evaluates a specific flow rates parameter, as well as some empirical equations for impeller geometrical data. The introduced modelling equations based on the blade shape configurations provide the pump characteristic lines. The method is validated by applying it to various commercial centrifugal pumps with known performance maps produced by their developers. From the cases examined, it can be stated that the present model can be applied to predict performances of centrifugal pumps of any diameter, particularly at all efficiency regions that the pump is supposed to operate according to its geometrical data. As a result the proposed method provides a satisfactory approximation of industrial centrifugal pumps performance curves, constituting a potential tool for pump researchers and manufacturers.

Rotating Shaft

Impeller

Hub Plate

Outflow

Blade

Inflow

KeywordsCentrifugal pump, performance map, empirical method, impeller, characteristic line, numerical prediction.

1 INTRODUCTION

Centrifugal pump description

Centrifugal pumps are the most common type of pumps used to move fluids through a piping system. They are devices that find extensive applications particularly in mining, chemical and mechanical industry.The fluid enters the pump impeller along or near to the rotating axis and is accelerated by the impeller, flowing radially outward or axially into a diffuser or volute chamber, from where it exits into the downstream piping system. The main parts of a centrifugal (or radial) pump are shown in Fig.1.1.

Casing

Eye

Fig.1.1. Main components of a centrifugal pump (Taken from [47])

Impeller's geometry

The basic component of a centrifugal pump is a rotor (or impeller) in which a number of blades is attached (see Fig.1.2). Fig.1.3 shows the impeller geometry explaining its main geometrical characteristics.

Fig. 1.2: Schematic view for a five-bladed impeller model

Fig. 1.3: Impeller's main geometrical characteristics (Taken from [48]) D1=impeller inlet diameter, D2=impeller exit diameter, b1=blade angle at impeller leading edge, b2=blade angle at impeller trailing edge, w=wrap

angle.

Previous Design efforts

Centrifugal pumps are designed in order to be fitted to installations and work over a wide range of operating conditions. In such cases the prediction of performance is of primary importance for safe and effective operation of pumps and constitutes an important challenge for the pump designer. The challenge becomes particularly difficult when it is necessary to predict the performance of different types of centrifugal pumps varying from low to high volume flow rates.

Improving pump efficiency needs a good understanding of its components behavior at design point and off-design conditions. As measurement technologies for more accurate flow field studies are limited and they are expensive, due to complicated nature of geometry and the flow itself, numerical models have been developed and extended for performance prediction and primary design steps. However,

numerical results would be more reliable if more experimental data were available for validation.

Last years the trend in pump industry [1-6] is to emphasize to the blade shape configuration. The reason behind this trend is the need for an effective way to achieve fast predictions methods for pump maps construction avoiding multiple steps in the test rig, [7-11] .

The principal objective of centrifugal pump design effort, according to [12-18] is the effective matching between pump operation and impeller geometry. It is important to choose such a centrifugal pump morphology that the pumps operating line falls within the pumps high-efficiency region [19-27] . Therefore, a variety of approximate methods or numerical precedures have been developed aiming to predict centrifugal pump performance characteristics [28,33] .

Predictions of centrifugal pump performance maps were also obtained by using advanced 3D Computational Fluid Dynamics (CFD) techniques, most commonly solving the Reynolds Averaged Navier Stokes (RANS) equations, coupled with a turbulence model. However there are a number of disadvantages when using CFD methods except their high computational cost and the lack of experimental data for validation.

Sometimes for example, prediction of centrifugal pump performance constitutes an important challenge when a pump has to be manufactured in order to be fitted in a given installation and to work over a wide range of operating conditions, [30], [31], [32]. The challenge becomes particularly difficult when is needed to predict the performance of different types of centrifugal pumps varying from low to high volume flow rates Pfeiderer [7]. In all these cases characteristic curves are not always available to evaluate the adequacy of the pumps performance for a particular situation and CFD methods canot help adequately. As a result there is a need for fast practical but accurate methods as the one presented in this work.

The current approach

The present study presents a fast method to estimate pump performance characteristics requiring only a few pump geometrical data. Centrifugal pump performance curves were produced using polynomial functions while the experimental data points were known by the pumps test rig measurements. By using efficiency curves from six different known pumps,we tried to find a closed form for the coefficients of the head performance that approximates the performance maps. Alternative curve fitting methods using exponential as well as polynomial functions for the blade angle =f() or the volute radius =g() were proposed too.

The first step of method presented here, (named The Trojan horse method) is to propose algorithms for the internal geometry of the impeller blades. As a second step calculates the distribution of the degree of efficiency as a function of both the nq and the morphology of the blade curvature. From the proposed efficiency polynomial results by integration-with appropriate approaches-the polynomial of manometer characteristic distribution. That is, that starting from internal blading we finally end up by obtaining the final pump characteristics. Due to the fact that starting with a few internal pump features we finally get to know the characteristics of the

pump as a whole we have called our method: The Trojan horse method.

The overall efficiency is estimated, not only the head. The numerical predictions are compared to experimental data for centrifugal pumps delivering low, medium and high volume flows. The data was either obtained in the test rig or found in the literature.

The results show that the proosed method can be used as a tool to the pump designer in order to obtain a quick assessment of performance curves.

Taking into account the challenging complexity of centrifugal pump performance map prediction, the present article aims to simplify and generalize to the extent that it is possible a method focused on sizing centrifugal pumps, predicting their performance map from very low to very high volume flow rate, requiring only a minimum number of geometrical data that is readily available.

Most of the authors cited in the literature survey above have analyzed several pumps geometries in their studies. In the present article these pumps are examined as applications to evaluate the increased applicability of the method, called Trojan horse method.

CALCULATIONS METHODOLOGY

General Assumptions

According to the present design procedure, we assume that there is a casing that encloses the outer circumference of the radial impeller.The design approach assumes that a vortex phenomenon is mainly responsible for the fluid transport[34- 38]. The present method employs a group of empirical euations in terms of a polynomial algorithm in order to estimate the pump performances.The input data required are the following:

Impeller inlet diameter, D1,

Impeller exit diameter, D2

Number of impeller blades, z

Blade angle at impeller trailing edge, 2

The basic assumptions of the present method are the following:

(i) Empirical equations for max are developed for different impeller's rotational speeds,

(ii) The prediction method takes into account the flow in the pump impeller as well as the effect of volute.

Methodology

The main steps of the pump performance map prediction procedure are the following :

Step 1: Estimation of the operable volume flow rate employing novel empirical relations and using impeller geometrical data.

Step 2: Approximation of the performance curves for various impeller speeds by means of novel empirical functions.

The originality of the present method lies on:

the calculation of the shut-off maximum head attained by the pump,

the derivation of a set of equations that estimate the volume flow rates at each constant speed characteristic line of the centrifugal pump and

the suggestion of novel empirical relations for the shape of the characteristic curves of the performance map such as a(,wrap angle) and b ( nq ).

2.2.1 Distribution Strategy

All the above described equations contain an algorithm for the pump impeller geometrical data, its blade number and its rotational speed.

When designing centrifugal impeller, the pump impeller and the geometric parameters of the design point must achieve the pump design conditions and maximum efficiency, max. However max varies with w. We used Eq.3.6 to examine the effect of wrap angle on max. The results are shown in Table

and Fig.2.1. We see that.max increases until w=1000, then decreases. Thus, each centrifugal pump with exact parameters at the design point has perfect max at a specific wrap angle.

Table 2.1 Calculations of efficiency as a function of wrap angle

w

max=1.6[-2(10-6)w3+0.00045w2-0.03w+1.1] [28]

50

0.76

55

0.7656

60

0.7808

65

0.8032

70

0.8304

75

0.86

80

0.8896

85

0.9168

90

0.9392

95

0.9544

100

0.96

105

0.9536

110

0.9328

115

0.8952

120

0.8384

125

0.76

1,2

1

0,8

0,6

0,4

0,2

0

40 50 60 70 80 90 100 110 120 130

w

1,2

1

0,8

0,6

0,4

0,2

0

40 50 60 70 80 90 100 110 120 130

w

n max

n max

Fig.2.1.Efficiency as function of wrap angle

,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,

,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,

Convenient forms of the blade angle are shown below in Tables 2.2 and 2.3 where: (r1) = 1 , (r2) = 2 or r(1) = r1 , r(2) = r2 . The constants and c in Tables 2.2 and 2.3 are calculated by curve fitting experimental results.

In the case that we know r and try to find then any distribution pattern for r from Table 2.2 can be chosen. In the case however that we only choose the algorithmic status for a parameter or r, then the combinations may be variegally complex or they can only come from the linear patterns. This

means that when a model function is selected, the other one -0 2

must be linear.

Table 2.2: The blade angle distribution models

= (r)

Conditions : 1=(r1), 2=(r2)

= r+c/r

1= r 1+ c/ r 1 , 2= r 2+ c/ r 2

= r ecr

= r ecr1 = r ecr2

1 1 , 2 2

= ecr

= ecr1 = ecr2

1 , 2

= r2 + c

= r 2+ c = r 2+ c

1 1 , 2 2

In the case that we know and try to find r then any distribution pattern for from Table 2.2 can be chosen..

In addition to Sigloch [40], Menny [41] or Bowade [42] fluid dynamic models, practical applications allows the presentation of functions that can analogously approximate the curvature that clearly results from the impeller geometry.

Table 2.3: The blade mean line configuration models

r = r ()

Conditions: r1=r(1), r2=r(2)

r = r e( a + c) 2

r = r e(a 1+ c ) r = r e(a 2+ c ) 1 2 , 2 2

r = r 1+ c

r 1= r 1+ c1 , r 2= r 2+ c2

r = (/ ) ec

r = ( / )ec r = ( / )ec

1 1 , 2 2

r = e(a 2+ c)

r = e( a 2+ c ) r = e( a 2+ c)

1 2

1 , 2

Criteria for optimum pump design

For a certain number of revolutions, both the angle selection as well as the manometer head and the flow rate should give a tracking match to the blade curvature (Fig.2.2) for the different methods (Sigloch [40] , Menny [41] , Bowade

[42] ) for = max = w [39].Fig.2.2 : Representation of a present study model

The details of the calculations and a summary of the various impeller geometrical constants are tabulated in Table 2.4.

r=*e(c/)

Xr=rcos(w/180)

<>Yr=rsin(w/180) w (linear)

0W

0.0391612

0.03916121

0

0

0.0431289

0.04140874

0.012058868

14.68679

0.0473821

0.04015604

0.025150709

29.37359

0.0519305

0.03509634

0.038275598

44.06038

0.0567829

0.02621048

0.050371751

58.74718

0.061948

0.01376966

0.060398232

73.43397

0.0674334

-0.0016848

0.067412309

88.12077

0.0732463

-0.019381

0.070635657

102.8076

0.0793932

-0.0383706

0.069505244

117.4944

0.0858798

-0.0575919

0.063706558

132.1812

r=*e(c/)

Xr=rcos(w/180)

Yr=rsin(w/180)

w (linear)

0W

0.0391612

0.03916121

0

0

0.0431289

0.04140874

0.012058868

14.68679

0.0473821

0.04015604

0.025150709

29.37359

0.0519305

0.03509634

0.038275598

44.06038

0.0567829

0.02621048

0.050371751

58.74718

0.061948

0.01376966

0.060398232

73.43397

0.0674334

-0.0016848

0.067412309

88.12077

0.0732463

-0.019381

0.070635657

102.8076

0.0793932

-0.0383706

0.069505244

117.4944

0.0858798

-0.0575919

0.063706558

132.1812

Table 2.4 Blade drawing positions

360 n 3.793 2 n + 434.9 2

w= [ 0.5793exp( q ) + 2.158exp( q ) ]

z 89.87

1087

,(2.1)

The basic function of a blade (Fig.2.2, Table.2.4) is to guide the flow. This means, that to impose a certain direction to the flow, the blade angle at a certain radius (r) should be imposed to the flow, guiding the flow to follow the same direction (Fig.2.2, Table.2.4 ).

Volume flow estimation

The pump total head depends on the tangential velocity at impeller outlet (see Fig.2.3) expressed as:

u2=D2n

Fig.2.3: Velocity triangle at impeller

There is a variety of available correlations for the slip factor introduced in and verified against experimental data. The correlation adopted here, is the one suggested:

Head estimation

The present method proposes an original formula to calculate the head attainable, where a correction coefficient depending on the exit diameter D2 is adopted.

Equation approximates the maximum efficiency of centrifugal pump as a function of the blade tip configuration , based on the flow analysis performed.

The maximum head obtained by a centrifugal pump corresponds to throttling conditions, where the volume flow is zero.

Efficiency performance curves estimation

It is intended to approximate the shape of characteristic lines by means of a function which starts from a head high value when the ratio is close to zero (corresponding to shut – off) and decays asymptotically as the ratio approaches the value 1 (corresponding to maximum value of the efficiency). The challenge of equations is to approximate the different shapes of centrifugal pump characteristic lines varying from low to high rotational speeds.

Volute design

The volute contour of the investigated pumps [9] when operating at a design volume flow rate, was found by applying some models that appear in Table. 2.5. In this Table, refers to the volute wall curvature and 0 is the diffusor's angle. A program of experiments to see the formation of the flow, particularly through the region of the fixed pump blades was carried out in.

Table 2.5: The volute shape models

R=R() the volute shape

The shape design

The Archimedes' Shaped Spiral

for 2 < < 4

Model

X=Rcos(1.57/90)

for 0 < < 2

y=Rsin(1.57/90)

R=ar2+c

The Logarithmic-Shaped

for 2 < < 4

Spiral Model

X=R*cos(1.57*/90)

for 0 < < 2

y=R*sin(1.57*/90)

R = 1.1r2ea+c

Diffusor shape

Coordinates positions of the blade tip

x y

1 0

2

for 2 < < 4, 1 < < 1.95

1 = r2e[(ln2)/360-ln2]

2=r2(1.01+0.08nq/100+0.07HBEP/

1000)

Diffusor wall

diffusor angle 0<<5 = 1+2tan(1.57/90)

The form and the constructional details of the volute geometry model are illustrated in Fig.2.4.

Fig.2.4: Typical arrangement of the casing with diffusor

RESULTS

In order to evaluate the accuracy of the model described above, various commercial pumps were considered as test cases, since their performance maps are available by their manufacturers. These maps were digitized using an appropriate software in order to be compared with the present methods results. For these pumps there were found numerical predictions in the literature, according to the authors knowledge. Table 3.1 summarizes the most important data of the pumps chosen for the performance map prediction.

Pump name | b2 (mm) | D2 (mm) | 2 | z | QBEP (m3/h) | HBEP (m) | n (RPM) |

[1] | 10 | 200 | 490 | 9 | 62.5 | 62 | 3000 |

[4] | 9 | 200 | 240 | 5 | 45.68 | 46,41 | 2900 |

[2] | 11 | 160 | 230 | 7 | 25 | 7 | 1450 |

[5] | 15 | 130 | 200 | 6 | 3.6 | 6 | 1500 |

[3] | 30 | 300 | 220 | 6 | 200 | 20 | 1450 |

[6] | 26 | 240 | 250 | 6 | 98 | 68 | 2900 |

Pump name | b2 (mm) | D2 (mm) | 2 | z | QBEP (m3/h) | HBEP (m) | n (RPM) |

[1] | 10 | 200 | 490 | 9 | 62.5 62 | 3000 | |

[4] | 9 | 200 | 240 | 5 | 45.68 | 46,41 | 2900 |

[2] | 11 | 160 | 230 | 7 | 25 | 7 | 1450 |

[5] | 15 | 130 | 200 | 6 | 3.6 | 6 | 1500 |

[3] | 30 | 300 | 220 | 6 | 200 | 20 | 1450 |

[6] | 26 | 240 | 250 | 6 | 98 | 68 | 2900 |

Table 3.1: pumps characteristics at BEP

The calculational procedure for predicting the pump characteristic Curves, has as follows:

Qo=QBEP/q (3.2)

Qmax =1.6Qo (3.3)

J=Qi/QBEP (3.4)

The coefficient j modulates the empirical equations to the real performance curves of the pump geometry (Fig.2.3 ). The coefficient j, represents the time interval of pumping time required for the volume flow to reach volume flow at BEP .

1=[-32+145log(nq)-41(log(nq))2]0.01 (3.5)

2=[-2(10-6)w3+0.00045w2-0.03w+1.1]1.6 (3.6)

1,9 0,1 0,5

1,9 0,1 0,5

max=(1 2 ) (3.7)

=0.86max[(z/7)0,1)(1.33j0,7-0.3j3)] (3.8)

H =Hu[1-0.1j1,7-0.035j4] (3.9)

0,66

0,66

2 1

2 1

Hu=HBEP/h, Hth=(u 2-u 2)/9.81, Hshut off =HBEP/m (3.10) where

q=1/[1+0.28/nq ] is the volumetric efficiency (3.11) m=1/[1+1/nq] is the mechanical efficiency (3.12)

Grapsas-Anagnostopoulos-Papantonis [1]

70

50

40

50

40

HeadH(m)

HeadH(m)

60

grapsas et al

grapsas et al

30

20

10

0

0 20 40 60 80 100

Flow rate Q(m3/h)

Figure 3.2 Using present model to predict pump [1] ead characteristics

Zhang Yongxue, Zhou Xin, Ji Zhongli and Jiang Cuiwei [4]

h=1-0.071/QB 0,25 is the hydraulic efficiency (3.13)

EP

is the blade number effect factor [34-38,43-46] as

=-0.013z2+0.1725z+0.51 (3.14)

is the overall efficiency and nq is the specific speed

We used Eq.3.8 and Eq.3.9 to obtain head and efficiency for different numbers of blades (z=5-9). The simulation (Fig.3.1) showed strong effect of the number of blades on . The blade effect factor (Fig.3.1) increases obviously with a specific number of blades, then decreases. Thus, each centrifugal pump with exact parameters at the design point has perfect head and efficiency performances at a specific blade number.

70

60

HeadH(m)

HeadH(m)

50

Zhang Yongxue et al

Zhang Yongxue et al

40

30

20

10

0

20 25 30 35 40 45 50 55 60 65 70

Flow rate Q(m3/h)

Blade numper effect factor

Blade numper effect factor

1,2

1

0,8

0,6

0,4

0,2

0

0 1 2 3 4 5 6 7 8 9 10

1,2

1

0,8

0,6

0,4

0,2

0

0 1 2 3 4 5 6 7 8 9 10

Figure 3.3 Using present model to predict pump [4] ead characteristics

Lei Tan, Shuliang Cao, Yuming Wang and Baoshan Zhu [2]

Blade numper Z

Blade numper Z

Fig.3.1. The relation between blade number effect factor and the

number of blades z.

Figs.3.2-3.7 present the calculational results of the above equations (3.2-3.14) compared to the real pump data of Table

3.1. In these figures the two procedures (experimental vs predicted) are shown as poly. As we can see, (Figs.3.2-3.7) the approach with the new prediction method is very good, since the curves created are close to the actual operating points of the pumps given by the six authors [1-6].

10

9

8

lei tan et al

lei tan et al

7

HeadH(m)

HeadH(m)

6

5

4

3

2

1

0

5 10 15 20 25 30 35

Flow rate Q(m3/h)

Figure 3.4a Using present model to predict pump [2] ead

characteristics

0,9

0,8

0

0

0

0

Efficiency

Efficiency

0,7

,6

,5

0,4

0,3

0,2

0,1

0

5 10 15 20 25 30 35

Flow rate Q(m3/h)

3.5 Xin Zhou, Yongxue Zhang, Zhongli Ji, and Long Chen [3]

25

15

10

15

10

lei tan et al

lei tan et al

HeadH(m)

HeadH(m)

20

xin zhou et al

xin zhou et al

5

Figure 3.4b Using present model to predict pump [2] efficiency

A. Farid Ayad Hassan, H. M. Abdalla, A. Abou El- Azm Aly [5]

6

5

HeadH(m)

HeadH(m)

4

A. Farid Ayad et al

A. Farid Ayad et al

3

2

1

8 10 12 14

8 10 12 14

0

0

0,9

0,8

0,7

0

0

0

0

0

0

Efficiency

Efficiency

,6

,5

,4

0,3

0,2

0,1

0

50 100 50 300

150 200 2

150 200 2

Flow rate Q(m3/h)

Figure 3.6a Using present model to predict pump [3] ead

characteristics

xin zhou

xin zhou

2 4 6

Flow rate Q(m3/h)

16 18 20

50 100 50 300

150 200 2

150 200 2

Flow rate Q(m3/h)

Figure 3.5a Using present model to predict pump [5] ead

characteristics

Figure 3.6b Using present model to predict pump [3] efficiency

3.6 Tahani & Pourheidari [6]

0,9

0,8

0,7

0

0

0

0

0

0

Efficiency

Efficiency

,6

,5

,4

0,3

0,2

0,1

0

8 10 12 14

8 10 12 14

A. Farid Ayad et al

A. Farid Ayad et al

2 4 6 16 18 20

Flow rate Q(m3/h)

90

80

70

60

40

30

40

30

HeadH(m)

HeadH(m)

50

20

10

0

80 90 100110120

80 90 100110120

40 50 60 70 130140

Flow rate Q(m3/h)

tahani et al

Figure 3.5b Using present model to predict pump [5] efficiency

Figure 3.7a Using present model to predict pump [6] ead

characteristics.

0,8

0,7

0,6

0

0

0

0

0

0

Efficiency

Efficiency

,5

,4

,3

0,2

0,1

0

40 60

80 100

80 100

Flow rate Q(m3/h)

120 140

tahani et al

performance curves of six pumps. The pump characteristics performances have been tested on different types of centrifugal pumps and the obtained results prove the accuracy and high capability of the implemented method.

Conclusively, the proposed method could be beneficial to the pump industry in the early design stages. Future work will cope with the extension of the method to predict the head and the efficiency of more than 100 centrifugal pumps.

td>

Impeller angular velocity

Nomenclature

D

Impeller diameter

n

Rotational speed

H

Pump Head

z

Blade number

Q

Volumetric flow rate

b

Blade width

R

Radius

Greeks

Blade angle

Efficiency

Inclination angle

Fluid density

Subscripts

1

Impeller inlet

2

Impeller outlet

Nomenclature

D

Impeller diameter

n

Rotational speed

H

Pump Head

z

Blade number

Q

Volumetric flow rate

b

Blade width

R

Radius

Greeks

Blade angle

Efficiency

Inclination angle

Fluid density

Impeller angular velocity

Subscripts

1

Impeller inlet

2

Impeller outlet

Figure 3.7b Using present model to predict [6] pump efficiency

CONCLUSIONS

In this study, an original methodology for the challenging topic of centrifugal pump performance prediction for industrial applications was presented. The motivation for this work was to introduce a fast engineering assessment tool which is able to predict both the operating limits of centrifugal pumps in terms of volume flow rates as well as the shape of its performance curves. The model is based on a pioneering methodology using empirical relations of the mean flow upstream and downstream of the centrifugal impeller. The minimum and the maximum volume flows are approximated by means of genuine fist issued empirical equations. The Head and the efficiency that any pump can reach is estimated by means of the impeller outlet diameter and the local max.

Two original functions are introduced in order to generate the shape of the performance curves of the pump, namely polynomial functions. Evaluation of the method is done by considering commercial pumps with known performance maps from their researchers, of various speeds. The results obtained can be summarized in the following Table 4.1:

Good agreement at intermediate RPM

Better agreement with measurements when the polynomial functions are used

Good agreement at intermediate RPM

Better agreement with measurements when the polynomial functions are used

Table 4.1: Evaluation of results

The simple and fast method presented here was validated succesfully against some models of centrifugal pumps for which experimental data were found in the literature or from the test rig. Comparisons between numerical and experimental data obtained in the test rig show that the proposed model can satisfactorily predict performance characteristics of centrifugal pumps, for the cases examined.

As it could be observed, the approximation between performance curves estimated via experiments test or via empirical mathematical formulas is quite good. In this way the performance of a typical centrifugal pump could be mathematically provided without the need of the expensive experimental procedure.

The development of the present study calculational procedure is due to the fact that centrifugal pumps have an important contribution in industrial world. Accordingly to the international standards of centrifugal pumps, the present study presents an analysis system for obtaining the

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