Numerical Investigation of Double-Volute Balancing in Centrifugal Pumps

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Numerical simulations were conducted on centrifugal pumps with both single- and double-volute configurations while ensuring extended computational domains in the inlet and outlet sections for improved flow characteristics. Experimental validation was performed to validate the numerical findings and provide additional evidence of the efficiency of the technique used in the simulations. The simulations demonstrated a notable decrease in the radial hydraulic forces with the implementation of the double-volute configuration. The pressure differentials between the single- and double-volutes played a critical role in counteracting the unbalanced forces generated by the impeller. Consequently, adopting a double-volute centrifugal pump design resulted in a substantial reduction in impeller-induced forces and the forces exerted on the bearings, resulting in an approximate 50% decrease in radial forces. Centrifugal Pump Double volute Numerical simulation Radial hydraulic force Impeller-Induced Forces Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 1. INTRODUCTION Centrifugal pumps are mechanical devices designed to generate fluid flow by converting rotational energy into fluid kinetic energy [1]. They are extensively used in various industries, such as oil/gas, water treatment, power generation (nuclear and fossil), hydrocarbon processing, chemical processing, and paper production [2]. The fundamental principle of centrifugal pumps involves the transfer of mechanical energy from a motor to the fluid. This energy transfer is achieved through a rotating component called the impeller, which is enclosed within a stationary casing or volute. The fluid receives energy from the rotating impeller, which creates fluid flow and transports fluids to desired locations [3]. The impeller comprises curved blades or vanes mounted on a central shaft. The fluid is forced radially outward from the impeller’s center because of the centrifugal force produced by the rotating impeller [4]. This outward motion generates a low-pressure zone near the impeller’s center and a high-pressure zone near the outer edges. The stationary volute is an essential component that surrounds the impeller in a centrifugal pump. The primary role of the volute design is to transform the fluid discharged from the impeller, which possesses high velocity and low pressure, into a flow characterized by high pressure and low velocity [5]. The volute is designed as a spiral-shaped chamber that gradually enlarges as it extends away from the impeller. This expansion provides the necessary space for the fluid to decelerate, thereby converting its kinetic energy into pressure energy [2]. By directing the fluid flow tangentially to the pump outlet, the volute ensures smooth and efficient discharge of the fluid from the pump. This design optimizes pump performance and maximizes the conversion of mechanical energy into hydraulic energy. Centrifugal pumps have certain disadvantages in addition to their benefits. One important problem is the generation of radial hydraulic forces, which can have a negative impact on pump performance, as shown in Fig. 1. The figure illustrates the radial forces acting on the impeller in both single- and double-volute configurations. It can be observed in the literature that the single-volute pump experiences significantly higher radial forces than the double-volute configuration [6], as shown in Fig. 1. Radial hydraulic forces in centrifugal pumps are primarily caused by the rotation of the impeller, resulting in an uneven distribution of pressure around the impeller’s periphery [7]. This imbalance leads to axial thrust and radial hydraulic forces acting on the pump shaft, negatively affecting the pump’s performance and longevity. These forces cause excessive vibration, resulting in mechanical stress, bearing wear, and potential failure [8]. The double-volute balancing technique offers a potential solution for reducing these radial hydraulic forces. This technique incorporates a secondary volute in addition to the traditional single-volute configuration. The secondary volute is designed to counterbalance the radial forces acting on the impeller by redirecting the fluid flow in a way that promotes a more symmetrical distribution of forces, as shown in Fig. 1b. Implementation of this technique can enhance pump performance, minimize vibration and noise levels, and extend the lifespan of pump components. In addition to their advantages, double volute pumps have some disadvantages. One of the main reasons is their high cost [9]. Double volute designs are generally more expensive because of their more complex structure and the use of more materials. The manufacturing process and material costs contribute to the increased cost of this design. In addition, assembly and maintenance require more time and resources. Double volute designs have complicated parts and ducts, making cleaning and maintenance processes difficult. These parts can also be more challenging to access than single-volute designs. Because of their construction, they are generally manufactured in larger sizes than single volutes. This can be a disadvantage for systems that have limitations in the application space or are intended for use in tight spaces. Because double-volute designs contain more parts and channels, the risk of wear and tear can also increase. Consequently, this may lead to higher long-term maintenance costs and time requirements. Extensive research has been conducted to enhance the performance of centrifugal pumps [10–12]. The double-volute balancing technique for centrifugal pumps has recently gained considerable attention as a means of addressing imbalanced forces and enhancing pump performance. To assess the efficacy of this technique, researchers, including [13], conducted both experimental and computational tests. Their investigation revealed that the incorporation of a double-volute design in centrifugal pumps significantly reduced the unbalanced forces within the system. The impeller-generated imbalance forces were successfully counterbalanced by using the pressure variations between the double-volutes. Furthermore, simulations demonstrated that the adoption of a double-volute design resulted in improved pump operation and increased overall efficiency. Wang and Li [14] investigated the impact of different volute designs on a centrifugal aviation fuel pump, with a focus on achieving high efficiency and minimal vibration. Their research revealed that double-volute designs significantly improved the fuel pump’s large flow efficiency when compared to single-volute designs, with an increase in efficiency ranging from 20% to 30%. Alemi et al. [15] performed a numerical analysis to explore methods for reducing radial forces in centrifugal pumps by developing innovative multi-volute casing geometries. The researchers highlighted that high radial forces in pumps can lead to issues such as noise, vibration, and increased bearing load. To address this concern, they proposed a new multi-volute casing geometry that demonstrated a 30% reduction in the radial force compared with a conventional volute casing. The proposed geometry involved the incorporation of two volutes with different areas and angles. The experiment led the researchers to conclude that, across the entire range of flow rates, a concentric volute with a 270° arrangement produced the lowest radial force. Li et al. [16] conducted a study on the redesign of a centrifugal pump with a double-suction design to reduce vibration and noise. The researchers achieved this by enhancing flow homogeneity at the impeller discharge through modifications to design elements such as the number of impeller blades, radial gap, and staggered arrangement. Numerical simulations were performed to analyze the characteristics of the pump model. The results of the study demonstrated that the amplitudes of pressure fluctuations at the frequency of blade passing and their harmonics had significantly decreased, indicating lower vibration levels. In addition, more uniform pressure distributions were observed in both the impeller and volute. Dehghan and Shojaeefard [17] investigated the efficiency of the volute casing in centrifugal pumps. They conducted both experimental and numerical analyses, examining the impact of various parameters such as cross-section shape, volute throat area, design theories, cutwater angle and diameter, diffuser length and outlet, volute inlet width, base circle diameter, and linear/quadratic diffusers. The study concluded that among the evaluated designs, the circular cross-section shape had the maximum efficiency. They also found that the conservation of the angular momentum theory was the most suitable for volute design. The results emphasize the crucial role of volute design in achieving improved hydraulic efficiency and head in centrifugal pumps. Mina et al. [6] conducted a study on single-, double-, and triple-volute centrifugal pumps at several speeds. The outcomes indicated that the non-dimensional performance curves were similar for all three pumps. However, the incorporation of partition vanes and multiple volutes resulted in increased efficiency and reduced radial thrust. The study conducted by [6] demonstrated that during shut-off conditions, the two-volute configuration reduced radial thrust by 54% compared to a single-volute pump, while the three-volute configuration achieved a larger reduction of 72%. These findings highlight the effectiveness of using multiple volutes in mitigating radial thrust and improving pump performance. Shim and Kim [18] investigated the hydraulic efficiency and radial thrust force of a double-volute centrifugal pump for various volute geometries. They compared the experimentally obtained hydraulic performance and radial thrust with the validated numerical data. The findings indicated that the cross-sectional area of the volute casing, the expansion rate of the rib structure, and the diameter of the volute inlet significantly influenced both hydraulic efficiency and radial thrust. They discovered that the optimal design for hydraulic efficiency reduced the radial thrust force by 67.4%. They also revealed that the most suitable design for radial thrust decreased the average radial thrust by 75.4%. In the numerical investigation conducted by [19], the performance of the pump was validated through extensive performance tests. The findings demonstrated that the symmetrical double volute effectively mitigates radial forces, with the highest radial force and vibration velocity recorded at 0.6 Q among the tested flow rates. In addition, the frequencies associated with notable amplitudes in vibration and radial forces were primarily correlated with the blade passing frequency of the impellers and the shaft frequency. These results strongly suggest a significant association between the unsteady radial force and the radial vibration of the centrifugal pump. Boehning et al. [20] examined the hydraulic radial forces on the impeller in a centrifugal blood pump and conducted experiments with three volute types: single, double, and circular. They stated that the radial forces and efficiencies of the impeller differ for different pumps. In the literature, low specific speed pumps are generally examined to investigate single and double volute centrifugal pumps, although a few reviews also explore higher specific speed pumps. This study provides a detailed examination of the single and double volute configurations of a centrifugal pump with a specific speed of 19. The article presents a numerical investigation aimed at elucidating the mechanism behind the double-volute balancing technique and evaluating its impact on reducing radial hydraulic forces in centrifugal pumps. Through numerical simulations, the study explores the influence of the double-volute configuration on the force distribution within the pump. Additionally, experimental validation was conducted to confirm the numerical findings, utilizing available experimental data from the literature. This experimental validation offers real-world evidence of the technique's efficacy. The study also investigates various flow characteristics, such as pressure distribution, vortex structures, and velocity profiles, within the centrifugal pump. By analyzing these factors, the study not only addresses the reduction of radial forces but also provides a comprehensive understanding of flow behavior and its implications for pump performance. By reducing radial hydraulic forces and improving operational stability, the double-volute balancing technique offers the potential for enhanced pump performance and extended service life. This study contributes to the literature by providing a more detailed understanding of the efficiency of the double-volute configuration in reducing hydraulic forces and improving pump performance. 2. NUMERICAL METHOD The purpose of this study was to investigate how radial hydraulic forces affect centrifugal pump bearings. This was accomplished by comparing similar single- and double-volute centrifugal pump domains. Two computational domains were used to perform numerical simulations for the single- (Fig. 2a) and double-volute (Fig. 2b) pumps. Numerical calculations were performed using Simcenter Star CCM+. Unsteady simulations were conducted to evaluate the flow within the centrifugal pump. The flow solver used in Star CCM + for analyzing unsteady incompressible flow is based on the SIMPLE algorithm, which effectively handles the coupling between pressure and velocity. A segregated solver was employed to solve the conservation equations for momentum and continuity as a system of simultaneous equations. First-order discretization with implicit integration was applied to discretize the equations both spatially and temporally, ensuring the stability and convergence of the simulations [ 21 ]. To capture and model turbulent flow characteristics in the analysis, the k-ε turbulent method was used. This approach provides an effective means of accounting for the turbulent nature of the fluid and accurately predicting the flow behavior within the centrifugal pump domains [ 22 ]. The time step was defined as 1.10 -5 , resulting in an angular rotation of 0.174° for the impeller in each time step. This adjustment significantly enhanced the reliability of both the analysis and the transport equation within the interfaces. The continuity and momentum equations are fundamental equations used in numerical calculations of fluid flow in centrifugal pumps. These equations describe the conservation of mass and momentum, respectively, and are solved simultaneously to simulate the flow behavior. The continuity equation can be expressed as $$\frac{\partial \varvec{\rho }}{\partial \varvec{t}}+\nabla .\left(\varvec{\rho }\varvec{V}\right)=0$$ 1 where ρ is the density of the fluid and V is the velocity vector. The momentum equations, also known as the Navier–Stokes equations, describe the conservation of momentum within the fluid. It can be written in component form as follows: $$\frac{\partial \left(\rho u\right)}{\partial t}+\nabla .\left(\rho u\otimes u\right)=-\nabla p+\nabla .\tau +\rho g-{S}_{f}$$ 2 where u is the velocity, µ is the dynamic viscosity and ρ is the density of the fluid, p is pressure, t is time, g is gravity's acceleration, τ is the stress tensor (Eq. ( 3 )), F f is the fictitious force. To account for the effects of rotational motion, fictitious forces associated with Coriolis (Eq. ( 4 )) and centrifugal Eq. ( 5 )) effects are incorporated on the right-hand side of the momentum equations when analyzing motion in a rotating reference frame. $$\varvec{\tau }=\varvec{\mu }\left(\nabla \varvec{u}+{\left(\nabla \varvec{u}\right)}^{\varvec{T}}-\frac{2}{3}\left(\nabla .\varvec{u}\right)\varvec{I}\right)$$ 3 $${\varvec{S}}_{\varvec{c}\varvec{o}\varvec{r}}=2\varvec{\rho }\varvec{w}\times \varvec{u}$$ 4 $${S}_{cen}=\rho w\times \left(w\times r\right)$$ 5 In this study, within the context of computational fluid dynamics (CFD) simulations, the force acting on a surface is calculated as follows: The total force ( F ) acting on the surface is obtained by combining the pressure force F pressure and the shear force F shea r , aligned with the user-defined direction vector n f : $$F=\left({F}_{pressure}+{F}_{shear}\right)\bullet {n}_{f}$$ 6 The pressure force is calculated based on the discrepancy between the face static pressure p f and the reference pressure p ref , multiplied by the face area vector af $${F}_{pressure}=\left({p}_{f}-{p}_{ref}\right)\bullet {a}_{f}$$ 7 Similarly, the shear force F shear is determined by the stress tensor T f acting on the face f and the face area vector a f : $${F}_{shear}={T}_{f}\bullet {a}_{f}$$ 8 2.1. Computational Domain The centrifugal pump consists of four main components: the inlet duct, volute, impeller, and outlet duct, as shown in Fig. 3 . The inlet section was extended by 150 mm (inlet duct) to achieve smoother flow at the pump inlet. An interface was defined between the inlet duct and the rotating impeller. Similarly, the outlet section was extended by 200 mm to ensure smooth flow at the pump outlet. In the CFD analysis of centrifugal pumps, it is common practice to extend the inlet and outlet sections to improve flow characteristics. These extensions are necessary to ensure smooth and stable flow. The extension of the inlet section, also known as the inlet duct, helps achieve a more uniform flow entering the pump, resulting in improved flow adaptation and reduced turbulence. Similarly, extending the outlet section of the pump improves the smoothness of the flow exiting the pump. This promotes a reduction in pressure fluctuations and ensures flow stability. The extended outlet section allows the flow to decelerate gradually and minimizes energy losses. By extending the inlet and outlet sections, the accuracy of the CFD analysis is improved, yielding results that are closer to real-world conditions. These extensions also contribute to optimizing pump performance by improving flow uniformity. The following design factors were considered for the centrifugal pump analysis in this study. Flow rate ( Q ): 300 l/min, head ( H ): 25 m, rotation speed ( n ): 3000 rpm. Table 1 Geometric properties of the validated pump. Parameters Value Specific speed 19 Flowrate 300 l/min Rotation speed 3000 rpm Head 25 m Impeller suction diameter 58.0 mm Impeller discharge diameter 149.70 mm Impeller outlet width 8.68 mm Volute diameter 149.70 mm Outlet flange diameter 52.58 mm The specific speed ( n q ) of a centrifugal pump is a dimensionless parameter that provides insight into its geometric similarity and performance characteristics. It is calculated using Eq. ( 3 ). $${\varvec{n}}_{\varvec{q}}=\frac{\varvec{n}\sqrt{\varvec{Q}}}{{\left(\varvec{H}\right)}^{3/4}}$$ 3 where n q is the specific speed, n is the rotation speed of the pump (rpm), Q is the flow rate (l/min), and H is the head generated by the pump (m). 2.2. Boundary Conditions and the Mesh Domain In this study, the following boundary conditions were used to accurately model the hydraulic behavior of the pump (Fig. 4 ): Inlet Boundary: At the pump’s inlet, a stagnation inlet boundary condition was specified. This condition assumes that the flow entering the pump is in a state of stagnation, where the velocity is zero and the pressure is at its maximum. Applying the stagnation inlet boundary condition enables the simulation to capture the effect of fluid entering the pump with a high-pressure head, mimicking real operating conditions. Outlet Boundary: A mass flow outlet boundary condition was applied at the pump outlet. This condition allows the fluid to exit the computational domain. A negative value was assigned to the mass flow rate to indicate the direction of flow out of the pump. Wall Boundary: The walls of the pump, including the inlet/outlet ducts, impeller, and volute, were assigned no-slip boundary conditions. This condition assumes that the fluid velocity at the wall is zero, indicating no relative motion between the fluid and solid surfaces. Interface Boundary: Interface boundary conditions were defined at the intersection of the impeller inlet and the inlet duct outlet, as well as the impeller outlet and the volute inlet, employing a repeating interface. This approach ensures proper communication and transfer of flow properties between the two domains. The interface boundary condition allows for a seamless flow passage and prevents any artificial disturbances or reflections at the interface. In this study, a polyhedral mesh was used for the computational domains, as shown in Fig. 5 . The mesh density was carefully evaluated to ensure mesh independence during the simulation. The head of the single-volute pump was chosen as the reference variable to evaluate mesh independence. The simulated pump head demonstrated stability once the number of mesh elements reached approximately 1.0 and 1.2 million for single- and double-volute pumps, respectively. Therefore, considering the computational efficiency, meshes with 1.07 and 1.26 million cells were employed for the single- and double-volute pump simulations, respectively. Various mesh diagnostics and quality measures are available to evaluate the quality of a mesh before conducting simulations. These diagnostics assist in identifying potential issues and ensuring the accuracy and reliability of the results. One such measure is the skewness angle, which is defined as the maximum angle between the face normal and the vector connecting the face centroid to the cell centroid. A smaller skewness angle indicates better mesh quality. Cells with a skewness angle exceeding 85° are generally considered "bad" cells or have poor mesh quality. Another metric used to assess mesh quality is the minimum volume change, which should ideally be close to zero. Negative-volume cells also indicate poor mesh quality and should be avoided. In this study, the maximum boundary skewness angle is 69°, and the minimum volume change is 0.0129. The mesh is topologically valid and does not contain any negative volume cells. 2.3. Validation Study The simulated results were compared with the experimental data obtained from [ 13 ] to validate the simulations. The simulations were conducted using the same pump design and operating conditions as described in their study, as shown in Fig. 6 . The properties of the pump are given in Table 2 . The total head values were calculated using the simulations by varying the flow rates. Three different normalized flow rates were employed, corresponding to Q/Q N values of 0.75, 1.00, and 1.25, to perform validation. The focus of the validation process was to compare the resulting head ratio values ( H/H N ) obtained from the simulations. Table 2 Geometric properties of the validated pump, as described by [ 13 ]. Parameters Value Flowrate 30 m 3 /h Rotation speed 2900 rpm Head 18 m Impeller suction diameter 65 mm Impeller discharge diameter 130 mm Impeller outlet width 9.5 mm Volute diameter 140 mm Outlet flange diameter 50 mm The simulation results were subsequently compared with the corresponding experimental data, as depicted in Fig. 7 . This comparative analysis evaluated the accuracy and reliability of the simulation model in predicting pump performance. A high degree of agreement was obtained between the experimental and simulation results. The similarity in the outcomes provided compelling evidence for the accuracy and reliability of the simulation model in predicting pump performance across various flow rates. The validation process confirmed that the simulated total head values were closely aligned with the measurements obtained in the experiments. The results obtained from the simulation process conducted to validate the model are presented in Fig. 8. The percentage error rates obtained for the values of 0.75, 1, and 1.25 of Q/Q N , which are 5.56, 6.53, and 2.64, respectively, are considered acceptable within the context of the study. 3. RESULTS AND DISCUSSIONS Figures 9a and 9b present the pressure contours for the single- and double-volute pumps, respectively. These contours offer valuable insights into the pressure distribution within the pump and reveal specific characteristics. A low-pressure zone is observed near the impeller inlet, indicating a region of reduced pressure. Furthermore, negative pressure occurs at the blade inlet, suggesting a suction effect. At the outlet of the volute, the maximum pressure is typically observed. Additionally, the pressure distribution on the pressure side of the blade is generally higher than that on the suction side. This discrepancy in pressure distribution between the two surfaces is a common phenomenon in centrifugal pumps and is influenced by blade geometry and fluid flow behavior, as illustrated in Fig. 9. In a double-volute pump, the high-pressure region formed by the volute is primarily situated on the outer side of the divided section. The area with the highest pressure in the volute is located away from the impeller. As a result, the hydraulic radial force exerted on the impeller is significantly lower in the double-volute design than in the single-volute configuration. Figure 9 shows the directions in which the resulting radial force acts for both pumps. The resultant radial force acts at an angle of approximately 42° to the horizontal in the single-volute pump, while it acts at an angle of approximately 8° in the double-volute pump. Figure 10 illustrates the radial forces generated in centrifugal pumps with single- and double-volute configurations. The volute walls shown in Fig. 3 were used as the region affected by the radial force. In a single-volute centrifugal pump, the magnitudes of the forces in the x and y directions are similar. However, in a double-volute pump, the forces in the x-direction are significantly lower, whereas most forces are concentrated in the y-direction. This finding suggests that a double-volute pump achieves substantial balancing in the x direction, which corresponds to the flow outlet direction. Specifically, a single-volute setup results in a hydraulic radial force of 48.35 N, whereas a double-volute setup generates a radial force of 25.12 N. Consequently, employing a double-volute centrifugal pump reduces impeller-induced forces. Therefore, the forces exerted on the bearing are approximately 50%. The moment exerted on the impellers of both pumps is approximately the same, measuring 5.2 Nm. The simulation shows that using the double-volute configuration significantly reduces the radial hydraulic forces. The pressure differences between the single- and double-volutes, which successfully counterbalance the imbalanced forces produced by the impeller, are principally responsible for this reduction. The reduced radial hydraulic forces observed in the double-volute configuration have major effects on the efficiency of the pump, the stability of its operation, and its general dependability. The results show how the double-volute balancing method can improve these crucial facets of centrifugal pump functioning. The concept of vortex formation is a central focus in the Q criterion approach, emphasizing the impact of fluid rotation, especially the strength of vortices, within the vortex region of a centrifugal pump. This approach is instrumental in identifying and analyzing areas where vortices significantly influence flow behavior. By prioritizing the assessment of vortex strength, the Q criterion offers valuable insights into the intricate flow patterns and characteristics of centrifugal pumps [ 23 ]. In this context, Fig. 11 provides a comprehensive representation of the 3D structure of vortices in single- and double-volute centrifugal pumps using the Q criterion. Notably, in a double-volute pump, an observable vortex formation occurs at the terminus of the divided section at the diffuser outlet. However, the vortices in other regions exhibit relatively similar characteristics between the two pump configurations. The presence of uneven velocity distribution in the impeller blade passage is a primary contributor to the emergence of vortices, with one being attached to the volute and the other along the blade [ 24 ]. It is imperative to acknowledge that vortex structures significantly contribute to energy losses due to pressure fluctuations [ 16 ]. Figure 12 shows the velocity vectors in different sections of the volute. The velocity of the liquid is highest near the impeller and gradually decreases as it moves toward the outer regions of the volute. This velocity distribution indicates that the liquid flows faster in the proximity of the impeller and experiences deceleration as it progresses toward the outer sections of the volute. The flow distribution toward the exit of the double-volute pump, as shown in Fig. 12b, is not significantly altered by the division of the volute into two sections. The liquid continues to flow smoothly and evenly in both the single- and double-volute configurations. However, in the double-volute configuration, particularly at the entrance of the divided section, the velocity is slightly higher than that in the single-volute case. This variation can be attributed to the specific geometry and flow patterns in that region. Overall, the division of the volute into two sections does not significantly affect the flow distribution or the general performance of the double-volute centrifugal pump. The flow remains well distributed, and the pump operates efficiently in both configurations. Figure 13 depicts the pressure distribution within the volute section. In the case of a double-volute configuration, a noteworthy observation is that the pressure near the impeller is lower than that on the outer side of the volute, as shown in Fig. 13b. This pressure reduction near the impeller is a consequence of the liquid undergoing acceleration in this region. As the liquid is accelerated by the impeller, its kinetic energy increases while its pressure decreases in accordance with Bernoulli’s principle. This pressure reduction near the impeller has significant implications. This decreases the pressure force exerted by the fluid on the impeller in this specific region. By reducing the pressure force, the double-volute configuration helps mitigate the radial hydraulic forces acting on the impeller. The pressure distribution within the double-volute pump demonstrates the effectiveness of this configuration in reducing the radial hydraulic forces experienced by the impeller. The lower pressure near the impeller contributes to the improved operational stability and enhanced performance of the centrifugal pump. 4. CONCLUSION In this study, a numerical investigation is conducted to assess the effectiveness of the double-volute balancing technique for centrifugal pumps. Based on the results and discussions, the following conclusions can be drawn: Radial hydraulic forces in centrifugal pumps are significantly reduced when a double-volute arrangement is used. This is accomplished by using the pressure differential between the single and double volutes to balance out the impeller’s imbalanced forces. The reduction in radial forces enhances the operating stability, lowers vibration levels, and increases component longevity, all of which have a favorable effect on pump performance. The use of double-volute pump results in a remarkable decrease in the hydraulic radial force exerted on the bearings, with a reduction of approximately 50%. This reduction is achieved without adversely affecting the flow behavior or the moment exerted on the impeller. In conclusion, the numerical analysis confirms the efficacy of the double-volute balancing technique in reducing radial hydraulic forces and improving centrifugal pump performance. 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J Appl Fluid Mech 14:1133–1146 Huang S, Wei Y, Guo C, Kang W (2019) Numerical simulation and performance prediction of centrifugal pump’s full flow field based on OpenFOAM. Processes 7:605. https://doi.org/10.3390/pr7090605 Zhang YL, Li JF, Zhu ZC (2023) The acceleration effect of pump as turbine system during starting period. Sci Rep 13:. https://doi.org/10.1038/s41598-023-31899-9 Song X, Qi D, Xu L, et al (2021) Numerical simulation prediction of erosion characteristics in a double-suction centrifugal pump. Processes 9:. https://doi.org/10.3390/pr9091483 Cite Share Download PDF Status: Published Journal Publication published 22 Aug, 2024 Read the published version in Meccanica → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3539446","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":254046743,"identity":"6598aacb-f75a-4fc0-8e1e-f499f7c74a14","order_by":0,"name":"Ali Kibar","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYDACdsZmhsQGBgZ+HjDXAkQZ4NfCDNUi2QPmSkC1JODTAoSMQC0GZyBaGAhq4W9mbjZ4uMMu3/jM4YMfflRIyDCwN2+TYPxxD6cWicOMzQmJZ5Itt51tS5bsOQN0GM+xMgmGhGLc1gC1HEhsYzYwO89jxsDbBtQikWMG1ILbZfIQLfUGxv08Zox//wG1yL/Br8UA7LC2wwYGvD1mzLwNIFt48GsxBGoxSDxz3EDizLFkaZljEjxsPGnFFglpuLXIHW9/LPlzR7UBf0/ywY9vamzs+dkPb7zxwQa3FkzABiJI0TAKRsEoGAWjABMAAOoySaZpqZjtAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-2310-1088","institution":"Kocaeli University: Kocaeli Universitesi","correspondingAuthor":true,"prefix":"","firstName":"Ali","middleName":"","lastName":"Kibar","suffix":""},{"id":254046744,"identity":"ed75bd73-dab3-4d03-bbdd-293f0ef85353","order_by":1,"name":"Kadri Suleyman Yigit","email":"","orcid":"","institution":"Kocaeli University: Kocaeli Universitesi","correspondingAuthor":false,"prefix":"","firstName":"Kadri","middleName":"Suleyman","lastName":"Yigit","suffix":""}],"badges":[],"createdAt":"2023-11-02 01:26:17","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false,"coiExplicitlySet":false},"doi":"10.21203/rs.3.rs-3539446/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3539446/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11012-024-01870-7","type":"published","date":"2024-08-23T00:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":47450576,"identity":"660da295-6b41-4d8c-b5f5-065edf7335d1","added_by":"auto","created_at":"2023-12-01 17:13:09","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":43122,"visible":true,"origin":"","legend":"\u003cp\u003eHydraulic radial force distributions on the impeller for a) single- and b) double-volute centrifugal pumps.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/5f250fdad9b341e363859b5f.png"},{"id":47450578,"identity":"b1d86048-6054-4c4c-aa6d-b3089fe71821","added_by":"auto","created_at":"2023-12-01 17:13:09","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":70287,"visible":true,"origin":"","legend":"\u003cp\u003ea) Single- and b) double-volute domains.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/9796f9dfcd13ae9453933f50.png"},{"id":47452508,"identity":"3fb91199-bead-45e5-aecd-fd27b2418c33","added_by":"auto","created_at":"2023-12-01 17:21:09","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":128467,"visible":true,"origin":"","legend":"\u003cp\u003eComponents of a centrifugal pump domain.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/414de325c81a55f4fe2d0a00.png"},{"id":47450586,"identity":"29726488-31aa-4f6a-adbb-b73537ba7e5e","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":154497,"visible":true,"origin":"","legend":"\u003cp\u003eBoundary conditions and interface of the centrifugal pump.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/5bb788fe6f837bb530a6b8b2.png"},{"id":47452510,"identity":"311e96b4-52b8-4713-a4f0-7be6f33b36ba","added_by":"auto","created_at":"2023-12-01 17:21:10","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":166762,"visible":true,"origin":"","legend":"\u003cp\u003eMesh of the computational domain of the centrifugal pump.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/e6d6d53f554380222b2c5ef9.png"},{"id":47450579,"identity":"28e67286-e3de-4a89-9dd1-5c79704b3c8b","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":175818,"visible":true,"origin":"","legend":"\u003cp\u003eGeometrical and mesh domains of the study conducted by [13].\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/64f050a6c9850a6e40fae0e0.png"},{"id":47452509,"identity":"2b7325ce-6f96-4d42-8153-faae81a56df1","added_by":"auto","created_at":"2023-12-01 17:21:10","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":19069,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of numerical and experimental results [13].\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/e95ebf6bb155e7ae2485aa4c.png"},{"id":47450581,"identity":"b6f9689a-5002-418a-8e27-6455c682031d","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":112892,"visible":true,"origin":"","legend":"\u003cp\u003ePressure distribution in the plane section of the pump. a) \u003cem\u003eQ/Q\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e : 0.75, b) \u003cem\u003eQ/Q\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e : 1, c) \u003cem\u003eQ/Q\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e : 1.25.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/7d498b9a6156ae16d95c0b78.png"},{"id":47450584,"identity":"9185ca6b-7868-4945-a588-7c3d88571bea","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":102074,"visible":true,"origin":"","legend":"\u003cp\u003ePressure distributions of a) single- and b) double-volute pumps.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/c1d62454dbc93a5a38c2e293.png"},{"id":47453635,"identity":"22bb8def-1f55-46f2-80cc-9d44beac2b6d","added_by":"auto","created_at":"2023-12-01 17:29:10","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":22478,"visible":true,"origin":"","legend":"\u003cp\u003eThe radial hydraulic forces in the x and y directions.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/ad8955f71a6ec2e4ad9bf66e.png"},{"id":47450587,"identity":"0864bd21-005a-40ab-8976-9effe9933073","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":183839,"visible":true,"origin":"","legend":"\u003cp\u003eVortex formation in a) single-volute and b) double-volute centrifugal pumps.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/230a11a5850c4edd7110b882.png"},{"id":47452512,"identity":"d572e541-0999-43b9-a9da-bd9c93c894c0","added_by":"auto","created_at":"2023-12-01 17:21:10","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":149678,"visible":true,"origin":"","legend":"\u003cp\u003eVelocity vector in several sections of the volute in a) single- and b) double-volute centrifugal pumps.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/341703c925fefaba8c039ce8.png"},{"id":47450588,"identity":"d039a258-2dc0-491d-9c49-d713a21d7a61","added_by":"auto","created_at":"2023-12-01 17:13:10","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":136501,"visible":true,"origin":"","legend":"\u003cp\u003ePressure distribution in the volute for a) single- and b) double-volute centrifugal pumps.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/4771dc58f899070190873b2e.png"},{"id":63288354,"identity":"cc62a98e-b56e-4361-bda9-a0cff415d81a","added_by":"auto","created_at":"2024-08-26 13:56:25","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1758086,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3539446/v1/9a2833d6-5325-4675-923d-148e70b7b418.pdf"}],"financialInterests":"","formattedTitle":"Numerical Investigation of Double-Volute Balancing in Centrifugal Pumps","fulltext":[{"header":"1. INTRODUCTION","content":"\u003cp\u003eCentrifugal pumps are mechanical devices designed to generate fluid flow by converting rotational energy into fluid kinetic energy\u0026nbsp;[1]. They are extensively used in various industries, such as oil/gas, water treatment, power generation (nuclear and fossil), hydrocarbon processing, chemical processing, and paper production\u0026nbsp;[2]. The fundamental principle of centrifugal pumps involves the transfer of mechanical energy from a motor to the fluid. This energy transfer is achieved through a rotating component called the impeller, which is enclosed within a stationary casing or volute. The fluid receives energy from the rotating impeller, which creates fluid flow and transports fluids to desired locations\u0026nbsp;[3].\u003c/p\u003e\n\u003cp\u003eThe impeller comprises curved blades or vanes mounted on a central shaft. The fluid is forced radially outward from the impeller’s center because of the centrifugal force produced by the rotating impeller\u0026nbsp;[4]. This outward motion generates a low-pressure zone near the impeller’s center and a high-pressure zone near the outer edges. The stationary volute is an essential component that surrounds the impeller in a centrifugal pump. The primary role of the volute design is to transform the fluid discharged from the impeller, which possesses high velocity and low pressure, into a flow characterized by high pressure and low velocity\u0026nbsp;[5]. The volute is designed as a spiral-shaped chamber that gradually enlarges as it extends away from the impeller. This expansion provides the necessary space for the fluid to decelerate, thereby converting its kinetic energy into pressure energy\u0026nbsp;[2]. By directing the fluid flow tangentially to the pump outlet, the volute ensures smooth and efficient discharge of the fluid from the pump. This design optimizes pump performance and maximizes the conversion of mechanical energy into hydraulic energy.\u003c/p\u003e\n\u003cp\u003eCentrifugal pumps have certain disadvantages in addition to their benefits. One important problem is the generation of radial hydraulic forces, which can have a negative impact on pump performance, as shown in Fig. 1. The figure illustrates the radial forces acting on the impeller in both single- and double-volute configurations. It can be observed in the literature that the single-volute pump experiences significantly higher radial forces than the double-volute configuration\u0026nbsp;[6], as shown in Fig. 1.\u003c/p\u003e\n\u003cp\u003eRadial hydraulic forces in centrifugal pumps are primarily caused by the rotation of the impeller, resulting in an uneven distribution of pressure around the impeller’s periphery [7]. This imbalance leads to axial thrust and radial hydraulic forces acting on the pump shaft, negatively affecting the pump’s performance and longevity. These forces cause excessive vibration, resulting in mechanical stress, bearing wear, and potential failure [8]. The double-volute balancing technique offers a potential solution for reducing these radial hydraulic forces. This technique incorporates a secondary volute in addition to the traditional single-volute configuration. The secondary volute is designed to counterbalance the radial forces acting on the impeller by redirecting the fluid flow in a way that promotes a more symmetrical distribution of forces, as shown in Fig. 1b. Implementation of this technique can enhance pump performance, minimize vibration and noise levels, and extend the lifespan of pump components.\u003c/p\u003e\n\u003cp\u003eIn addition to their advantages, double volute pumps have some disadvantages. One of the main reasons is their high cost\u0026nbsp;[9]. Double volute designs are generally more expensive because of their more complex structure and the use of more materials. The manufacturing process and material costs contribute to the increased cost of this design. In addition, assembly and maintenance require more time and resources. Double volute designs have complicated parts and ducts, making cleaning and maintenance processes difficult. These parts can also be more challenging to access than single-volute designs. Because of their construction, they are generally manufactured in larger sizes than single volutes. This can be a disadvantage for systems that have limitations in the application space or are intended for use in tight spaces. Because double-volute designs contain more parts and channels, the risk of wear and tear can also increase. Consequently, this may lead to higher long-term maintenance costs and time requirements.\u003c/p\u003e\n\u003cp\u003eExtensive research has been conducted to enhance the performance of centrifugal pumps\u0026nbsp;[10–12]. The double-volute balancing technique for centrifugal pumps has recently gained considerable attention as a means of addressing imbalanced forces and enhancing pump performance. To assess the efficacy of this technique, researchers, including\u0026nbsp;[13], conducted both experimental and computational tests. Their investigation revealed that the incorporation of a double-volute design in centrifugal pumps significantly reduced the unbalanced forces within the system. The impeller-generated imbalance forces were successfully counterbalanced by using the pressure variations between the double-volutes. Furthermore, simulations demonstrated that the adoption of a double-volute design resulted in improved pump operation and increased overall efficiency.\u003c/p\u003e\n\u003cp\u003eWang and Li\u0026nbsp;[14]\u0026nbsp;investigated the impact of different volute designs on a centrifugal aviation fuel pump, with a focus on achieving high efficiency and minimal vibration. Their research revealed that double-volute designs significantly improved the fuel pump’s large flow efficiency when compared to single-volute designs, with an increase in efficiency ranging from 20% to 30%. Alemi et al.\u0026nbsp;[15]\u0026nbsp;performed a numerical analysis to explore methods for reducing radial forces in centrifugal pumps by developing innovative multi-volute casing geometries. The researchers highlighted that high radial forces in pumps can lead to issues such as noise, vibration, and increased bearing load. To address this concern, they proposed a new multi-volute casing geometry that demonstrated a 30% reduction in the radial force compared with a conventional volute casing. The proposed geometry involved the incorporation of two volutes with different areas and angles. The experiment led the researchers to conclude that, across the entire range of flow rates, a concentric volute with a 270° arrangement produced the lowest radial force.\u003c/p\u003e\n\u003cp\u003eLi et al.\u0026nbsp;[16]\u0026nbsp;conducted a study on the redesign of a centrifugal pump with a double-suction design to reduce vibration and noise. The researchers achieved this by enhancing flow homogeneity at the impeller discharge through modifications to design elements such as the number of impeller blades, radial gap, and staggered arrangement. Numerical simulations were performed to analyze the characteristics of the pump model. The results of the study demonstrated that the amplitudes of pressure fluctuations at the frequency of blade passing and their harmonics had significantly decreased, indicating lower vibration levels. In addition, more uniform pressure distributions were observed in both the impeller and volute.\u003c/p\u003e\n\u003cp\u003eDehghan and Shojaeefard\u0026nbsp;[17]\u0026nbsp;investigated the efficiency of the volute casing in centrifugal pumps. They conducted both experimental and numerical analyses, examining the impact of various parameters such as cross-section shape, volute throat area, design theories, cutwater angle and diameter, diffuser length and outlet, volute inlet width, base circle diameter, and linear/quadratic diffusers. The study concluded that among the evaluated designs, the circular cross-section shape had the maximum efficiency. They also found that the conservation of the angular momentum theory was the most suitable for volute design. The results emphasize the crucial role of volute design in achieving improved hydraulic efficiency and head in centrifugal pumps.\u003c/p\u003e\n\u003cp\u003eMina et al.\u0026nbsp;[6]\u0026nbsp;conducted a study on single-, double-, and triple-volute centrifugal pumps at several speeds. The outcomes indicated that the non-dimensional performance curves were similar for all three pumps. However, the incorporation of partition vanes and multiple volutes resulted in increased efficiency and reduced radial thrust. The study conducted by\u0026nbsp;[6]\u0026nbsp;demonstrated that during shut-off conditions, the two-volute configuration reduced radial thrust by 54% compared to a single-volute pump, while the three-volute configuration achieved a larger reduction of 72%. These findings highlight the effectiveness of using multiple volutes in mitigating radial thrust and improving pump performance.\u003c/p\u003e\n\u003cp\u003eShim and Kim\u0026nbsp;[18]\u0026nbsp;investigated the hydraulic efficiency and radial thrust force of a double-volute centrifugal pump for various volute geometries. They compared the experimentally obtained hydraulic performance and radial thrust with the validated numerical data. The findings indicated that the cross-sectional area of the volute casing, the expansion rate of the rib structure, and the diameter of the volute inlet significantly influenced both hydraulic efficiency and radial thrust. They discovered that the optimal design for hydraulic efficiency reduced the radial thrust force by 67.4%. They also revealed that the most suitable design for radial thrust decreased the average radial thrust by 75.4%.\u003c/p\u003e\n\u003cp\u003eIn the numerical investigation conducted by\u0026nbsp;[19], the performance of the pump was validated through extensive performance tests. The findings demonstrated that the symmetrical double volute effectively mitigates radial forces, with the highest radial force and vibration velocity recorded at 0.6 Q among the tested flow rates. In addition, the frequencies associated with notable amplitudes in vibration and radial forces were primarily correlated with the blade passing frequency of the impellers and the shaft frequency. These results strongly suggest a significant association between the unsteady radial force and the radial vibration of the centrifugal pump.\u0026nbsp;Boehning et al.\u0026nbsp;[20]\u0026nbsp;examined the hydraulic radial forces on the impeller in a centrifugal blood pump and conducted experiments with three volute types: single, double, and circular. They stated that the radial forces and efficiencies of the impeller differ for different pumps.\u003c/p\u003e\n\u003cp\u003eIn the literature, low specific speed pumps are generally examined to investigate single and double volute centrifugal pumps, although a few reviews also explore higher specific speed pumps. This study provides a detailed examination of the single and double volute configurations of a centrifugal pump with a specific speed of 19. The article presents a numerical investigation aimed at elucidating the mechanism behind the double-volute balancing technique and evaluating its impact on reducing radial hydraulic forces in centrifugal pumps. Through numerical simulations, the study explores the influence of the double-volute configuration on the force distribution within the pump. Additionally, experimental validation was conducted to confirm the numerical findings, utilizing available experimental data from the literature. This experimental validation offers real-world evidence of the technique's efficacy. The study also investigates various flow characteristics, such as pressure distribution, vortex structures, and velocity profiles, within the centrifugal pump. By analyzing these factors, the study not only addresses the reduction of radial forces but also provides a comprehensive understanding of flow behavior and its implications for pump performance. By reducing radial hydraulic forces and improving operational stability, the double-volute balancing technique offers the potential for enhanced pump performance and extended service life. This study contributes to the literature by providing a more detailed understanding of the efficiency of the double-volute configuration in reducing hydraulic forces and improving pump performance.\u003c/p\u003e"},{"header":"2. NUMERICAL METHOD","content":"\u003cp\u003eThe purpose of this study was to investigate how radial hydraulic forces affect centrifugal pump bearings. This was accomplished by comparing similar single- and double-volute centrifugal pump domains. Two computational domains were used to perform numerical simulations for the single- (Fig. 2a) and double-volute (Fig. 2b) pumps.\u003c/p\u003e\n\u003cp\u003eNumerical calculations were performed using Simcenter Star CCM+. Unsteady simulations were conducted to evaluate the flow within the centrifugal pump. The flow solver used in Star CCM\u0026thinsp;+\u0026thinsp;for analyzing unsteady incompressible flow is based on the SIMPLE algorithm, which effectively handles the coupling between pressure and velocity. A segregated solver was employed to solve the conservation equations for momentum and continuity as a system of simultaneous equations. First-order discretization with implicit integration was applied to discretize the equations both spatially and temporally, ensuring the stability and convergence of the simulations [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e]. To capture and model turbulent flow characteristics in the analysis, the \u003cem\u003ek-\u0026epsilon;\u003c/em\u003e turbulent method was used. This approach provides an effective means of accounting for the turbulent nature of the fluid and accurately predicting the flow behavior within the centrifugal pump domains [\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e]. The time step was defined as 1.10\u003csup\u003e-5\u003c/sup\u003e, resulting in an angular rotation of 0.174\u0026deg; for the impeller in each time step. This adjustment significantly enhanced the reliability of both the analysis and the transport equation within the interfaces.\u003c/p\u003e\n\u003cp\u003eThe continuity and momentum equations are fundamental equations used in numerical calculations of fluid flow in centrifugal pumps. These equations describe the conservation of mass and momentum, respectively, and are solved simultaneously to simulate the flow behavior. The continuity equation can be expressed as\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$$\\frac{\\partial \\varvec{\\rho }}{\\partial \\varvec{t}}+\\nabla .\\left(\\varvec{\\rho }\\varvec{V}\\right)=0$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026rho;\u003c/em\u003e is the density of the fluid and \u003cem\u003eV\u003c/em\u003e is the velocity vector.\u003c/p\u003e\n\u003cp\u003eThe momentum equations, also known as the Navier\u0026ndash;Stokes equations, describe the conservation of momentum within the fluid. It can be written in component form as follows:\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e$$\\frac{\\partial \\left(\\rho u\\right)}{\\partial t}+\\nabla .\\left(\\rho u\\otimes u\\right)=-\\nabla p+\\nabla .\\tau +\\rho g-{S}_{f}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eu\u003c/em\u003e is the velocity, \u003cem\u003e\u0026micro;\u003c/em\u003e is the dynamic viscosity and \u003cem\u003e\u0026rho;\u003c/em\u003e is the density of the fluid, \u003cem\u003ep\u003c/em\u003e is pressure, \u003cem\u003et\u003c/em\u003e is time, \u003cem\u003eg\u003c/em\u003e is gravity\u0026apos;s acceleration, \u0026tau; is the stress tensor (Eq. (\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e)), \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e is the fictitious force. To account for the effects of rotational motion, fictitious forces associated with Coriolis (Eq. (\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e)) and centrifugal Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e)) effects are incorporated on the right-hand side of the momentum equations when analyzing motion in a rotating reference frame.\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e$$\\varvec{\\tau }=\\varvec{\\mu }\\left(\\nabla \\varvec{u}+{\\left(\\nabla \\varvec{u}\\right)}^{\\varvec{T}}-\\frac{2}{3}\\left(\\nabla .\\varvec{u}\\right)\\varvec{I}\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e$${\\varvec{S}}_{\\varvec{c}\\varvec{o}\\varvec{r}}=2\\varvec{\\rho }\\varvec{w}\\times \\varvec{u}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e$${S}_{cen}=\\rho w\\times \\left(w\\times r\\right)$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn this study, within the context of computational fluid dynamics (CFD) simulations, the force acting on a surface is calculated as follows:\u003c/p\u003e\n\u003cp\u003eThe total force (\u003cem\u003eF\u003c/em\u003e) acting on the surface is obtained by combining the pressure force \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003epressure\u003c/em\u003e\u003c/sub\u003e and the shear force \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eshea\u003c/em\u003er\u003c/sub\u003e, aligned with the user-defined direction vector \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e$$F=\\left({F}_{pressure}+{F}_{shear}\\right)\\bullet {n}_{f}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe pressure force is calculated based on the discrepancy between the face static pressure \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e and the reference pressure \u003cem\u003ep\u003c/em\u003e\u003csub\u003e\u003cem\u003eref\u003c/em\u003e\u003c/sub\u003e, multiplied by the face area vector \u003cem\u003eaf\u003c/em\u003e\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e$${F}_{pressure}=\\left({p}_{f}-{p}_{ref}\\right)\\bullet {a}_{f}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSimilarly, the shear force \u003cem\u003eF\u003c/em\u003e\u003csub\u003e\u003cem\u003eshear\u003c/em\u003e\u003c/sub\u003e is determined by the stress tensor \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e acting on the face \u003cem\u003ef\u003c/em\u003e and the face area vector \u003cem\u003ea\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e$${F}_{shear}={T}_{f}\\bullet {a}_{f}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003e2.1. Computational Domain\u003c/h2\u003e\n \u003cp\u003eThe centrifugal pump consists of four main components: the inlet duct, volute, impeller, and outlet duct, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. The inlet section was extended by 150 mm (inlet duct) to achieve smoother flow at the pump inlet. An interface was defined between the inlet duct and the rotating impeller. Similarly, the outlet section was extended by 200 mm to ensure smooth flow at the pump outlet. In the CFD analysis of centrifugal pumps, it is common practice to extend the inlet and outlet sections to improve flow characteristics. These extensions are necessary to ensure smooth and stable flow. The extension of the inlet section, also known as the inlet duct, helps achieve a more uniform flow entering the pump, resulting in improved flow adaptation and reduced turbulence. Similarly, extending the outlet section of the pump improves the smoothness of the flow exiting the pump. This promotes a reduction in pressure fluctuations and ensures flow stability. The extended outlet section allows the flow to decelerate gradually and minimizes energy losses. By extending the inlet and outlet sections, the accuracy of the CFD analysis is improved, yielding results that are closer to real-world conditions. These extensions also contribute to optimizing pump performance by improving flow uniformity.\u003c/p\u003e\n \u003cp\u003eThe following design factors were considered for the centrifugal pump analysis in this study. Flow rate (\u003cem\u003eQ\u003c/em\u003e): 300 l/min, head (\u003cem\u003eH\u003c/em\u003e): 25 m, rotation speed (\u003cem\u003en\u003c/em\u003e): 3000 rpm.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eGeometric properties of the validated pump.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eValue\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eSpecific speed\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFlowrate\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e300 l/min\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRotation speed\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3000 rpm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eHead\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e25 m\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller suction diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e58.0 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller discharge diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e149.70 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller outlet width\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.68 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eVolute diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e149.70 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eOutlet flange diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52.58 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n \u003cp\u003eThe specific speed (\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003eq\u003c/em\u003e\u003c/sub\u003e) of a centrifugal pump is a dimensionless parameter that provides insight into its geometric similarity and performance characteristics. It is calculated using Eq.\u0026nbsp;(\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e\n \u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e$${\\varvec{n}}_{\\varvec{q}}=\\frac{\\varvec{n}\\sqrt{\\varvec{Q}}}{{\\left(\\varvec{H}\\right)}^{3/4}}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003eq\u003c/em\u003e\u003c/sub\u003e is the specific speed, \u003cem\u003en\u003c/em\u003e is the rotation speed of the pump (rpm), \u003cem\u003eQ\u003c/em\u003e is the flow rate (l/min), and \u003cem\u003eH\u003c/em\u003e is the head generated by the pump (m).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n \u003ch2\u003e2.2. Boundary Conditions and the Mesh Domain\u003c/h2\u003e\n \u003cp\u003eIn this study, the following boundary conditions were used to accurately model the hydraulic behavior of the pump (Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e):\u003c/p\u003e\n \u003cp\u003eInlet Boundary: At the pump\u0026rsquo;s inlet, a stagnation inlet boundary condition was specified. This condition assumes that the flow entering the pump is in a state of stagnation, where the velocity is zero and the pressure is at its maximum. Applying the stagnation inlet boundary condition enables the simulation to capture the effect of fluid entering the pump with a high-pressure head, mimicking real operating conditions.\u003c/p\u003e\n \u003cp\u003eOutlet Boundary: A mass flow outlet boundary condition was applied at the pump outlet. This condition allows the fluid to exit the computational domain. A negative value was assigned to the mass flow rate to indicate the direction of flow out of the pump.\u003c/p\u003e\n \u003cp\u003eWall Boundary: The walls of the pump, including the inlet/outlet ducts, impeller, and volute, were assigned no-slip boundary conditions. This condition assumes that the fluid velocity at the wall is zero, indicating no relative motion between the fluid and solid surfaces.\u003c/p\u003e\n \u003cp\u003eInterface Boundary: Interface boundary conditions were defined at the intersection of the impeller inlet and the inlet duct outlet, as well as the impeller outlet and the volute inlet, employing a repeating interface. This approach ensures proper communication and transfer of flow properties between the two domains. The interface boundary condition allows for a seamless flow passage and prevents any artificial disturbances or reflections at the interface.\u003c/p\u003e\n \u003cp\u003eIn this study, a polyhedral mesh was used for the computational domains, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. The mesh density was carefully evaluated to ensure mesh independence during the simulation. The head of the single-volute pump was chosen as the reference variable to evaluate mesh independence. The simulated pump head demonstrated stability once the number of mesh elements reached approximately 1.0 and 1.2 million for single- and double-volute pumps, respectively. Therefore, considering the computational efficiency, meshes with 1.07 and 1.26 million cells were employed for the single- and double-volute pump simulations, respectively.\u003c/p\u003e\n \u003cp\u003eVarious mesh diagnostics and quality measures are available to evaluate the quality of a mesh before conducting simulations. These diagnostics assist in identifying potential issues and ensuring the accuracy and reliability of the results. One such measure is the skewness angle, which is defined as the maximum angle between the face normal and the vector connecting the face centroid to the cell centroid. A smaller skewness angle indicates better mesh quality. Cells with a skewness angle exceeding 85\u0026deg; are generally considered \u0026quot;bad\u0026quot; cells or have poor mesh quality. Another metric used to assess mesh quality is the minimum volume change, which should ideally be close to zero. Negative-volume cells also indicate poor mesh quality and should be avoided. In this study, the maximum boundary skewness angle is 69\u0026deg;, and the minimum volume change is 0.0129. The mesh is topologically valid and does not contain any negative volume cells.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e2.3. Validation Study\u003c/h2\u003e\n \u003cp\u003eThe simulated results were compared with the experimental data obtained from [\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e] to validate the simulations. The simulations were conducted using the same pump design and operating conditions as described in their study, as shown in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. The properties of the pump are given in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. The total head values were calculated using the simulations by varying the flow rates. Three different normalized flow rates were employed, corresponding to \u003cem\u003eQ/Q\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e values of 0.75, 1.00, and 1.25, to perform validation. The focus of the validation process was to compare the resulting head ratio values (\u003cem\u003eH/H\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e) obtained from the simulations.\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eGeometric properties of the validated pump, as described by [\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e].\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"2\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eValue\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFlowrate\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30 m\u003csup\u003e3\u003c/sup\u003e/h\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRotation speed\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2900 rpm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eHead\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e18 m\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller suction diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller discharge diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e130 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eImpeller outlet width\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.5 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eVolute diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e140 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eOutlet flange diameter\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50 mm\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n \u003cp\u003eThe simulation results were subsequently compared with the corresponding experimental data, as depicted in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e. This comparative analysis evaluated the accuracy and reliability of the simulation model in predicting pump performance. A high degree of agreement was obtained between the experimental and simulation results. The similarity in the outcomes provided compelling evidence for the accuracy and reliability of the simulation model in predicting pump performance across various flow rates. The validation process confirmed that the simulated total head values were closely aligned with the measurements obtained in the experiments. The results obtained from the simulation process conducted to validate the model are presented in Fig. 8. The percentage error rates obtained for the values of 0.75, 1, and 1.25 of \u003cem\u003eQ/Q\u003c/em\u003e\u003csub\u003e\u003cem\u003eN\u003c/em\u003e\u003c/sub\u003e, which are 5.56, 6.53, and 2.64, respectively, are considered acceptable within the context of the study.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"3. RESULTS AND DISCUSSIONS","content":"\u003cp\u003eFigures 9a and 9b present the pressure contours for the single- and double-volute pumps, respectively. These contours offer valuable insights into the pressure distribution within the pump and reveal specific characteristics. A low-pressure zone is observed near the impeller inlet, indicating a region of reduced pressure. Furthermore, negative pressure occurs at the blade inlet, suggesting a suction effect. At the outlet of the volute, the maximum pressure is typically observed. Additionally, the pressure distribution on the pressure side of the blade is generally higher than that on the suction side. This discrepancy in pressure distribution between the two surfaces is a common phenomenon in centrifugal pumps and is influenced by blade geometry and fluid flow behavior, as illustrated in Fig. 9. In a double-volute pump, the high-pressure region formed by the volute is primarily situated on the outer side of the divided section. The area with the highest pressure in the volute is located away from the impeller. As a result, the hydraulic radial force exerted on the impeller is significantly lower in the double-volute design than in the single-volute configuration. Figure 9 shows the directions in which the resulting radial force acts for both pumps. The resultant radial force acts at an angle of approximately 42\u0026deg; to the horizontal in the single-volute pump, while it acts at an angle of approximately 8\u0026deg; in the double-volute pump.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e illustrates the radial forces generated in centrifugal pumps with single- and double-volute configurations. The volute walls shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e were used as the region affected by the radial force. In a single-volute centrifugal pump, the magnitudes of the forces in the x and y directions are similar. However, in a double-volute pump, the forces in the x-direction are significantly lower, whereas most forces are concentrated in the y-direction. This finding suggests that a double-volute pump achieves substantial balancing in the x direction, which corresponds to the flow outlet direction. Specifically, a single-volute setup results in a hydraulic radial force of 48.35 N, whereas a double-volute setup generates a radial force of 25.12 N. Consequently, employing a double-volute centrifugal pump reduces impeller-induced forces. Therefore, the forces exerted on the bearing are approximately 50%. The moment exerted on the impellers of both pumps is approximately the same, measuring 5.2 Nm.\u003c/p\u003e\n\u003cp\u003eThe simulation shows that using the double-volute configuration significantly reduces the radial hydraulic forces. The pressure differences between the single- and double-volutes, which successfully counterbalance the imbalanced forces produced by the impeller, are principally responsible for this reduction. The reduced radial hydraulic forces observed in the double-volute configuration have major effects on the efficiency of the pump, the stability of its operation, and its general dependability. The results show how the double-volute balancing method can improve these crucial facets of centrifugal pump functioning.\u003c/p\u003e\n\u003cp\u003eThe concept of vortex formation is a central focus in the \u003cem\u003eQ\u003c/em\u003e criterion approach, emphasizing the impact of fluid rotation, especially the strength of vortices, within the vortex region of a centrifugal pump. This approach is instrumental in identifying and analyzing areas where vortices significantly influence flow behavior. By prioritizing the assessment of vortex strength, the \u003cem\u003eQ\u003c/em\u003e criterion offers valuable insights into the intricate flow patterns and characteristics of centrifugal pumps [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e]. In this context, Fig. 11 provides a comprehensive representation of the 3D structure of vortices in single- and double-volute centrifugal pumps using the \u003cem\u003eQ\u003c/em\u003e criterion. Notably, in a double-volute pump, an observable vortex formation occurs at the terminus of the divided section at the diffuser outlet. However, the vortices in other regions exhibit relatively similar characteristics between the two pump configurations. The presence of uneven velocity distribution in the impeller blade passage is a primary contributor to the emergence of vortices, with one being attached to the volute and the other along the blade [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e]. It is imperative to acknowledge that vortex structures significantly contribute to energy losses due to pressure fluctuations [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eFigure 12 shows the velocity vectors in different sections of the volute. The velocity of the liquid is highest near the impeller and gradually decreases as it moves toward the outer regions of the volute. This velocity distribution indicates that the liquid flows faster in the proximity of the impeller and experiences deceleration as it progresses toward the outer sections of the volute. The flow distribution toward the exit of the double-volute pump, as shown in Fig. 12b, is not significantly altered by the division of the volute into two sections. The liquid continues to flow smoothly and evenly in both the single- and double-volute configurations. However, in the double-volute configuration, particularly at the entrance of the divided section, the velocity is slightly higher than that in the single-volute case. This variation can be attributed to the specific geometry and flow patterns in that region. Overall, the division of the volute into two sections does not significantly affect the flow distribution or the general performance of the double-volute centrifugal pump. The flow remains well distributed, and the pump operates efficiently in both configurations.\u003c/p\u003e\n\u003cp\u003eFigure 13 depicts the pressure distribution within the volute section. In the case of a double-volute configuration, a noteworthy observation is that the pressure near the impeller is lower than that on the outer side of the volute, as shown in Fig. 13b. This pressure reduction near the impeller is a consequence of the liquid undergoing acceleration in this region. As the liquid is accelerated by the impeller, its kinetic energy increases while its pressure decreases in accordance with Bernoulli\u0026rsquo;s principle. This pressure reduction near the impeller has significant implications. This decreases the pressure force exerted by the fluid on the impeller in this specific region. By reducing the pressure force, the double-volute configuration helps mitigate the radial hydraulic forces acting on the impeller. The pressure distribution within the double-volute pump demonstrates the effectiveness of this configuration in reducing the radial hydraulic forces experienced by the impeller. The lower pressure near the impeller contributes to the improved operational stability and enhanced performance of the centrifugal pump.\u003c/p\u003e"},{"header":"4. CONCLUSION","content":"\u003cp\u003eIn this study, a numerical investigation is conducted to assess the effectiveness of the double-volute balancing technique for centrifugal pumps. Based on the results and discussions, the following conclusions can be drawn:\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eRadial hydraulic forces in centrifugal pumps are significantly reduced when a double-volute arrangement is used. This is accomplished by using the pressure differential between the single and double volutes to balance out the impeller’s imbalanced forces.\u003c/li\u003e\n \u003cli\u003eThe reduction in radial forces enhances the operating stability, lowers vibration levels, and increases component longevity, all of which have a favorable effect on pump performance.\u003c/li\u003e\n \u003cli\u003eThe use of double-volute pump results in a remarkable decrease in the hydraulic radial force exerted on the bearings, with a reduction of approximately 50%. This reduction is achieved without adversely affecting the flow behavior or the moment exerted on the impeller.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIn conclusion, the numerical analysis confirms the efficacy of the double-volute balancing technique in reducing radial hydraulic forces and improving centrifugal pump performance. This method offers promising answers for dealing with problems caused by radial forces.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCONFLICTS OF INTEREST\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo conflict of interest was declared by the authors.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eNyi N, Ko A, Khaing CC (2019) Design and Performance Analysis of Double-Suction Centrifugal Pump. Int J Sci Res Publ 9:p92129. https://doi.org/10.29322/ijsrp.9.08.2019.p92129\u003c/li\u003e\n \u003cli\u003eSulzer Pumps (2010) Principal Features of Centrifugal Pumps for Selected Applications. Centrif Pump Handb 251\u0026ndash;283. https://doi.org/10.1016/B978-0-7506-8612-9.00009-7\u003c/li\u003e\n \u003cli\u003eKarassik IJ, Messina JP, Cooper P, et al (2008) Pump Handbook. McGraw-Hill Education\u003c/li\u003e\n \u003cli\u003eSi Q, Ali A, Yuan J, et al (2019) Flow-Induced Noises in a Centrifugal Pump: A Review. Sci Adv Mater 11:909\u0026ndash;924. https://doi.org/10.1166/SAM.2019.3617\u003c/li\u003e\n \u003cli\u003eShamsuddeen MM, Ma SB, Kim S, et al (2021) Flow Field Analysis and Feasibility Study of a Multistage Centrifugal Pump Designed for Low-Viscous Fluids. Appl Sci 2021, Vol 11, Page 1314 11:1314. https://doi.org/10.3390/APP11031314\u003c/li\u003e\n \u003cli\u003eMina EM, Abdelmessih RN, Matbouly ME (2019) Reduction of radial thrust by using triple-volute casing. Ain Shams Eng J 10:721\u0026ndash;729. https://doi.org/10.1016/J.ASEJ.2019.03.003\u003c/li\u003e\n \u003cli\u003eHao Y, Tan L, Liu Y, et al (2017) Energy performance and radial force of a mixed-flow pump with symmetrical and unsymmetrical tip clearances. Energies 10:1\u0026ndash;13. https://doi.org/10.3390/en10010057\u003c/li\u003e\n \u003cli\u003eZeng G, Li Q, Wu P, et al (2020) Investigation of the impact of splitter blades on a low specific speed pump for fluid-induced vibration. J Mech Sci Technol 34:2883\u0026ndash;2893. https://doi.org/10.1007/s12206-020-0620-7\u003c/li\u003e\n \u003cli\u003eMohammadi Z, Heidari F, Fasamanesh M, et al (2023) Centrifugal pumps. Transp Oper Food Mater within Food Factories Unit Oper Process Equip Food Ind 155\u0026ndash;200. https://doi.org/10.1016/B978-0-12-818585-8.00001-5\u003c/li\u003e\n \u003cli\u003e\u0026Ouml;zbey M, G\u0026uuml;rb\u0026uuml;z M, Karakurt U (2021) Experimental investigation of the effects of hydrophobic impeller surfaces on the centrifugal pump performance. J Fac Eng Archit Gazi Univ 36:267\u0026ndash;274. https://doi.org/10.17341/gazimmfd.551887\u003c/li\u003e\n \u003cli\u003eSakran HK, Abdul Aziz MS, Abdullah MZ, Khor CY (2022) Effects of Blade Number on the Centrifugal Pump Performance: A Review. Arab J Sci Eng 47:7945\u0026ndash;7961. https://doi.org/10.1007/s13369-021-06545-z\u003c/li\u003e\n \u003cli\u003eZhou L, Shi W, Wu S (2013) Performance optimization in a centrifugal pump impeller by orthogonal experiment and numerical simulation. Adv Mech Eng 2013:385809. https://doi.org/10.1155/2013/385809\u003c/li\u003e\n \u003cli\u003eYuan Y, Yuan S, Tang L (2019) Numerical Investigation on the Mechanism of Double-Volute Balancing Radial Hydraulic Force on the Centrifugal Pump. Process 2019, Vol 7, Page 689 7:689. https://doi.org/10.3390/PR7100689\u003c/li\u003e\n \u003cli\u003eWang W, Li Z (2021) Influence of different types of volutes on centrifugal aviation fuel pump. Adv Mech Eng 13:1\u0026ndash;13. https://doi.org/10.1177/16878140211005202\u003c/li\u003e\n \u003cli\u003eAlemi H, Nourbakhsh SA, Raisee M, Najafi AF (2015) Development of new \u0026ldquo;multivolute casing\u0026rdquo; geometries for radial force reduction in centrifugal pumps. Eng Appl Comput Fluid Mech 9:1\u0026ndash;11. https://doi.org/10.1080/19942060.2015.1004787\u003c/li\u003e\n \u003cli\u003eLi Q, Li S, Wu P, et al (2021) Investigation on Reduction of Pressure Fluctuation for a Double-Suction Centrifugal Pump. Chinese J Mech Eng (English Ed 34:. https://doi.org/10.1186/s10033-020-00505-8\u003c/li\u003e\n \u003cli\u003eDehghan AA, Shojaeefard MH (2022) Experimental and numerical optimization of a centrifugal pump volute and its effect on head and hydraulic efficiency at the best efficiency point. Proc Inst Mech Eng Part C J Mech Eng Sci 236:4577\u0026ndash;4598. https://doi.org/10.1177/09544062211056019\u003c/li\u003e\n \u003cli\u003eShim HS, Kim KY (2017) Numerical investigation on hydrodynamic characteristics of a centrifugal pump with a double volute at off-design conditions. Int J Fluid Mach Syst 10:218\u0026ndash;226. https://doi.org/10.5293/IJFMS.2017.10.3.218\u003c/li\u003e\n \u003cli\u003eCui B, Li X, Rao K, et al (2018) Analysis of unsteady radial forces of multistage centrifugal pump with double volute. Eng Comput (Swansea, Wales) 35:1500\u0026ndash;1511. https://doi.org/10.1108/EC-12-2016-0445\u003c/li\u003e\n \u003cli\u003eBoehning F, Timms DL, Amaral F, et al (2011) Evaluation of Hydraulic Radial Forces on the Impeller by the Volute in a Centrifugal Rotary Blood Pump. Artif Organs 35:818\u0026ndash;825. https://doi.org/10.1111/J.1525-1594.2011.01312.X\u003c/li\u003e\n \u003cli\u003eKorkmaz YS, Kibar A, Yiǧit KS (2021) Experimental and Numerical Investigation of Flow in Hydraulic Elbows. J Appl Fluid Mech 14:1133\u0026ndash;1146\u003c/li\u003e\n \u003cli\u003eHuang S, Wei Y, Guo C, Kang W (2019) Numerical simulation and performance prediction of centrifugal pump\u0026rsquo;s full flow field based on OpenFOAM. Processes 7:605. https://doi.org/10.3390/pr7090605\u003c/li\u003e\n \u003cli\u003eZhang YL, Li JF, Zhu ZC (2023) The acceleration effect of pump as turbine system during starting period. Sci Rep 13:. https://doi.org/10.1038/s41598-023-31899-9\u003c/li\u003e\n \u003cli\u003eSong X, Qi D, Xu L, et al (2021) Numerical simulation prediction of erosion characteristics in a double-suction centrifugal pump. Processes 9:. https://doi.org/10.3390/pr9091483\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Centrifugal Pump, Double volute, Numerical simulation, Radial hydraulic force, Impeller-Induced Forces","lastPublishedDoi":"10.21203/rs.3.rs-3539446/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3539446/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates the impact of radial hydraulic forces on centrifugal pump bearings and assesses the effectiveness of the double-volute balancing technique in mitigating these forces. Numerical simulations were conducted on centrifugal pumps with both single- and double-volute configurations while ensuring extended computational domains in the inlet and outlet sections for improved flow characteristics. Experimental validation was performed to validate the numerical findings and provide additional evidence of the efficiency of the technique used in the simulations. The simulations demonstrated a notable decrease in the radial hydraulic forces with the implementation of the double-volute configuration. The pressure differentials between the single- and double-volutes played a critical role in counteracting the unbalanced forces generated by the impeller. Consequently, adopting a double-volute centrifugal pump design resulted in a substantial reduction in impeller-induced forces and the forces exerted on the bearings, resulting in an approximate 50% decrease in radial forces.\u003c/p\u003e","manuscriptTitle":"Numerical Investigation of Double-Volute Balancing in Centrifugal Pumps","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-12-01 17:13:05","doi":"10.21203/rs.3.rs-3539446/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e58c7606-66da-412f-9af5-8e8e0f2934e8","owner":[],"postedDate":"December 1st, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-08-26T13:56:19+00:00","versionOfRecord":{"articleIdentity":"rs-3539446","link":"https://doi.org/10.1007/s11012-024-01870-7","journal":{"identity":"meccanica","isVorOnly":false,"title":"Meccanica"},"publishedOn":"2024-08-23 00:00:00","publishedOnDateReadable":"August 23rd, 2024"},"versionCreatedAt":"2023-12-01 17:13:05","video":"","vorDoi":"10.1007/s11012-024-01870-7","vorDoiUrl":"https://doi.org/10.1007/s11012-024-01870-7","workflowStages":[]},"version":"v1","identity":"rs-3539446","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3539446","identity":"rs-3539446","version":["v1"]},"buildId":"J0_U0BvcaRcwD8yVFaRlm","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}