Numerical study on the hydrodynamic characteristics of multi-stage centrifugal pumps influenced by impeller-guide vane clocking effect

Abstract

A high-speed multi-stage centrifugal pump building upon a transient investigation into the hydrodynamic characteristics, is conducted for 16 sets of clocking positions. The most significance of the pump head has a maximum increase of 3.6% at 0.8Qd. Phase differences emerge in the pressure pulsations at the impeller-guide vane gaps across stages. It results in waveform distortions that are primarily influenced by impeller clocking effects. Innovative average energy flux density analysis shows that impeller-guide vane clocking positions have the greatest influence on pressure pulsations near their gaps. The cumulative propagation of pressure pulsations has a relatively minor effect on fluid domains. Worst stability is observed when impellers rotated with a 3/4 maximum offset angle and guide vane positions remain unchanged. The best clocking position solution was adapted in a prototype pump for performance and operational stability measuring, achieving an efficiency of 53.67% and 1 mm/s vibration. This paper aims to provide valuable insights for enhancing the energy efficiency and operational safety of multi-stage pumps, offering beneficial guidance for engineering practices within the field of fluid machinery applications.

KEYWORDS:

• Clocking effects
• multi-stage centrifugal pumps
• pressure pulsations
• dynamic performance

Nomenclature

A	=	The amplitude of the wave
a	=	The distance from the pulsation source
CRMS	=	Square pressure pulsation coefficients
Cp	=	The intensity of pressure fluctuations
ηdy	=	Guide vane losses
ηimp	=	Impeller losses
f0	=	The rotational frequency
fZ	=	The blade-passing frequency
H	=	Head of delivery
Himp	=	Head of delivery in impeller
N	=	The number of samples
$\overline{\mathrm{p}}$	=	The time-averaged static pressure
ΔPimp	=	Impeller inlet–outlet pressure drop
Pin	=	The guide vane inlet pressure
Pout	=	The guide vane outlet pressure
$\overline{p}$	=	The time-averaged static pressure
ρ	=	The density of the conveyed medium
Q	=	Flow rate
$\overline{u_i}$	=	Reynolds-averaged velocity in the i direction
u2	=	The circumferential velocity
ω	=	The angular frequency

1. Introduction

1.1. Background

High-speed multi-stage pumps, known for their characteristics such as high head, wide flow range, and compact structure, have extensive applications in fields like forest firefighting, urban emergency drainage, and industrial water circulation (Lindler, 1998; Zhu et al., 2016). As the demand for multi-stage centrifugal pumps continues to grow, their annual energy consumption has increased, resulting in significant energy losses and maintenance costs due to performance disparities, which are incongruent with the requirements of the ‘dual carbon’ (peak carbon and carbon neutrality) strategy (Zhou et al., 2018; Zhu et al., 2022). The relative position changes of different rotors in rotating machinery can impact the overall performance of the system. Moreover, the presence of phase differences during the installation of rotors can lead to rotor clocking effects, causing pressure pulsations and excitatory forces, thereby affecting the stability of multi-stage centrifugal pumps (Liu et al., 2020). The clocking effect refers to the influence of circumferential positions between stationary components or rotating components on the performance of rotating machinery (Zhang et al., 2021).

1.2. Literature review

The concept was initially proposed by Walker in 1972 in his research on compressors (Walker et al., 1997). By the late 1990s, there was controversy among scholars regarding the clocking effect's influence on the performance of turbine machinery. One viewpoint suggested that the impact of the clocking effect on compressor performance is relatively small, especially for low-speed compressors, where its effect is comparable to manufacturing and assembly errors. Another viewpoint acknowledged that unsteady effects have a certain impact on compressors. At the end of the twentieth century, in-depth studies on the clocking effect of rotating machinery such as compressors and turbines revealed that different clocking positions have a particularly noticeable impact on overall efficiency under high-speed operating conditions (Huber et al., 1996). Adjusting the circumferential position of blade rows or moving blades can slightly improve device efficiency and reduce blade pressure loading (Cizmas & Subramanya, 1998; Saren et al., 1998). Similarly, at low Reynolds numbers, the clocking effect of rotating components has a more significant impact on device efficiency (Arnone et al., 2002). Currently, in engineering, there is a particular emphasis on the impact of the clocking effect on rotating machinery. For example, in applications such as high-power mechanical pumps in thermal power plants, visual analysis of losses caused by clocking positions through entropy production theory and computational fluid dynamics revealed that clocking positions have a greater impact on the total pressure loss coefficient of cylindrical casings, and introduce additional flow losses (Gu et al., 2019). Similarly, in the study of centrifugal pumps, it has been shown that the clocking effect affects the generation of radial forces and hydraulic performance through its influence on total pressure loss (Guan et al., 2021; Hongyu et al., 2021; Jiang et al., 2021; Lai et al., 2020). Many scholars have studied the effects of the clocking effect on the performance of compressors, turbines, and centrifugal pumps, but there is less research focused on multi-stage centrifugal pumps, lacking quantitative analysis of the effects of clocking positions at each stage and how adjusting the circumferential position of blade rows or movable guide vanes in multi-stage centrifugal pumps affects device efficiency and blade pressure loading.

The influence of the clocking effect on rotating machinery is not only reflected in performance but also manifests in its effects on internal flow, especially for multi-stage rotating machinery, where downstream blade rows within impellers can generate a swirl effect and interfere with upstream blade rows (Pinelli et al., 2022; Wei et al., 2022). The interference caused by the clocking effect on internal flow has multiple disturbing factors and high discernibility. By applying nonlinear harmonic methods between adjacent stages of a two-stage compressor, it was found that the wake of upstream moving blades significantly affects downstream moving blades (He et al., 2002). Different positions of guide vanes affect circumferential position variation and radial forces of impellers, leading to differences in main frequency pressure amplitude and pressure pulsation intensity (Gu et al., 2019; Qu et al., 2016; Wei et al., 2016). The frequency at which blades pass is also influenced by the clocking effect, with different impeller structures and angles affecting blade-passing frequencies (Fu et al., 2018; Hongyu et al., 2021), especially at certain configuration where the clocking effect is found to have the greatest impact on guide vane blade-passing frequencies at 1 and 2 times (Key et al., 2010). The interaction of wakes is entirely due to pressure changes caused by the relative motion of blade rows. The interaction between shed wakes of upstream and downstream blade rows constitutes the main component of blade instability forces. Previous research has focused on the interaction between individual rotating or stationary blade rows. With the increase in the number of stages in turbine machinery, there is a need for further study of the cumulative effects of wakes and blade interactions. Especially in the 14-stage pump studied in this paper, where the interstage distance is relatively close, pressure pulsations effectively demonstrate interference between the relative motion of upstream and downstream blade rows.

1.3. Contributions

High-speed multi-stage pumps are widely used in emergency rescue water supply and industrial water circulation processes. Changing the circumferential relative positions of stationary vanes can impact the aerodynamic efficiency and pressure pulsations of the impeller, leading to operational deviations and safety incidents. As the number of stages in impeller machinery increases, the cumulative effects of wake transmission make the interactions between wakes and between wakes and blades highly complex. Therefore, research on the clocking effects of multi-stage pumps is crucial. The paper conducted experimental studies on the performance of a simplified three-stage centrifugal pump derived from a fourteen-stage pump. This was complemented with numerical simulations to further analyze pressure pulsations, energy flux density, etc.

The main contributions of this paper include quantifying the impact of clocking effects on performance, demonstrating the cumulative effects of wake transmission, and analyzing the interference between upstream and downstream blade rows in relative motion. A summary was provided in conjunction with the results of simplified experiments, full-scale experiments, and numerical simulations.

1.4. Paper organisation

In this paper, 16 different configurations were designed by considering the different stages of relative positions between moving and stationary blades among multiple stages. The clocking effects on performance and internal flow patterns of the multi-stage centrifugal pump were investigated. The energy flux density of pressure pulsations was defined using an analogy with wave intensity. Through analysis of energy losses, time-domain, frequency-domain, and energy flow density at critical flow components, the study revealed the influence of impeller and guide vane clocking effects on the transient flow characteristics of the multi-stage centrifugal pump.

2. Research methods

2.1. Pump parameters and computational domain

The research object of this paper is the emergency rescue multi-stage pump (Figure 1) used in mountainous areas and remote disaster areas. The structure of 14 impellers is 7 symmetrically arranged as a group (Table 1).

Figure 1. The emergency rescue multi-stage pump in 14 stages.

Table 1. Hydraulic parameters of the multistage pump.

Geometry	Value	Geometry	Value
Design flowrate Qd	36m^3/h	Impeller outlet diameter D2	233 mm
Design head Hd	1500m	Impeller Outlet Width b2	6.5 mm
Rotational speed n	3800 r/min	Blade outlet placement angle β2	30°
Pump inlet diameter Ds	80mm	Blade wrap angle φ2	136.4°
Pump outlet diameter Dd	65mm	Base diameter D3	236 mm
The gap between oral rings Da	0.27mm	Positive guide vane inlet width b3	11.2 mm
Balde number of the Impeller Z1	5	Positive guide vane outlet diameter D3	304.5 mm
Balde number of the diffuser Z2	4	Inlet width of anti-guide vanes b4	15 mm
hub diameter of the impeller Dh	80mm	Inlet diameter of anti-guide vanes D4	260 mm
Impeller inlet diameter D1	96 mm	Outlet diameter of anti-guide vanes D5	96 mm

Since the inlet of the first-stage impeller of the multi-stage pump can be considered as a flow coupled with the inlet pipe, each subsequent impeller The inlet is close to the outlet of the anti-guide vane, all of which are swirling flow, so the performance of the entire multi-stage pump can be predicted by selecting a reasonable finite number of stages (Wang et al., 2013). In this paper, the target pump is simplified to a three-stage pump model for numerical simulation to reflect the performance of the entire multi-stage pump. The fluid domain model of the three-stage pump comprises sections for inlet and outlet, impellers, and guide vanes for each stage. The computational domain for the entire flow field is depicted in Figure 2.

Figure 2. Calculation domain of three-stage centrifugal pump.

The design scheme for clocking positions in this study involves maintaining the circumferential positions of the first-stage impeller and guide vanes while altering the clocking positions of the second-stage impeller, final-stage impeller, second-stage guide vanes, and final-stage guide vanes. Given that the impeller has 5 blades, and the guide vanes have 4 blades, considering blade thickness, the maximum angular misalignment for impeller blades is set at 68°, and for guide vane blades, it is set at 90°. As such, four clocking positions were designated for both impellers and guide vanes (impeller rotation angles of 0°, 17°, 34°, and 51°, and guide vane rotation angles of 0°, 22.5°, 45°, and 67.5°), employing a four-factor, four-level orthogonal experimental design to reduce the number of design scenarios and conserve computational resources. The detailed combination of schemes is presented in Table 2, comprising a total of 16 schemes designated as L1 to L16. Scheme L1 serves as the initial configuration with no circumferential position changes for impellers and guide vanes. Figure 3 illustrates clocking positions for impellers and guide vane. Figure 3(b) provides a three-dimensional view of the impeller and guide vane clocking positions, using scheme L8 as an example. All design schemes share the same computational domain and grid discretization, with the only variation being the clocking positions of impellers and guide vanes. Unsteady-state numerical calculations and analyses were conducted for different scheme combinations.

Figure 3. Clocking position of impeller and guide vane.

Table 2. Clocking position scheme of the impeller and guide vane.

Scheme numbers	The second impeller θ/°	The last impeller θ/°	The second guide vanes θ’/°	The last guide vanes θ’/°
L1	0	0	0	0
L2	0	17	22.5	22.5
L3	0	34	45	45
L4	0	51	67.5	67.5
L5	17	0	22.5	45
L6	17	17	0	67.5
L7	17	34	67.5	0
L8	17	51	45	22.5
L9	34	0	45	67.8
L10	34	17	67.5	45
L11	34	34	0	22.5
L12	34	51	22.5	0
L13	51	0	67.5	22.5
L14	51	17	45	0
L15	51	34	22.5	67.5
L16	51	51	0	45

2.2. Numerical simulation methods

In this study, the Shear Stress Transport (Menter) k-ω (SST k-ω) turbulence model is chosen to perform numerical computations for incompressible flow. It considers turbulent shear stress, offering a hybrid function that is capable of modelling both near-wall viscous flows and far-field free flows (Ramadhan Al-Obaidi, 2019). This makes it applicable across the entire boundary layer, producing accurate results. The expression for this model is given by: (1) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\rho k}\right)+\mathrm{\nabla }\cdot \left(\mathrm{\rho k}\overline{u_i}\right)\\ & =\mathrm{\nabla }\cdot \left[\left(\mu +{\sigma }_k{\mu }_t\right)\mathrm{\nabla k}\right]+G_k-\rho {\text{β}}^{\ast }\mathrm{k\omega }\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\mathrm{\rho \omega }\right)+\mathrm{\nabla }\cdot \left(\mathrm{\rho \omega }\overline{u_i}\right)\\ & =\mathrm{\nabla }\cdot \left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right)\mathrm{\nabla \omega }\right]+G_{\omega }-\rho \text{β}{\omega }^2\end{array}$(2) (3) ${\mu }_t=\frac{\mathrm{\rho k}}{\omega }$(3) (4) $G_k=P_k+P_b+P_{\mathrm{nl}}$(4)

Where $\overline{u_i}$ represents the Reynolds-averaged velocity in the i direction. The model coefficient is ${\sigma }_k={\sigma }_w=0.5$. ${\text{β}}^{\ast }=0.09$ is the free shear correction factor. $\text{β}=0.075$ is the turbulence production correction factor. P_k represents the turbulent production term. P_b represents the buoyancy term. P_nl represents the non-linear pressure-strain term. P_w represents the dissipation rate. D_w represents the cross-diffusion term.

Star CCM + is used to mesh the computational domain. The grid discretization for the computational domain of the model pump with three stages was performed by polyhedral meshes, which with high accuracy and faster calculations than tetrahedral meshes for the same complex geometry (Fišer & Jícha, 2013). The grid details are illustrated in Figure 4.

Figure 4. Compute domain partial grid.

As observed from Figure 5, with an increase in the number of grids, the head variation of the single-stage pump model remains minimal when the grid count reaches 6.4 million, with fluctuations satisfying the requirement of being within 2%. Thus, it can be concluded that a grid count of 6.4 million is the minimum viable grid size. The maximum nondimensional wall distance, y + < 10, was obtained in the complete flow field, which could satisfy the requirement of all turbulence modelling methods used in this paper.

Figure 5. Grid independence.

The numerical simulation of the model pump flow field is carried out with the following boundary conditions. The inlet section, pump chamber, guide vanes, mouth ring, and outlet section were set as stationary computational domains. The impeller is set as a rotating computational domain, with interfaces established between the impeller and the front mouth ring, as well as between the impeller and the pump chamber. The working medium is water at 25°C, with a reference pressure of 0 atm. A stagnation inlet boundary condition is employed, and an inlet pressure of 1 atm is specified. The outlet boundary condition is set as a mass flow inlet, with a negative sign added to the mass flow rate value to indicate an outlet condition. Wall surfaces are defined as no-slip, smooth, adiabatic viscous walls with zero relative velocity. For near-wall flows, a standard wall function is employed. Based on the steady simulation results, transient simulations are performed for each timing scheme under rated operating conditions, using the steady-state results as initial values. Each time step is set at 0.000131578 s, corresponding to the time taken for the impeller to rotate by 3 degrees. A maximum of 20 iterations are set per time step, and a single impeller rotation cycle is set at 0.01579 s. A total of 9 impeller rotation cycles are simulated, resulting in a total simulation time of 0.142105s. Convergence residuals are maintained under or equal to 10⁻⁵ RMS (Jiantao Zhaoa, 2022). Some of the parameters are based on empirical data from some peers (Yandong Gu et al., 2024).

Pressure pulsations were normalised using the pressure fluctuation coefficient C_p, which characterises the intensity of pressure fluctuations. Additionally, the root mean square of the pressure fluctuation coefficient C_RMS was introduced to analyze the circumferential patterns of pressure pulsations at the gaps between impellers and guide vanes. The specific calculation formulas are as follows. (5) $C_p=\frac{p-\overline{p}}{0.5\rho u_2^2}$(5) (6) $C_{\mathrm{RMS}}=\frac{\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(p_i-\overline{p}\right)}^2}}{0.5\rho u_2^2}$(6) Where p represents the static pressure at a monitoring point at a certain moment, in Pa. $\overline{p}$ represents the time-averaged static pressure at the monitoring point within one impeller rotation cycle, in Pa. ρ represents the density of the conveyed medium in the multi-stage centrifugal pump, in kg/m³. $u_2$ represents the circumferential velocity at the impeller outlet, m/s. N represents the number of samples.

For the studied object with a rated speed of 3800 r/min, the rotational frequency can be calculated as f₀=  n/60 = 63.33 Hz, and the blade-passing frequency f_Z = Z f₀=  316.66 Hz.

To accurately characterise the energy propagation characteristics of pressure pulsations within the multi-stage centrifugal pump, an analogy is drawn between the intensity of the wave and the energy flow density of pressure fluctuations (Derong et al., 2020). The energy flow density is defined as follows:

The energy flux of a wave, $\overline{W}$, is defined as the energy passing through a certain cross-section per unit of time. The average energy flux density I represents the mean energy flux of pressure pulsations within one cycle. Because the pressure pulsation signal contains various frequencies, the total energy flux density of pressure pulsations is used to represent the intensity of the pressure pulsation signal. (7) $\overline{W}=\overline{\mathrm{ϵu}}\mathrm{\Delta S}$(7) (8) $\int I\text{d}f=\int \frac{\overline{W}}{\mathrm{\Delta S}}\text{d}f=\int \frac{1}{2}\rho A^2{\omega }^{2}u\text{d}f$(8)

2.4. Validation of the numerical simulation methods

In Figure 6, The test rig is illustrated to produce pump performance measurements for different operating conditions of the three-stage centrifugal pump, which implements ISO 9906:2012 standard. it encompasses inlet and outlet pipelines, electrical control valves, electromagnetic flow metres, pressure sensors, and other equipment, with the accuracy of the flow rate measurements is ±2.5%, the head is ±3%, the torque is ±2.5%, and the rotation is ±1%. The rated flow rate of the three-stage test pump is 36 m³/h, with a design head of 360 m.

Figure 6. Test bench.

As depicted in Figure 7, a comparison is made between the numerical simulation and experimental results for the performance curve of the three-stage centrifugal pump at its rated speed. Observations from the graph include: both the experimental and numerical efficiency curves exhibit an initial increase followed by a decrease, both favouring higher flow rates, which aligns with the design expectations. The numerical computation yields a head value lower than the experimental value, with a maximum deviation of 11.81 m (3.1% relative error) occurring at 0.6 times the design flow rate of 0.6Q_d. Additionally, the numerical computation overestimates the efficiency, showing a maximum deviation of 3.7% (6.5% relative error) at 1.0 Q_d. Overall, the general trend of the curves matches closely with the experimentally obtained curves, validating the reliability of the numerical simulation results.

Figure 7. Different speed test pump characteristic curve and the simulation contrast.

3. Results analysis

3.1. Clocking effect on pump performance

Numerical simulations were conducted for the three-stage centrifugal pump at 16 different clocking schemes under three flowrates. For a more intuitive comparison between the various clocking schemes and the original scheme, non-dimensional coefficients H/H₀ and η/η₀ are introduced, which respectively represent the relative change in head and efficiency with respect to the initial scheme L1. As shown in Figure 8, only two clocking schemes, namely L2 and L12, exhibit a decrease in efficiency compared to the initial scheme under the 0.8Q_d. Notably, the efficiency improvement is most prominent in schemes L4, L6, and L7, with a maximum increase of 1.9%. Similarly, schemes L3, L4, and L8 result in the most significant head increase, with a maximum improvement of 3.6%. Moving to Figure 8 (b), under the 1.0Q_d operating condition, efficiency improves in four clocking schemes: L4, L6, L10, and L15, compared to the initial scheme. Only scheme L14 experiences a slight decrease in head relative to the initial scheme. Among these, scheme L6 achieves the highest efficiency increase, with a maximum improvement of 0.29%. On the other hand, schemes L3 and L10 exhibit the most noticeable head improvement, with a maximum increase of 1.3%. Turning to Figure 8 (c), it becomes evident that only clocking schemes L6 and L16 maintain efficiency levels similar to the initial scheme, while other schemes experience a slight efficiency decrease. Among the head variations, only schemes L2 and L11 show a slight decrease relative to the initial scheme. Notably, schemes L4, L8, and L9 exhibit the most substantial improvement in head performance, each showing a maximum increase of 1.2%.

Figure 8. Changes in pump performance of different schemes under different working conditions.

Through a comprehensive comparison of the changes in various clocking schemes’ characteristics relative to the initial scheme under three different operating conditions, it is apparent that altering the timing of the rotor and guide vanes has the greatest impact on pump characteristics at low flow rates. The variation in the rotor and guide vane timing has a relatively minor impact on the hydraulic performance of the three-stage centrifugal pump at the design point condition. Because in the speed triangle of the pump, a decrease in flow rate will lead to an increase in the tangential component of velocity. According to Euler's theorem, with a constant rotational speed, the head increases as the tangential component of velocity increases. During partial load operation, the already increased head fluctuation will be even greater. Most of the clocking schemes enhance the pump's head at different operating conditions, yet they do not significantly improve its efficiency. Depending on whether the optimisation goal is to enhance efficiency or head, different optimal combinations of clocking schemes emerge under different operating conditions. In conclusion, based on a comprehensive analysis, scheme L6 emerges as the optimal clocking scheme. It consistently improves both the head and efficiency of the three-stage centrifugal pump compared to the initial scheme across various operating conditions.

3.2. Clocking effects on pump internal flow field

Figure 9 illustrates the static pressure distribution in the middle section of the last-stage impeller and guide vane under rated operating conditions. It is evident that certain clocking schemes, namely L3, L5, L8, L10, and L16, exhibit localised high-pressure regions near the junction of the positive guide vane and the pressure chamber. These regions notably surpass the pressure levels of other clocking scheme combinations. The occurrence of local high-pressure zones at the exit of the positive guide vane could potentially lead to increased hydraulic losses during the process of fluid separation as it enters the negative guide vane. This phenomenon is also corroborated by the analysis of losses in the last-stage guide vane (as discussed in Section 3.1), where L3, L5, L10, and L16 are identified as having the highest losses. It's noteworthy that these schemes (L3, L5, L10, and L16) share the same guide vane clocking position, with a rotation of 45° relative to the initial position. This positioning elevates the static pressure at the entrance of the pump's pressure chamber, contributing to a higher pump head compared to other schemes. With an increase in the number of pump stages, the static pressure at the same impeller radius progressively increases, peaking at the exit of the positive guide vane. The areas with the most significant variations in static pressure distribution concentrate around the impeller exit, the gap between the impeller and guide vane, and the inlet of the guide vane. This asymmetry in flow and instability in these regions contribute to the notable differences in performance among various clocking schemes.

Figure 9. Static pressure distribution in the middle section of the last stage impeller and guide vane.

Figure 10 displays the distribution of turbulence kinetic energy in the middle section of the last-stage impeller and guide vane under rated operating conditions. It shows that the majority of clocking schemes exhibit a notable reduction in turbulence kinetic energy. This decrease indicates that the flow becomes more stable after passing through the preceding guide vane, resulting in a more uniform flow distribution in the circumferential direction at the impeller's exit. However, schemes L11 and L13 exhibit regions of significantly higher turbulence kinetic energy. These regions are distributed near the inlet of the guide vane blades and attachments on the pressure side of the impeller's exit. These areas align with regions of high-pressure gradients, suggesting the presence of severe interference and substantial energy losses. To illustrate the impact, consider the pairs of second and last-stage schemes (L1 and L1, L2 and L13, L3 and L5, L4 and L9). In these cases, the relative clocking positions of the second and last-stage impeller and guide vane are the same. Consequently, the distribution of turbulence kinetic energy is quite similar between these pairs. However, the magnitudes of turbulence kinetic energy differ, with the maximum value in the last stage being higher than that in the second stage. Excluding the maximum values, the gradient of turbulence kinetic energy in the last stage is smaller than that in the second stage. This observation suggests that the distribution of turbulence kinetic energy is primarily determined by the relative clocking positions of the impeller and guide vane. The areas with elevated turbulence kinetic energy result from the interaction between moving and stationary components. Improvements in flow stability help mitigate turbulence energy in most areas. By comparing all 16 clocking schemes, scheme L1 exhibits a smaller maximum turbulence kinetic energy and distribution area compared to most other schemes. This finding suggests that altering the clocking positions of the impeller and guide vane increases the potential for increased turbulence dissipation losses within the flow passages. This observation aligns with the results obtained from the external characteristic analysis in Section 3.1, where the efficiency of most clocking schemes under rated conditions is slightly lower than that of the initial scheme, L1.

Figure 10. Turbulent kinetic energy distribution in the middle section of the last stage impeller and guide vane.

3.3. Influence of clocking effects on pressure pulsations

The monitoring points are labelled using A, B, and C to denote the first, second, and third stages of the centrifugal pump. The gaps between the impeller and guide vanes are indicated by the letter Q, followed by a combination of letters and numbers representing specific monitoring points within each section. For example, in the first-stage flow domain, A1, A2, and A3 represent pressure pulsation monitoring points within the impeller passages, while A4, A5, and A6 represent points within the guide vanes. QA1 to QA40 indicate 40 pressure pulsation monitoring points along the gaps between the impeller and guide vanes, evenly spaced at 9-degree intervals. The monitoring points associated with the impeller move with its rotation, while the positions of the other monitoring points remain fixed. In total, there are 46 pressure pulsation monitoring points across different sections. The placement of these monitoring points is illustrated in Figure 11.

Figure 11. Location of monitoring points for primary overcurrent components.

3.2.1. Analysis of circumferential pressure pulsations in the gap between impeller and guide vanes

To delve into the relationship between pressure pulsation patterns in the gaps between the impeller and guide vanes and the clocking effects, pressure pulsation data, collected from the 40 evenly distributed monitoring points situated in the gaps between the impeller and guide vanes, is transformed into root mean square pressure pulsation coefficients (C_RMS) for each of the 16 clocking configurations (L1 to L16).

As shown in Figure 12, the distribution of root mean square pressure pulsation coefficients (C_RMS) exhibits four peaks in the circumferential pressure pulsation distribution in the absence of clocking effects on impeller and guide vanes (L1). These peaks are aligned across different stages, independent of the number of impellers. This indicates that the pressure pulsation patterns are primarily governed by the dynamic interaction between guide vanes and water gaps, with the influence of impellers being second.

Figure 12. Circumferential pressure pulsation of water body between impellers and guide vanes in different clocking positions.

When the second impeller and third impeller rotate by the same angle (as in L1, L6, L11, and L16), the C_RMS peaks and troughs formed by the second and third impeller pressure coefficients exhibit the same phase. However, the amplitudes of these C_RMS variations differ among the configurations. Similarly, when the second guide vanes and final guide vanes rotate by the same angle (as in L1, L2, L3, and L4), the C_RMS peaks and troughs formed by the second and third impeller pressure coefficients have distinct phases, but their amplitudes are closer. This implies that the circumferential pressure pulsation phase variations in the gaps between impellers and guide vanes are primarily influenced by the clocking positions of impellers, while the amplitude variations are mainly influenced by the clocking positions of guide vanes. This is because the change in the position of the guide vane changes the relative flow angle of the pump, which affects the circumferential velocity closely related to the head. However, the change in the position of the impeller does not affect the change of the triangle of the pump outlet velocity, and the influence of dynamic and static interference can only be reflected in the phase.

When both second and third impellers and guide vanes exhibit clocking position variations, the C_RMS curves of primary-stage impeller pressure coefficients differ significantly from the L1 configuration. Multiple less distinct peaks appear although the original peaks remain present. Some configurations display less distinct peaks and troughs in the C_RMS curves of second and third impeller pressure coefficients, forming an overall profile of five peaks, matching the number of blades. This indicates that the pressure pulsation variations caused by the clocking effects of impellers and guide vanes are not confined to specific stages; the pulsations propagate and interact throughout the pump, revealing that the influence of guide vanes on the dynamic interaction between impellers and guide vanes diminishes while the impellers’ influence strengthens under impeller clocking effects.

3.2.2. Frequency domain analysis of pressure pulsations

To further investigate the impact of impeller and guide vanes clocking on pressure pulsations, the pressure pulsation signals in the time domain are transformed into the frequency domain using Fast Fourier Transform (FFT) analysis. As shown in Figure 13, it presents the frequency domain characteristics of pressure pulsation coefficients (C_p) at monitoring points within the primary impeller passage. The horizontal axis represents the ratio of frequency to the shaft frequency (f/f₀), where f/f₀= 1 corresponds to 1 time the shaft frequency, and f/f₀= 5 corresponds to 1 time the blade-passing frequency. For monitoring points A1 to A3 in the vicinity of the entrance of the primary impeller, the frequency domain pressure pulsation signals are mainly composed of shaft frequency, blade-passing frequency, and their harmonics. Because the monitoring point A1 is close to the impeller inlet, the axial frequency occurs due to dynamic and static interference with the suction chamber, and the impeller inlet cuts the inlet water flow to form the blade frequency and the harmonics of the blade frequency.

Figure 13. Frequency domain characteristics of pressure pulsation in primary flow components of different clocking positions.

As fluid flows along the primary impeller passage from monitoring point A1 to A3, the amplitude of frequency domain signals increases. The primary frequency becomes consistent at 1 time the blade-passing frequency, and the secondary frequency shifts to 2 times the blade-passing frequency. In the high-frequency range (10 times the blade-passing frequency), distinct frequency domain signals emerge. At monitoring point A3, the frequency domain signals of different clocking configurations are very similar, with the primary frequency corresponding to a C_p of approximately 0.04 and the secondary frequency corresponding to a C_p of approximately 0.02. However, at point A1, the maximum C_p for the primary frequency is about 0.038 (in configuration L8), while the minimum C_p for the primary frequency is about 0.01 (in configuration L6). This indicates minor differences in pressure pulsations at the impeller inlet among various clocking configurations within the primary impeller passage. The complex low-frequency signals generated by the impeller and guide vanes’ clocking effects dominate the interference in regions farther from the pulsation source. The high similarity of pressure pulsations at monitoring point A3 (exit of the primary impeller) is due to the insignificance of low-frequency signals from other sources that affect areas close to the pulsation source.

Comparing monitoring points A4 to A6: For different clocking configurations, the main and secondary frequencies in the guide vanes region are the same as those in the impeller region. The amplitude of C_p for each region's pressure pulsation increases from the entrance of the guide vanes to the junction of positive and negative guide vanes and then decreases. Monitoring point A5 exhibits a frequency domain graph similar to that of impeller internal monitoring point A3. The maximum C_p value for the primary frequency, across all configurations, is approximately 0.05 (in configuration L1). The amplitudes of the blade-passing frequency harmonics gradually decrease as frequency increases. Figure 13 (g) presents the frequency domain characteristics of pressure pulsations at monitoring points within the gap between the primary impeller and guide vanes (point QA). For various clocking configurations, the frequency domain graphs are similar at this point. The main frequency and its harmonics dominate the signals, and the amplitude of C_p is approximately 0.03, exhibiting both high and low-frequency components. This suggests that the pressure pulsations are more intense in the QA region compared to other regions of the same stage. The gap between the impeller and guide vanes in this region is the primary source of pulsations, and its effects are largely independent of other pulsation sources.

It displays the frequency domain characteristics of pressure pulsations within second and last-stage impellers in Figure 14. Seen from it, the amplitude of pressure pulsation main frequency increases with the stage number for the same configuration at the same monitoring point. The primary and secondary frequencies of pressure pulsations in the impeller region are unaffected by the impeller and diffuser vane clocking. Across different clocking configurations, the primary frequency is consistently 1 time the blade-passing frequency, and the secondary frequency is consistently 2 times the blade-passing frequency. At the exit of the second impeller (monitoring point B3), the amplitudes of the primary frequency in configurations L3, L4, L7, L12, L14, and L15 are all smaller than that in configuration L1. Configuration L15 has the smallest amplitude at 0.029. Similarly, at the exit of the last-stage impeller (monitoring point C3), the amplitudes of the primary frequency in configurations L5, L6, L7, L8, L9, and L11 are all smaller than that in configuration L1. Configuration L11 has the smallest amplitude at 0.029. These findings suggest that a suitable combination of impeller and guide vane clocking positions can weaken the influence of dynamic and static interference on pressure pulsations in these areas. This is due to the mutual interference of the phase difference, which causes the amplitude of the main frequency to be smaller than the initial scheme, but because the phase difference is not the dominant factor of pressure pulsation, it does not greatly affect the amplitude change.

Figure 14. Frequency domain characteristics of pressure pulsation in second and last-stage impellers with different clocking positions.

Figure 15 illustrates the frequency domain characteristics of pressure pulsations within second and last-stage guide vanes. Seen from it, the amplitude of pressure pulsation at different monitoring points increases initially and then decreases along the path from the inlet of the leading guide vanes to the junction of the leading and trailing guide vanes within the same stage fluid domain. At monitoring points B4 and B5, there are three configurations that have smaller primary frequency amplitudes compared to configuration L1 (L4, L5, and L14). At monitoring point C4, configurations L2, L9, and L16 have smaller primary frequency. Among them, configuration L2 has the smallest amplitude at around 0.01. At monitoring point C5, configurations L2, L6, L8, L9, L11, L14, and L16 have smaller primary frequency amplitudes. Configuration L6 has the smallest amplitude at around 0.016. These findings indicate that monitoring points near the source of pulsation in second or last-stage guide vanes are significantly affected by the impeller and guide vane clocking. Several clocking configurations can weaken the effects of dynamic and static interference on pressure pulsations. Since the change of the guide vane at the end of the secondary stage will change the relative flow angle, which will directly affect the performance of this stage, even if the guide vane has dynamic and static interference with the subsequent moving vane or outlet, the large change in the amplitude of pressure pulsation can also affect it.

Figure 15. Frequency domain characteristics of pressure pulsation in second and last-stage impellers with different clocking positions.

3.4. Influence of impeller and guide vane clocking on multi-stage centrifugal pump stability

When utilising frequency domain analysis to examine the pressure pulsation characteristics of different clocking positions in the previous analysis, it was observed that the strength of the main frequency component of pressure pulsations in certain regions is similar to that of the secondary frequency component or is close to the various harmonic frequencies. Using only the amplitude of the main frequency to represent the intensity of pressure pulsations does not align with the corresponding time domain characteristics of pressure pulsations. Due to the energy propagation nature of pressure pulsations, the propagation process of pressure pulsations within the multi-stage centrifugal pump can be visualised as a cyclic propagation and transformation of kinetic energy and potential energy within the flow passage.

Figure 16 presents the average energy flow density at monitoring points within the three stages of the pump for different clocking schemes. The energy flow density of pressure pulsations takes into account the amplitude variations of different frequency signals caused by impeller rotation speed and pressure pulsations. This metric enables easier analysis of energy loss regions under different operating conditions and clocking effects in a multi-stage pump. Larger energy flow density amplitudes indicate greater instability of flow, resulting in more energy losses due to pressure pulsations per unit volume and poorer operational stability. From the figure, it can be observed that the amplitude of average energy flow density under different clocking schemes follows a cyclical pattern with respect to the pump stages. It tends to increase initially and then decrease. The minimum energy flow density occurs at the impeller inlet, while the maximum amplitude is at the gap between the impellers and guide vanes. The difference in amplitude between the maximum and minimum values spans several orders of magnitude, indicating that the most severe flow instability occurs in the gaps between various stages’ impellers and guide vanes. On the other hand, flow at areas like the impeller inlet and guide vane outlet is more stable. The flow state in the gaps between impellers and guide vanes is a crucial factor in determining the operational stability of the multi-stage pump.

Figure 16. Average energy flow density of monitoring points with different clocking schemes.

Across various clocking schemes, there are no significant differences in the average energy flow density in the first stage area. As impeller and guide vane clocking positions change, distinct differences in the extreme values of average energy flow density become evident in second and last-stage areas. Configuration L16 exhibits the highest average energy flow density in the second-stage area, while configuration L12 demonstrates the highest average energy flow density in the last-stage area. Interestingly, configurations L16 and L12 share the same impeller and guide vane clocking position combination, where the impeller is rotated by 3/4 of its maximum misalignment angle, while the guide vane position remains unchanged. This suggests that the pump's operational stability is compromised when the impeller is rotated by 3/4 of its maximum misalignment angle, and the guide vane position remains constant.

Taking into account the average energy flow density in the gaps between second and last-stage impellers and guide vanes, configurations L5 and L10 perform the best. In these configurations, the maximum values of average energy flow density in second and last-stage areas are much smaller than in configuration L1. However, in configuration L10, the average energy flow density in the gaps between impellers and guide vanes is lower than at the impeller outlet or guide vane inlet. This indicates that the matching of the impeller and guide vane clocking positions is more reasonable, leading to improved suppression of local unstable flow in that region. Configuration L10 effectively enhances the operational stability of the pump. Additionally, scheme 10 increases the pump head the most and the efficiency remains almost unchanged compared to the original plan based on the aforementioned pump performance calculation results. Configuration L10 appears to be the best solution for balancing the pump performance and operational stability.

3.5. Experimental verification for 14-stage pump

In order to verify the effect of clocking effect on pump performance and operational stability, a 14-stage centrifugal pump was made out for testing. The same type of experimental test rig is illustrated to process pump performance measurements for 14 operating conditions, which implement ISO 9906:2012 standard. At the same time, the pump vibration test was also carried out. As is shown in Figure 17, the pump head is 1501.6 m with an efficiency of 53.67% at the design flow rate, which achieves energy-saving values according to Chinese standards GB/T13007-2011. Moreover, the vibration intensity of the pump is 1 mm/s, which means it is the best class A product according to the Chinese standard GB/T 29531-2013.

Figure 17. Test results of 14-stage pump.

4. Conclusion and perspective

This paper chooses a simplified three-stage centrifugal pump derived from a fourteen-stage pump as the research subject. 16 different configurations were designed, using an orthogonal method, by considering the different combinations of relative positions between moving and stationary blades among multiple stages. The steady and transient calculations are performed for each scheme under various operating conditions. The clocking effects on performance and internal flow patterns of the multi-stage centrifugal pump were investigated. The energy flux density of pressure pulsations was defined using an analogy with wave intensity. The following conclusions are drawn:

1. Most clocking schemes can improve the head of multi-stage centrifugal pumps under different operating conditions. Sequence L8 improves the most significant of the pump head with a maximum increase of 3.6% at 0.8Q_d. After comprehensive consideration, scheme L6 is the optimal scheme because both the head and efficiency have been improved compared to scheme L1 under different operating conditions.

2. Phase differences in pressure pulsation peaks were observed between rotor and guide vane stages due to time variations in their positions, disrupting waveform patterns. The rotor clocking effects had a greater influence on the rotor-guide vane interference than the guide vane clocking effects, as indicated by the matching number of peaks and close CRMS values. High-frequency pressure pulsation signals increased in amplitude near the source, with the main frequency (C_p≈0.05) much larger than that of the secondary frequency. Sequence L5 showed the lowest amplitude at second and third-stage monitoring points. Away from dynamic and static interference, the initial scheme L1 exhibited smoother pressure pulsation without temporal position change. The main frequency amplitude at monitoring points B3 and C3 was less than that of scheme L1, with the minimum being 0.029.

3. Average energy flux density analysis showed that matching rotor and guide vane clocking positions had the most impact on pressure pulsations near the rotor-guide vane stage gap. The multi-stage pump’s stability was poorest when the rotor rotated at 3/4 of its maximum offset angle with unchanged guide vane positions. Mitigating rotor-guide vane interference through appropriate clocking positions improved the operational stability of the multi-stage centrifugal pump. Sequences L10 and L5 showed good comprehensive performance, with L10 having the best rotor-guide vane clocking position match.

In this paper, it has centred on investigating the impact of clocking effects on the performance and internal flow characteristics of multistage centrifugal pumps. However, there is a lack of an evaluation system based on physical mechanism, which combines and analyzes changes in performance, pressure pulsation, energy flow density, etc., to form a closely interlocking derivation process. In future research, it should considera broader spectrum of contributing factors. For instance, exploring the implications of clocking effects on energy transfer, where variations in entropy production are anticipated across different blade and guide vane configurations. Moreover, it is noteworthy that clocking effects can extend their influence on pump vibration and noise levels. To this end, incorporating more comprehensive experimental data will be pivotal in providing a robust assessment of these aspects. Looking ahead, there is a pressing need to validate the efficacy of optimal phase difference solutions in practical engineering applications. Subsequent design refinements should be pursued with a keen focus on achieving heightened efficiency and stability requirements.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author, Peng Wang, upon reasonable request.

Additional information

Funding

This work was supported by the National Key Research and Development Program of China: [Grant Number 2022YFC3204603]; the engineering projects for significant scientific and technological achievements of Wuhu City: [Grant Number 2022zc07]; Postgraduate Research &amp; Practice Innovation Program of Jiangsu Province: [Grant Number KYCX22_3647]; State Key Laboratory of Mechanical System and Vibration: [Grant Number MSV202201].