Experimental and numerical study of transient flow in a ...

Journal of Mechanical Science and Technology 25 (3) (2011) 749~757

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0107-7

Experimental and numerical study of transient flow

in a centrifugal pump during startup† Zhifeng Li1,2, Peng Wu1,2, Dazhuan Wu1,2,* and Leqin Wang1,2

1Institute of Process Equipment, Department of Chemical and Biological Engineering, Zhejiang University, Hangzhou, 310027, P. R. China 2Engineering Research Center of High Pressure Process Equipment and Safety, Ministry of Education, Hangzhou, 310027, P. R. China

(Manuscript Received September 7, 2010; Revised December 3, 2010; Accepted December 30, 2010)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Transient characteristics and flows in a centrifugal pump during its starting period were experimentally and numerically investigated.

The two-dimensional particle image velocimetry technique was used to capture transient flow evolutions in the pump’s diffuser. A new dynamic slip region method that combines the dynamic mesh method with the non-conformal grid boundaries is proposed to resolve the transient flows caused by the started impeller. Numerical self-coupling was realized by establishing a circulation pipe system along with the pump model equivalent to the experimental pump system. Numerical and experimental results agree well in both explicit characteris-tics and internal transient flow structures, confirming the validity of the proposed method. Analysis of the instantaneous flow in the im-pellers indicates that for the early stage of the startup, the transient vortex evolution between blades is the main reason for the transient head coefficient being lower than the steady state value. The reversed flow at the blade inlet is a less important reason for this effect. In later stages, the weakening of the intensity of the spatial vortex visible on S2m and the main flow stream are the main reasons for the ex-plicit performance slowly rebounding to the steady value.

Keywords: Numerical simulation; Experimental test; Transient flow; Centrifugal pump; Starting period ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Transient processes of hydraulic turbo machineries are cur-rently encountered in various applications such as the rapid startup of a pump used for underwater weapons launch [1], the startup and shutdown of a liquid fuel feed pump used in aero-space propulsion systems [2], and the frequent startup, shut down, and phase modulation of a pump turbine. During hy-draulic turbo machinery transient processes, transient pressure fluctuation and load impact may lead to facility or power sys-tem destruction. Thus, in addition to operation efficiencies, ensuring transitional reliabilities requires exploring the un-steady flow mechanism during transient operations.

During the past few decades, there have been several stud-ies on the transient characteristics of turbo machinery transient operation. Tsukamoto et al. [3-5] experimentally explored the explicit transient characteristics of low specific speed cen-trifugal pumps whose rotational speed experienced starting, stopping, and fluctuating periods. Experiments and analysis based on two-dimensional linear cascade theories demon-

strated that the lag in recirculation growth around the impeller vanes was the main reason for the difference between transient and quasi-steady characteristics. Experiments conducted by Thanapandi and Prasad [6] showed that change of pump be-havior during normal startup and stopping is quasi-steady. They also originally analyzed dynamic performances by ex-amining characteristics.

For a centrifugal pump with relatively high specific speed, Lefebvre and Barker [1] associated performance decline with the large flow incidence angle at the blade’s leading edge by conducting transient operation experiments. Other related experiments were carried out by Saito [7], Ghelici [8], and Bolpaire [9]. The transient behavior of a cavitating centrifugal pump was investigated by Tanaka and Tsukamoto [10-12]. To date, however, no experimental transient flow visualizations have been reported.

Wang et al. [13-15] experimentally and theoretically per-formed a series of experiments on transient performances during the rapidly starting period of centrifugal pumps. Most recently, Dazin et al. [16] presented a theoretical expression of the instantaneous head, which included terms of angular ac-celeration, flow acceleration, and velocity profile variation by solving transient angular momentum and energy equations. They demonstrated that instantaneous velocity profile varia-

†This paper was recommended for publication in revised form by Associate EditorByeong Rog Shin

*Corresponding author. Tel.: +86 571 8795 2406, Fax.: +86 571 8795 2406 E-mail address: [email protected]

© KSME & Springer 2011

750 Z. Li et al. / Journal of Mechanical Science and Technology 25 (3) (2011) 749~757

tion was another important factor determining transient char-acteristics.

From the perspective of numerical simulation, Wu [14] nu-merically studied transient flow during the startup of a cen-trifugal pump. The additional source forces in a rotational accelerating reference frame were described by user defined functions (UDF) hooked with the FLUENT solver for each updated time step. Li et al. [17] performed similar attempts on a Francis turbine’s runaway process. Although acceleration effect was taken into account, simulation of the transient flow evolution was not dynamic because the model itself was static (or rather the impeller phase relative to the volute tongue had not changed). Based on the above discussion, there is still a lack of an effective three-dimensional numerical method for simulating transient flow in the accelerated impeller.

Although the computational fluid dynamics (CFD) method and experimental fluid visualization technique are well ad-vanced, the unsteady flow mechanisms in turbo machineries during transient operations remain unclear. The objective of this study is to investigate the transient characteristics and the details of the flow evolutions through experiments and nu-merical simulations. Instantaneous flow-rate, head rotational speed, and power were tested during the impeller’s rapid startup period. Simultaneously, two-dimensional instantane-ous flow structures in the diffuser were recorded using particle image velocimetry (PIV) system. More importantly, a three-dimensional numerical model equivalent to the space of an experimental water loop was originally established, and a dynamic slip region (DSR) method was proposed to simulate transient flow evolutions during startup operations.

2. Experiment

2.1 Test Equipment and Method

The arrangement of the experimental setup and data acqui-sition system is schematically illustrated in Fig. 1.

Fig. 2 shows the physical photo of the test system. The pump, pipes, electromagnetic flow meter, valves, and tanks constitute the closed loop recirculation system. Several circuit loops can be controlled by the gate valves (A, B, and C). Flow rates are regulated by the pneumatic ball valve driven by an

air compressor. To ensure that cavitation did not occur, the water level in the small tank was maintained at 0.8 m above the pump inlet.

The pump flow passage was formed by the impeller, dif-fuser, and pump body. The major geometry dimensions of the impeller are provided in Fig. 3. The impeller has four blades and its principal design specifications are listed in Table 1. To capture the flow field, the pump body was designed to be flat. Transparent Plexiglas was the material of the pump’s inlet, outlet, front, top, and back side faces (Fig. 4).

The impeller was driven by the variable frequency motor controlled and triggered by the frequency converter. Rota-tional speed and starting time could then be controlled by the

Gate valveElectromagnetic Flowmeter

Plexiglass pipe

Pump body

Piezoelectric pressure transducers

Pneumatic ball valve

Small water tank

Large water tank

CCD camera

Variable frequency motor

Speed and torque measuring instrument

Signal conversion

circuit

IPCFrequency converter

A

B

C

Fig. 1. Schematic view of experimental setup and instrumentationsystem.

Fig. 2. General view of the experimental system.

Fig. 3. Geometry of axis cross-section.

Fig. 4. Photograph (left) and numerical model (right) of the pump.

Z. Li et al. / Journal of Mechanical Science and Technology 25 (3) (2011) 749~757 751

frequency converter. The transient rotational speed and torque measuring instru-

ment was located between motor and impeller shafts. Strain Bridge and photoelectric tachometric transducer were the principle measuring elements of the torque and rotational speed, respectively. The circumrotating transformer inside the instrument enabled non-contact energy and signal transmis-sion. Frequency signals were then converted into a current signal by an F/I converter. An optical isolation was used to isolate the converter’s signals from outside interference.

Instantaneous flow rates were obtained by the electromag-netic flow meters installed at the inlet and outlet sections of the pump body (Fig. 2). Instantaneous static pressures were measured by the piezoelectric pressure transducers installed at the inlet and outlet pipes. The transient head was calculated as follows:

2

4 41 1 1=

2 900o i t

o i

p p QHg g d dρ π

⎛ ⎞− ⎛ ⎞+ −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

. (1)

All current signals (4–20 mA) obtained from the above sen-

sors were unified to the standard voltage signals (1–5 V) by passing through a signal conversion circuit composed of 250 Ω precision resistors. Finally, the voltage signals were con-verted to digital signals by the A/D board installed on the In-dustrial PC’s PCI slot. To ensure the stability of signal acqui-sition, the shell of the IPC, the signal conversion circuit, and the sensors were all connected to the ground. Actual data ac-quisition frequency was 100 KHz. To eliminate environmental noise and smooth data, each data was taken from an average of 1000 points.

The pump, tank, and flow passage were filled with water with density ρ equal to 1000 kg/m3 and dynamic viscosity μ equal to 0.001 Pa•s. Glass hollow beads with diameters of 8–12 μm were chosen as tracer particles whose densities were very close to the water’s so they can uniformly suspend in it. For the velocity measurements, the LaVision® produced FlowMaster 2D PIV system running DaVis 7.2 was employed to capture and process instantaneous flow fields. Illumination source for the recording of PIV images was an Nd:YAG twin cavity laser system capable of producing 2 × 200 mJ pulses of 200 ns duration.

Scattered light from the tracer particles was recorded on a 2048 × 2048 pixel digital CCD camera at 7 Hz. The digital images were processed by cross-correlation within DaVis 7.2. Resulting data were then imported into the Tecplot for further image processing.

For the experiment of internal flow visualization, the inner

circulatory loop was used with the gate valves A, C closed, and B opened. The impeller acceleration mode was set to lin-ear by the frequency inverter controller. Rotational speed after acceleration was 1475 rpm. Startup time was 1.14 s and angu-lar acceleration was 135.5 rad·s-2.

3. Numerical simulation

3.1 The DSR Method

The authors validated the dynamic mesh (DM) method in solving transient flow induced by an impulsively starting [18] or translate-rotating circular cylinder [19]. The DM method was extended to solve the two-dimensional transient flow around an impulsively started straight blade [20] and centrifugal pump [21]. In these studies, meshes were dynamically updated with the motion of the boundary using spring analogy [22] and local remeshing method [23] generally suitable for unstruc-tured meshes.

Numerous attempts show that mesh quality will deteriorate when the boundary is under fast rotational motion. Thus, time consumption on mesh regenerations will increase and accu-racy of results will decline. Guaranteeing dynamic mesh up-date quality is an important issue in solving the transient flow caused by rapid accelerations of the rotating impeller.

The problem can be handled by the dynamic slip region (DSR) method in which the flow domain is divided into sepa-rate regions: dynamic and stationary (Fig. 4). The dynamic region is accelerated as an entirety whose rotational speed is assigned with a prescribed curve at present. Governing equa-tions in the dynamic region are described in the arbitrary La-grangian and Eulerian way (or the dynamic mesh conservation equations described in the FLUENT Users’ Guide [24] or by Demirdžić and Perić [25]), whereas the stationary region is governed by ordinary N-S equation. Different regions are separately discretized, so the non-conformal grids between adjacent regions are linked by a pair of grid interfaces de-scribed in the FLUENT Users’ Guide [24].

For the fluid flow in the dynamic region, the integral form of the dynamic mesh conservation equations for space, mass, and momentum in an arbitrary control volume V bounded by a closed surface S can be written as

uV S S V

V Vt

dd dS dS d

d+ ⋅ = ⋅ +∫ ∫ ∫ ∫Q F D S (2a)

where the conserved variables Q, the convective fluxes F, the diffusion term D, and the source term of momentum Su are given by

1

uρρ

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

Q , Fb

( )b( )b

ρρ

⎡ ⎤−⎢ ⎥

−⎢ ⎥⎢ ⎥−⎣ ⎦

u

= u u

u u u

,

2

0D 0

μ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥∇⎣ ⎦

=

u

,00

pgrad

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥−⎣ ⎦

uS (2b)

Table 1. Rating hydraulic specifications of the pump.

Rotational speed (rpm) 1450

Flow rate (m3/h) 360

Total head (8 m) 8


where t designates the time, ρ represents the fluid density, u is the fluid velocity, ub is the boundary velocity of the moving mesh, μ is the dynamic viscosity of the fluid, and p is the pres-sure of the fluid. The solution method of Eq. (2) was intro-duced in detail in the FLUENT Users’ Guide [24] and by Demirdžić and Perić [25]. 3.2 Numerical model and solution condition

Taking the unsteady boundary conditions into consideration, a three-dimensional closed-loop numerical model was set up to realize numerical self-coupling. Therefore, there was no need to specify any inlet or outlet boundary conditions. The numerical model of impeller and pump is compared with the real object in Fig. 4. The numerical model is meshed into un-structured hybrid tetrahedrons and hexahedrons as shown in Fig. 5. Similar to actual conditions, the transient flow is aroused by the starting impeller whose rotational speed is externally specified, and the flow rate is regulated by adjusting the rotation angle φ of the numerical ball valve.

The code used is FLUENT 6.2, a commercial finite volume based CFD code with a broad range of computational capa-bilities. The time-dependent term scheme is first order implicit. The pressure-velocity coupling is calculated through SIMPLE algorithm. Second order upwind scheme with numerical un-der-relaxation is applied for the discretization of convection term and central difference schemes for diffusion terms.

For steady state computations, the impeller rotational speed is fixed at 1475 rpm. The Reynolds number based on the im-peller’s outer diameter (Re=ρω∞D2/2μ) is up to 4.4×106. The RANS-based realizable k-ε model is used to simulate steady turbulent flow.

For startup simulation, according to the experimental condi-tions, the transient angular speed is specified as follows:

, 1.14 ;, 1.14 .tr

s

a t tt

ωωω

≤⎧= ⎨ >⎩

(3)

The transition turbulence behavior at high Reynolds num-

bers in pumps during startup is still an issue under investiga-tion. Tutar and Hold [26] and Benim et al. [27] pointed out that the RANS model is not appropriate for computations of tran-sient turbulent separated flow. The compromised Detached

Eddy Simulation (DES) turbulence model is used to simulate the transient flow at present because the grid requirements and computation time for the Large Eddy Simulation (LES) are very large.

After the transient simulation results are obtained, the calcu-lation method of transient head is similar to Eq. (1), in which p0 and p1 represent the area weighted average static pressure at the inlet and outlet sections, respectively, and Qt represents the transient flow rate through the inlet or outlet sections. The transient power of impeller is calculated as follows:

Ptr=ωtr·Mtr . (4)

4. Results and discussion

4.1 Numerical flow rate test

Although there is no need to specify inlet and outlet bound-ary conditions for the closed-loop numerical model, similar to the actual experiments, the flow rate depends on the ball valve angle φ. The relationship between numerical flow rate and ball valve angle φ is predicted by varying φ from 0 to 90 ° while the impeller rotational speed is fixed at 1500 rpm.

As shown in Fig. 6, flow rates at inlet and outlet sections nearly coincide with each other, confirming the numerical accuracy. The maximum flow rate is equal to 251 m3·h-1 with the valve fully opened, and the flow is completely cut off when φ =78 °. Corresponding to the experimental test flow rate of 186 m3·h-1, and referring to Fig. 6, the ball valve an-gle φ is fixed at 38.5 ° for the following transient flow simula-tion. 4.2 Grid and temporal independence test

Unsteady grid independence study was conducted by solv-ing the transient flow for five different grid numbers varied from 5.8×105 to 3.2×106, respectively. The temporal interval was fixed as 0.001. To verify temporal independence, three time steps of 0.5×10-3, 1.0×10-3, and 2.0×10-3 were used to simulate the transient flow. The mesh number was fixed at 11×105. In Figs. 7 and 8, the flow rate and head are compared to verify grid independence and temporal influence. To a cer-tain extent, the results deviate from each other insignificantly, which demonstrates that the transient computational process is

Fig. 5. Mesh of the closed-loop model.

/°ϕ

Fig. 6. Flow rate versus ball valve angle curves (N∞ = 1500 rpm).


independent of grid number and time interval at the present range.

A brief view of the computational time is summarized in Table 2, which illustrates that the simulations are resource consuming, and computations on Linux platform are faster than on Windows platform.

4.3 Numerical and experimental comparisons of the tran-

sient explicit characteristics

The predicted instantaneous flow rate and head are com-pared with the experimental data in Figs. 9 and 10, respec-tively. The flow rate growths are in the same tendency, which shows a cubic-like trait. Quantitatively, before 0.5 s, the pre-dicted flow rate curve agrees very well with the tested data. After that, the numerical flow rate is slightly over-estimated.

Table 2. Summary of CPU time.

1000 iterations for steady state (One serial CPU of 1.8 GHz

on Windows platform)

1000 time steps for startup simulation on a PC cluster of parallel Intel Xeon

processors of 3.2 GHz on Linux platform (Δt =0.001 s, 40 iterations each time step)

Mesh number /105 CPU time /h Mesh number

/105 CPU time /h

6.8 4.7 5.8 38.5 (2 CPUs)

11.8 9.5 7.9 53.8 (2 CPUs)

13.9 10.9 11.0 104.5 (2 CPUs)

24.2 19.0 11.0 34.0 (7 CPUs)

51.9 45.0 16.8 73.5 (4 CPUs)

32.2 146.5 (4 CPUs)

t / s

Fig. 7. Results of grid independence test (Δt = 0.001 s).

t / s

Fig. 8. Results of time step independence test (Nm = 11×105).

t / s

Fig. 9. Comparison between numerical and experimental transient rotational speeds and flow rates.

t / s

Fig. 10. Comparison between numerical and experimental transient rotational speeds and heads.

t / s

Fig. 11. Comparison between numerical and experimental transient rotational speeds and powers.

0 20 40 60 80 100 120 140 160 1800123456789

101112

P (k

W)

Q (m3 h-1)

Pexp

Pnum

Fig. 12. Comparison between numerical and experimental transient performance (P-Q) curve.


Qualitatively, the numerical and experimental heads are in good agreement with each other. Quantitatively, the predicted transient head is lower than the experimental data at the outset, and then it eventually exceeds it.

The flow rate and head deviations are mainly due to the volume leakage through the mechanical seal, gap between the impeller’s shroud, and the pump body, which was not taken into account.

A comparison of instantaneous shaft power is represented in Fig. 11, in which the curves are very close to each other. Small fluctuations can be observed for both the numerical and experimental shaft power around the moment of 1.0 s. From then on, the two curve waves advance in the same level.

From the transient P-Q curve shown in Fig. 12, the transient power smoothly increases with the flow rate during the start-ing period. The tested P-Q curve is above the calculated one, which may be caused by neglecting the volume leakage and mechanical frictions in numerical simulations during the start-ing period. However, their tendencies are similar and their final values are exactly the same.

Overall comparisons of the transient explicit characteristics mutually verified the accuracy of the numerical simulation and the experimental test results.

Fig. 13 presents the comparison between numerically pre-dicted and experimental quasi-steady head coefficient Ch dur-ing starting period. Clear differences can be found during startup period. The transient head coefficient rapidly falls from very large values to a minimum value of about 0.4 in 0.3 s. After 1.3 s, it rebounds to the steady-state level, whereas the quasi-steady head coefficient is stable around 0.8. Explana-tions for this transient characteristic from the perspective of internal flow will be given in the following sections. 4.4 Transient flow evolutions and comparisons in the test

section The numerical and experimental results were further com-

pared from the perspective of transient flow evolutions. Two typical moments of instantaneous velocity field of PIV test and numerical simulation are presented and compared in Fig. 14. For convenience of analysis, numerical time is taken as the benchmark.

For both experiments and simulations, there forms a stream

of main flow brought by the starting impeller at the center of the test section from 0.5 to 1.0 s, which represents the feature of the flow in the test section. At 0.5 s, there is a pair of vor-tices rotating in the opposite directions clearly captured at the top left and lower right of the test section for both PIV test and numerical simulation. At 1.0 s, the simulated top left vortex becomes wide and flat, and so does the PIV test result. For both 0.5 and 1.0 s, the simulated lower right vortex compares well with the PIV test in aspects of size, position, and velocity magnitude.

The experimental and simulated instantaneous velocity field evolutions in test section have good comparisons in flow structure and velocity magnitude, which further confirms the numerical method and the results of the present study.

4.5 Transient flow evolutions in the impeller The experimental observations of transient flow between

blades cannot be captured because of the non-transparent shrouded impeller. For the analysis of transient flow mecha-nism, only the simulated results are given. Fig. 15 shows the relative velocity evolutions on the middle stream surfaces S1m and S2m

[28] during 0.1≤t≤1.6. The reference frame is fixed at the rotating impeller at each moment.

At t=0.1 s, there is a reversed flow with large incidence an-gle (≈90°) visible at the blade inlet on S1m and a spatial vortex in the flow passage reflected from both S1m and S2m. A saddle

t / s

Fig. 13. Comparison between steady and numerical transient head coefficients.

Numerical simulations t=0.5 s t=1.0 s

y/mm

0

50

100

150

185

0 20 40 60 68

y/mm

0

50

100

150

185

0 20 40 60 68 PIV tests t=4/7 s t=7/7 s

Fig. 14. Comparisons between numerical and PIV test velocity evolu-tion in the axis cross-section across the pump body (N∞=1475 rpm, t*=1.14 s, Qe=186 m3/h).

Ch


point is formed at their junction. At this moment, Ch declines to about 76 percent of the steady value.

When t=0.3 s, the spatial vortex is further strengthened in size and strength. At the same time, the incidence angle at the blade inlet decreases. The vortex, together with the large inci-dence angle, causes the transient head coefficient to reach its

lowest point at t=0.3s. When t=0.5 s, the relative flow on S1m is quite smooth, ex-

cept for the area adjacent to the blade pressure side. At the same time, a spatial vortex formed near the hub inlet is visible on S2m. This flow feature continues even at t=1.0 s. The shape and rotating direction of the vortex on S2m continuously changes and its intensity gradually decreases after t=0.5 s (Fig. 14).

A main flow stream from the middle of adjacent blades be-gins to form from t=0.7 s. The intensity weakening of the vor-tex visible on S2m and the main flow stream are the main rea-sons causing Ch to rise slowly below the steady value from 0.3 to 1.6 s.

Finally, the flows on both S1m and S2m become smooth, as shown at t=1.6 s. The smoothed flow also verifies the design rationality of the blade.

Based on the above discussion, the transient flow character-istics during startup period behave as evolutions of the vortex in the impeller and the reversed flow at blade inlet in the early time. In the latter stage (t≥0.5 s), although flow at the blade inlet becomes smooth, change in rotating direction occurs for the spatial vortex visible on S2m.

5. Conclusions

In this paper the transient characteristics of a centrifugal pump during its starting period are numerically and experi-mentally investigated. Both simulated and experimental test transient performances and instantaneous flow evolutions are presented.

The use of the close-loop numerical model can realize un-steady self-coupled computations. There is no need to provide any unsteady inlet and outlet boundary conditions externally, for example, from the experimental test. The extra error from externally specified unsteady boundary conditions can be avoided. The transient flow rate is regulated via adjusting the ball valve angle φ. The simulation procedure is fully dynamic and approximates actual situations. The results are thus more accurate.

Comparisons of both explicit characteristics and internal transient flow evolutions between numerical simulation and experiments are found to be satisfactory. Results indicate that the proposed DSR method is capable of solving transient flow in centrifugal pump under transient operation conditions.

Most importantly, analysis on the transient flow evolutions in the impeller is presented. From the transient flow point of view, for the early stage of the startup, the transient vortex evolution between blades is the main reason leading to the transient head coefficient being lower than the steady state value. Reversed flow at the blade inlet is a secondary reason for this occurrence. In the later stages, the transient flow fea-tures are the spatial vortex visible on S2m and the main flow stream, which are the main reasons causing Ch to rise slowly below the steady value.

The content and conclusions of the current work can pro-

XY Z t = 0.1 s

XY Z t = 0.3 s

XY Z t = 0.5 s

XY Z t=0.7 s

XY Z t=1.0 s

XY Z t=1.6 s

Fig. 15. Transient relative velocity evolutions on the middle streamsurfaces S1m (left) and S2m (right) between blades (Reference frame isfixed on the impeller, 0.1 s≤t≤1.6 s.).


vide references for the performance prediction, design optimi-zation, and fluid control of hydraulic turbo machineries used in a variety of transient process.

Future work aims to investigate the influences of various acceleration rates and pipe resistances using numerical meth-ods and experiments. Further efforts are needed to explore the transient flow inside the impeller, which is believed to be more attractive. It is foreseeable that the use of high-speed PIV and photography techniques can provide faster and more precise results in the research on unsteady flow evolutions in hydraulic turbo machineries under transient operations.

Acknowledgment

This study was carried out as part of the Fundamental Re-search Funds for the Central Universities, the National Natu-ral Science Foundation of China (Project Numbers 50979095 and 50906074) and the China Postdoctoral Science Founda-tion (No. 20100471697). The support is gratefully acknowl-edged. The authors thank Associate Professor Wu Xuecheng and doctoral candidate Tian Chen from the Institute for Ther-mal Power Engineering of Zhejiang University for their tech-nical support on the PIV experimental test.

Nomenclature------------------------------------------------------------------------

b2 : Width of the impeller outlet (m) Cq : Flow coefficient: Q/ (πd2b2u2) Ch : Head coefficient: 2gH/ (u2

2) d2 : Average diameter of the impeller outlet (m) di : Inlet pipe diameter (m) do : Outlet pipe diameter (m) H : Total head (m) Mtr : Transient total moment on the impeller, resulting from both pressure and viscous forces exerted on the impeller (N·m) N∞ : Impeller rotational speed after startup (rpm) n : Impeller rotational speed (rpm) Ptr : Transient power (W) p : Pressure of the fluid (Pa) pi : Area weighted averaged static pressure on the inlet section (Pa) po : Area weighted averaged static pressure on the outlet section (Pa) Qt : Transient volume flow rate (m3/h) Qe : Experimental steady volume flow rate (m3/h) t : Time (s) t* : Startup time of the impeller (s) Δt : Time step size (s) u2 : Outlet peripheral velocity of the impeller: πd2n/60 (m·s-1) u : Fluid velocity (m·s-1) ub : Boundary velocity of the moving mesh (m·s-1) aω : Angular acceleration (rad·s-2)

μ : Dynamic viscosity (Pa·s) ρ : Fluid density (kg·m-3)

φ : Ball valve open angle (°) ωtr : Transient angular speed (rad·s-1) ω∞ : Angular speed after startup (rad·s-1) Re : Reynolds number: Re:ρω∞d2

2/2μ

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Zhifeng Li is currently a Post Doctor in the Department of Chemical and Bio-logical Engineering, Zhejiang University. He obtained his B.Sc. degree in 2004 and his Ph.D. in 2009 from Zhejiang University in P. R. China. His major research interests include design, opti-mization, control, and engineering appli-

cations of various fluid machineries, pressure vessels, and mechanical equipment.

Dazhuan Wu is currently an associate professor in the Department of Chemical and Biological Engineering, Zhejiang University. He obtained his B.Sc. degree in 1999 and his Ph.D. in 2004 from Zhejiang University in P. R. China. His major research interests include optimization and its engineering

applications, cavitation, flow noise, unsteady flow, and control. He has published more than 30 scientific publications in the field of fluids engineering.

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