Comparative analysis of internal flow characteristics and particle trajectories of centrifugal pumps with different impeller structures under solid-liquid two-phase flow conditions

The disc pump is a non-conventional centrifugal pump that utilizes centrifugal force and boundary layer effects to transport fluids. This pump is widely used in industries such as petrochemicals. However, due to the presence of particles in the solid-liquid two-phase flow, it causes wear and performance degradation of the pump’s flow components. However, due to the presence of particles in the solid-liquid two-phase flow, it can cause wear and performance degradation of the pump components. To study the internal flow characteristics of solid-liquid flow and the particle trajectory in a bladed disc pump, numerical simulation method was adopted to perform steady-state numerical simulation of spherical particles with different diameters (0.1 mm, 0.25 mm, 0.4 mm) in the bladed disc pump. The results indicate that compared to conventional centrifugal pumps, the performance of the bladed disc pump is lower. The variation in particle diameter has a minor influence on the intensity of the internal flow field in the bladed disc pump. The particle trajectories inside the impeller of the bladed disc pump are more chaotic, with larger diameter particles exhibiting a wider range of trajectory variations, while smaller diameter particles maintain relatively stable motion trajectories.


Introduction.
With the rapid development of industrialization, there is an increasing demand for petroleum resources, highlighting the importance of offshore oil drilling technology.In deepwater drilling projects, a pump system is required for subsea mud lifting operations.Centrifugal pumps, due to their small size and high efficiency, are typically used as power devices to provide lift for subsea mud transportation.In this process, it is necessary to ensure the stability of the centrifugal pump when conveying the two-phase flow medium of solid-liquid.However, the presence of a large number of particles in the mud can cause wear to the components of the centrifugal pump, thereby reducing its service life.In such cases, a special pump design has emerged, known as the co-rotating bladed disc pump, which is commonly used in subsea mud lifting systems [1][2][3][4][5].This pump was proposed by Nikola Tesla [6] and was first patented.The impeller structure consists of a group of smooth discs without blades (see Figure 1(a)).Max [6,9] made improvements based on Nikola Tesla's design by enlarging the spacing between the discs and adding radial straight blades on the discs, as shown in Figure 1(b) [10].This pump is also known as a bladed disc pump, which is the type of disc pump studied in this paper.Hasinger et al. [11] initially conducted an analysis of the viscous forces, internal flow field, and performance of the disc pump.The studies by Murata et al. [12][13] and Roddy et al. [14] indicate that the performance of this pump is generally inferior to that of conventional centrifugal pumps, but the internal flow field is more stable and the inter-disk streamline is more axially symmetric.Wang et al. [15] found through analysis that the main losses of the Tesla disc pump occur downstream of the impeller.Cheremushkin et al. [16] optimized the volute of the Tesla disc pump model, resulting in the optimized model achieving better efficiency than the original model.Some scholars have studied the solid-liquid two-phase flow inside the bladed disc pump from different perspectives, mainly studied the influence of particle size, particle density, and particle concentration on the wear parts, particle concentration distribution, and head efficiency inside the pump.For example, Gao et al. [2] performed numerical calculations on the dynamic characteristics of the solidliquid two-phase flow inside the pump using the DPM model and analyzed the distribution of pressure and velocity fields inside the impeller.It was found that under low particle content, the influence of particles on the distribution of the liquid phase was not significant, and the flow field distribution inside the pump had a great similarity.Li et al. [17] studied the disc pump with impeller structure improvements to determine the effect of particle concentration on pump performance.The results showed that particles were mainly concentrated on the working surface of the impeller blades, which accelerated the wear and destruction of the impeller blades, and with the increase of particle concentration, the head and efficiency of the disc pump slightly increased.Yin et al. [18,19] modified the impeller structure and conducted steady-state studies on the solid-liquid two-phase flow inside the pump using the standard k-ε model and Eulerian model.It was found that particle distribution mainly concentrated in the bladeless area of the impeller, and the modification could reduce the wear of the impeller caused by particles.Zhou et al. [20] used the Eulerian model to study the effect of particle size, particle concentration, and particle density on the disc pump.The results showed that the larger the particle density and particle size, the more difficult for particles to be accelerated by the liquid phase and stay in the bladeless area where the relative velocity of the liquid phase is slower.With the increase of particle density and particle size, the change of particle concentration distribution on the impeller surface became slower, and the particle concentration had a relatively small influence on the distribution of particles in the bladeless area and the blade area.
From the above studies, it can be observed that research on the flow characteristics and wear of solidliquid two-phase flow in bladed disc pumps has made some progress, but there are still shortcomings.Currently, the research on wear under the solid-liquid two-phase flow in bladed disc pumps mainly focuses on the impact of particle parameters (density, particle size, concentration) on particle distribution, in order to predict the wear of pump flow components.However, the study of individual particle trajectory variations is still insufficient, and the particle trajectory directly affects the wear situation inside the pump.Therefore, it is necessary to conduct relevant research to clarify the influence of particle size variations on particle trajectory inside the pump and further establish the underlying mechanisms between particle size variations and wear inside the pump.This paper presents a numerical study on the particle-fluid flow in a bladed disc pump.Using the Lagrange discrete phase model, the internal flow patterns and particle trajectories of solid-liquid twophase flow in a bladed disc pump under different diameter conditions are investigated.A comparison is made with conventional centrifugal pumps to explore the advantages of bladed disc pumps in solidliquid two-phase transportation.

Geometry and meshes.
The main research objects of this paper are the bladed disc pump and the IS centrifugal pump.The bladed disc pump is obtained by modifying the impeller component of the existing IS100-65-200 centrifugal pump.The basic parameters of the bladed disc pump and the centrifugal pump impeller are shown in Table 1, and their 3D models are shown in Figure 2.  .Impeller 3D model.Using Unigraphics NX, a 3D geometric model was built to construct the computational domain for fluid analysis.The computational domain consists of an inlet extension, an impeller, a volute, and an outlet extension.In order to minimize backflow in the inlet and outlet pipes and reduce simulation errors, the length of both the inlet and outlet extensions were extended to five times their pipe diameters, as shown in Figure 3.This study focuses on the structural characteristics of bladed disc pumps and centrifugal pumps, and proposes different mesh partition strategies to improve the accuracy and computational efficiency of numerical simulations.Due to the relatively simple structure of bladed disc pumps, a structured hexahedral mesh is employed to discretize the entire flow field.For centrifugal pumps, which have a higher degree of impeller distortion, an unstructured mesh partitioning method is employed.In addition, for areas with complex flow passages such as radial straight blades and volute tongues, local mesh refinement and wall boundary layer refinement techniques are applied to optimize the mesh quality and enhance the accuracy of the simulation results, as shown in Figure 4.Under the design conditions of the pump, simulations were conducted for the steady-state flow of clear water in both the bladed disc pump and the centrifugal pump, and predictions of pump head and efficiency were made using five different grid quantities, as shown in Figure 5. Based on the data analysis in Figure 5, it can be observed that when five different grid quantities were applied in the calculation of the bladed disc pump and centrifugal pump, the head error of the bladed disc pump was less than 2%, and the efficiency error was less than 1.5%.On the other hand, the head error of the centrifugal pump is less than 0.5% and the efficiency error is less than 0.2%.It is worth noting that regardless of whether it is a bladed disc pump or a centrifugal pump, their results are basically the same after the second calculation.Considering the balance between computational resources and computational accuracy, the third grid scheme is selected for calculation.This means that the bladed disc pump adopts a grid quantity of 3.93 million, and the centrifugal pump adopts a grid quantity of 3.76 million.

Numerical methods and boundary conditions.
Numerical simulation of the solid-liquid two-phase flow inside the pump was performed using Fluent 19.0.The numerical calculations were based on the Discrete Phase Model (DPM) to consider the interactions between the fluid flow and solid particles.In the simulation, the fluid flow was treated as the continuous phase, while the particles were treated as the discrete phase.The DPM model not only considers the interactions between the particles and the fluid, but also the collision and wear between particles and the wall.The settings for the DPM model are presented in Table 2. Additionally, clean water was selected as the continuous phase, and the Realizable k-ε turbulence model was employed for steady-state simulation.The inlet boundary condition used the average velocity, the outlet boundary condition adopted the static pressure, and the volume fraction distribution of particles at the inlet was assumed to be uniform.The standard wall function was applied for the near-wall region, and the impeller and volute were implemented with the no-slip solid wall boundary condition.The escape boundary condition was used for particles at the inlet and outlet, while the surface boundary condition inside the components was set as reflect and elastic collision.

Simulation accuracy verification and experimental analysis.
Under the clear water medium, numerical calculations were performed on the flow at the rated speed of 2900 rpm and different flow conditions (0.2Qd, 0.4Qd, 0.6Qd, 0.8Qd, 1.0Qd, 1.2Qd).The numerical simulation results were compared with experimental results, as shown in Figure 6.The comparison between the measured values and simulated values in the figure shows that as the flow rate increases, the changing trends of both are basically consistent.The maximum relative error of head for the bladed disc pump is 6.5%, and the maximum relative error of efficiency is 8.8%, while the maximum relative error of head for the centrifugal pump is 7.9%, and the maximum relative error of efficiency is 8.4%.This is due to the power loss caused by volume leakage and mechanical friction, which were not considered in the numerical calculations.In addition, due to the influence of experimental uncertainty, the experimental curve exhibits slight fluctuations, while the calculated curve remains relatively smooth.Therefore, the comparison results prove that the simulation reliability of clear water flow field is high, and can provide support for further analysis of solid-liquid two-phase flow field.From the figure, it can be further observed that the head of the bladed disc pump decreases relatively smoothly within the flow rate range of Q = 0~80 m 3 /h.When the flow rate Q is greater than 80 m 3 /h, the decreasing trend increases; while its efficiency curve rises first and then decreases, reaching the maximum value of 42.1% at around Q= 75 m 3 /h.On the other hand, the maximum efficiency of the centrifugal pump of the same size is 68.8%, corresponding to a flow rate of 102 m 3 /h.When the flow rate exceeds 115 m 3 /h, the efficiency and head curves of the centrifugal pump decrease sharply.Comparing the performance curves of the bladed disc pump and the conventional centrifugal pump shows that the performance of the bladed disc pump is far lower than that of the conventional centrifugal pump.At the design flow rate, there is an efficiency difference of 30.5%; at Q = 120 m 3 /h, the maximum efficiency difference reaches 37.2%.In terms of head, the head of the bladed disc pump is 24.8 m lower than that of the centrifugal pump at the design flow rate, less than half of the centrifugal pump.At Q = 120 m 3 /h, the maximum head difference is 25.7 m.

Streamline distribution.
At a particle volume fraction Cv = 1%, under the rated flow conditions, the influence of particle diameter on the streamlines on the pump horizontal plane is shown in Figure 7.When the particle diameter increases, the variation of particle diameter has little influence on the streamlines inside the pump.This is because the number of particles is small, the volume fraction is low, and the particle diameter is also small.The flow field is mainly dominated by the liquid phase, and the particles have strong followability, resulting in a small impact on the internal flow field and no significant local resistance.Therefore, the effect on the distribution of streamlines is small, and it exhibits a similar state to single-phase flow field.Section Acquisition: The impeller width of the studied model is 16 mm, using the backwall of the impeller as the relative position Z= 0 mm and the front shroud wall as the relative position Z= 16 mm.According to the structural parameters of the bladed disc pump impeller, when Z= 0~4, 12~16 mm, there is a bladed area; and Z= 4~12 mm is a bladeless area.The schematic diagram of the bladed disc pump impeller structure is as shown in Figure 8. Make a suitable section in the bladeless area within the bladed disc pump impeller, Z= 8 mm refers to the center section of the bladeless area of the impeller.Similarly, take the same section at position Z= 8 mm in the centrifugal pump impeller and conduct a comparative analysis of both.Under different particle diameter conditions, the distribution of mixed phase streamlines in the impeller inner center section of centrifugal and bladed disc pumps is shown in Figure 9. From the figure, it can be observed that the variation of streamline distribution in the impeller of the bladed disc pump is similar to that of the centrifugal pump, and the change in particle diameter has a relatively small impact on the streamlines, which is consistent with the analysis result in Figure 7.With the increase in particle diameter, With the increase of particle diameter, the streamline inside the impeller of centrifugal pump is relatively smooth, which indicates that there is a good internal flow condition, there is no obvious disruption in the streamline, and energy loss is small.Unlike the centrifugal pump, in the bladeless area of the bladed disc pump impeller, the distribution of fluid flow trajectories is uneven and disorderly, with vortices appearing in each flow path near the suction surface, resulting in larger energy losses.This is due to the structure of the bladed disc pump impeller: in the bladed area, the fluid movement is driven by the blades, the fluid velocity is larger and the direction is more biased towards the circumferential direction.While in the bladeless area, the viscous forces dominate, the constraint on the fluid is weaker, the fluid velocity is relatively lower and the direction differs significantly from that of the bladed area.The difference in fluid velocity and direction in the two areas results in the existence of strong convection between the two, leading to the uneven distribution of the internal streamlines.Through comparative analysis, it is found that there are similar internal flow patterns in the solidliquid two-phase flow transportation of bladed disc pumps and centrifugal pumps.However, due to the flow differences between the bladed area and the bladeless area inside the impeller of the bladed disc pump, the flow trajectory distribution of the fluid is more chaotic, and obvious vortex phenomena occur near the suction side.Therefore, compared with the centrifugal pump, the bladed disc pump has a greater energy loss, which is one of the reasons for its lower efficiency than the centrifugal pump.

Particle trajectory variations under different particle diameters.
Figure 10 shows the motion trajectory of particles with different diameters inside the impeller, at a flow rate of 1.0Qd and a particle volume fraction of Cv = 1%.
From Figure 10, it can be seen that under the same flow field conditions, particles of different diameters exhibit different trajectory trends.Compared to the trajectory of particles in the impeller of a centrifugal pump, the trajectory of particles in the impeller of a bladed disc pump is more chaotic but still shows certain regularity.Inside the bladed disc pump, particles tend to move towards the suction side of the blades after entering the impeller.The degree of this tendency, from strong to weak, follows the order of particle diameters: 0.4 mm, 0.25 mm, and 0.1 mm.As the particles move within the impeller, they gradually deviate from the suction side of the blades and turn towards the next pressure side of the blades.This is because larger particles experience a more noticeable effect from the liquid phase forces and the radial inertia of the impeller, causing them to deviate further from the suction side, resulting in the particles gradually deflecting towards the pressure side of the blades, leading to collisions with the blades and causing surface wear.Unlike the movement trajectory of particles inside the impeller of a centrifugal pump, the trajectories of particles with larger diameters inside the bladed disc pump impeller exhibit a greater range of variation.In comparison, particles with smaller diameters are less affected by the liquid phase force and the radial inertia of the impeller, maintaining a more stable movement trajectory.At the same time, the variation in trajectory length of particles inside the impeller of a bladed disc pump follows a similar pattern to that of a centrifugal pump.The larger the particle diameter, the longer the trajectory length.This is due to the increased effects of factors such as gravity and inertia, resulting in poorer particle followability during movement with the liquid phase.

Conclusions.
Experimental and CFD methods were used to compare the overall performance and internal flow characteristics of a bladed disc pump and a centrifugal pump in this paper.The influence of particle diameter on the internal flow field and trajectory of the pump was investigated, and the following conclusions were reached: (1) Compared to conventional centrifugal pumps, the performance of the bladed disc pumps is lower.Under the design flow rate, the head of the bladed disc pump is 24.8 m lower than that of the conventional centrifugal pump, and its efficiency is also 30.5% lower.
(2) Similar to the influence of particle diameter on the streamline in a centrifugal pump, as the particle diameter increases, the influence of particles on the distribution of streamlines in the bladed disc pump is not significant.Due to the smaller number of particles, lower volume fraction and small particle diameter, the flow field is mainly dominated by the liquid-phase flow, and the particles have strong following, exerting minimal influence on the internal flow field and causing no significant local resistance, thus presenting a state similar to a single-phase flow field.Compared with centrifugal pumps, due to the flow differences between the bladed area and the bladeless area inside the impeller of the bladed disc pump, the flow trajectory distribution of the fluid is more chaotic, and obvious vortex phenomena occur near the suction side.This flow state leads to significant energy loss in bladed disc pumps, which is one of the reasons for its lower efficiency compared to centrifugal pumps.
(3) Compared to the trajectory of particles within the impeller of a centrifugal pump, the trajectory of particles within the impeller of a bladed disc pump is more chaotic, but still displays a certain regularity.As the particle diameter increases, the particle's motion trajectory gradually tilts toward the pressure side of the blade.Unlike the particle motion trajectory within the impeller of a centrifugal pump, the trajectory of larger particle diameters inside the impeller of a bladed disc pump has a wider range of variation, while particles of smaller diameters maintain relatively stable motion trajectories.The variation in trajectory length of particles within the impeller of a bladed disc pump follows a similar pattern to that of a centrifugal pump.

Figure 1 .
Figure 1.Evolution of disc pump impeller.Hasinger et al.[11] initially conducted an analysis of the viscous forces, internal flow field, and performance of the disc pump.The studies by Murata et al.[12][13] and Roddy et al.[14] indicate that the performance of this pump is generally inferior to that of conventional centrifugal pumps, but the internal flow field is more stable and the inter-disk streamline is more axially symmetric.Wang et al.[15] found through analysis that the main losses of the Tesla disc pump occur downstream of the impeller.Cheremushkin et al.[16] optimized the volute of the Tesla disc pump model, resulting in the optimized model achieving better efficiency than the original model.Some scholars have studied the solid-liquid two-phase flow inside the bladed disc pump from different perspectives, mainly studied the influence of particle size, particle density, and particle concentration on the wear parts, particle concentration distribution, and head efficiency inside the pump.For example, Gao et al.[2] performed numerical calculations on the dynamic characteristics of the solidliquid two-phase flow inside the pump using the DPM model and analyzed the distribution of pressure and velocity fields inside the impeller.It was found that under low particle content, the influence of particles on the distribution of the liquid phase was not significant, and the flow field distribution inside the pump had a great similarity.Li et al.[17] studied the disc pump with impeller structure improvements to determine the effect of particle concentration on pump performance.The results showed that particles were mainly concentrated on the working surface of the impeller blades, which accelerated the wear and destruction of the impeller blades, and with the increase of particle concentration, the head and efficiency of the disc pump slightly increased.Yin et al.[18,19] modified the impeller structure and conducted steady-state studies on the solid-liquid two-phase flow inside the pump using the standard k-ε model and Eulerian model.It was found that particle distribution mainly concentrated in the bladeless area of the impeller, and the modification could reduce the wear of the impeller caused by particles.Zhou et al.[20] used the Eulerian model to study the effect of particle size, particle concentration, and particle density on the disc pump.The results showed that the larger the particle density and particle size, the more difficult for particles to be accelerated by the liquid phase and stay in the bladeless area where the relative velocity of the liquid phase is slower.With the increase of particle density and particle size, the change of particle concentration distribution on the impeller surface became slower, and the particle concentration had a relatively small influence on the distribution of particles in the bladeless area and the blade area.From the above studies, it can be observed that research on the flow characteristics and wear of solidliquid two-phase flow in bladed disc pumps has made some progress, but there are still shortcomings.Currently, the research on wear under the solid-liquid two-phase flow in bladed disc pumps mainly focuses on the impact of particle parameters (density, particle size, concentration) on particle distribution, in order to predict the wear of pump flow components.However, the study of individual particle trajectory variations is still insufficient, and the particle trajectory directly affects the wear

Figure 2
Figure 2. Impeller 3D model.Using Unigraphics NX, a 3D geometric model was built to construct the computational domain for fluid analysis.The computational domain consists of an inlet extension, an impeller, a volute, and an outlet extension.In order to minimize backflow in the inlet and outlet pipes and reduce simulation errors, the length of both the inlet and outlet extensions were extended to five times their pipe diameters, as shown in Figure3.

Figure 3 .
Figure 3.A 3D model of the computational fluid domain.This study focuses on the structural characteristics of bladed disc pumps and centrifugal pumps, and proposes different mesh partition strategies to improve the accuracy and computational efficiency of numerical simulations.Due to the relatively simple structure of bladed disc pumps, a structured hexahedral mesh is employed to discretize the entire flow field.For centrifugal pumps, which have a higher degree of impeller distortion, an unstructured mesh partitioning method is employed.In addition, for areas with complex flow passages such as radial straight blades and volute tongues, local mesh refinement and wall boundary layer refinement techniques are applied to optimize the mesh quality and enhance the accuracy of the simulation results, as shown in Figure4.

Figure 5 .
Figure 5. Mesh independency analysis.Based on the data analysis in Figure5, it can be observed that when five different grid quantities were applied in the calculation of the bladed disc pump and centrifugal pump, the head error of the bladed disc pump was less than 2%, and the efficiency error was less than 1.5%.On the other hand, the head error of the centrifugal pump is less than 0.5% and the efficiency error is less than 0.2%.It is worth noting that regardless of whether it is a bladed disc pump or a centrifugal pump, their results are basically the same after the second calculation.Considering the balance between computational resources and computational accuracy, the third grid scheme is selected for calculation.This means that the bladed disc pump adopts a grid quantity of 3.93 million, and the centrifugal pump adopts a grid quantity of 3.76 million.

( a )Figure 6 .
Figure 6.Performance curves under clear water conditions.From the figure, it can be further observed that the head of the bladed disc pump decreases relatively smoothly within the flow rate range of Q = 0~80 m 3 /h.When the flow rate Q is greater than 80 m 3 /h, the decreasing trend increases; while its efficiency curve rises first and then decreases, reaching the maximum value of 42.1% at around Q= 75 m 3 /h.On the other hand, the maximum efficiency of the centrifugal pump of the same size is 68.8%, corresponding to a flow rate of 102 m 3 /h.When the flow rate exceeds 115 m 3 /h, the efficiency and head curves of the centrifugal pump decrease sharply.Comparing the performance curves of the bladed disc pump and the conventional centrifugal pump shows that the performance of the bladed disc pump is far lower than that of the conventional centrifugal pump.At the design flow rate, there is an efficiency difference of 30.5%; at Q = 120 m 3 /h, the maximum efficiency difference reaches 37.2%.In terms of head, the head of the bladed disc pump is 24.8 m lower than that of the centrifugal pump at the design flow rate, less than half of the centrifugal pump.At Q = 120 m 3 /h, the maximum head difference is 25.7 m.

Figure 7 .
Figure 7. Streamline distribution in the horizontal plane.Section Acquisition: The impeller width of the studied model is 16 mm, using the backwall of the impeller as the relative position Z= 0 mm and the front shroud wall as the relative position Z= 16 mm.According to the structural parameters of the bladed disc pump impeller, when Z= 0~4, 12~16 mm, there is a bladed area; and Z= 4~12 mm is a bladeless area.The schematic diagram of the bladed disc pump impeller structure is as shown in Figure8.Make a suitable section in the bladeless area within the bladed disc pump impeller, Z= 8 mm refers to the center section of the bladeless area of the impeller.Similarly, take the same section at position Z= 8 mm in the centrifugal pump impeller and conduct a comparative analysis of both.

Figure 9 .
Figure 9. Streamline distribution of impeller central section.Through comparative analysis, it is found that there are similar internal flow patterns in the solidliquid two-phase flow transportation of bladed disc pumps and centrifugal pumps.However, due to the flow differences between the bladed area and the bladeless area inside the impeller of the bladed disc pump, the flow trajectory distribution of the fluid is more chaotic, and obvious vortex phenomena occur near the suction side.Therefore, compared with the centrifugal pump, the bladed disc pump has a greater energy loss, which is one of the reasons for its lower efficiency than the centrifugal pump.

Figure 10 .
Figure 10.Comparison of particle trajectories for single particles at different diameters.

Table 1
Main geometric parameters of impeller.

Table 2
Simulation settings of discrete phase model.