Numerical simulation of unsteady flow in a multistage ...

Progress in Computational Fluid Dynamics, Vol. 10, No. 4, 2010

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Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique Si Huang* School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China E-mail: [email protected] *Corresponding author

A.A. Mohamad Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, T2N 1N4, Canada E-mail: [email protected]

K. Nandakumar Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA E-mail: [email protected]

Z.Y. Ruan and D.K. Sang School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou 510640, China E-mail: [email protected] E-mail: [email protected] Abstract: In this work, three-dimensional, unsteady Reynolds-Averaged Navier–Stokes (RANS) equations with standard k-İ turbulence models are solved by employing sliding mesh technique within an entire stage of a multistage diffuser pump to investigate transient flow field and pressure fluctuations due to the interaction between impeller and diffuser vanes. Sliding mesh calculation is carried out as the impeller zones slide (i.e., rotate) relative to diffuser zone along the grid interface in discrete time steps. The complicated time-periodic and spatial-periodic characteristics as well as rotor–stator interaction phenomena inside the pump stage are simulated and analysed to understand dynamic variation of the interior flow field and interference and finally to aim at optimal design. Keywords: multistage centrifugal pump; sliding mesh; 3D flow field; numerical simulation. Reference to this paper should be made as follows: Huang, S., Mohamad, A.A., Nandakumar, K., Ruan, Z.Y. and Sang, D.K. (2010) ‘Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique’, Progress in Computational Fluid Dynamics, Vol. 10, No. 4, pp.239–245. Biographical notes: Si Huang finished his PhD in Mechanical Engineering at Tohoku University (Japan) in March 1995. He is an Associate Professor at the School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, China. Prior to it he worked at the Sulzer Pumps Inc. (Canada). A.A. Mohamad finished his PhD in School of Mechanical Engineering at Purdue University (USA) in 1992. He is Professor of Mechanical Engineering at the Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta, Canada. He has held International Conferences on Computational Heat and Mass Transfer since 1999.

Copyright © 2010 Inderscience Enterprises Ltd.

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S. Huang, A.A. Mohamad, K. Nandakumar, Z.Y. Ruan and D.K. Sang K. Nandakumar has joined the Cain Department of Chemical Engineering at the Louisiana State University, USA, as the Gordon A. and Mary Cain Endowed Chair Professor in August 2009. Prior to it he held the position of GASCO Chair Professor at the Petroleum Institute during 2007–2009 in the Chemical Engineering Program. He has had a academic career at the University of Alberta, Edmonton, Canada since 1983 in both teaching and research. Zhiyong Ruan finished his Bachelor Degree in Mechanical Engineering at Maoming University (China) in 2007. He is Graduate Student at the School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, China. Dike Sang finished his Bachelor Degree in Mechanical Engineering at Maoming University (China) in 2007. He is Graduate Student at the School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, China.

1

Introduction

Compared with single-stage pumps, multistage centrifugal pumps are more complex in geometrical configurations. Especially, the gaps between impeller blades and diffuser vanes are so narrow that flow fields in each part strongly affect each other and produce pressure fluctuations downstream of the impeller. These fluctuations not only generate noise and vibration that cause unacceptable levels of stress and reduce component life due to fatigue, but also introduce unfavourable characteristics of pump performance even at or near the design point. Therefore, there is a need to understand the sources of unsteadiness to control the pressure fluctuations and to improve the overall pump performance and reliability. In recent years, many investigations have contributed to the understanding of the complex three-dimensional flows, performance prediction and the technique of pump optimisation design, by means of both experimental and numerical approaches. In particular, due to the joint evolution of computer power and the accuracy of numerical methods, it is now feasible to use CFD codes for the prediction of the complex three-dimensional unsteady turbulent flow in the entire pump and impeller–volute (or diffuser) interaction. Comparatively, the numerical approaches have been much less reported for either steady or unsteady flows in multistage pumps than those in single-stage pumps. Goto and Zangeneh (2002) developed a 3D inverse design system for both vanes of impellers/diffusers. Cao et al. (2005) presented a combined approach of inverse method and direct flow analysis for the hydrodynamic design of gas–liquid two-phase flow axial pump impeller. Huang et al. (2006) performed 3D steady flow simulation within an entire stage of a multistage diffuser pump with a multiple frame of reference (MRF) that the grids of the impeller and the diffuser were connected by means of a frozen-impeller interface. However, the interaction between impeller and diffuser requires an unsteady solution process to calculate the time-dependent terms in the equations. This can be partially accomplished in a quasi-steady way similar to Asuaje’s work (Asuaje et al., 2004) in a single-stage pump: calculating the steady solution with different impeller/volute positions according to the angular velocity of the impeller. Nonetheless, it is always much better if the code is able to

perform an unsteady flow calculation for rotating machines. Sliding mesh technique has proven to be a powerful tool to investigate the transient flow field inside a centrifugal pump including the dynamic effects as well as the interference between impeller and volute (see e.g., Shi and Tsukamoto, 2001; Gonzalez et al., 2002; Gonzalez and Santolaria, 2006; Majidi, 2005). Thus, in this work, a fully unsteady model in FLUENT 6.2 commercial code that includes the sliding mesh technique is used as analysis tool to simulate the transient flow in the multistage centrifugal pump. Full RANS equations with k-İ turbulence model are solved within a diffuser pump stage. These computations will enhance our understanding of the flow mechanism of impeller–diffuser interaction in diffuser pumps and provide us some valuable ideas for pump design.

2

Model description and computational method

Proposition 1: Governing equations Reynolds Averaged Navier-Stokes equation is applied in this transient flow simulation to describe the unsteady turbulent flow. The current rotational speed of the impeller is 1450 rpm and the blade number of the impeller is Zi = 7, it means that the time period of a whole impeller revolution is T = 0.0414 s and the blade passing frequency is f = 1/(T ⋅ Z i ) = 169.2 Hz. To make appropriate choice of turbulence model for such high-frequency unsteady problem, the simulations by Shi and Tsukamoto (2001), Gonzalez et al. (2002), Gonzalez and Santolaria (2006) and Majidi (2005) could be referred. In their cases, the blade passing frequencies were f = 172.2, 189.0 and 123.5 Hz, respectively, which are quite close to the present value f = 169.2 Hz. The standard k-İ model in commercial codes was chosen, validated by experimental data in these four papers. This indicates that the Navier–Stokes code with the standard k-İ model is capable of predicting the current highly unsteady problem well in centrifugal pumps. In this work, therefore, the standard k-İ turbulent viscosity model is adopted for analysing unsteady turbulent flow. The time-dependent term scheme is second order, implicit. The pressure–velocity coupling is calculated

Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique through the SIMPLEC algorithm. Second order, upwind discretisations have been used for convection terms and central difference schemes for diffusion terms. Proposition 2: Configuration of the pump stage Figure 1 is the one stage in the pump studied both in the previous steady work (Huang et al., 2006) and in this unsteady simulation. Fluid starts from inlet of impeller (position 1), in turn passes by outlet of impeller (position 2), front vane inlet of diffuser (position 3), front vane outlet of diffuser (position 4), back vane inlet of diffuser (position 5), finally leaves the back vane outlet of diffuser (position 6) and enters next stage of the pump. Therefore, a stage element is isolated from the multistage pump to form a computational domain that starts at position 1 and ends at position 6. The main configuration dimensions of each part in the pump stage is presented in Figure 1 and Table 1. Figure 1

View of one stage in a multistage centrifugal pump (see online version for colours)

Table 1

Configuration dimensions of pump stage

Part Impeller

Diffuser

Parameter

Value

Zi

7

D0 (mm)

0.16

D2 (mm)

0.36

B2 (mm)

0.023

β1 (deg)

20

β2 (deg)

25

n (rpm)

1450

n5

75

Zd

8

D3 (mm)

0.362

D4 (mm)

0.514

b3 (mm)

0.03

b4 (mm)

0.046

α3 (deg)

10

α4 (deg)

9

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Proposition 3: Generation of zone and sliding mesh The pre-processor Gambit in FLUENT is performed to form the computational geometry zone and grids. Figure 2 shows the 3D geometry creations of impeller and diffuser. Figure 2

Geometry creations of impeller and diffuser

For current complex pump geometries, unstructured tetrahedron mesh is used for grid generation. The unsteady flow is generally solved using a sliding mesh technique, which has been applied to turbomachinery flows. The sliding mesh technique provided by FLUENT allows that relative motion of the impeller grid with respect to the inlet and the diffuser during unsteady simulation but it is computationally more demanding than the MRF model. A ring zone between impeller outlet and diffuser vane inlet is defined as the sliding grid interface. During the sliding mesh calculation, the cell zones slide (i.e., rotate) relative to one another along the grid interface in discrete time steps. Node alignment of both cells along the grid interface is not required. Furthermore, in such non-deforming zones grids merely move but keep same for all iterative calculations so sliding mesh model is usually faster and more stable than dynamic mesh model that requires grid regeneration while entering next time step. Figure 3 shows the appearance of sliding mesh of the stage composed of impeller (6076 nodes, 21,883 elements) and diffuser (20,048 nodes, 78,414 elements) at t = 0 and 0.15 s. Although grid size is not adequate to investigate local boundary layer variables, global ones are well captured. Figure 3

Sliding mesh of the computational domain: (a) t = 0 s (b) t = 0.15 s (continues on next page)

(a)

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S. Huang, A.A. Mohamad, K. Nandakumar, Z.Y. Ruan and D.K. Sang

Figure 3

Sliding mesh of the computational domain: (a) t = 0 s (b) t = 0.15 s (continued)

(b)

Proposition 4: Boundary conditions In the panel of FLUENT software, the rotating impeller zone is set to ‘moving mesh’ in motion type and the diffuser to ‘stationary’. The boundary condition at pump inlet is prescribed by mass flow rate. Outflow boundary is imposed both at exit of domain and the gap between impeller exit and diffuser entrance, in which flow leakage takes place. Flow rates at different exits are allocated according to leakage quantity. Turbulent flow condition at domain inlet is given with intensity (5%) and hydraulic diameter, while zero gradients of k and e are specified at the outlet. Wall boundaries are set on the surfaces adjacent to solid regions. Standard wall functions, based on the logarithmic law, are applied for near-wall treatment. The sliding interface between the impeller and diffuser is treated as an additional fluid zonal boundary in transient calculations with sliding meshes, and is updated implicitly after the interior of computational domain has moved. Proposition properties

5:

Operating

parameters

and

material

Some flow operating parameters and fluid material properties used in current calculation are listed in Table 2, where the flow rate is the design point of the pump. Table 2

3

Operating conditions and material properties

M (kg/s)

Pa (Pa)

ρ (kg/m3)

v kg/(m·s)

75

101325

998.2

0.001003

Numerical results and discussion

Proposition 6: Outline of flow field Prior to the unsteady analysis, the steady numerical computation of turbulent flow in the entire stage of the multistage centrifugal pump was performed. A comparison for the numerical and experimental performance data for

the tested pump was executed. Both data, experimental and numerical, were obtained after averaging the unsteady values. The head obtained in design points matched well with the experimental ones. More details on the flow structure analysis and comparison between numerical results and experimental head were already presented in Huang et al. (2006). Here, some more results are supplemented. Figure 4 shows the whole 3D view of streamlines in the pump stage at rotational time t = 0.15 s. Although the flow rate is set at the design point, large separation vortexes still take place on the corner where flow turns over from the front side to back side of the diffuser. Similar big vortexes on pump diffuser wall were also experimentally visualised and numerically simulated in Goto et al. (2002) even at 130% of the design flow rate. The presence of such kinds of big vortexes is mainly caused by conventional design of diffuser and treated as a major factor generating hydraulic loss and affecting pump performance. Figure 4

3D streamlines in pump stage t = 0.15 s (Left: impeller and front diffuser; right: back diffuser) (see online version for colours)

Proposition 7: Time-periodic solutions For unsteady calculation, the convergent solution in each time step is reached by setting residual criterion of each governing equation. Properly specifying the time step size ∆t is important to capture the necessary time resolution with less computational demanding. Since the FLUENT formulation is fully implicit, there is no stability criterion that needs to be met in determining ∆t. However, to model transient phenomena properly, it is necessary to set ∆t at least one order of magnitude smaller than the time scale 1/f in the simulation being modelled. For a rotor/stator model, more than 20 time steps between each blade passing is recommended by FLUENT Manual for a stable, efficient calculation, so the present time step is set to ∆t = T 20Z i and the Courant number (C = V ∆t ∆x , where V is the estimated local velocity, and ∆x, the corresponding local mesh dimension) is kept below 20. It would be most interesting to track time-periodic solutions Kof some global quantities, such as hydrodynamic forces F acting on the impeller and the diffuser (Figs. 5–8), of some local quantities, such as area-weighted average static pressure at inlet and outlet of pump stage, respectively (Figs. 9, 10). As apparently

Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique shown in Figure 10, there is a start-up phase that is about one revolution longer before the flow starts to exhibit time-periodic behaviour. Among these figures, the dimensionless hydrodynamic forces are defined as: Fi ′ = Fi

1 ρ AVin2 , 2

Figure 8

Time-periodic solution of Fy acting on the diffuser

Figure 9

Time-periodic solution of p at inlet of pump stage

243

i = x, y and z

where A is the total area of flow passage surface of the studied object. Vin is the flow velocity at the inlet of the pump stage. Subscripts x, y and z represent x-, y- and z-coordinates in Cartesian system. Figure 5

Figure 6

Time-periodic solution of Fx acting on the impeller

Time-periodic solution of Fy acting on the impeller Figure 10 Time-periodic solution of p at outlet of pump stage

Figure 7

Time-periodic solution of Fx acting on the diffuser

Comparing the data in Figures 5 and 6 with those in Figures 7 and 8, the amplitude of hydrodynamic force fluctuation acted on the diffuser is considerably higher than that acted on the impeller because the flow has received more kinematical energy due to energy conversion while entering the diffuser. Regarding time-periodic flow behaviour, apparently, there are seven peak values of hydrodynamic force either acted on impeller or diffuser during one revolution of the impeller corresponding to the number of impeller blades (Zi = 7). In another word, the frequency components of the hydrodynamic force fluctuations either acted on impeller or diffuser

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S. Huang, A.A. Mohamad, K. Nandakumar, Z.Y. Ruan and D.K. Sang

are composed mainly of the impeller blade passing frequency. For the time-periodic behaviour of area-weighted average static pressure both at inlet and outlet of pump stage (Figs. 9, 10), however, the number of peak values of p tends to be eight during one revolution of the impeller corresponding to the number of the diffuser vane (Zd = 8). This indicates that the diffuser vanes could produce a considerable influence on fluctuating of static pressure that are spread to the discharge and reflected to the inlet of stage.

Figure 11 Spatial-periodic solution of p with angular coordinate at the impeller blade inlet (r = 80 mm, t = 0.15 s)

Proposition 8: Spatial-periodic solutions Besides time-periodic impulse phenomena, the presence of a spatial-periodic fluctuation pattern is also one of significant unsteady flow features encountered in turbomachinery. The pressure field in function of the blade position gives rise to the different blade loadings, which appear to be predominant for the instantaneous torque over the other possible effects. Figures 11–13 show the spatial-periodic solution of static pressure with angular coordinate at the fixed moment t = 0.15 s near the impeller blade inlet (r = 80 mm), the impeller blade outlet (r = 180 mm) and near the diffuser vane inlet (r = 200 mm) corresponding to Figure 1. Among Figures 11–13, the static pressure varies periodically with angular coordinate from –180° to 180°. In passages of the impeller (r = 80 mm, Fig. 11), the static pressures decrease regularly with angular coordinate from the pressure side to the suction one of the impeller blade. The spatial period of the static pressure curve is seven and meets the number of impeller blades, indicating that the flow at this position is scarcely affected by the diffusers. In passages of the diffuser (r = 200 mm, Fig. 12), on the other hand, the spatial period of the static pressure curve with angular coordinate is eight and equal to the number of diffuser vanes, meaning that the influence of impellers is neglected. Figure 13 shows the numerical solution of the static pressure p along with circumferential angle at the impeller outlet (r = 180 mm), where is the interface of impeller and diffuser. The flow at this position is unstable due to the interference between impeller blades and diffuser vanes, and the pressure fluctuations of fluid along with circumferential angle become irregular. Among one half range of circumferential angle (θ = –180° to 0°), the spatial fluctuation pattern of static pressure is similar to that of static pressure in the impeller channel (r = 80 mm). Meanwhile, the spatial fluctuation pattern of static pressure in the other half range of circumferential angle (θ = 0 to 180°) resembles that of static pressure at the diffuser passages (r = 200 mm). This result represents that both impeller and diffuser are equally important for contributing their higher harmonics to frequency components of the static pressure fluctuations at the interface between both sides.

Figure 12 Spatial-periodic solution of p with angular coordinate at the diffuser vane inlet (r = 200 mm, t = 0.15 s)

Figure 13 Spatial-periodic solution of p with angular coordinate at the impeller blade outlet (r = 180 mm, t = 0.15 s)

Numerical simulation of unsteady flow in a multistage centrifugal pump using sliding mesh technique

4

Conclusions

A 3D unsteady RANS equations combined with the sliding mesh technique in this work are computed for the whole stage of a multistage diffuser pump. The numerical procedure has been used to instantaneously analyse different flow phenomena, both local and global variables inside the pump stage. The calculated results provide insights towards a better understanding of both complicated time-periodic and spatial-periodic characteristics as well as rotor–stator interaction phenomena, which could be used in a design process. It is found from the computation that there is a start-up phase about one revolution longer before the flow displays time-periodic behaviour. Blade passing frequency of impeller is a major source contributing to the fluctuation frequency of global quantities like hydrodynamic force acted on the pump, whereas the number of diffuser vanes significantly affects impulse frequency of static pressure in the stage. As a result of rotor–stator interaction, the pattern of pressure fluctuations along with angular coordinate at the interface between impeller and diffuser are equally composed of the higher harmonics from both sides.

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Gonzalez, J., Fernandez, J., Blanco, E. and Santolaria, C. (2002) ‘Numerical simulation of the dynamic effects due to impeller-volute interaction in a centrifugal pump’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 124, No. 2, pp.348–355. Gonzalez, J. and Santolaria, C. (2006) ‘Unsteady flow structure and global variables in a centrifugal pump’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 128, No. 5, pp.937–946. Goto, A. and Zangeneh, M. (2002) ‘Hydrodynamic design of pump diffuser using inverse design method and CFD’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 124, pp.319–328. Goto, A., Nohmi, M., Sakurai, T. and Sogawa, Y. (2002) ‘Hydrodynamic design system for pumps based on 3-D CAD, CFD, and inverse design method’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 124, pp.329–335. Huang, S., Islam, M.F. and Liu, P. (2006) ‘Numerical simulation of 3D turbulent flow through an entire stage in a multistage centrifugal pump’, International Journal of Computational Fluid Dynamics, Vol. 20, No. 5, pp.309–314. Majidi, K. (2005) ‘Numerical study of unsteady flow in a centrifugal pump’, Journal of Turbomachinery, Transactions of the ASME, Vol. 127, No. 2, pp.363–371. Shi, F. and Tsukamoto, H. (2001) ‘Numerical study of pressure fluctuations caused by impeller-diffuser interaction in a diffuser pump stage’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 123, No. 3, pp.466–474.