CFD Analysis of Counter-Rotating Impeller Performance in Mixed-Flow Pumps

Abstract

This study presents a computational fluid dynamics (CFD) analysis of the performance of counter-rotating impeller systems in mixed-flow pumps. The analysis evaluates the impact of varying rotor velocity ratios and blade geometry on head rise, efficiency, and hydraulic losses. Through detailed CFD simulations, the counter-rotating system demonstrates significant improvements in head and efficiency at low flow rates, with model B achieving up to 20% higher efficiency near the best efficiency point (BEP). However, increased hydraulic losses offset efficiency gains at higher flow rates. While the findings provide valuable insights for optimizing the design of counter-rotating systems in mixed-flow pumps, experimental validation is needed to confirm the results and ensure real-world applicability. The study lays the groundwork for future work in refining counter-rotating pump designs to minimize hydraulic losses and slippage.

1. Introduction

The field of turbomachinery has a rich and significant history, evolving over the past two centuries with notable advancements occurring in the last seven decades. The concept of centrifugal impellers, first envisioned by Leonardo Da Vinci in the 16th century, laid the foundation for modern turbomachinery by utilizing centrifugal force for fluid movement. A significant breakthrough occurred in 1689 when Denis Papin developed a forced vortex system using blades within a circular or spiral housing. In 1754, Euler’s theoretical work applying Newton’s laws to centrifugal impellers, commonly known as Euler’s law, provided a mathematical basis for further developments in fluid transport mechanisms. These innovations led to extensive research on centrifugal impellers from 1910 to 1930, enhancing theoretical and experimental understanding of flow mechanisms. [1,2].

In recent decades, the field of turbomachinery has expanded significantly, driven by the development of computational tools that enhance performance. Turbomachines play a critical role in various industries, including oil and gas (O&G), petrochemical, and renewable energy sectors, where they serve as vital components in compressors, turbines, and pumps [3,4]. Centrifugal pumps, in particular, dominate industrial applications due to their efficiency, low capital (CAPEX) and operating costs (OPEX), and versatility across a wide range of operational conditions [5]. The global centrifugal pump market is on a rapid growth trajectory, with the U.S. market projected to reach USD 8.6 billion by 2020 and China’s market expected to hit USD 7.6 billion by 2027. This promising growth trend underscores the increasing demand for and potential opportunities in centrifugal pumps [6].

A 2021 report from Cognitive Market Research forecasts substantial growth in the global mixed-flow impeller pumps market, with projections showing a revenue increase of 130% in 2015 to 530% by 2027, driven mainly by demand in South America. This growth is expected to be particularly notable in the water treatment, chemical, and metallurgical industries, which are projected to grow by at least 35% by 2027 [7,8,9].

This study builds upon previous research on counter-rotating systems primarily focused on axial turbomachinery. However, the adaptation of such systems to centrifugal pumps presents unique challenges. Previous studies have utilized experimental and computational methods to examine counter-rotating systems in various turbomachines, including axial pumps and fans.

Table 1. Experimental and CFD studies for counter-rotative systems in turbomachines.

Study	System	CFD Code/Software	Physics Modeling/Methodology	Mesh Configuration	Ref.
Kanemoto (2004)	Counter-rotative axial pump	[-]	Experimental	[-]	[15]
Momosaki et al. (2010)	Counter-rotative axial pump	ANSYS CFX-12.0	[-]	1535180 Nodes	[16]
Cao et al. (2014)	Counter-rotative axial pump	[-]	Experimental	[-]	[17]
Tosin et al. (2015)	Counter-rotative–mixed-flow pump	ANSYS CFX	RANS and Rayleigh-Plesset |multiphase model	12 × 106 Nodes	[18]
Tosin et al. (2015)	Counter-rotative–mixed-flow pump	ANSYS CFX	RANS SST turbulence model	2.1 million Nodes	[19]
Friebe et al. (2018)	Counter-rotative axial fan	CFturbo	RANS	[-]	[20]
Nguyen (2020)	Counter-rotative centrifugal compressor	STAR-CCM+	RANS	3.4–11 million cells (polyhedral)	[21]

summarizes fundamental experimental and computational fluid dynamics (CFD)-based studies on counter-rotating systems across different turbomachines. These studies highlight the potential benefits of counter-rotating designs, including improved efficiency, increased pressure capacity, and enhanced flow dynamics [1,10,11,12]. The benefits of counter-rotating centrifugal pumps include improved efficiency, reduced size and weight, and the ability to explore new designs through CFD at relatively low costs [13,14].

This study explores the design and performance of counter-rotating centrifugal pumps by conducting preliminary calculations of velocity triangles under typical crude extraction conditions. The pump’s geometry will be designed with blade variations and rotor-speed adjustments, and the performance of the counter-rotating system will be compared with traditional pumps through dimensional analysis and CFD simulations. This comparison will provide critical insights into the potential applications of counter-rotating systems in industries such as oil and gas, petrochemicals, and renewable energy.

While counter-rotating systems have been explored extensively in axial turbomachinery, their application in mixed-flow pumps presents distinct challenges and opportunities that must be thoroughly addressed in the existing literature. Unlike axial systems, where counter-rotating designs primarily focus on enhancing axial flow efficiency, mixed-flow pumps require a more complex interplay of radial and axial components, leading to different flow dynamics and performance considerations. The complexity of mixed-flow systems, including the interaction between centrifugal and axial forces, necessitates a tailored approach to counter-rotating designs, as traditional methods used in axial systems may not be directly transferable. This study fills a critical gap by investigating the specific impact of counter-rotating systems in mixed-flow pumps through a combination of CFD simulations and performance analysis. By optimizing the rotor speed, blade geometry, and flow dynamics, this research extends the state of the art in counter-rotating turbomachinery, offering valuable insights for industries that rely on high-performance mixed-flow pumps. This study’s results can improve pump efficiency and performance in applications such as oil and gas, petrochemical, and renewable energy, thereby contributing to advancements in industrial pump design.

2. Application Experiments

2.1. Experimental Data

This study uses the BCS GN5200 centrifugal pump model, as detailed in Monteverde’s research [22]. The pump operates at various rotational speeds, with the numerical model validation focusing on data at 3500 rpm. At this speed, the pump achieves a Best Efficiency Point (BEP) of 36.5 m³/h and a head of 7.78 m per stage, with a specific speed of 1.32. Figure 1 displays the performance curves of the BCS GN5200 operating at 3500 rpm, as provided by the manufacturer.

The impeller and stator geometrical characteristics for each stage are listed in

Table 2. Principal dimensions of the GN5200 impeller and diffuser.

Parameter	Units	Impeller Valor	Diffuser Valor
N° of blades	[-]	7	7
Inlet inner diameter	[mm]	30.2	79.9
Inlet external diameter	[mm]	65.3	93.3
Outlet inner diameter	[mm]	79.9	30.2
Outlet external diameter	[mm]	93.3	65.3

, showing key dimensions that were used in the simulations. These parameters are critical for understanding the pump’s geometry and performance characteristics:

• Number of Blades: The impeller and diffuser each have seven blades. The number of blades is critical in determining the pump’s flow patterns and efficiency.
• Inlet Inner Diameter: The impeller’s inlet inner diameter is 30.2 mm, while the diffuser’s is larger, at 79.9 mm. This variation helps manage the fluid’s transition from the impeller to the diffuser, affecting the fluid velocity and pressure distribution.
• Inlet External Diameter: The impeller’s inlet external diameter is 65.3 mm, while the diffuser’s is 93.3 mm. This difference reflects the change in the flow passage area, which influences the pump’s ability to handle different flow rates.
• Outlet Inner Diameter: The impeller’s outlet inner diameter is 79.9 mm, whereas the diffuser’s is 30.2 mm. This decrease in the diffuser’s outlet diameter helps to maintain pressure as the fluid moves out of the pump.
• Outlet External Diameter: The external diameter at the impeller’s outlet is 93.3 mm, while the diffuser’s is 65.3 mm. This difference contributes to the energy conversion as the fluid moves from the impeller to the diffuser.

Monteverde’s study involved three different working fluids—water, a glycerin–water solution, and pure glycerin—covering a wide range of viscosities and temperatures [22,23]. For this study, five specific flow rates were chosen from the pump’s operating curves: 2.21, 4.55, 6.68, 8.89, 10.89, and 12.55 L/s. Additional input data for model convergence included an inlet pressure of 1 × 10⁸ Pa, assuming negligible temperature variations and constant density and viscosity for each fluid.

2.2. Geometry and Numerical Modeling

This study employed numerical simulations and geometry modeling to assess the performance of the BCS GN5200 pump. These two processes are distinct but interconnected components of the overall analysis.

2.2.1. Geometry Modeling

Three models were developed for the study: a base impeller model and two counter-rotating impeller models. The geometry was created using Autodesk Inventor 2021, which includes the impeller, stator (each with seven vanes), and pump casing.

• Base Impeller Model: This model represents a standard impeller and stator configuration, consisting of an impeller with seven vanes and a stator with an equal number of vanes. The entire assembly, including the pump casing, was modeled for analysis.
• Counter-Rotative Models: Two counter-rotating designs were introduced, each varying in the point at which the first rotor transitions to the second rotor. These designs are based on the methodology proposed by Nguyen, illustrated in Figure 2, and mathematically expressed in Equation (1).

  $$LR={\displaystyle \frac{L_{1st}}{L_m}}$$

  (1)

  where $LR$ is the ratio of the top length of the rotor shroud $\left(L_{1st}\right)$ to the meridional plane contour length of the base impeller ($L_m$). Two specific cases were analyzed:

• Model A: LR = 0.65
• Model B: LR = 0.32

This formulation defines how the two rotors are geometrically split in the counter-rotating designs, with Figure 3 providing a visual representation.

Figure 2. Geometrical representation of contour length for LR relation.

Figure 2. Geometrical representation of contour length for LR relation.

2.2.2. Numerical Modeling

Numerical simulations were carried out using STAR-CCM+ v19 (Siemens, Germany), which utilizes CFD techniques to evaluate the pump’s performance. These simulations focused on fluid flow and mechanical interactions within the pump under various operating conditions.

Boundary conditions were applied at the pump’s inlet and outlet regions, with flow rates and pressure conditions based on experimental data [22]. The numerical analysis included unsteady simulations to capture transient flow characteristics, and the Rigid Body Motion (RBM) model was used to simulate the pump’s moving parts accurately.

Key performance metrics such as head, efficiency, power, and torque were calculated and analyzed, emphasizing the counter-rotating models’ behavior.

The computational domain for the BCS GN5200 pump is shown in Figure 4. Three regions and boundary conditions were considered:

• The pump inlet was modeled as a mass flow inlet, where the inlet velocity was calculated using the equation $Q={\mathrm{v}}_i\cdot a^{\ast }$, with $a$ being the cross-sectional area at the inner rotor inlet.
• The pump outlet was treated as a pressure outlet, with pressure values taken from Monteverde [22]. Standard atmospheric pressure (1 atm) was used as a reference.
• The stator zone was static, and its walls were assigned a no-slip condition.

All performance data presented in the results were derived from detailed CFD simulations, with no experimental data used. Simulations were conducted at varying rotor speeds, with the “one rotor” configuration representing the internal rotor operating at 3500 RPM while the external rotor remained stationary. In counter-rotating configurations, the internal and external rotors were run at speeds between 3500 and 4375 RPM. Discrete simulation points were collected at specific flow rates, and continuous lines in the performance graphs were interpolated to highlight general trends.

The impeller was modeled using unsteady Rigid Body Motion (RBM) to accurately simulate moving components and capture transient flow effects, which are critical for studying rotating parts. The Courant–Friedrichs–Lewy (CFL) number was evaluated at a flow rate of 6.68 L/s, representing a mid-range point on the pump’s performance curve. The internal and external rotors were simulated at 3500 rpm, confirming timestep independence under these conditions. Figure 5 shows the Courant number distribution, ranging from 1.32 × 10⁻⁵ and $10$.

A sensitivity analysis was performed by halving and doubling the timestep size relative to the initial value to ensure timestep independence. Head rise, efficiency, and pressure distributions were compared across these simulations to verify solution convergence. The results varied by less than 1%, confirming timestep independence. While the CFL number guided timestep selection, these additional tests validated the robustness of the solution and ensured accurate capture of transient flow effects unaffected by timestep variations.

Performance metrics, including head, efficiency, power, and torque, were calculated. Torque was measured using a momentum report based on impeller rotation in the Z and -Z directions. Efficiency was determined by summing torque contributions, assuming a single shaft and mechanical power input [18,19].

2.2.3. Physical Model Specification

The flow was treated as turbulent due to the high rotational velocity of the impellers.

Table 3. Re values for the two cases.

Impeller diameter (D) (m)	0.093
$\mathrm{Density} \left(\rho \right)$ (kg/m3)	1000
$\mathrm{Viscosity} \left(\mu \right)$ (Pa s)	0.003
Rotation rate (rpm)	3500	4375
Re	1.17 × 107	1.46 × 107
Regime	Turbulent	Turbulent

presents the Reynolds numbers calculated using Equation (2), indicating the turbulent nature of the flow for both models.

$$Re={\displaystyle \frac{\rho N{D}^2}{\mu }}$$

(2)

The properties and values are shown in Table 3.

Water was used as the working fluid for all simulations. A single-phase, unsteady RANS (U-RANS) approach was employed, utilizing the SST κ-ω model turbulence model to solve for eddy viscosity due to its superior performance in boundary layer predictions under adverse pressure gradients.

2.2.4. Mesh Generation

The pump geometry was discretized using a polyhedral mesh for central regions and prismatic layers for the surfaces, ensuring adequate mesh resolution. A base size of 14 mm was selected, with a total prism layer thickness of 0.1 mm. Figure 6 presents the y⁺ distribution for the base impeller (a), Model A (b), and Model B (c), all evaluated with the same flow rate and Reynolds number. The variations in y⁺ values are attributed to localized differences in flow behavior near the wall boundaries, particularly in flow separation and turbulence regions. Although the overall Reynolds numbers and mesh configurations were consistent, the geometry of the impellers and rotor speed differences led to distinct flow interactions, resulting in variations in y⁺. The mesh was designed to ensure y⁺ < 1, with six prism layers near the walls and a stretching ratio of 1.2 to maintain acceptable aspect ratios.

Mesh Independence Analysis

Mesh independence was tested by varying the cell count by ±30%. The water head’s mean square error (MSE) was calculated to evaluate mesh quality using (Equation (3)).

$$\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l} \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d} \mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{{\sum }_i^N{\left(\frac{H_{Ex{p}_i}-H_{CF{D}_i}}{H_{Ex{p}_i}}\right)}^2}{N}}}$$

(3)

Results indicated that the base mesh provided the best compromise between accuracy and computational efficiency, as shown in Figure 7.

A Grid Convergence Index (GCI) was computed to validate further mesh independence using refinement conditions, apparent order, and error estimations derived from Equations (4)–(8).

$$p_0={\displaystyle \frac{1}{\ln \left(r_{21}\right)}}\left|\ln \left|{\displaystyle \frac{{\mathsf{\epsilon }}_{32}}{{\epsilon }_{21}}}\right|+q(p)\right|$$

(4)

$$q\left(p_0\right)=\ln \left({\displaystyle \frac{r_{21}^{p_0}-S}{r_{32}^{p_0}-S}}\right)$$

(5)

$$S=1·sgn\left({\displaystyle \frac{{\epsilon }_{32}}{{\epsilon }_{21}}}\right)$$

(6)

$$e_a^{21}=\left|{\displaystyle \frac{{\phi }_2-{\phi }_1}{{\phi }_1}}\right|$$

(7)

$$\mathrm{G}\mathrm{C}{\mathrm{I}}_{fine}^{21}={\displaystyle \frac{1.25e_a^{21}}{r_{21}^{p_0}-1}}$$

(8)

The extrapolated curve for the water head (Equations (9) and (10)) confirmed that the base mesh was a good solution, with a maximum deviation of 3%, as shown in Figure 8.

$${\phi }_{ext}^{21}={\displaystyle \frac{r_{21}^{p_0}{\phi }_1-{\phi }_2}{r_{21}^{p_0}-1}}$$

(9)

$$e_{ext}^{21}=\left|{\displaystyle \frac{{\phi }_{ext}^{21}-{\phi }_1}{{\phi }_{ext}^{21}}}\right|$$

(10)

The 30% variation in cell count provided a reasonable compromise between mesh resolution and simulation time. Although a more significant variation could yield a more detailed analysis, increasing the cell count beyond 30% would significantly extend the simulation time and computational resources required. The results showed that key performance metrics—head, efficiency, and slip factor—remained consistent across different mesh sizes, with deviations of less than 1%, indicating that further refinement would minimally affect accuracy while substantially increasing computational costs

3. Results and Discussion

This section presents the performance curve results for the BCS GN5200 pump modified with a counter-rotating blade system. The analysis was conducted at six flow rates: 2.21, 4.55, 6.68, 8.89, 10.89, and 12.55 L/s. Two models (A and B) with different internal diameter ratios were examined, and three velocity ratios between the internal and external rotors were analyzed: 3500:3500 rpm, 3500:4375 rpm, and 4375:3500 rpm.

3.1. CFD Model Validation

Figure 9 compares simulated and experimental values for the base model with an unmodified impeller to validate the CFD model. The results show close agreement, with deviations in total head and efficiency within ±10%, except for one efficiency point. Experimental data from Monteverde [22] served as a benchmark, confirming the accuracy of the CFD simulations in terms of total head rise and efficiency across various flow rates. The deviations are within an acceptable range, demonstrating the robustness of the CFD model.

3.2. Contra-Rotative Design Model Performance

The performance improvements of the counter-rotating impeller system were quantified through CFD simulations using standard pump performance parameters such as total head, efficiency, and slip factor. The quantification of these results involved the following steps:

• Head Rise ($H$):
  The total head rise was calculated by measuring the difference between the pressure at the pump inlet and outlet, normalized by the fluid density and gravitational acceleration, as per this Equation:

  $$H={\displaystyle \frac{(P_{outlet}-P_{inlet})}{\rho g}}$$

  (11)

  where $P_{outlet}$ and $P_{inlet}$ are the pressure values at the outlet and inlet, $\rho$ is the fluid density, and $g$ is the gravitational acceleration. The head values were then plotted as a function of flow rate and compared across different rotor velocity ratios.
• Efficiency ($\eta$):
  Efficiency was calculated as the ratio of hydraulic power to mechanical input power. The hydraulic power was determined from the head rise and flow rate, while the mechanical power was obtained by measuring the torque applied to the rotors:

  $$\eta ={\displaystyle \frac{\rho gQH}{T\omega }}$$

  (12)

  where $Q$ is the flow rate, $T$ is the torque applied to the impellers, and $\omega$ is the angular velocity. The CFD simulations provided the torque and flow rate data necessary for these calculations, allowing us to track efficiency improvements as a function of rotor speed and flow rate.
• Slip Factor:
  The slip factor quantifies the actual flow angle deviation from the impeller exit’s ideal flow angle. It was calculated based on the velocity triangle formed by the relative and absolute velocities at the impeller outlet, with the slip increasing as the flow rate rises. The slip factor was measured for each rotor configuration to determine how closely the flow followed the idealized design path.

Figure 10 illustrates the total head results for the BCS GN5200 pump with counter-rotating impeller models A and B across various velocity ratios. At flow rates above 10 L/s, models A3535 and B3535 closely match the baseline model in head values but show steeper head rise slopes, highlighting the performance advantages of the counter-rotating design. Configurations with higher velocity ratios, such as A3543 and B4335, demonstrate superior performance, achieving a head increase of up to 7 m at lower flow rates and maintaining a 5 m advantage near the baseline model’s BEP.

The head increase observed in models A3543 and B3543 is due to the optimal rotor velocity ratios, where higher external rotor speeds enable more efficient energy transfer at low flow rates. In Figure 10 and Figure 11, linear interpolation has been applied to the discrete CFD data points to illustrate general trends in total head and efficiency across various flow rates and rotor configurations. This interpolation is solely for visualization purposes and does not generate new data. The original CFD data points are also shown for reference.

At higher flow rates, however, these efficiency gains diminish due to increased hydraulic losses, particularly in model A, where turbulence and recirculation become more pronounced. The higher slip factor in model A4335 further contributes to the decline in efficiency at elevated flow rates.

A detailed convergence analysis was performed to ensure the accuracy of the results for the A3543 and A4335 configurations at a flow rate of 10.5 L/s. While the initial residuals for continuity, momentum, and turbulence equations fell below the convergence threshold of 10⁻⁴, oscillations in the pressure and momentum equations suggested potential unsteady flow behavior. A timestep independence test revealed sensitivity in the A4335 case at this specific flow rate. By refining the timestep size, the head rise and efficiency results aligned more closely with surrounding data points. Additionally, a mesh independence test confirmed that the original mesh resolution was sufficient, with variations within 1%. The refined results for these cases are now consistent with overall trends and are reflected in the current figures.

The analysis shows that configurations with longer blades enhance performance, especially at higher rotational speeds. Model B, in particular, outperforms Model A at lower flow rates but experiences more significant head degradation as flow rates increase. The shift in head performance inversion points suggests that specific operational settings can optimize or reduce pump efficiency depending on the velocity ratio.

Comparing these results with previous studies, such as Kanemoto [15] and Cao [17], reveals similar efficiency improvements in counter-rotating axial systems. However, this study extends those findings to mixed-flow pumps, where the interaction of radial and axial forces presents additional challenges. With optimized rotor velocity ratios, mixed-flow pumps can achieve efficiency gains comparable to or greater than axial systems, though careful design is necessary to manage the complexity.

Figure 11 shows that models B3543 and B3535 exhibit up to 20% higher efficiency near the BEP than the base model. This improvement is due to model B’s optimized velocity distribution and longer blades, which enhance energy transfer, especially at lower flow rates. Conversely, models with higher inlet rotor velocities (A4335 and B4335) show less efficiency gain due to increased turbulence and hydraulic losses, leading to flow instabilities and reduced energy transfer efficiency at higher flow rates.

3.3. Slip and Pump Losses

In models with higher rotational speeds, the slip factor increase was more pronounced, leading to greater energy dissipation and reduced overall efficiency. Model B, with its optimized blade length and rotor velocity ratio, exhibited a more gradual increase in slip factor, indicating better flow alignment. Hydraulic losses were highest at low flow rates due to recirculation but decreased as the flow rate approached the pump’s BEP.

Figure 12 shows the theoretical slip factor as a function of flow rate for various models and velocity ratios. As expected, the slip factor increases with the flow rate in all models. Models A3535 and A4335 exhibit higher slip factors associated with lower efficiency due to poor flow alignment at higher flow rates. In contrast, model B3543 demonstrates the lowest slip factor across most flow rates, indicating better performance and reduced energy loss. This lower slip factor corresponds to the superior efficiency observed in model B3543, particularly at lower flow rates.

Hydraulic losses were analyzed by observing the relative velocity patterns in models A and B. Figure 13 and Figure 14 illustrate that recirculation losses are more prominent at low flow rates (e.g., 2.21 L/s), particularly in model A. In contrast, model B exhibits better flow control with fewer recirculation zones. As the flow rate increases, turbulence becomes the dominant source of hydraulic losses, especially at higher flow rates (e.g., 10.89 L/s). Model B consistently shows smoother velocity distributions, reducing turbulence and improving overall efficiency, particularly in the 3500:4375 rpm and 4375:3500 rpm velocity ratios. These results confirm that model B’s blade configuration and velocity ratios are more optimized for reducing hydraulic losses, leading to better performance than model A.

The increased slip factor in model A, especially in the A4335 configuration, results from the higher inlet rotor speed and shorter blades. Higher rotational velocity exacerbates flow instability, leading to more significant slippage. These hydraulic losses, including recirculation and turbulence, are more critical at low flow rates, contributing to overall energy inefficiency.

Momosaki observed similar slip-related inefficiencies in axial systems [16]. Our findings extend this understanding to mixed-flow pumps, where the interaction between radial and axial forces amplifies flow misalignment and slippage. Managing these forces in counter-rotating mixed-flow pumps presents unique challenges, underscoring the need for optimized designs to minimize slip and associated losses.

3.4. Head Analysis Based on Euler Theory

Figure 15 illustrates the incidence losses at the rotor outlets for different flow rates and models. The beta angle (β2), which represents the flow’s exit angle relative to the rotor’s tangential direction, was observed by comparing the actual flow direction to the designed flow angle. Significant β2 deviations were seen at higher velocities, particularly in models A3535 and B3535 at lower flow rates, where the misalignment led to increased slippage and reduced pump performance. Model B3535 consistently showed better flow alignment, particularly at high flow rates, with less deviation from the designed β2, aligning more closely with theoretical predictions and improving pump performance.

Figure 16 and Figure 17 compare the theoretical Euler head, slip head, and CFD data for both models at different velocity ratios. Model B only achieved a higher theoretical head when operating with the internal rotor due to its more efficient design and reduced slippage. However, with both rotors in operation, Model A’s performance improved, as the second rotor compensated for the initial slippage, bringing its performance closer to Model B. The CFD data closely matched the slip head predictions at lower flow rates but deviated at higher flow rates due to increased energy losses. These results suggest that optimizing rotor design and velocity ratios can significantly enhance pump performance in counter-rotating systems.

Significant β2 deviations were observed at higher velocities in models A4335 and B4335, leading to increased slippage and reduced efficiency, particularly at low flow rates. As flow rates increased, β2 values aligned more consistently with theoretical predictions, especially in Model B.

Nguyen reported similar β2 angle deviations [21] in counter-rotating centrifugal compressors, where misalignment was a vital issue. In mixed-flow pumps, radial forces exacerbate these deviations, emphasizing the need for optimized rotor designs to maintain flow alignment and minimize energy losses.

4. Conclusions

This study used CFD analysis to explore counter-rotating systems’ performance in a mixed-flow pump. Two impeller models (A and B) with different internal diameter ratios and operating at three velocity ratios were assessed for their head rise, efficiency, slip factor, and incidence losses. The CFD model was validated with experimental data, confirming its accuracy in representing the pump’s hydraulic performance.

Key findings from this study include:

• Head and Efficiency Improvements: The introduction of counter-rotating rotors significantly enhanced the total head at low flow rates. Model B, operating at a velocity ratio of 3500:4375 rpm, demonstrated the highest head gains, particularly at low flows, achieving values up to 17 m at the BEP. However, this increase in head did not consistently translate into higher efficiency. Model B exhibited the best efficiency around the BEP, with up to 75% efficiency, whereas Model A showed diminished performance due to increased slippage at higher flow rates.
• Slip and Hydraulic Losses: The counter-rotating configuration introduced additional slip at both rotors, particularly in Model A, which showed a steeper increase in slip factor as the flow rate increased. Model B’s configuration managed slip more effectively, showing a more gradual rise in slip factor and better overall hydraulic performance. Hydraulic losses, such as recirculation and turbulence, were more significant at low flow rates but diminished at higher flows, aligning with theoretical expectations based on Euler’s pump theory.
• Incidence Losses and β Angle Deviations: Significant β angle deviations were observed in models A4335 and B4335 at higher velocities, contributing to increased slippage and reduced pump efficiency. Model B3535 exhibited the slightest deviation, with performance more closely aligned with theoretical predictions. The analysis highlighted the importance of controlling incidence losses to optimize pump performance, especially at higher operational velocities.

This study offers valuable insights into the design and operation of counter-rotating systems in mixed-flow pumps. The results demonstrate that optimizing head and efficiency through appropriate rotor speed and geometry adjustments is achievable. Future research should focus on refining rotor designs to minimize slippage and further enhance the efficiency of counter-rotating systems in centrifugal and mixed-flow pumps, expanding their industrial applications.

While this study primarily addresses the steady-state performance of counter-rotating impeller systems, time-dependent analyses could provide additional insights into the transient behavior and the time required for the flow to reach a stable state. However, conducting such analyses would require new transient simulations, which fall outside the scope of this work. Future studies will aim to incorporate transient simulations to explore the time evolution of the flow field in greater detail.