Particle distribution and two-phase flow characteristics of twin-screw pump during solid-liquid transfer

The twin-screw pump is a kind of positive displacement machinery with smooth pressure and flow pulsation and strong self-priming ability, which is widely used in the fields of oilfield production, marine engineering, and other fields. However, in the process of conveying liquid-phase media, it is difficult to avoid the inclusion of solid impurities and grit, resulting in wear of the pump casing and rotor, and the pump’s pressurization capacity is reduced or even shut down. At present, the solid-liquid two-phase flow characteristics within the twin-screw pump have not been clarified, which hinders the study of particle wear in the pump. In this paper, a solid-liquid two-phase flow study is carried out on a twin-screw pump through numerical simulation and experimental testing. The distribution of particles with different diameters in the rotor domain with the liquid-phase transport process is explored by simulating the two-phase flow under different differential pressures and particle concentrations. The results show that the particles in the rotor domain are mainly distributed in the region away from the male and female rotor engagement, and the number of particles near the engagement is small but the velocity is higher under the influence of the clearance jets; the elevation of the differential pressure and the particle concentration increase the risk of the particles to collide with the rotor more frequently and at higher velocity; at different particle concentrations, new localized vortices are caused by particle conveying which compared to pure liquid transport, but there are also cases where the original vortices are suppressed. The results will provide a theoretical basis for further research on particle wear characteristics in twin-screw pumps.


Introduction
The twin-screw pump is a kind of positive displacement pressurization equipment, which has the advantages of smooth flow and pressure pulsation, strong self-priming ability, high efficiency, etc., and is widely used in petroleum, water conservancy, marine, chemical industry, and other fields [1].In the fluid pressurization and transportation system, the wear problem caused by destructive particles has been the research focus of hydraulic machinery.When the fluid is mixed with unavoidable hard particles, the rotor and casing of the twin-screw pump will be subjected to different degrees of wear, resulting in performance degradation and shortened life, and even rapid shutdown [2].
In previous studies, scholars have explored the causes of wear and wear prediction of twin-screw pumps.Vetter [3] proposed the corresponding particle wear theory for the special clearance structure of screw pump in 1996, and successively carried out wear prediction, whole machine and simplified wear tests.In his later research, he found that the wear of the circumferential clearance was the most serious through the measurement of the wear rotor in the field.At the same time, he used the Rotating Clearance Tribometer (RCT) designed based on the structure of the twin-screw rotor to carry out the test and put forward the particulate wear mechanism and wear prediction mathematical model of the twin-screw pump [4].Due to the long cycle and high cost of real machine testing, Schmidt [5] also conducted a study using a rotary wear instrument with a similar structure to the tooth chamber of the twin-screw pump, and obtained the quantitative relationship between the weight of wear loss and different parameters such as particle parameter, material hardness, rotational speed, then fitted the prediction equations for the wear of the twin-screw pump, but the prediction errors were large when comparing the results with those of the field tests.Dorembos et al. [2] reported four cases of twin-screw pump failing due to particulate wear in oilfield extraction projects in SPE2001 and demonstrated a typical form of rotor subjected to wear, i.e., the high-pressure side of the rotor shows an inverted rounding at the top of the rotor teeth due to excessive wear.
However, the specific movement form and distribution characteristics of particles in the twin-screw pump are still not clear, and in the actual engineering, the use of high hardness, wear-resistant materials, or anti-wear coating treatment of the rotor has become the main measure to temporarily solve the wear problem of the twin-screw pump.When the solid-liquid two-phase conveying characteristics of twinscrew pump are clarified, the anti-wear design of twin-screw pump can be fundamentally guided, the manufacturing cost is reduced, and the conveying efficiency will be improved.
In this paper, a double-suction twin-screw pump is taken as the research object, and a solid-liquid two-phase transient numerical simulation based on the Euler-Lagrange method is carried out using the computational fluid dynamics method.The distribution characteristics of multiple particle diameters in the pump are analyzed, the factors affecting the particle distribution are studied by changing the outlet pressure of the pump and the particle concentration, and the influence of the solid-containing particle transport conditions on the liquid-phase flow field is paid attention to, so as to clarify the solid-liquid two-phase conveying characteristics of the twin-screw pump, and to provide theoretical bases for the study of the wear of the twin-screw pump.

Physical model
The test pump used for this study and its structure are shown in figure 1, where two sets of left and right screw rotors with opposite rotations and meshing with each other are present.The geometrical parameters of the single set of screw rotors are shown in table 1.  Theoretical volume/L 0.2664 Because of the symmetry of the structure of the double suction twin-screw pump, previous studies have shown that its internal flow field is characterized by left-right symmetry [6,7], and in the solidliquid transient numerical simulation in which the particles are regarded as discrete phases, the iterative calculation of a large number of particles and the storage of the data increase the simulation time, and in order to improve the efficiency of the study, the numerical simulation is carried out by constructing a single-side rotor fluid domain as in figure 2.

Mathematical model
This part discusses the solved governing equations for the solid-liquid two-phase.The incompressible viscous fluid flowing through the twin-screw pump is regarded as a continuous phase, and the continuity equation is shown in equation (1).The liquid flow in twin-screw pump is highly rotating, hence the SST k-ω turbulence model is used in this paper to better capture the flow characteristics in the clearances and near-wall surfaces inside the tooth chamber.The solid-liquid two-phase is coupled in two ways, and therefore it is necessary to add an additional force, i F , to represent the effect of the particles on the fluid flow in the momentum equation of the solved fluid, and the momentum equation is shown in the equation (2).
where u represents the liquid-phase velocity, p represents the static pressure,  represents the liquid-phase density,  represents the liquid-phase dynamic viscosity coefficient, t  represents the turbulent viscosity coefficient, and i F represents the solid-to-liquid interphase force, which is solved as an additional source term in the N-S equation.
The particles moving in a continuous medium are regarded as discrete phases, and the forces affecting the motion of the particles mainly include buoyancy caused by gravity, drag force, virtual mass force, pressure gradient force, etc., and the expression of the motion of the particles is shown in equation (3).
Where p m represents the mass of the particles, the right side of the equation is all the forces acting on particles, where B F is the buoyancy force caused by gravity, D F is the drag force acting on particles, P F is the pressure gradient force exerted on particles due to the liquid-phase acceleration that causes pressure gradient around the particles, and VM F is the virtual mass force of the particles.Due to the low content of particles in the simulated condition, the pressure gradient force, virtual mass force, and other external forces can be neglected in general [8], and in this paper, we mainly consider the effects of buoyancy and drag force on the motion of particles.The Schiller Naumann drag force model suitable for the sparse discrete phase is adopted, and the drag force formula and drag coefficient expressions are shown in equations ( 4) and (5).
Where l  represents the liquid phase density, p d represents the particle diameter, p u represents the particle velocity tensor component, l  represents the liquid phase velocity tensor component, and Re represents the particle Reynolds number.
Considering that the particles will collide and rebound with the wall in the process of moving with the fluid in the pump, the collision model proposed by Grant and Tabakoff [9] based on the collision test is adopted, and equations ( 6) and ( 7) denote the rate of change of normal and tangential momentum of the particles after collision with the wall, respectively.49 where  represents the angle of collision of the particle with the wall.

Mesh generation and independence verification
In this paper, the computational domain of the twin-screw pump is divided into rotor domain, inlet domain and outlet domain, in which the rotor domain is the main focus of this paper, with the rotation of the male and female rotors, the position of the rotor domain is constantly changing, so it is necessary to combine with the dynamic mesh to realize the transient numerical simulation.In this paper, the structured mesh of the rotor domain is drawn using SCORG, a dynamic mesh generation software for screw machinery developed based on the adaptive superlinear difference method.figure 3(a) shows the axial cross-section mesh of the screw rotor, and the body mesh of the rotor domain is drawn based on the face mesh under each corner, as shown in figure 3(b).The structured mesh of the rotor domain can ensure the quality of the axial clearance and circumferential clearance meshes and avoid excessive distortion of the mesh to affect the numerical simulation accuracy.
Figure 3. Screw rotor domain mesh (a) rotor axial section mesh (b) rotor domain mesh.SolidWorks software is used to construct the inlet domain and outlet domain based on the test pump, and the nonstructural mesh is drawn in ANSYS Mesh, as shown in figure 4, with the number of mesh in the inlet domain being 493494 and the number of mesh in the outlet domain being 249934.The mesh of the rotor domain, the inlet domain, and the outlet domain is imported into ANSYS CFX software for the cross-interface setup, and a numerical model that can realize the data transfer between domains is obtained.Since the inlet and outlet domains are not the focus of attention in this paper, the independence of the rotor domain mesh is mainly verified, and four sets of rotor domain meshes with the quantities of 3.44×10 4 、4.20×10 4 、5.12×10 4 、6.25×10 4 are plotted at a multiplicity of 1.22 for numerical simulation under the same conditions, which is shown in figure 5 that the volumetric efficiency remains almost unchanged when the number of rotor domain meshes is larger than 5.12×10 4 .Finally, the mesh scheme with 5.12×10 4 rotor domain meshes and 1.25×10 5 total meshes was selected.

Boundary conditions
The rated operating parameters of the test twin-screw pump are as follows: differential pressure is 0.8 MPa and rotational speed is 1450 r/min.This paper focuses on the solid-liquid two-phase transport under lower particle concentration conveyed by the twin-screw pump, so the calculation working conditions shown in table 2 are developed.Used for numerical simulation of the twin-screw pump inlet boundary condition is pressure inlet, outlet boundary condition is pressure outlet, the male and female rotor wall and the casing are set to no-slip wall.In ANSYS CFX, the data interface is established by using the Fortran language program to realize the calling of the dynamic mesh and the periodic rotation of the rotor.
Table 2. Twin-screw pump solid-liquid two-phase calculation working condition settings.

Physical quantity Value
Outlet pressure (P out ) /MPa 0.4, 0.6, 0.8 Particle concentration (C v ) /% 1%, 3%, 6% Particle diameter ratio (D p ) /mm 0.03(30%), 0.09(40%), 0.12(30%) For the settings of particle, the particle diameter is formulated based on the dimensions of the circumferential clearance (0.1 mm) and the radial clearance (0.1 mm) of the twin-screw pump, and the main focus is on particles of three sizes, much smaller than the clearance, approximately equal to the clearance, and larger than the clearance.The particles are injected at a fixed mass flow rate at the inlet position of the pump, and the release velocity is aligned with the inlet velocity of the liquid phase.Twoway coupling is used between the solid-liquid phases.

Experimental verification
In order to verify the rationality of the numerical model, the performance test platform of the twin-screw pump was designed and constructed, as shown in figure 6.Based on the calculation conditions adopted in the numerical simulation, the rotational speed of the screw pump was adjusted to 1450 r/min and the inlet pressure was set to 0 MPa under the pure liquid condition, and the tests were carried out with the outlet pressures of 0.2 MPa, 0.4 MPa, 0.6 MPa, 0.8 MPa, and 1.0 MPa respectively by adjusting the motorized valves of the outlet pipeline, so that the volumetric and hydraulic efficiencies obtained by the As can be seen from figure 7, the trends of the simulated volumetric and hydraulic efficiencies with outlet pressure are consistent with the experimental results, and the maximum errors of volumetric and hydraulic efficiencies appear in the condition where the outlet pressure is 1.0 MPa, which are 2.53% and 4.78%, respectively.Because it is more difficult to monitor the particle state in the twin-screw pump, to verify the reasonableness of the solid-liquid two-phase numerical model, the experimental model of the horizontal circular pipe conveying particles flow in the literature [10] is used as a validation object, and the solidliquid two-phase numerical method of the twin-screw pump established in this paper is utilized to carry out a solid-liquid two-phase conveying numerical simulation for the horizontal circular pipe in the literature, and the numerical simulation model of the circular pipe is presented in figure 8(a).The monitoring passes through the particle velocities at eight positions in figure 8(b), and the monitoring values are compared with the experimental measurements for validation.The validation results are shown in figure 9, where y/R indicates the relative radial position of the monitoring point in the circular tube, and v/V is the ratio of the particle velocity to the liquid-phase velocity.It can be seen that the particle velocities calculated by the solid-liquid two-phase numerical model established in this paper are consistent with the overall trend of the experimental values, with a maximum absolute error of 0.104 m/s and a relative error of 9.6%.In summary, the solid-liquid two-phase simulation values of the twin-screw pump are more consistent with the test results, and the overall error is within a reasonable range, so the numerical model can be used for further research.Figure 10 shows the distribution of three particle diameters injected into the pump by the twin-screw pump under the design condition (P out = 0.8MPa, Rev = 1450rpm), at different concentrations.The number of particles in the rotor domain increases with the concentration, but the distribution of particles in the male and female rotor domains are not completely uniform, mainly because, in the double-suction twin-screw pump used in this paper, the position of the male rotor is closer to the inlet end of the inlet pump compared with that of the female rotor, which makes the particles entering into the pump more likely to be sucked in by the male rotor firstly, which leads to the imbalance of the distribution of particles in the male and female rotor domains.In order to obtain the distribution characteristics of particles in the rotor domain, figure 11 shows the particle velocity of three diameters along the rotor axial direction when P out = 0.8MPa and C v = 3%, the velocity trend of particles of different diameters is similar in all levels of the tooth chamber, and the number of particles distributed in the area near the engagement of the male and female rotors is much less than that far away from the engagement, and the area near the engagement is affected by the high speed of the flank and radial clearance.The region near the engagement is affected by the high-speed jet flow from the clearance, and the liquid-phase flow is more turbulent, which leads to the higher velocity of particles near the region.Comparing the distributions for D p = 0.03mm and D p = 0.15 mm, it can be found that the number of small-sized particles in the rotor domain is more than that of largesized particles when different particle diameters are injected with the same occupancy ratio.In order to obtain the radial distribution characteristics of the particles in the rotor domain, figure 12 shows the velocity distributions of the three particle diameters along the radial direction of the rotor at P out = 0.8MPa and C v = 3%.In addition to the high-speed particles in the region near the rotor engagement, the velocity of the particles near the outside of the tooth chamber is larger than that of the particles near the main shaft on the whole, which is because the outside of the tooth chamber is susceptible to the influence of the high-speed reflux of the circumferential clearance, resulting in the high velocity of the particles close to the outside while the particles close to the main shaft follow the undisturbed, low-speed main flow; it can be seen in D p = 0.15mm that, under the effect of the centrifugal force, the larger particles have the characteristic and tendency to aggregate towards the outside of the tooth chamber, while the smaller particles show a more uniform suspension.

Characterization of particle distribution in the rotor domain
In D p = 0.03mm and D p = 0.09mm of figure 12, the behavior of particles entering the flank clearance and circumferential clearance is highlighted, and the velocity of particles entering the clearance is much higher than the other particles, which is consistent with the characteristics of high-velocity liquid phase leakage in the clearance, whereas the particles with D p = 0.15mm do not show the behavior of entering the circumferential clearance and radial clearance because their diameters are larger than the dimensions of above two clearances.Particles with D p = 0.15mm still have the risk of entering the flank clearance because the size of that is larger than the other two clearances.

Analysis of factors affecting particle distribution
This section focuses on the effects of differential pressure and particle concentration on particle distribution, with attention to the specific distribution characteristics of particles of various sizes under different influencing factors.Figure 13 shows the velocity distribution of the tooth chamber cross-section under three kinds of differential pressure, and it can be seen from figure 13(b) that the velocity inside the clearance of the twin-screw pump rises with the increase of the differential pressure, and the maximum liquid-phase velocity inside the clearance reaches 29.29m/s at P out = 0.8MPa, appearing in the position of the radial clearance, and the maximum liquid-phase velocity inside the radial clearance is 19.79m/s at P out = 0.4MPa.The rise in clearance leakage flow velocity also causes disturbance to the main flow in the tooth chamber, as seen in figure 13(a), the liquid-phase flow rate at most locations in the tooth chamber increases as a whole with increasing differential pressure, thus also leading to an increase in the velocity of the particles in the pump.
Figure 14.Liquid phase velocity compared to particle velocity in the male rotor domain at different differential pressures (a) liquid velocity of sections in male chamber under different outlet pressure (b) particle velocity in male chamber under different outlet pressure.Figure 14 compares the liquid and particle velocity in the tooth chamber of the male rotor domain at different differential pressures of C v = 3%, and the particle velocity inside the tooth chamber maintains a high degree of consistency with the liquid velocity, which is the lowest at P out = 0.4 MPa.Therefore, it is clear that the rise in differential pressure increases the velocity of the particles moving inside the tooth chamber, and the result is that the particles exist in a more turbulent form inside the tooth chamber, and at the same time, the particles gain higher kinetic energy from the liquid phase to hit the rotor wall, which increases the risk of wear.From figure 14, it can be observed that the velocity of the particles gradually increases to the upper and lower sides at a low speed from near the center axis, which is due to the fact that the upper and lower ends are closer to the rotor engagement region, and the motion of the particles is more high-speed and turbulent.The P out = 0.8MPa condition with the highest particle velocity is selected to analyze the distribution of particles in the rotor domain by the law of particle concentration, figure 15 shows the velocity distribution of all particle sizes in the rotor domain under different concentrations, to avoid the analysis being affected by the display of the particles, all the particle sizes in the figure are shown with the same size.With the increase in particle concentration, the particles in the rotor domain change from a dispersed suspension state to a concentrated distribution with localized agglomerations.The velocity of the particles does not change significantly with concentration, and the particles near the engagement of the male and female rotors show higher velocity than the other regions at different concentrations, especially at C v = 6%, the velocity range of the particles near the engagement is wider, and it can be observed through the transient results that the particles are more likely to enter the flank clearance and follow the liquid-phase reflux in high concentrations, which increases the chances of the high-speed collision between the particles and the rotor.

Analysis of particle effects on liquid-phase flow
Figure 16.Velocity distribution of cross-section in the tooth chamber at different differential pressures (a) P out = 0.4MPa (b) P out = 0.6MPa (c) P out = 0.8MPa.The solid-phase condition of C v = 6% was selected to analyze the liquid-phase transport in the twinscrew pump affected by particles under different differential pressures.figure 16 compares the liquidphase velocity distribution in the cross-section inside the tooth chamber at C v = 0% and C v = 6%.The presence of particles has a small effect on the liquid-phase flow inside the tooth chamber, which still presents high liquid velocity inside the clearance under all operating conditions; the presence of particles has a small inhibitory effect on the localized high velocity induced by clearance jets, but it causes perturbations in the low-velocity flow away from the engagement region, which leads to a small or localized elevation of liquid velocity.The P out = 0.4MPa condition, in which the flow field is more obviously affected by particles, is selected to analyze the degree of influence of different particle concentrations on the liquid-phase flow.figure 17 compares the velocity and streamline distribution of the cross section in the tooth chamber of the male and female rotors under different concentrations, and the liquid phase velocity in the tooth chamber is not significantly affected by the increase in particle concentration.It can be seen from the streamline that the flow of the liquid phase is affected by two types of influences when transporting different particle concentrations, the red-boxed position indicates the local vortex phenomenon induced by the particles, and the black-boxed position is the local vortex in the pure liquid condition which is suppressed when transporting the particles, and the vortex suppression and aggravation phenomena induced by the particles mainly occurs in the region around the rotor engagement, and no linear correlation between this phenomenon and the particle concentration was found from the results.

Conclusion
In this paper, transient numerical simulations based on Euler-Lagrange were carried out for the conveying of solid-liquid two-phase flow by twin-screw pump to analyze the characteristics of the particle distribution and its influence on the two-phase flow under different working conditions and particle parameters, the specific conclusions are as follows: (1) Analyzed the overall distribution characteristics of particles in the rotor domain of the twin-screw pump.With the increase of particle concentration, the number of particles remaining in the rotor domain increases, compared with other regions, the distribution of particles distributed near the engagement of male and female rotor shows a smaller number of particles but higher speed; particles smaller than the clearance are more likely to enter the sealing clearance, and flow with the high-speed reflux; under the action of the centrifugal force, the larger the size of the particles, the more obvious the tendency of the aggregation of particles to the outer side of the rotor domain.
(2) The effect of differential pressure and particle concentration on the particle distribution is discussed.As the differential pressure increases, the rate of clearance leakage in the pump rises, which intensifies the effect on the liquid-phase flow in the tooth chamber, resulting in a consequent increase in the velocity of particles in the tooth chamber; the change in concentration does not have a significant effect on the velocity distribution of the particles in the rotor domain, but at C v =6%, the velocity range of the particles is wider and the particles are more likely to follow the reflux into the clearance.Taken together, the increase in both differential pressure and particle concentration increases the risk of frequent and high-velocity particle-wall impacts.
(3) Characteristics of two-phase flow at different pressure differentials and the effect of particle concentration on liquid-phase flow are analyzed.Solid-liquid two-phase transportation has a greater impact on small differential pressure conditions, which is due to the lower clearance leakage flow velocity and the presence of a higher risk of low-velocity aggregation of particles in the tooth chamber; changes in particle concentration do not have a significant effect on the liquid-phase velocity, but the presence of particles interferes with the flow of the liquid, and there are mainly two opposite effects characterized by the exacerbation of local vortices and the suppression of the original vortices.

Figure 1 .
Figure 1.Schematic diagram of (a) prototype for test and (b) structure of twin-screw pump.

Figure 2 .
Figure 2. Calculation domain for single-side twin screw pump.

Figure 4 .
Figure 4. Mesh of inlet and outlet domain.Since the inlet and outlet domains are not the focus of attention in this paper, the independence of the rotor domain mesh is mainly verified, and four sets of rotor domain meshes with the quantities of 3.44×10 4 、4.20×10 4 、5.12×10 4 、6.25×10 4 are plotted at a multiplicity of 1.22 for numerical simulation under the same conditions, which is shown in figure5that the volumetric efficiency remains almost unchanged when the number of rotor domain meshes is larger than 5.12×104 .Finally, the mesh scheme with 5.12×10 4 rotor domain meshes and 1.25×10 5 total meshes was selected.

Figure 6 .
Figure 6.Twin-screw pump test bench.As can be seen from figure7, the trends of the simulated volumetric and hydraulic efficiencies with outlet pressure are consistent with the experimental results, and the maximum errors of volumetric and hydraulic efficiencies appear in the condition where the outlet pressure is 1.0 MPa, which are 2.53% and 4.78%, respectively.

Figure 7 .
Figure 7.Comparison of numerical calculation and pump external characteristic test results.

Figure 8 .
Figure 8. Schematic of numerical model for two-phase test (a) model of simulated pipe (b) monitoring points of pipe.Because it is more difficult to monitor the particle state in the twin-screw pump, to verify the reasonableness of the solid-liquid two-phase numerical model, the experimental model of the horizontal circular pipe conveying particles flow in the literature[10] is used as a validation object, and the solidliquid two-phase numerical method of the twin-screw pump established in this paper is utilized to carry out a solid-liquid two-phase conveying numerical simulation for the horizontal circular pipe in the literature, and the numerical simulation model of the circular pipe is presented in figure8(a).The monitoring passes through the particle velocities at eight positions in figure8(b), and the monitoring values are compared with the experimental measurements for validation.The validation results are shown in figure9, where y/R indicates the relative radial position of the monitoring point in the circular tube, and v/V is the ratio of the particle velocity to the liquid-phase velocity.It can be seen that the particle velocities calculated by the solid-liquid two-phase numerical model established in this paper are consistent with the overall trend of the experimental values, with a maximum absolute error of 0.104 m/s and a relative error of 9.6%.

Figure 9 .
Figure 9.Comparison of solid-liquid two-phase numerical simulation and experimental results.

Figure 10 .
Figure 10.The overall distribution of particles in the rotor domain at different concentrations.Figure10shows the distribution of three particle diameters injected into the pump by the twin-screw pump under the design condition (P out = 0.8MPa, Rev = 1450rpm), at different concentrations.The number of particles in the rotor domain increases with the concentration, but the distribution of particles in the male and female rotor domains are not completely uniform, mainly because, in the double-suction twin-screw pump used in this paper, the position of the male rotor is closer to the inlet end of the inlet pump compared with that of the female rotor, which makes the particles entering into the pump more likely to be sucked in by the male rotor firstly, which leads to the imbalance of the distribution of particles in the male and female rotor domains.

Figure 11 .
Figure 11.Velocity of particles with different diameters in the rotor domain.In order to obtain the distribution characteristics of particles in the rotor domain, figure11shows the particle velocity of three diameters along the rotor axial direction when P out = 0.8MPa and C v = 3%, the velocity trend of particles of different diameters is similar in all levels of the tooth chamber, and the number of particles distributed in the area near the engagement of the male and female rotors is much less than that far away from the engagement, and the area near the engagement is affected by the high speed of the flank and radial clearance.The region near the engagement is affected by the high-speed jet flow from the clearance, and the liquid-phase flow is more turbulent, which leads to the higher velocity of particles near the region.Comparing the distributions for D p = 0.03mm and D p = 0.15 mm,

Figure 12 .
Figure 12.Velocity of particles with different diameters in the radial direction of the rotor.In order to obtain the radial distribution characteristics of the particles in the rotor domain, figure12shows the velocity distributions of the three particle diameters along the radial direction of the rotor at P out = 0.8MPa and C v = 3%.In addition to the high-speed particles in the region near the rotor engagement, the velocity of the particles near the outside of the tooth chamber is larger than that of the particles near the main shaft on the whole, which is because the outside of the tooth chamber is susceptible to the influence of the high-speed reflux of the circumferential clearance, resulting in the high velocity of the particles close to the outside while the particles close to the main shaft follow the undisturbed, low-speed main flow; it can be seen in D p = 0.15mm that, under the effect of the centrifugal force, the larger particles have the characteristic and tendency to aggregate towards the outside of the tooth chamber, while the smaller particles show a more uniform suspension.In D p = 0.03mm and D p = 0.09mm of figure12, the behavior of particles entering the flank clearance and circumferential clearance is highlighted, and the velocity of particles entering the clearance is much higher than the other particles, which is consistent with the characteristics of high-velocity liquid phase leakage in the clearance, whereas the particles with D p = 0.15mm do not show the behavior of entering the circumferential clearance and radial clearance because their diameters are larger than the dimensions of above two clearances.Particles with D p = 0.15mm still have the risk of entering the flank clearance because the size of that is larger than the other two clearances.

Figure 13 .
Figure 13.Liquid-phase velocity in the cross-section of the tooth chamber at different outlet pressure (a) liquid velocity of sections in chamber under different outlet pressure (b) liquid velocity in radial clearance and flank clearance under different outlet pressure.Figure13shows the velocity distribution of the tooth chamber cross-section under three kinds of differential pressure, and it can be seen from figure13(b) that the velocity inside the clearance of the twin-screw pump rises with the increase of the differential pressure, and the maximum liquid-phase velocity inside the clearance reaches 29.29m/s at P out = 0.8MPa, appearing in the position of the radial clearance, and the maximum liquid-phase velocity inside the radial clearance is 19.79m/s at P out = 0.4MPa.The rise in clearance leakage flow velocity also causes disturbance to the main flow in the tooth chamber, as seen in figure13(a), the liquid-phase flow rate at most locations in the tooth chamber increases as a whole with increasing differential pressure, thus also leading to an increase in the velocity of the particles in the pump.

Figure 15 .
Figure 15.Velocity distribution in the rotor domain for all particle diameters at different concentrations.From figure14, it can be observed that the velocity of the particles gradually increases to the upper and lower sides at a low speed from near the center axis, which is due to the fact that the upper and lower ends are closer to the rotor engagement region, and the motion of the particles is more high-speed and turbulent.The P out = 0.8MPa condition with the highest particle velocity is selected to analyze the distribution of particles in the rotor domain by the law of particle concentration, figure15shows the velocity distribution of all particle sizes in the rotor domain under different concentrations, to avoid the analysis being affected by the display of the particles, all the particle sizes in the figure are shown with the same size.With the increase in particle concentration, the particles in the rotor domain change from a dispersed suspension state to a concentrated distribution with localized agglomerations.The velocity of the particles does not change significantly with concentration, and the particles near the engagement of the male and female rotors show higher velocity than the other regions at different concentrations, especially at C v = 6%, the velocity range of the particles near the engagement is wider, and it can be observed through the transient results that the particles are more likely to enter the flank clearance and follow the liquid-phase reflux in high concentrations, which increases the chances of the high-speed collision between the particles and the rotor.

Figure 17 .
Figure 17.Cross-sectional velocity distribution and streamlines in the tooth chamber for different particle concentrations.The P out = 0.4MPa condition, in which the flow field is more obviously affected by particles, is selected to analyze the degree of influence of different particle concentrations on the liquid-phase flow.figure17compares the velocity and streamline distribution of the cross section in the tooth chamber of the male and female rotors under different concentrations, and the liquid phase velocity in the tooth

Table 1 .
Main geometrical parameters of the test pump.