Separation characteristics and structure optimization of double spherical tangent double-field coupling demulsifier

The demulsification and dewatering of the W/O emulsion are widely used in petrochemical industry, oilfield exploitation, and resource and environmental engineering. However, efficiently treating emulsion via traditional single methods. In this study, a new double-field coupling demulsification and dewatering device is proposed, where the conical structure of the device is double spherical tangential type. The numerical model for double-field coupling is established, especially, the population balance model (PBM) is used to simulate the coalescence and breakup of dispersed droplets under the double-field coupling action. And the effects of three conical structures on the internal flow and separation efficiency are analyzed. Results show that the conical structure has a significant effect on the coalescence of droplets, especially the double spherical tangential cone is more conducive for improving the coalescence ability of small droplets and improving the separation efficiency of the device. After optimization, the optimal R value of the double spherical tangential coupling device is 300 mm, and the separation efficiency can be up to 96.32%, which is 6.13% higher than the separation efficiency of the straight double-cone coupling device.

Vacuum dielectric constant, F·m−1 I Unit tensor ɛ o , ɛ w Oil and water phase dielectric constant, F·m−1 K i , K j Number densities, kg·m−3 ζ Arc angle of big cone, °L i , L j Two colliding droplet diameters, mm µ o , µ w Oil and water phase viscosity, mpa· s L s Overflow pipe length, mm ρ Density of emulsion, kg·m−3 L t Cone length, mm ρ o , ρ w Oil and water density, kg·m−3 L u Underflow pipe length, mm σ o , σ w Oil and water phase conductivity, s·m−1 L 1 , L 2 Small cone and big cone length, mm w Emulsion velocity, m·s−1

Introduction
Water in oil (W/O) emulsion generally exists in petrochemical industry, oilfield exploitation, development of heavy oil reservoirs, and resource and environmental engineering fields Lv et al. 2020). The W/O emulsion is a colloidal dispersion of oil and water, where the water phase is dispersed in the oil phase (Ghasemi et al. 2020). At present, the global energy demand is continuously increasing (Almeida-Trasvina et al. 2021). The recycling of emulsion is particularly important, and demulsification and dewatering are the most critical step in the process of resource recycling (Praporgescu and Mihăilescu 2011;Lam et al. 2016;Raya et al. 2020). Physical, chemical, and biological methods are used for emulsion treatment (Kang et al. 2018;Long et al. 2013;Gu et al. 2015). The most common methods are physical methods, including electric field, rotary displacement, gravity settlement, and vacuum heating methods (Qian et al. 2017;Mhatre et al. 2015;Li et al. 2015;Qi et al. 2018).
In general, it is difficult to achieve demulsification and dewatering of emulsion efficiently by using a single processing method (Liu and Wang 2016;Jung et al.2019;Hamza et al. 2020;Qiu et al. 2020;Gong et al. 2018). The demulsification and dewatering can be well achieved by using various processing methods or by coupling an integrated unit operation, which is also the key direction of future development (Eow and Ghadiri 2002;Vashahi et al. 2019;Xu et al. 2020;Motin and Bénard 2017). The electric field method refers to the stretching deformation of dispersed droplets in oil under the action of electric field, which aggravates the collision of droplets, promotes the coalescence of droplets in emulsion, and increases the size of droplets (Mozaffari et al. 2021;Bijesh et al. 2018). However, this method cannot achieve the effect of rapid and efficient separation . Centrifugation is generally applied in the separation process that involves the different densities of oil and water in a separator to effectively and quickly achieve the oil-water separation. The oil phase with a small density flows out from the center area of the device to the overflow orifice; the water phase with a high density moves along the wall of the device to the underflow orifice. Therefore, the coupling of electric and swirl centrifugal fields can achieve a better effect of droplet size enlargement and rapid and effective separation. They complement each other's advantages, thereby improving the demulsification and dewatering efficiency of the emulsion. For example, Bailes and Watson (1992) investigated the demulsification of water in crude oil emulsion by a high-voltage pulsed DC electric field and proposed a continuous rotating electrostatic demulsifier. Therefore, emulsion can be effectively dewatered and purified under the combined action of electric field and centrifugal fields. Ghadiri and Eow (2003) studied the demulsification and dewatering treatments of emulsion under the combined action of pulsed direct current electric and centrifugal fields. They proposed a centrifugal electric coalescence separator to make the droplets coalesce quickly under the action of double-field coupling and improve the separation efficiency. Adamski et al. (2018) investigated the agglomeration and separation system and the process of emulsion under the action of centrifugal force and electric field to make the droplets in the emulsion coalesce under the action of a double field. Then, they used the separator for effective separation.
Studies have focused on the influence of inlet velocity, voltage amplitude, and inlet structure on the separation efficiency of the device, ignoring the important influence of the cone structure on the separation efficiency of the cyclone device. For example, Zhang et al. (2020) investigated the influence of different voltages and inlet flow rates on the separation efficiency of emulsion under the coupling effect of electric and swirl centrifugal fields and obtained the optimal voltage amplitude and inlet velocity. Noik and Trapy (2004) studied the influence of device inlet structure type on emulsion separation efficiency under the action of electric and swirl centrifugal field and found that cone-shaped inlet type was more conducive to emulsion swirl separation. Al-Kayiem et al. (2019) studied the effects of different inlet forms on the separation performance of the device and found that the flow field of the dual inlet was more stable than that of the single inlet, which was more conducive to improving the separation efficiency of the device. Al-Kayiem et al. (2020) investigated the influence of the vane's deflection angle in an oil/water axial inlet hydrocyclone separator on the separation efficiency. The study found that the separation efficiency obtained by using the 45° swirl generator was higher than other angles. However, the influence of the conical structure on the separation performance of the device has not been studied. Therefore, in this paper, the electric and centrifugal fields coupling device is taken as the research object and research and analysis are conducted from the cone type of the device. A straight double-cone structure ( Fig. 1) does not have a smooth connection and has a gradient shape, which causes the vibration and the instability of the flow field, affecting the separation efficiency. This study proposes a new type of double-field coupling device where the double spherical tangent cone is applied (Fig. 2). The double spherical tangent structure (Fig. 3) achieves a smooth connection in the double-cone section. Consequently, the flow field becomes more stable and can effectively eliminate the adverse factors. In this paper, under the different cone conditions, numerical calculations were conducted on the emulsion using computational fluid dynamics (CFD) and the population balance model (PBM) to obtain the coupling device's oil-water 1 3 separation performance. The cone structure of the coupling device with two spherical tangents is optimized. In addition, this present study can explore a new structure with higher separation performance and provide insightful references for the design of a new double-field coupling device. Figure 4a shows the structure of the double-sphere tangent-type double-field coupling separation device, which is mainly composed of a double tangential inlet, a swirl chamber, a double-sphere tangent section, and a straight  pipe section near an underflow outlet. An electric field is embedded in the swirl chamber (i.e., the outer wall of the swirl chamber is grounded, and the outer wall of the overflow pipe is connected to the positive pole of the power supply). Under the action of the electric field, small emulsion droplets stretch and deform in the emulsion, intensify the collision of droplets, promote the coalescence of emulsion droplets, increase the droplet size, and then achieve rapid and effective separation under the action of centrifugal force. Figure 4b illustrates the schematic of the traditional straight double-cone structure, and Table 1 lists the structural parameters of the coupling separation device.

Governing equations
The continuity equation and momentum conservation equation are expressed as follows (Cernecky.and Plandorova 2013):

Electric field equation
In this study, a user-defined function (UDF) is used to couple the electric field governing equation with the flow field governing equation . The electric field force of the droplet in the two-field coupling element is described as follows:

Coalescence and breakup equations
Considering the aggregation and fragmentation of droplets under the action of electric field force, the Luo model is adopted for the analysis in this study . Then, the volume formation rate of particles generated via secondary collision is expressed as follow: where f (L i , L j ) is the collision frequency, and p (L i , L j ) is the coalescence probability. The collision frequency is determined on the basis of the relative velocity, particle size, and bulk density of the two droplets, so f (L i , L j ) is defined as: When the relative collision velocity of the centerline of the two droplets is lower than the critical value, they will not coalescence (Howarth 1964). In addition, the contact time after oil-water collision determines the coalescence of droplets. When t d > t c , the two droplets cannot coalesce. In turbulence, the contact time of colliding droplets is (Motin 2015):   where F is the force between the droplets; μ w is the viscosity of liquid droplets; h i is the initial thickness of liquid film; h f is the final thickness of liquid film rupture. In this study, the probability of droplet coalescence is described as follows: The electric field accelerates the cracking of oil film and effectively promotes the droplet coalescence. However, if the electric field strength is too high, the droplet may be broken, thereby causing electric dispersion. Atten (1993) gave the expression of the critical electric field, which leads to droplets to break: According to the formula, when the electric field intensity is 10 6 V/m (approximately 14 kV), the droplet with a diameter larger than 1.16 mm will be broken. Admittedly, the droplet size is less than 1.16 mm in the experiment. Therefore, the electric field dissipation is ignored in this study.
Based on the Hagesaether method (Hagesaether et al. 2002), the breaking formula of the discrete method is described as follows: where In this study, we found that when the shear force of the droplet is lower than the pressure difference between the inside and the outside of the interface film, the model cannot predict the droplet breaking process. The critical conditions are: where τ c is the critical shear stress for droplet breaking; σ is the tension coefficient; L w is the droplet diameters.

Volume fraction equation of dispersed phase
According to Rusche (2003) and (Oliveira and Issa 2003), the volume fraction equation of the dispersed phase is expressed as follows Cheng et al. (2018):

Solution method of population balance equations (PBE)
The PBE could predict droplet size distribution well (Saeed- Mozaffari et al. 2019) and is solved by discrete method and moments method. In this study, to obtain the droplet size distribution directly, we applied the discrete method developed by Ramkrishna. In ANSYS Fluent, the discrete PBE is expressed as (Ansys Incorporation 2015 and Gong et al. 2020): where a ij is the special coalescence frequency between two droplets with size of x i and x j ; N i and N j are the volumetric number concentration of droplets with volume V i and V j ; g v is the integral droplet volume fraction at underflow; ϛ kj is the criteria function to judge whether the droplet formed by the combination of droplet k and j reaches a certain droplet size interval (i), and is defined as: V ag is the particle volume resulting from the aggregation of particles k and j, and is defined as: where x kj is the distribution function, and is defined as: Also, in order to preserve the properties of breakage droplets, introduce the function φ i,k to measure the fractional distribution of breakage droplets to the nearby characteristic sizes, and is defined as:

Grid independence analysis
The device model is divided into three different mesh numbers: 134615, 388,405, and 711,769. The static pressure and tangential velocity of different mesh numbers at the section z = 700 mm are shown in Fig. 5. The optimal number of grids is found by comparing the distribution of tangential velocity and static pressure. In Fig. 5, the simulation results are almost the same when the number of grids is 388405 and 711,769. Considering the time consumption and calculation results, the appropriate number of meshing is 388405.

Boundary conditions
The physical parameters of the emulsion used in the experiment are shown in Table 2. The normal velocity of inlet is v = 10 (m‧s −1 ), the volume fraction of oil in the emulsion is α = 90%, the turbulence intensity at the inlet of the coupling separation device is T = 5%, the split ratio of the upper outlet of the coupling device is O FRW = 0.9, and the split ratio of the lower outlet of the coupling separation device U FRW = 0.1. For the electric field region of the device, the outer wall of the overflow pipe is added with 11 kV high voltage, and the other boundary surfaces are 0; the inlet type is set as the velocity-inlet, and the overflow and underflow orifices type are set to outflow. The near-wall surface is treated with the standard wall function, and the wall surface is set as the nonslip boundary condition.
The Reynolds stress model (RSM) is chosen in the simulation because it makes the N-S equation to be closed and reduces the Reynolds pressure in the equation and the dissipation rate. In RSM, rapid changes in streamline bending, vortex, swirl, and tension are considered more strictly, and the model has a more accurate prediction ability for a complex flow. In simulation analysis, a pressure-based solver is adopted, and the "SIMPLEC" algorithm is adopted to realize the coupling effect of velocity and pressure (Chu et al. 2017;Vakamalla et al. 2014). In spatial discretization, gradient, pressure, momentum, volume fraction, turbulent kinetic energy, turbulent dissipation rate, and Reynolds stress are analyzed by using the least squares element and PRESTO!, QUICK (Chu et al. 2017;Vakamalla et al. 2014). Because the least square element has the characteristics of high precision and high efficiency on an irregular unstructured

Experimental materials and devices
The emulsion used in the experiments is prepared by injecting water into No. 25 transformer oil and mixing it at a ratio of 1:9. The physical parameters of the emulsion are shown in Table 2 and all parameter values are measured under the condition of 20°. In the preparation, span-80 emulsifier was added at a concentration of 5 g/L to maintain the stability of the emulsion, and an electric stirrer (MGD699) was set for stirring for 15 min. After the mixture is allowed to stand for 24 h, no free water precipitation is present, so it meets the experimental requirements. The prepared emulsion is placed in the storage tank for standby. The separation device used in this experiment is shown in Fig. 6. The main components are as follows: single pump, coupling device with straight double-cone structure, high-voltage power supply, pressure gauge, overflow tank, underflow tank, tee, and valves. The purpose of the tee is to ensure that the two-way inlet flow velocity is the same and that it is cut in to enter the device so that the separation efficiency is higher under the effect of the centrifugal field. During the operation of the device, the emulsion enters the device, and the oil-water separation is carried out under the coupling effect of the highvoltage electric field and the swirl centrifugal field. The high-concentration oil is discharged from the overflow port into the overflow tank, and the high-concentration water is discharged from the underflow port of the device into the underflow tank. After a certain period, the sampling valve of the overflow tank and the bottom flow tank is opened to extract a small amount of liquid. The moisture content of the sample droplets is measured with a petroleum hygrometer (SYD-2122C), and the separation efficiency of the emulsion is calculated by using the formula (20). During the experiment, the pressure change at any time should be explored to avoid the low pressure affecting the separation efficiency and high-pressure, causing safety accidents.
Here f wo is the water volume fraction discharged from overflow orifice, f w is water volume fraction of prepared emulsion.

Experimental procedure
Before running the device, complete the following steps: (1) place the equipment stably and ensure that there is power supply around the coupling device; (2) check the tightness and smoothness inside the device; (3) ensure that the line is connected, the part embedded in the high-voltage electric field is insulated, and the power supply voltage gear is set to zero.
When the power is turned on and the device is running, the following steps are operated: first, turn on the single screw pump. Then, after the device works stably, turn on the high-voltage pulse power supply and set various parameters Fig. 6 a experimental of double-field coupling demulsification and dewatering and b schematic diagram of double-field coupling demulsification and dewatering for the experiment. Finally, after the experiment, disconnect the high-voltage pulse power supply, and close each valve and the main power supply after the emulsion is drained.

Model validation
The separation efficiency obtained via the experimental method and numerical simulation is compared to verify the rationality of the numerical model (Fig. 7). The experimental data in Fig. 7 are the average of five experiments. In Fig. 7, the experimental results are in agreement with the numerical results. However, the numerical results are slightly larger than the experimental results because numerical simulation is carried out under relatively ideal conditions, ignoring the change in the temperature e of the material in the experiment, the influence of the friction of the mechanical operation on the emulsion, and other factors. However, the difference between the numerical and experimental results is acceptable, indicating that the model is feasible. Therefore, the model is suitable for the double-field coupling separation device.

Structure optimization
The structure of the swirl device can be determined only by confirming a sphere radius R because the distance value of the double-cone section along the z-axis of the swirl device is certain, and the double spheres, the sphere, and the straight pipe section are tangent. The range of the sphere radius R is 100-3000 mm. The corresponding dewatering and deoiling rate are shown in Fig. 8. At a voltage of U = 11 kV and an inlet velocity of v = 10 m/s, only single factor R changes. In Fig. 8, the optimal sphere radius R is 300 mm.

Static pressure distribution
Static pressure distribution not only plays an important role in the flow of emulsion in the swirling device but also affects the separation performance and energy loss of the swirling device (Tian et al. 2020). The static contours of pressure distribution at x = 0 mm section under different cone structure is shown in Fig. 9. According to the figure, static pressure distribution is centrosymmetric because the device is double tangential inlet. The static pressure decreases with the decrease of z-axis value. The reason is that the fluid rotates in the device and overcomes the pressure loss caused by the friction of fluid viscosity and fluid steering. The static pressure is positively correlated with the radius. As the radius decreases, the static pressure decreases, which is consistent with the distribution trend of the static pressure inside the coupling device . In the overflow pipe region, the negative pressure of the double spherical tangent coupling device is smaller than that of the straight double-cone coupling device, which is more conducive to the flow with higher oil content flowing out of the overflow port. The radial distribution of static pressure at different sections is shown in Fig. 9c-f. It can be seen from the figure that the negative pressure value of the optimized double spherical tangent coupling device near the underflow port (z = 130 mm) and the axis is large, which is 132.15% higher than that of the straight double-cone coupling device, which is conducive to the discharge of liquid with higher water content from the underflow port. Near the overflow port (z = 840 mm), the negative pressure value of the optimized double spherical tangent coupling device in the axial region is small, which is 20.58% lower than that of the straight double-cone coupling device, which is conducive to the flow with higher oil content discharges from the overflow port and improve the separation efficiency. In the cone region, the static pressure value and pressure gradient of the optimized double spherical tangential coupling device are large, which indicates that the internal energy loss of the device is small and the energy consumption is low. The pressure difference Δp 1 between the device inlet and the overflow port and the pressure difference Δp 2 between the feed port and the underflow port of the coupling device is shown in Fig. 10. Δp 2 is more important because it indicates the amount of the pressure loss of the liquid flowing through the coupling device. The pressure drop is defined as follows . In Fig. 10, Δp 2 of the three structures is greater than Δp 1 , that is, the pressure drop ratio (Δp 2 /Δp 1 ) is greater than 1, which is consistent with the research results of Chen et al. (2015). Δp 2 of the optimized double spherical tangent structure is reduced by 33.7% compared with Δp 1 of the straight double-cone structure because the optimized double spherical tangent structure reduces the emulsion flow resistance. A reasonable device structure should have as little pressure drop as possible, because the lower the pressure drop, the smaller the energy loss. In addition, the pressure is beneficial to drive the flow of the emulsion (Mozaffari et al. 2017), which is more conducive to the oil-water separation of emulsion in the swirl device. Therefore, the optimized structure is more conducive to the oil-water separation of emulsion. (21)

Tangential velocity distribution
Under the action of high-speed rotation in the separation device, different centrifugal forces are produced because of the density difference between oil and water, to achieve the effect of oil-water separation. The greater the tangential velocity in the same section is, the greater the centrifugal force on the droplets in the swirling device is, which is conducive to the droplet migration to the wall area of the swirling device, and more conducive to the swirling separation . However, a larger tangential velocity does not indicate a better outcome. If the tangential velocity is too large, droplets become deformed and broken by shear force, which aggravates the emulsification of oil and reduces the separation efficiency. Therefore, the tangential velocity is controlled within a certain range as much as possible. The radial distribution curve of tangential velocity on different sections is shown in Fig. 11. It shows with, with the increase of radius, the tangential velocity first increases and then decreases to zero. The tangential velocity of the three structures is small near the axis and wall. With the decrease Fig. 11 Tangential velocity distribution of three structures at different sections a z = 130 mm, b z = 650 mm, c z = 780 mm, d z = 840 m of z coordinate value, the tangential velocity decreases, and the surface along the z coordinate value decreases, which leads to energy loss. As can be seen from Fig. 11, near the underflow outlet (z = 130 mm section), the tangential velocity of the optimized double spherical tangent coupling device is higher than that of the straight double-cone type and the non-optimized double spherical tangent coupling device, which indicates that the droplets are more likely to move toward the wall due to greater centrifugal force, facilitating the separation of oil and water. In the cone region, the tangential velocity distribution of the straight double-cone coupling device and the non-optimized double spherical tangent coupling device have obvious deviation, and the tangential velocity distribution of the optimized double spherical tangent coupling device is symmetrical, indicating that the internal flow of the structure is stable, which is conducive to oil-water separation. At z = 780 mm section, the tangential velocity of the optimized double spherical tangent coupling device is larger in the range of 0.5 <|r/R|< 0.9, which is 25.03% higher than that of the straight double-cone coupling device, indicating that the droplet is subjected to greater centrifugal force, which promotes the droplet to move toward the wall and is more conducive to the separation of oil and water. Near the overflow port (z = 840 mm section), the tangential velocity of the optimized double spherical tangential coupling device is larger than that of the straight biconical coupling device, and the tangential velocity of the optimized double spherical tangent coupling device is 58.90% higher than that of the straight double-cone coupling device in the axial region, which reduces the water content in the axial region. It is favorable for the flowing liquid with high oil content to flow out from the overflow port and promote the oil-water separation.

Oil volume fraction distribution
The volume distribution of the oil volume fraction can reflect the movement of the W/ O emulsion during the separation in the coupling device. The oil volume fraction distributions of the three structures are shown in Fig. 12. In Fig. 12, the volume fraction of the oil volume fraction in the area near the inner wall of the swirl device decreases as the z-axis decreases and reaches the maximum value near the overflow port and the minimum value near the underflow port. The oil volume fraction decreases as the radius increases, and the oil phase is concentrated in the central area of the axis so that a stable oil core is formed in the central area. At this time, the maximum value is reached at the center of the axis, which is consistent with the oil phase distribution studied by Gong et al. (2020).
In Fig. 12, near the overflow port (z = 840 mm section), the high oil volume fraction, could promote the emulsion with high oil content to flow out from the overflow port, to achieve a high separation efficiency. It can be seen from the figure that at the cross section of z = 840 mm, the oil volume fraction of the optimized double spherical tangent coupling device in the axial region (0 <|r/R|< 0.2) is 0.64% higher than that of the straight double-cone coupling device, indicating that the oil volume fraction of the optimized structure near the overflow port is large, which is more conducive to the discharge of the fluid with higher oil content from the overflow port and improves the separation efficiency of the device. Near the underflow port (z = 130 mm section), the oil volume fraction of the non-optimized double spherical tangent coupling device is larger than that of the straight double-cone and optimized double spherical tangent coupling device. In the area of 0.5 <|r/R|< 1, the oil volume fraction of the optimized double spherical tangential coupling device is 36.61% lower than that of the straight double-cone coupling device, indicating that the oil volume fraction of the liquid discharged from the underflow port by the optimized double spherical tangential coupling device is small and the separation effect of the device. At z = 650 mm section, the oil volume fraction difference of the optimized double spherical tangent coupling device is 0.39, which is 8.33% and 30% higher than that of the straight double-cone coupling device and the non-optimized double spherical tangent coupling device, which indicates that the optimized double spherical tangent coupling device is more conducive to oil-water separation and improves the separation efficiency.

Droplet diameter distribution
Droplet diameter plays a decisive role in the centrifugal force on the emulsion. The water in the emulsion is separated by the centrifugal force in the swirl device, and the droplet diameter is an important factor affecting the centrifugal force (Yang et al. 2012;Schütz et al. 2009). Figure 13 shows the droplet diameter distribution of the straight double-cone, the non-optimized double spherical tangent, and the optimized double spherical tangent coupling devices at x = 0 mm section. In Fig. 13a, small droplets collide and coalesce under the coupling effect of double fields, and the droplet diameter gradually increases along the z-axis of the distance from the coupling device. In the axial region, effectively coalesce the droplets in the emulsion is difficult because of the small centrifugal force on the droplets, so the droplet diameter is small in the axial region of the coupling device. The double-cone section area is the main area of separation. In Fig. 13b, the droplet size of the straight double spherical tangent coupling device is the smallest, which is the least conducive to the swirling separation of droplet. The droplet size of the optimized double spherical tangent coupling device is the largest, indicating that the droplet is subject to great centrifugal force, and the oil-water separation is more sufficient, which is more conducive to improve the separation efficiency. In Fig. 13c, near the underflow port, the droplet size of the nonoptimized double spherical tangent coupling device is larger than that of the straight double-cone coupling device. At the same time, the optimized double spherical tangent coupling device is larger than that of the non-optimized double spherical tangent coupling device, which indicates that the optimized double spherical tangent coupling device is conducive to the discharge of the fluid with higher water content from the underflow port and improves the deoiling rate. Figure 14 shows the droplet diameter distribution of the double-field coupling demulsification-dewatering device with three structures. It can be seen that the droplet diameter distribution of the optimized double spherical tangent coupling device shifts to a larger direction than that of the straight double-cone and non-optimized double spherical tangent coupling device, indicating that the optimized double spherical tangent coupling device is more conducive to the coalescence of droplets, and the droplet is subject  Figure 15 shows the separation efficiency under different structures obtained via the numerical method. It can be seen from Fig. 15 that the separation efficiency of the double spherical tangent coupling device is 96.32%, which is 6.13% and 1.82% higher than that of the straight double-cone coupling device and the non-optimized double spherical tangent coupling device, respectively, indicating that the optimized double spherical tangent coupling device has better separation performance. The reason is that the coupling device carries out a lot of oil-water separation in the double-cone region. Because the straight double-cone coupling device is not smooth in the cone region, the flow field will vibrate in the cone region, which is not conducive to the coalescence of droplets. The double spherical tangent coupling device can solve this problem well. In the cone region, the connection is smooth. Compared with the coupling device with straight double cone, for the device with double spherical tangent cone, the flow field is more stable, and the static pressure value is larger, the pressure drop is smaller, and the energy loss is low. The larger the tangential velocity is, the larger the centrifugal force is, which is conducive to the droplet migration to the wall. The oil volume fraction near the overflow port is larger, which is conducive to the discharge of emulsion with higher oil content from the overflow port. The oil volume fraction on the wall of the underflow port is smaller, which is conducive to the outflow of emulsion with higher water content from the underflow port. The larger droplet size is conducive to improve the separation efficiency of the device.

Conclusion
In this study, a new double-field coupling device with double spherical tangent cone structure is presented. In the numerical calculation, the droplet coalescence and breakup equations are added to the continuity equation, electric field governing equation, and momentum conservation equation to realize the coupling effect of CFD and PBM. The double spherical tangent cone structure of the coupling device is optimized. The flow field and separation efficiency of the double spherical tangent and the straight double-cone coupling device are analyzed, and confirmatory experiments Fig. 13 a Contours of droplet diameter at the section x = 0 mm (A: straight double-cone structure, B: not optimized double spherical tangent cone structure, and C: optimized double spherical tangent cone structure); b at x = 0 and 500 mm < z < 800 mm; C at x = 0 and 0 < z < 200 mm  are carried out. The results show that the numerical model can reasonably predict the separation performance of the coupling device. The separation efficiency of the optimized double spherical tangent coupling device is higher than that of the straight double-cone and non-optimized double spherical tangent coupling devices. Under the same conditions, the optimized double spherical tangent coupling device has a lower pressure drop and less energy loss. The tangential velocity on the same section is large, increasing the droplet size, promoting droplet migration to the wall, and enhancing the oil-water separation ability of the emulsion. The oil volume fraction of the optimized double spherical tangent coupling device is higher near the overflow port, and near the underflow, showing that the optimized double spherical tangent coupling device is more conducive to the discharge of fluid with high oil content from the overflow and the outflow of fluid with low oil content from the underflow port, which improves the separation efficiency of the coupling device. The separation efficiency of the optimized double spherical tangent coupling device is 6.13%and 1.82% higher than that of the straight double-cone and the non-optimized double spherical tangent coupling device, respectively. The optimized double spherical tangent coupling device has the best separation performance.