Effect of Internal Vortex-Finder on Classiﬁcation Performance for Double Vortex-Finder Hydrocyclone

: The double vortex-ﬁnder hydrocyclone formed by a coaxial insertion of an internal vortex-ﬁnder with a smaller diameter inside the conventional single vortex-ﬁnder used to obtain two kinds of products from the internal and external overﬂows in one classiﬁcation has attracted wide attention. To further improve the classiﬁcation performance of the hydrocyclone, the effects of the internal vortex-ﬁnder diameter and length on the classiﬁcation performance were studied by numerical simulation and response surface modeling with the behavior of ﬂuid and particle motion in the double vortex-ﬁnder hydrocyclone as the research object. The results showed that the split ratio and pressure drop of internal and external overﬂow increased with the diameter of the internal vortex-ﬁnder. The classiﬁcation performance was optimal when the diameter ratio of internal and external overﬂow was 0.88, the yield of − 20 µ m particles was more than 80.0%, and the highest was 95.0%. Increasing the internal vortex-ﬁnder length could reduce the coarse particle content and improve the classiﬁcation accuracy of the internal overﬂow product. When the length of the internal vortex-ﬁnder is larger than 80 mm, the +30 µ m yield was lower than 20.0%, and the maximum k value was 16.3%; the k is the signiﬁcant factor used to characterize the effectiveness of − 20 µ m particle collection. The response surface modeling revealed that the internal vortex-ﬁnder diameter was the most important factor affecting the distribution rate of internal overﬂow. This paper is expected to advance the development of the classiﬁcation industry.


Introduction
Hydrocyclone is a piece of simple-structure and efficient solid-liquid classification equipment with a history of more than 100 years [1]. With its excellent classification performance, hydrocyclone is widely used in mineral processing, the chemical industry, petroleum, etc. [2][3][4][5]. However, the mismatch phenomenon of coarse particles entrained in the overflow product and fine particles entrained in the bottom flow product has always existed in the hydrocyclone classification process [6,7].
To effectively improve the classification performance of hydrocyclone, researchers have conducted in-depth research on the structure of the vortex-finder to improve the classification efficiency of the hydrocyclone by changing the structure of the vortex-finder. Qiang et al. [8] considered that the diameter of the vortex-finder could significantly affect hydrocyclone classification performance, and the appropriate reduction of the vortex-finder diameter could reduce the cut size. Ghodrat et al. [9,10] considered that the diameter was the most influential element for classification efficiency among the vortex-finder parameters, and a too large or too small vortex-finder diameter could lead to the loss of hydrocyclone classification capacity. Cui et al. [11] consider that an extension of the diameter of the vortex-finder may lead to an outward shift of the zero-axis velocity envelope, resulting in a coarse overflow. Therefore, the vortex-finder diameter is an extremely valuable indicator affecting the hydrocyclone classification behavior.
The insertion depth of the vortex-finder is an important factor affecting the hydrocyclone classification efficiency and classification accuracy [12][13][14]. Liu et al. [15] concluded that the most efficient classification was achieved when the insertion depth of the vortexfinder was at the column-cone junction of the hydrocyclone. Hwang et al. [10,16] believed that with the increase in the insertion depth of the overflow pipe, the separation efficiency showed a trend of increasing first and then decreasing. Tian et al. [17] found through numerical simulation that the optimal length of overflow pipe should be 1.682.02 times of column length. Therefore, the vortex-finder length is crucial to the hydrocyclone classification efficiency. However, no unified qualitative conclusion has been obtained from many in-depth studies on the vortex-finder length.
In addition to numerical simulation methods, multi-factor experimental design has become an important means to study the influence of structural parameters on the hydrocyclone separation effect. The comprehensive influence of multiple structural factors on hydrocyclone classification efficiency was investigated using the response surface method and orthogonal experiments to seek the optimal structural size of hydrocyclone [18][19][20]. Tang et al. [10] studied the influence rules of the overflow tube diameter, insertion depth, and wall thickness on the classification efficiency of the cyclone by response surface method, and found that the overflow tube diameter was the most important factor affecting the classification efficiency. Liu et al. [15] found the comprehensive influence relationship of inlet flow rate, vortex-finder insertion depth, and vortex-finder wall thickness on hydrocyclone classification efficiency using orthogonal experiments.
To obtain two kinds of products from the internal and external overflows in one classification, Obeng et al. [21,22] designed a double vortex-finder hydrocyclone, a coaxial insertion of a smaller internal vortex-finder inside the conventional single vortex-finder, which received wide attention. However, the influence of structural parameters of dual overflow pipe on flow field and separation efficiency is not clear. The influence of the internal vortex-finder diameter and length on the hydrocyclone flow field was analyzed by numerical simulation to explore the influence of the structural parameters of the double vortex-finder hydrocyclone on the classification. The response surface model of the internal vortex-finder diameter, length, and classification accuracy was established by the central composite response surface method to improve the classification performance of the double vortex-finder hydrocyclone. This paper can advance the development of the classification industry.

Hydrocyclone Structure and Grid Division
The double vortex-finder hydrocyclone in this paper is evolved from the Φ75 conventional hydrocyclone by adding an internal overflow pipe. The structure diagram is shown in Figure 1a. D 0 is the internal overflow pipe diameter, and L 0 is the insertion depth of the internal overflow pipe. To facilitate the study of the influence law of overflow, the overflow pipe structure parameters on the hydrocyclone flow field, and classification efficiency, the ratio of vortex-finder diameter, insertion depth, and the corresponding parameters of cone section in the hydrocyclone are often studied as dimensionless factors [23][24][25]. Therefore, the ratio of the internal overflow pipe diameter D 0 to the external D and the ratio of the internal overflow pipe insertion depth L 0 to the external L were regarded as the influencing factors on flow distribution and investigated in study. When D 0 = (0.64-0.88) D, the internal vortex-finder is 80 mm, as shown in Figure 2. When L 0 = (0.86-1.26) L, the internal vortex-finder diameter is 20 mm, as shown in Figure 3.       The double vortex-finder hydrocyclone fluid domain is meshed using ICEM18.0 software, as shown in Figure 1b. To verify the irrelevance of the grid, the tangential velocity with different grid numbers of 150,000, 250,000, 300,000, and 400,000 are measured, as shown in Figure 4. The results show that the tangential velocity is the same when the mesh number is greater than 300,000, revealing the irrelevance of the grid. The grid number of the hydrocyclone fluid domain is determined to be 300,000 in consideration of the computational cost and simulation accuracy.
The double vortex-finder hydrocyclone fluid domain is meshed using ICEM18.0 software, as shown in Figure 1b. To verify the irrelevance of the grid, the tangential velocity with different grid numbers of 150,000, 250,000, 300,000, and 400,000 are measured, as shown in Figure 4. The results show that the tangential velocity is the same when the mesh number is greater than 300,000, revealing the irrelevance of the grid. The grid number of the hydrocyclone fluid domain is determined to be 300,000 in consideration of the computational cost and simulation accuracy.

Mathematical Model
The accuracy of the VOF model for predicting the clear water field in the hydrocyclone has been verified, and the model achieves the capture of the air core section by solving the momentum equation (Equation (1)) to predict the flow field characteristics [26,27].
where αq is the volume fraction of the q phase, vj is the velocity in the j direction of the q phase, and t and xj represent the time and length of the position, respectively.
The Mixture model can efficiently accomplish low concentration discrete phase classification prediction by calculating the mixed term momentum equation (Equation (2)) and the continuity equation (Equation (3)) [28,29].
where ρm is the density of the phases being mixed, vm is the average mixed-phase velocity, p is the pressure, g is the gravitational force, F is the bulk force, and αkρk is the k phase viscosity, and vdr,k is the k phase drift velocity. The RSM model meets the prediction of the complex flow field in the hydrocyclone, considering the various anisotropy assumptions in the turbulent flow field [30][31][32], and is solved by the transport equation (Equation (4)).

Mathematical Model
The accuracy of the VOF model for predicting the clear water field in the hydrocyclone has been verified, and the model achieves the capture of the air core section by solving the momentum equation (Equation (1)) to predict the flow field characteristics [26,27].
where α q is the volume fraction of the q phase, v j is the velocity in the j direction of the q phase, and t and x j represent the time and length of the position, respectively. The Mixture model can efficiently accomplish low concentration discrete phase classification prediction by calculating the mixed term momentum equation (Equation (2)) and the continuity equation (Equation (3)) [28,29].
where ρ m is the density of the phases being mixed, v m is the average mixed-phase velocity, p is the pressure, g is the gravitational force, F is the bulk force, and α k ρ k is the k phase viscosity, and v dr,k is the k phase drift velocity. The RSM model meets the prediction of the complex flow field in the hydrocyclone, considering the various anisotropy assumptions in the turbulent flow field [30][31][32], and is solved by the transport equation (Equation (4)).
where ρµ i ' µ j ' is the mean Reynolds stress, D τ,ij is the turbulent diffusion, D L,ij is the molecular diffusion, p ij is the pressure, G ij is the generation, ϕ ij is the pressure strain term, ε ij is the dissipation, ρ is the fluid density, and µ k is the fluid viscosity.

Model Parameters
The RSM model was adopted for the turbulence model, the VOF model and the Mixture model were adopted for the multiphase flow model. The entrance was set to Velocity-inlet, and the exit was set to Pressure-out. SIMPLE was adopted for pressurevelocity coupling, PRESTO was adopted for pressure discretization, and QUICK format was adopted for other discretization methods. The inlet velocity was 2.28 m/s, the reflux coefficient was 1, and the initial pressure of the flow field was 1 standard atmosphere. CaCO 3 particles with a volume fraction of 5.4% are adopted for the dispersion phase, and the particle properties are seen in Table 1. The flow imbalance error of each phase in unit time is less than 5 × 10 −3 s as the basis for the convergence of the calculation.

Model Validation
To verify the accuracy and reliability of the model, the VOF coupled with the RSM model was applied to predict the flow field inside the hydrocyclone. The predicted tangential velocity of the flow field is compared with data measured by Hsieh [33], as can be seen in Figure 5a. The results show the tangential velocity is basically the same, verifying the accuracy of the VOF model. The results of the classification experiments of CaCO 3 particles are compared with the prediction of the Mixture model, as shown in Figure 5b. The results show a high consistency, indicating that the Mixture multiphase flow model meets the calculation requirements. molecular diffusion, pij is the pressure, Gij is the generation, φij is the pressure strain εij is the dissipation, ρ is the fluid density, and μk is the fluid viscosity.

Model Parameters
The RSM model was adopted for the turbulence model, the VOF model an Mixture model were adopted for the multiphase flow model. The entrance was Velocity-inlet, and the exit was set to Pressure-out. SIMPLE was adopted for pre velocity coupling, PRESTO was adopted for pressure discretization, and QUICK f was adopted for other discretization methods. The inlet velocity was 2.28 m/s, the coefficient was 1, and the initial pressure of the flow field was 1 standard atmosp CaCO3 particles with a volume fraction of 5.4% are adopted for the dispersion phas the particle properties are seen in Table 1. The flow imbalance error of each phase i time is less than 5×10 −3 s as the basis for the convergence of the calculation.

Model Validation
To verify the accuracy and reliability of the model, the VOF coupled with the model was applied to predict the flow field inside the hydrocyclone. The pred tangential velocity of the flow field is compared with data measured by Hsieh [33], be seen in Figure 5a

Central Composite Response Surface Design
The optimal parameters of the study factors and significant factors were found in response surface analysis by fitting with multiple quadratic regression equations [34]. The central composite response design was one of the most commonly used response surface design methods [35], widely used in various multi-factor experiments. The comprehensive effect of the interaction of multiple structural factors on hydrocyclone classification efficiency was investigated using the response surface design method. The internal overflow pipe diameter D 0 and insertion depth L 0 were response factors. The optimized value of Separations 2022, 9, 88 6 of 15 steepness index K was chosen as the significant factor to effectively predict the effectively of the internal overflow pipe structure parameters on the classification efficiency of microfine particles with 20 µm in particle size and below, which is calculated as equation (Equation (5)). The design factor levels of the response surface are shown in Table 2.
where d 20 , d 30 are the classification rate of the corresponding particle size.  Figure 6 shows the diagram of the radial pressure at column and cone sections of the hydrocyclone with different diameters of internal vortex-finder. The variations in diameter have no significant impact on the radial pressure distribution in the flow field at the column section location (z = 200 mm). However, the radial pressure increases with the diameter at the cone section position near the wall area (z = 150 mm). Therefore, the increase in diameter can improve the radial force of the fluid near the wall of the cone section and strengthen the radial motion of the fluid, which is conducive to the particle classification.

Tangential Velocity
Tangential velocity affects the hydrocyclone classification performance. Figure 7 shows distribution curves of the tangential velocity of the double vortex-finder hydrocyclone at different heights. At z = 150 mm and z = 200 mm, within the diameter of 040 mm, the peak tangential velocity is maximum when the diameter is 0.64D, and the peak tangential velocity is minimum when the diameter is 0.88D. Within the diameter of 4075 mm, no evident difference in the tangential velocities shows that the tangential velocities are affected by the internal vortex-finder diameter. There are mainly microfine particles within the diameter of 040 mm and coarse particles within the diameter of 4075 mm. The internal vortex-finder diameter has no significantly effect on coarse particles, but internal vortex-finders with smaller diameters are beneficial to the classification of microfine particles. Accordingly, the internal vortex-finder diameter can be designed

Tangential Velocity
Tangential velocity affects the hydrocyclone classification performance. Figure 7 shows distribution curves of the tangential velocity of the double vortex-finder hydrocyclone at different heights. At z = 150 mm and z = 200 mm, within the diameter of 040 mm, the peak tangential velocity is maximum when the diameter is 0.64 D, and the peak tangential velocity is minimum when the diameter is 0.88 D. Within the diameter of 4075 mm, no evident difference in the tangential velocities shows that the tangential velocities are affected by the internal vortex-finder diameter. There are mainly microfine particles within Separations 2022, 9, 88 7 of 15 the diameter of 040 mm and coarse particles within the diameter of 4075 mm. The internal vortex-finder diameter has no significantly effect on coarse particles, but internal vortexfinders with smaller diameters are beneficial to the classification of microfine particles. Accordingly, the internal vortex-finder diameter can be designed according to particle size distribution characteristics of raw materials and the expected classification.
shows distribution curves of the tangential velocity of the double vortex-finder hydrocyclone at different heights. At z = 150 mm and z = 200 mm, within the diameter of 040 mm, the peak tangential velocity is maximum when the diameter is 0.64D, and the peak tangential velocity is minimum when the diameter is 0.88D. Within the diameter of 4075 mm, no evident difference in the tangential velocities shows that the tangential velocities are affected by the internal vortex-finder diameter. There are mainly microfine particles within the diameter of 040 mm and coarse particles within the diameter of 4075 mm. The internal vortex-finder diameter has no significantly effect on coarse particles, but internal vortex-finders with smaller diameters are beneficial to the classification of microfine particles. Accordingly, the internal vortex-finder diameter can be designed according to particle size distribution characteristics of raw materials and the expected classification.

Air Core
The shape of the air core affects the stability of the flow field. The air core morphology is displayed in Figure 8. The air core diameters all fluctuate between 2 and 6 mm. The diameters of air core increase significantly at the axial position of about 225 mm, forming a protrusion because the air from the overflow and underflow ports meets here and is restricted by the vortex-finder wall. The air core diameter is most stable when diameter D 0 of the internal vortex-finder is 0.72 D, indicating that too large or too small internal vortex-finder diameter will cause the air core to shake violently and is not beneficial to the stabilization of the flow field, thus the internal vortex-finder diameter should be considered in the design of hydrocyclone. The shape of the air core affects the stability of the flow field. The air core morphology is displayed in Figure 8. The air core diameters all fluctuate between 2 and 6 mm. The diameters of air core increase significantly at the axial position of about 225 mm, forming a protrusion because the air from the overflow and underflow ports meets here and is restricted by the vortex-finder wall. The air core diameter is most stable when diameter D0 of the internal vortex-finder is 0.72D, indicating that too large or too small internal vortex-finder diameter will cause the air core to shake violently and is not beneficial to the stabilization of the flow field, thus the internal vortex-finder diameter should be considered in the design of hydrocyclone.  Figure 9 illustrates the impact of internal overflow pipe diameter pressure drop and the split ratio of internal and external overflow. The pressure drop of the flow field increases with the internal overflow pipe diameter. Since a large pressure drop means a large energy consumption, the energy consumption in the flow field increases with the internal overflow pipe diameter. Moreover, the increase in the internal overflow pipe diameter increases the split ratio of internal and external overflow. The split ratio of  with the internal overflow pipe diameter. Since a large pressure drop means a large energy consumption, the energy consumption in the flow field increases with the internal overflow pipe diameter. Moreover, the increase in the internal overflow pipe diameter increases the split ratio of internal and external overflow. The split ratio of internal and external overflow is only 0.185 when the internal overflow pipe diameter is 16 mm (0.64 D) and up to 1.40 when the internal overflow pipe diameter is 20 mm (0.80 D). When the overflow pipe diameter is small, most of its space is occupied by the air core, causing the split ratio of internal and external overflow to increase rapidly as the internal overflow pipe diameter increases. Therefore, when the internal overflow pipe diameter is less than a certain value, its internal area will be occupied by air, and no product will be separated. In addition, an appropriate increase in the internal overflow pipe diameter will facilitate the overflow of microfine particles from the internal overflow pipe, improving the yield of microfine particles. To better reflect the response of the internal vortex-finder diameter change on the split ratio and pressure drop, the data in the figure was fitted to provide a reference for predicting the distribution of pressure drop and the split ratio of internal and external overflow in internal overflow pipes with different diameters.  Figure 10 shows the pulp movement trajectory of hydrocyclon to the internal vortex-finder, most of the short-circuit flow at the h box in Figure 10) flows into the external overflow area, reducing th circuit flow inside the internal vortex-finder. The classification accu the short-circuit flow is directly discharged without effective ce Therefore, the internal overflow pipe with a smaller diameter classification of internal overflow. However, more pulp in the inter smaller diameter overflows to the outside, reducing the accuracy o  Figure 10 shows the pulp movement trajectory of hydrocyclone column section. Due to the internal vortex-finder, most of the short-circuit flow at the hydrocyclone wall (red box in Figure 10) flows into the external overflow area, reducing the total amount of short-circuit flow inside the internal vortex-finder. The classification accuracy is reduced due to the shortcircuit flow is directly discharged without effective centrifugal classification. Therefore, the internal overflow pipe with a smaller diameter contributes to the fine classification of internal overflow. However, more pulp in the internal overflow pipe with smaller diameter overflows to the outside, reducing the accuracy of the external overflow. Figure 10 shows the pulp movement trajectory of hydrocyclone column section. Due to the internal vortex-finder, most of the short-circuit flow at the hydrocyclone wall (red box in Figure 10) flows into the external overflow area, reducing the total amount of shortcircuit flow inside the internal vortex-finder. The classification accuracy is reduced due to the short-circuit flow is directly discharged without effective centrifugal classification. Therefore, the internal overflow pipe with a smaller diameter contributes to the fine classification of internal overflow. However, more pulp in the internal overflow pipe with smaller diameter overflows to the outside, reducing the accuracy of the external overflow.  Figure 11 shows the static pressure distribution of the hydrocyclone. It can be found that the change of the length of the internal overflow pipe has the same effect on the pressure field at different heights. Meanwhile, at a certain height, the radial pressure gradient of the flow field is larger when the internal overflow pipe length is less than or equal to the external overflow pipe length, and the pressure gradient reaches the maximum when the insertion depth of the internal and external vortex finders is the same. Since the direction of the pressure gradient is from the high-pressure area to the lowpressure area, a larger pressure gradient is favorable for the migration of fine particles from the wall to the axis. Thus, when the internal overflow pipe length is less than or equal to the external overflow pipe length, the content of fine particles in the external  Figure 11 shows the static pressure distribution of the hydrocyclone. It can be found that the change of the length of the internal overflow pipe has the same effect on the pressure field at different heights. Meanwhile, at a certain height, the radial pressure gradient of the flow field is larger when the internal overflow pipe length is less than or equal to the external overflow pipe length, and the pressure gradient reaches the maximum when the insertion depth of the internal and external vortex finders is the same. Since the direction of the pressure gradient is from the high-pressure area to the low-pressure area, a larger pressure gradient is favorable for the migration of fine particles from the wall to the axis. Thus, when the internal overflow pipe length is less than or equal to the external overflow pipe length, the content of fine particles in the external hydrocyclone region will decrease, improving the classification accuracy of the hydrocyclone. hydrocyclone region will decrease, improving the classification accuracy of the hydrocyclone. Figure 11. Influence of the insertion depth of the internal overflow pipe on the pressure. Figure 12 shows the influence of insertion depth of the internal overflow pipe on tangential velocity. Internal overflow pipe length is 0.86 L, tangential velocity variation is concentrated in the range from −20 mm to 20 mm of hydrocyclone radius, and the tangential velocity reaches the maximum. When the internal overflow pipe length is equal to or huger than the outside, the tangential velocity variation decreases gradually. It can be concluded that the change of the internal overflow pipe length does not significantly affect the centrifugal field of the hydrocyclone when the internal overflow pipe length is equal to or greater than the external overflow pipe.  Figure 12 shows the influence of insertion depth of the internal overflow pipe on tangential velocity. Internal overflow pipe length is 0.86 L, tangential velocity variation is concentrated in the range from −20 mm to 20 mm of hydrocyclone radius, and the tangential velocity reaches the maximum. When the internal overflow pipe length is equal to or huger than the outside, the tangential velocity variation decreases gradually. It can be concluded that the change of the internal overflow pipe length does not significantly affect the centrifugal field of the hydrocyclone when the internal overflow pipe length is equal to or greater than the external overflow pipe. Figure 11. Influence of the insertion depth of the internal overflow pipe on the pressure. Figure 12 shows the influence of insertion depth of the internal overflow pipe on tangential velocity. Internal overflow pipe length is 0.86 L, tangential velocity variation is concentrated in the range from −20 mm to 20 mm of hydrocyclone radius, and the tangential velocity reaches the maximum. When the internal overflow pipe length is equal to or huger than the outside, the tangential velocity variation decreases gradually. It can be concluded that the change of the internal overflow pipe length does not significantly affect the centrifugal field of the hydrocyclone when the internal overflow pipe length is equal to or greater than the external overflow pipe.   Figure 13 shows the influence of internal overflow pipe length on the hydrocyclone pressure drop and the split ratio of internal and external overflow. The split ratio of internal and external overflow varies from 1.2 to 1.45 with internal vortex-finder lengths. When the insertion depths of internal and external overflow are the same, the split ratio is the largest. Furthermore, the pressure drop is the largest when the insertion depths of the internal and external vortex-finders are the same, indicating that the flush internal and external vortex-finders improve the classification of the flow field. The peak values of pressure drop and the split ratio of internal and external overflow were predicted with data fitting. The results show that the peak pressure drop and peak split ratio correspond to the same internal vortex-finder length, which provides a reference to optimize parameters of the internal vortex-finder insertion length.  Figure 13 shows the influence of internal overflow pipe length on the hydrocyclone pressure drop and the split ratio of internal and external overflow. The split ratio of internal and external overflow varies from 1.2 to 1.45 with internal vortex-finder lengths. When the insertion depths of internal and external overflow are the same, the split ratio is the largest. Furthermore, the pressure drop is the largest when the insertion depths of the internal and external vortex-finders are the same, indicating that the flush internal and external vortex-finders improve the classification of the flow field. The peak values of pressure drop and the split ratio of internal and external overflow were predicted with data fitting. The results show that the peak pressure drop and peak split ratio correspond to the same internal vortex-finder length, which provides a reference to optimize parameters of the internal vortex-finder insertion length.   Figure 14 shows the pulp movement trajectory of the hydrocyclone column section with different internal overflow pipe lengths. The axial movement distance of the fluid entering the internal vortex finder gradually increases with the internal overflow pipe length, which is conducive to the residence of the particles in the cyclone field and for particle classification. Meanwhile, the internal overflow pipe length is smaller than the external overflow pipe length, and the vortex is easily formed at the bottom of the vortex-finder, which is not conducive to the stability of the flow field or for the classification accuracy.

Model Design
To further explore the influence of internal overflow pipe diameter and insertion depth on the classification efficiency of the cyclone, an optimization model was established according to the central composite design and response surface methodology, as shown in Table 3.  Figure 15 shows the internal overflow rates of hydrocyclones with relevant response surfaces. Figure 15a-i correspond to the first nine combinations in Table 3, respectively. When the internal overflow pipe diameter is small, the classification efficiency of particles with the particle size of 20 µm and below is low, only around 30%, as shown in Figure  15a,c,e. When the internal overflow pipe diameter is large, the corresponding microfine particle classification efficiency can reach over 80%, as shown in Figure 15b,d,f. It should be noted that although increasing the diameter of internal overflow is beneficial to increase the classification efficiency of fine particles, some coarse particles will also be included in the fine particles, reducing the classification accuracy.

Model Design
To further explore the influence of internal overflow pipe diameter and insertion depth on the classification efficiency of the cyclone, an optimization model was established according to the central composite design and response surface methodology, as shown in Table 3.  Figure 15 shows the internal overflow rates of hydrocyclones with relevant response surfaces. Figure 15a-i correspond to the first nine combinations in Table 3, respectively. When the internal overflow pipe diameter is small, the classification efficiency of particles with the particle size of 20 µm and below is low, only around 30%, as shown in Figure 15a,c,e. When the internal overflow pipe diameter is large, the corresponding microfine particle classification efficiency can reach over 80%, as shown in Figure 15b,d,f. It should be noted that although increasing the diameter of internal overflow is beneficial to increase the classification efficiency of fine particles, some coarse particles will also be included in the fine particles, reducing the classification accuracy.

Optimization Model Validation
The response function obtained by the quadratic regression equation is as follows.
Analysis of variance and tests need to be performed on the fitted equations for accuracy and reliability [36], and the results are shown in Table 4. To some extent, p-value provides the reliability basis for our model [37,38]. The high significance indicates that the fitted regression equation meets the prediction requirements.

Optimization Model Validation
The response function obtained by the quadratic regression equation is as follows.
Analysis of variance and tests need to be performed on the fitted equations for accuracy and reliability [36], and the results are shown in Table 4. To some extent, p-value provides the reliability basis for our model [37,38]. The high significance indicates that the fitted regression equation meets the prediction requirements. The influence of the internal overflow pipe diameter and insertion depth on the K value is obtained with the central composite response surface optimization. As shown in Figure 16, when the internal overflow pipe diameter is small, despite the change of the insertion depth, the K value is less than 10%. When the internal overflow pipe diameter is large, the K value is above 12%. Therefore, the classification efficiency of the internal overflow pipe in the hydrocyclone increases with the internal overflow pipe diameter. Meanwhile, when the internal overflow pipe diameter is large, the interaction between the internal overflow pipe diameter and the internal overflow pipe insertion depth becomes obvious. The efficiency and accuracy of classification can remain high with an appropriate increase in the internal overflow pipe diameter and a reduction of the internal overflow pipe insertion depth.  The influence of the internal overflow pipe diameter and insertion depth on value is obtained with the central composite response surface optimization. As sh Figure 16, when the internal overflow pipe diameter is small, despite the change insertion depth, the K value is less than 10%. When the internal overflow pipe diam large, the K value is above 12%. Therefore, the classification efficiency of the i overflow pipe in the hydrocyclone increases with the internal overflow pipe dia Meanwhile, when the internal overflow pipe diameter is large, the interaction b the internal overflow pipe diameter and the internal overflow pipe insertion becomes obvious. The efficiency and accuracy of classification can remain high w appropriate increase in the internal overflow pipe diameter and a reduction of the i overflow pipe insertion depth.

Conclusions
With the numerical simulation of the double vortex-finder hydrocyclone a optimized response surface model of the internal vortex-finder diameter and in depth, the following conclusions are drawn.
(1) The flow field pressure drop and split ratio increase with the internal finder diameter in the double overflow pipe hydrocyclone. The increase in i overflow pipe length will intensify the flow field fluctuation, and when the intern external vortex-finder lengths are equal, the flow field pressure drop and split rati the maximum.
(2) The content of fine particles of the internal overflow is positively related internal overflow pipe diameter. However, when the ratio of internal and e overflow pipe diameters is over 0.88, the coarse particles in the internal overflo increase rapidly, reducing the classification accuracy. Moreover, the coarse p

Conclusions
With the numerical simulation of the double vortex-finder hydrocyclone and the optimized response surface model of the internal vortex-finder diameter and insertion depth, the following conclusions are drawn.
(1) The flow field pressure drop and split ratio increase with the internal vortex-finder diameter in the double overflow pipe hydrocyclone. The increase in internal overflow pipe length will intensify the flow field fluctuation, and when the internal and external vortexfinder lengths are equal, the flow field pressure drop and split ratio reach the maximum.
(2) The content of fine particles of the internal overflow is positively related to the internal overflow pipe diameter. However, when the ratio of internal and external overflow pipe diameters is over 0.88, the coarse particles in the internal overflow will increase rapidly, reducing the classification accuracy. Moreover, the coarse particle content in the internal overflow pipe gradually decreases with the increase in the insertion depth of the internal overflow pipe.
(3) According to the response surface analysis, when the ratio of internal and external overflow pipe diameters is in the range from 0.8 to 0.88, the interaction between the internal overflow pipe diameter and insertion depth on classification efficiency is obvious. An appropriate increase in the internal overflow pipe diameter and insertion depth improves the efficiency and accuracy of classification.
In this study, the effects of the internal overflow pipe diameter and insertion depth on the flow field and classification performance have been analyzed. However, the effect of the acting force of particles was ignored in the numerical simulations, which needs to be further studied.