A Novel Methodology to Predict Centrifugal Pump Characteristics Through Navier-Stokes Exact Solutions

A Novel Methodology to Predict Centrifugal Pump Characteristics Through Navier-Stokes Exact Solutions

Fatsis Antonios

Mechanical Engineering Department Technological University of Central Greece 34400 Psachna, Greece

Vlachakis Vassilios

ESM, Virginia Polytechnic Institute and State University Blacksburg, VA, USA

Panoutsopoulou Angeliki

Hellenic Defense Systems S.A.

1, Ilioupoleos Avenue, Hymettus, Greece

Vlachakis Nikolaos

Mechanical Engineering Department Technological University of Central Greece 34400 Psachna, Greece

AbstractA novel method for centrifugal pump performance prediction through exact solutions of the Navier- Stokes equations is presented. Velocity components and the static pressure were obtained using the Bessel functions. The volume flow rate is obtained using empirical relations taking into account geometrical and operational parameters of the pump. Extensive validation of the numerical predictions against experimental data for various centrifugal pumps has revealed that the proposed method is reliable and it produces accurately the pumps characteristic line.

Keywords C entrifugal pump; exact solution; characteristic line; manometric head; flow rate.

1. INTRODUCTION

   A wide variety of centrifugal pump types have been constructed and used in industrial and residential applications. Centrifugal pumps are widely used for irrigation, water supply plants, steam power plants, sewage, oil refineries, chemical plants, hydraulic power service, food processing factories and mines. Moreover, they are also used extensively in the chemical industry because of their suitability in practically any service related to it. Due to the vast applications it is very important that centrifugal pumps should work efficiently. In this context there have been continuous efforts to improve the performance of centrifugal pumps. The significant cost and time of the trial-and-error process by constructing and testing physical prototypes reduces the profit margins of the pump manufacturers. In recent years, a growing availability of computational resources and progress in the a accuracy of numerical methods brought Computational Fluid Dynamics (CFD) methods from pure research work into the competitive industrial markets. However, advanced CFD commercial software is not suited for a quick assessment of characteristic lines of a series of pumps, due to the detailed impeller blade camber, thickness and height distribution required for the grid generation. Alternatively, numerical models based on one- dimensional approach can provide very fast and accurate predictions of the pump performance.

2. LITERATURE SURVEY

   Many researchers developed simple one-dimensional computational models trying to predict the characteristic curves of centrifugal pumps. The one-dimensional model in

   [1] is simple and robust, giving accurate performance predictions. It is the basis for comparison of numerical predictions of characteristic lines, since is shows good agreement to manufacturers data and experimental results. It is important that prediction models should include correlations accounting for loss mechanisms, since it was demonstrated in [2] that predictions based on the Eulers method and airfoil theory, sometimes produce unrealistic results.

   The effect of blade exit angle variation on the head and efficiency of a centrifugal pump was investigated in [3] using the numerical model of [1]. It was concluded that the exit blade angle is influencing mainly at high volume flow rates the pump head and efficiency.

   A one-dimensional performance analysis, [4], has proved to be an effective and important approach on pump design. It was concluded that pump characteristics depend on geometrical data and on losses in different parts of pump. Various types of losses of the basic model [1] were introduced in [5]. The results obtained show a good agreement to experimental results [6] for that test case.

   A meanline pump flow modeling method has been developed in [7] aiming to provide a fast capability for predicting the performance of pumps at off-design operating conditions. The design-point rotor efficiency and slip factor were obtained from empirical correlations to rotor-specific speed and geometry.

   A growing availability of computer power and progress in accuracy of numerical methods, brought turbomachinery CFD methods from pure research work into the competitive industrial markets. State of the art CFD solutions consist the solution of the Reynolds Averaged Navier Stokes Equations (RANS), for impeller flows as well as for complete pump flows including the impeller volute interaction. In this context a promising alternative or supplement to standard Reynolds averaged Navier-Stokes (RANS) turbulence modeling is Large Eddy Simulation (LES).

   Many software packages are available in the market for numerical analysis such as FLUENT, CFX, Fidap, Polyflow, Phoenix, Star CD, Flow 3d, ESI/CFDRC, SCRYU.

   Many researchers, such [8], [9], [10], [11], using the CFX- TASCflow software carried out calculations of the flow inside the impeller, as well as considering the whole pump including the spiral casing. In these calculations, the standard k- two-equation turbulence model was used. The predictions were compared to performance experimental data showing a fairly good agreement.

   Many researchers, such as [12], [13], [14], [15], [16], [17], [18], [19], [20] carried out the analysis of centrifugal pump flows using the ANSYS-CFX package solving the Reynolds Averaged Navier-Stokes (RANS) equations. The results obtained agree to available experimental data. The turbulence models used were the k- standard turbulence model and the k- based SST model.

   The commercial FLUENT software was used by many researchers, such as [21], [22], [23], [24], [25], [26] to resolve the RANS equations for the whole pump. Numerical results compare well to experimental data and performance prediction data.

   Numerical predictions of the flow in pumps were also performed using the Fine Turbo flow solver [27]. The computational grid was generated using IGG for the volute and Autogrid for the impeller. Reynolds Averaged Navier- Stokes equations in rotating frames of reference coupled with various turbulence models and near-wall treatment for low- Reynolds modeling.

   Other researchers developed their own three-dimensional numerical tools to analyze the flow structure inside centrifugal pumps. A three-level optimization technique was developed in [28].

   In very low specific speed range the efficiency of centrifugal pump, designed by the conventional method, becomes remarkably low. The recently used centrifugal pumps have a higher rotational speed and smaller size [29]. The internal flow of a very low specific speed centrifugal impeller has large influence on the pump performance, thus detailed information is needed for the relation between the internal flow and performance in order to develop high performance pumps.

   A three dimensional incompressible thin-layer Navier-Stokes method has been developed in [30] for multistage unsteady turbomachinery flow calculations. The method is based on a

   Results show that when the blade outlet angle increases, the centrifugal pump performance handling viscous fluids improves.

   In order to improve the accuracy of the numerical simulations and to be able to analyze and understand more thoroughly the flow in centrifugal pumps, it is essential to advance from a steady state to an unsteady simulation technique. In this context a promising alternative or supplement to standard Reynols averaged Navier-Stokes (RANS) turbulence modeling is Large Eddy Simulation (LES).

   LES method was applied in a Francis turbine [33] and in a centrifugal pump impeller, at design and off-design conditions.

   Unsteady flow computations were done in a centrifugal pump impeller using the LES method [34]. It is thus found that LES provides an improved insight into the basic fluid dynamics with a satisfactory accuracy compared to experiments.

   The present numerical method is an alternative to all those previously described numerical methods. It is based on the exact solution of the Navier-Stokes equations inside the pump impeller. The Bessel functions of the first order are used to provide approximate forms for the fluid velocity components and for the static pressure. In the following, an approximate form is used to derive the pump volume flow rate taking into account the inlet and outlet impeller diameter, the number of blades and the angular velocity of rotation of the impeller. Exhaustive testing of the model in ten different centrifugal pumps including comparisons with available experimental data and reference solutions, leads us to conclude that the present model can be used as a sound assessment tool to predict impeller characteristic lines.

3. GOVERNING EQUATIONS

   1. General Assumptions

      The present numerical model consists of the Navier- Stokes equations written in cylindrical coordinates, r, , z ,

      better suited for axisymmetric, rotating geometries than the Cartesian ones. Additionally, it is assumed that: (a) the impeller flow field has reached steady state conditions; (b) the circumferential component u does not depend on the

      axial coordinate, z . Thus, the continuity equation is becoming:

      u u u

      combination between the pseudo-compressibility and the dual

      r r z 0

      (1)

      time stepping technique. Calculations were carried out for unsteady incompressible flow cases and the results show satisfactory agreement with well-established theoretical and experimental data. Sensitivity of the solution to various numerical variables was assessed for a centrifugal pump case. A numerical model was developed in [31] for the numerical

      r r z

      The system of the Navier-Stokes equations can be written:

      r -Momentum:

      solution of the RANS equations in the impeller of a centrifugal pump. It was applied for direct flow analysis and

      2

      u u u 1 p 2

      ur r uz r r 2 u

      2u 1 u

      r r

      parametric investigation of the effect of some impeller design

      r r

      z r

      r2

      r r

      details on its hydrodynamic characteristics.

      A numerical simulation of the three-dimensional unsteady pump flow taking into account the impeller-volute interaction with different outlet blade angles using the SIMPLEC algorithm and the k- turbulence model, was done in [32].

      u 2u

      • r r

      r2 z2

      (2)

      -Momentum:

      u u u

      1 1 P

      u2

      1 u u

      impeller by means of an empirical relation which will be presented below.

      Conservation of mass yields a relation between inlet and

      ur r

      2 ur

      discharge meridional velocities that involves the cross-

      r r

      z -Momentum:

      r r

      r2

      r r r2

      (3)

      sectional areas of the streamtube at these locations. The volume flow rate through the pump is related to the velocity distribution at any location by the integral [36]:

      R2 R2

      u u 1 p

      2u 1 u

      2u

      V Vi dri 2 r umdri

      (7)

      ur z uz z

      z

      z z

      (4)

      R1 R1

      r z

      z

      r2

      r r

      z2

      where

      um is the meridional component of the flow velocity

   2. Non-dimensionalisation

      and

      Vi

      is the volume flow through a streamtube from an

      The above system of partial differential equations can be written in non-dimensional form choosing the following parameters:

      intermediate radius ri to radius ri d r . Integrating for the whole length of each streamline between R1 and R2 gives the

      volume flow V .

      u * ur

      , u * u , u * uz

      z* z r* r

      The limiting case that corresponds to V 0 leads us to

      r

      R2

      R z

      2

      R2

      R2 R2

      determine the shut-off manometric head, Hshutoff .

      As it is suggested in [1], the volume flow is proportional to

      where R2 is a characteristic length of the geometry in consideration. The fluid pressure and the Reynolds number can then be written as:

      p R 2

      the third power of the exit pump impeller diameter. Trying to account for the effect of the rotational speed of the impeller, the number of impeller blades and the ratio of the inlet hub diameter to the outlet diameter, the maximum volume flow is approximated here by:

      p*

      R2

      ,

      2

      Re 2 ,

      2 n

      u g H

      n

      z D

      2

      V 0, 08 n D3 2 shut off

      1

   3. Solution strategy

   Resolving the system of equations (1) to (4) using the

   max 2

   u1

   0,8 u2

   1200 7

   D2

   z

   u

   r

   above non-dimensional quantities, it was found that the axial velocity u* , the radial velocity * and the tangential

   for

   D D2 , while

   (8a)

   velocity u* , can be expressed in terms of the functions: 1 2

   u* J

   ebz

   (5a)

   V 0, 08 n D3 g Hshut off n z D1

   z 0 max 2

   0,8 u2

   1200 7 D

   u* J

   ebz

   (5b)

   2

   2

   r 1 (8b)

   u* r

   (5c)

   for

   D D2 1 2

   where J 0 and J1 are the Bessel functions of the First kind

   [35].

   Substituting the non-dimensional velocity field to the Navier- Stokes equations (2) to (4), one obtains the static pressure

   where outlet:

   u2 is the tangential velocity at the pump impeller

   u2 D2 n

   field by solving the equation:

   D1,

   D2 is the impeller inlet and outlet diameter,

   1 2 2

   2bz *

   4 z

   respectively and z is the number of pump impeller blades.

   2 Re

   J1 J0 e p (r, z) C (6)

   where C is a constant. The determination of this constant involves some algebra and it will be omitted here. It is important to highlight that the methodology used, combines flow parameters (such as shut-off manometric height and maximum volume flow) and geometrical data of the pump

   Based on [3], the effect of the blade exit angle, 2, which is not taken into account in the current work, has little influence to pump performance in the expected range of operation (close to the design point) – and it is limited mainly in high flow rates.

   Having determined the pressure from equation (6), the manometric head can be calculated from:

   p*

   p

   2 2

   g H

   2

   g H

   2 1 D2

   present numerical data show the same trend as the experimental data. Slight differences occur at lower than the

   R2 R2 2 n 2 4 2 n2

   design point flow rates.

   2 2 2 4

   1 4bz

   8 z

   H J 2 J 2 e D2 n D 2 (9)

   1 0

   2

   2 Re D2

   where z is the axial gap between the impeller disk and casing.

4. RESULTS

   A first test case to assess the applicability of the present method is the prediction of the 7-bladed centrifugal pump in [37]. The hub diameter is Dh 0.3m , suction diameter is Ds 0.405m , impeller diameter is D2 0.169 and impeller width is b2 0.05m . The rotational speed is 2900rpm, the design point flow rate is 20m3/h, the head is 35m. The pump specific speed is nq 15, defined as:

   Fig. 2. Comparison between numerical results by the present method and experimental results in [38].

   A third test case is the 6-bladed centrifugal pump impeller analyzed in [39]. The pump examined operates at high volume rates, having at design point manometric head of 31.5m and a flow rate of 225.8m3/h. Comparisons between

   prediction using the present method and experimental data

   nq RPM

   V

   H 0,75

   are shown in Fig. 3. One can see a slight over-prediction between the present method and experimental data, without altering the behavior of the predicted distribution for this high

   A comparison between prediction of manometric head obtained using the present method (illustrated with solid line) and experimental results (illustrated by dots) found in [37] is shown in Fig. 1. From this figure, one can see that the results obtained using the present method show a very good agreement with the available experimental data.

   Fig. 1. Comparison between numerical results by the present method and experimental results in [37]

   A second test case is the pump studied in [38]. The centrifugal impeller consists of six two-dimensioned curvature backward swept blades of constant thickness. The

   flow rate centrifugal pump.

   Fig. 3. Comparison between numerical results by the present method and experimental results in [39]

   A fourth test case is a 6-bladed centrifugal impeller with specific speed nq 38.47 that studied in [3]. In Fig. 4, on can see numerical prediction obtained by the present method, compared using manufacturer data [3]. Prediction obtained

   using the present method show quite good agreement to manufacturers data for all flow rates examined.

   inlet height b1 is 15.13 mm and the outlet height b2 is 8.11

   mm and the angular speed is 1000 rpm. The centrifugal impeller was manufactured of acryl for PIV measurements and operates at flow rate of 2m3/h at design point.

   Fig. 2 shows the comparison between results of the present method and experimental data [38]. One can see that the

   Fig. 4. Comparison between numerical results by the present method and experimental results in [3]

   A fifth test case is a centrifugal oil pump analyzed in [40]. This is a single stage standard industrial centrifugal oil pump with side-suction entry, which has been applied to transport clean crude oil and other liquid petroleum products. The pump duty specifications are as follows: the nominal flow rate is 25m3/h, the head is 60 m, the rotating speed is 2950 rpm and the specific speed nq 41.6. The impeller was of

   Fig. 6. Comparison between numerical results by the present method and experimental results in [41].

   A seventh test case is a pump geometry found in the textbook of [42]. Figure 7 shows comparison between numerical results by the present method, experimental results and reference numerical results for the pump of [42]. The present method predicts accurately the characteristic of the pump. At high flow rates, one can observe numerical results predict slightly higher manometric height.

   closed type with five twisted three-dimensional blades. Fig. 5

   shows the comparison between prediction of the present method and experimental data found in [40]. Once more, numerical data show a very good agreement to experimental data.

   Fig. 5. Comparison between numerical results by the present method and experimental results in [40].

   A sixth test case is one of the impellers analysed by Fard [41]. It is a 6-bladed centrifugal pump impeller with backwards curved blades, running at 1450 rpm. Fig. 6 shows comparison between numerical results by the present method, experimental results for this pump [41]. From this figure, one can see that the present method shows a very good agreement to pump experimental data.

   Fig. 7. Comparison between numerical results by the present method and experimental results in [42]

   The FLUENT package was used in [43] to study the three- dimensional turbulent flow through a commercial water pump at design and off-design conditions. The selected 6- bladed commercial pump has a specific speed of nq 22.86.

   Fig. 8 shows the comparison between numerical results obtained by the present method, experimental data and numerical results solving the Navier-Stokes equations using a standard k- two-equation turbulence model [43]. The predicted results are presented in terms of pressure profiles, velocity vectors and performance curves. The computed performance characteristics are in good agreement with the manufacturer curves and almost coincide in the region of the design volumetric capacity. Deflecting from this point either to lower or to higher flow rates, the head prediction is either underestimated or overestimated with reference value the value of head at volume flow of 90m3/h.

   Fig. 8. Comparison between numerical results by the present method, experimental results and reference numerical results in [43]

   Figure 9 presents the comparison between experimental data and numerical prediction for the 7-bladed pump analyzed in [44]. This is a centrifugal pump with a specific speed of nq 24.17. Results obtained by the present method show a

   very good agreement to experimental data.

   Fig. 9. Comparison between numerical results by the present method and experimental results in [44]

   Figure 10 shows the comparison between numerical predictions and experimental data for a 5-bladed slurry pump, [45], with a specific speed of nq 85. One can see that the

   numerical results using the present method agree very well with experimental data over all flow regimes.

   Fig. 10. Comparison between numerical results by the present method and experimental results in [45]

5. CONCLUSIONS

In the present article, a novel approach to predict characteristic lines for centrifugal pumps is presented. This approach is based on exact solutions of the incompressible, steady state Navier-Stokes for the case of centrifugal pumps. Velocity components and pressure are obtained using the Bessel functions of the first kind. The flow rate is found by integrating the meridional velocity from hub to tip. In order to express the pump head in terms of the flow rate, empirical relations are used. In these relations, the most important parameters are the pump impeller rotational speed, the number of impeller blades and the ratio of the inlet hub diameter to the outlet diameter. Test cases of ten different centrifugal pump impellers with two-dimensional and three- dimensional blades are chosen to illustrate the ability of the method to predict the variation of the pump head in terms of the flow rate. In these test cases the specific speed is varying from 15 to 85 and the number of blades from 5 to 7. For all the cases examined, a very satisfactory agreement between numerical predictions and experimental data is found. The predicted results are in agreement to experimental data not only at the best efficiency point, but also at higher and lower flow rates. This validates the method and makes it useful for industrial applications. An opportunity to improve the predictions could be to include the effect of the exit blade angle to the model. The advantages of the present method are: (a) it requires a minimum of geometrical data and (b) it is simple and accurate and it can be used as a very quick global pump performance assessment tool, prior to a detailed investigation of the three-dimensional pump flow field, either numerically or experimentally.

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