Design and Analysis of an Axial Center-Piercing Hydrocyclone

Abstract

To enhance the separation efficiency of downhole oil–water hydrocyclones in co-produced wells, an axial center-piercing hydrocyclone structure is proposed. A mathematical relationship model between the structural parameters and separation efficiency of the axial center-piercing hydrocyclone is constructed based on the response surface methodology. The numerical simulation method is employed to analyze both the unoptimized and optimized hydrocyclone structures, and their separation performances are simulated under identical operational conditions. The results indicate that the optimal structural parameter values are as follows: main diameter D = 70.4 mm, large cone angle α₁ = 32.4°, small cone angle α₂ = 3.9°, bottom flow tube length L₃ = 311.7 mm, and inverted cone length L₄ = 166.0 mm. The optimal operation parameters of the optimized structure are also obtained. Under the same operating parameter conditions, the separation efficiency of the optimized structure is consistently higher than that of the unoptimized structure. The highest efficiency achieved by the optimized structure is 98.6%, which is a 2% improvement over the unoptimized state. Finally, experiments are conducted with the optimized hydrocyclone separator structure under different split ratios. This study significantly contributes to the field of injection and production in a single well, particularly in promoting the application of hydrocyclones.

1. Introduction

The rise in water content within oil fields has given rise to a spectrum of associated challenges [1]. These include environmental issues stemming from the use of chemical agents for wastewater treatment, as well as economic challenges related to oil field development. As researchers focus their attention on addressing the challenges posed by high water content in oil fields, a pivotal advancement has emerged in the form of single-well injection and production technology [2]. This technology can enable the dual-purpose utilization of a single well, thereby enhancing the efficiency of individual wells. Additionally, it offers a means to significantly curtail ineffective water circulation, consequently reducing the need for extensive surface pipeline infrastructure and yielding substantial production cost savings. Undoubtedly, this technology assumes a vital role in realizing energy conservation strategies and facilitating the provision of oil-based energy. The hydrocyclone, renowned for its simplicity in operation, compact equipment design, and cost effectiveness [3,4,5], plays a pivotal role in the implementation of single-well injection technology. Hence, researchers have extensively investigated hydrocyclones, delving into various aspects such as novel structural design [6,7], heightened separation efficacy [8], internal flow-field attributes [9,10], and expanding application domains [11]. Xing et al. [12] and Zhao et al. [13,14] innovatively employed a double-stage tandem connection for cyclones in downhole working conditions. This approach was combined with the optimization of structural parameters to improve the separation efficiency. Wang et al. [15] conducted numerical simulations of the internal flow field within the downhole liquid–liquid hydrocyclone around its axis. Their findings revealed that the alterations in tangential velocity are not symmetrical in cases of forward and reverse rotation. Consequently, they concluded that a mutually constraining relationship exists among tangential velocity, axial velocity, and radial velocity during rotational conditions. Bai et al. proposed a new type of axial hydrocyclone [16], and its indoor prototype experiments revealed an optimal shunt ratio of 0.45 and an optimal treatment capacity of 1.50 m³/h. Comparative analysis with the traditional tangential hydrocyclone demonstrated the superior separation performance of this device. Ding et al. [17] devised six variations of oil–water separation cyclones for offshore downhole oil–water separation equipment. By assessing the factors of cost and separation efficiency, they initially identified the optimal model. Through laboratory experiments and simulations, they further established the best structural parameters of the model. As a result, the highest achievable separation efficiency reached 97.23%. Gao et al. [18] investigated the oil–water separation process in the downhole working conditions of a pumping unit. They conducted a comparative study on the pressure drop of the oil–water separator, and analyzed its behavior under both pulsating and stable inlet flows. The results indicated that the pressure drop under pulsating flow (1.893 MPa) was higher than that under stable flow (0.142 MPa). Xu et al. [19] devised a novel spiral separator and employed a combination of numerical simulation and particle image velocimetry (PIV) testing to scrutinize its internal flow-field and pressure characteristics. Their results revealed the formation of a stable spiral flow field at the outlet section, and the pressure loss remained within 90 Kpa under the design conditions. Considering the significant challenges involved in downhole working conditions, such as the presence of gas in produced fluids, sand content in produced liquids [20], and the escalating societal demand for oil resources, there has been a notable focus on enhancing the crude oil recovery of downhole hydrocyclones. The separation efficiency of a hydrocyclone is influenced by its structural parameters, operational parameters, and physical properties of the separation medium [21,22,23]. Scholars frequently employ the single-factor method [24], as well as the response surface method, to optimize the structural parameters of hydrocyclones. Xing et al. [25] optimized the structure of a downhole cyclone and a micro-cyclone by the response surface method and carried out comparative experiments on the optimized structures and the initial structures. The experimental results showed that the quality and efficiency of the optimized structures were higher than those of the initial structures under the same parameters. Liu et al. [26] introduced a novel hydrocyclone structure and optimized its structural parameters, including the overflow tube, blade parameters, and cone angle, using the single-factor method. Subsequently, they further refined the optimized hydrocyclone structure through the application of the response surface method. The oil–water hydrocyclone separator finds extensive application in simultaneous well injection and production technology due to its straightforward operation, compact equipment, and cost-effective maintenance. However, within real downhole conditions, a double electric pump is employed, which often necessitates passage through a central shaft in the hydrocyclone. The structural aspects of the hydrocyclone under these conditions have seldom been investigated.

In this paper, we introduce a novel design for an axial center-piercing hydrocyclone and employ the response surface method to optimize its performance. By establishing a mathematical model that correlates structural parameters with separation efficiency, we identify the optimal configuration that ensures highly efficient separation. Simultaneously, utilizing computational fluid dynamics, we investigate the flow-field distribution characteristics and oil–water separation efficiency of the hydrocyclone with the optimal structural parameters under various operating conditions. This analysis aims to tailor the hydrocyclone for effective downhole application in central-shaft-piercing scenarios. Consequently, our study offers crucial support and assurance for the successful implementation of hydrocyclones in downhole applications.

2. Structure and Working Principle

Figure 1 illustrates the utilization of an axial center-piercing hydrocyclone in single-well injection and production technology. This single-well injection and production system includes several key components, namely the electric potential centrifugal production pump, central shaft, axial center-piercing hydrocyclone, and motor (responsible for powering both the production pump and the reinjection pump). The operational principle of the process system can be outlined as follows. The fluid produced is introduced into the hydrocyclone from the inlet, and it undergoes interaction with the high-speed rotating flow field within the hydrocyclone. As a result, the fraction of the produced liquid with a substantial oil content is propelled upward to the surface within the overflow cavity, facilitated by the production pump. Conversely, the portion with a lower oil content travels through the bottom flow chamber and is subsequently reinjected into the water layer through the dedicated reinjection pump. Consequently, the process encompasses the entirety of an injection–production procedure, involving the separation of an oil–water two-phase liquid, the elevation of the oil phase, and the subsequent reinjection of the water phase. The operational mechanism for oil–water separation within the hydrocyclone is delineated as follows. The incoming produced liquid enters the hydrocyclone by a tangential inlet. Here, the lighter oil phase amalgamates to create an oil core at the periphery of the central shaft due to the influence of centrifugal force. Under the initial structure is shown in Figure 2, the specific parameters are shown in

Table 1. Main structural parameters of the fluid domain of an axial center-piercing hydrocyclone.

Parameters	D	L1	L2	L3	L4	α1	α2
Numerical value	78 mm	80.0 mm	535.0 mm	300.0 mm	166.0 mm	40.0°	3.0°

. The main diameter D = 78 mm, the large cone length L₁ = 80 mm, the small cone length L₂ = 535 mm, the bottom flow tube length L₃ = 300 mm, the inverted cone length L₄ = 166 mm, the large cone angle α₁ = 40°, and α₂ = 3°.

3. Research Methods

3.1. Mathematical Model Method

ANSYS-Fluent 2021R1 software was utilized to conduct numerical simulations and analyses to discern the flow-field characteristics and separation efficiency of the axial center-piercing hydrocyclone. To tackle the intricate flow patterns within the high-speed rotating environment of the axial center-piercing hydrocyclone, the Reynolds stress model (RSM) was employed. This model was chosen for its precise accuracy and its capability to mitigate computational intensity. Moreover, the RSM effectively accounts for fluid swirl and near-wall influences, enabling a more comprehensive representation of fluid turbulent behaviors.

In the process of numerical simulation, the continuity equations of water and oil are as follows:

$$\frac{\partial }{\partial t}({\alpha }_{\mathrm{c}}{\rho }_{\mathrm{c}})+\nabla ({\alpha }_{\mathrm{c}}{\rho }_{\mathrm{c}}{u}_{\mathrm{c}})=0$$

(1)

$$\frac{\partial }{\partial t}({\alpha }_{\mathrm{d}}{\rho }_{\mathrm{d}})+\nabla ({\alpha }_{\mathrm{d}}{\rho }_{\mathrm{d}}{u}_{\mathrm{d}})=0$$

(2)

For incompressible flows, the time-averaged Navier-Stokes equation was employed as per the literature [27].

$$\frac{\partial \overline{u_j}}{\partial x_j}=0$$

(3)

$$\rho \frac{\partial \overline{u_i}}{\partial t}+\rho \frac{\partial (\overline{u_i}\overline{u_j})}{\partial x_j}=-\frac{\partial \overline{p}}{\partial x_{\mathrm{i}}}+\frac{\partial }{\partial x_j}(\mu \frac{\partial \overline{u_i}}{\partial x_j}-\rho \overline{u^{\prime }{}_i{u}^{\prime }{}_j})$$

(4)

In the above formula, x_i and x_j are coordinate components, $\overline{u_i}$ and $\overline{u_j}$ are time-averaged velocity components, $\overline{p}$ is the time-averaged pressure in Pa, μ is the hydrodynamic viscosity in (Pa·s)/m², ρ is the fluid density in kg/m³, and $\overline{u^{\prime }{}_i{u}^{\prime }{}_j}$ is the Reynolds stress component. The transport equation in the Reynolds stress model is:

$$\partial \frac{\left(\rho \overline{u^{\prime }{}_i{u}^{\prime }{}_j}\right)}{\partial t}+\partial \frac{\left(\rho u_k\overline{u^{\prime }{}_i{u}^{\prime }{}_j}\right)}{\partial x_k}=D_{ij}+P_{ij}+{\Phi }_{ij}+{\epsilon }_{ij}$$

(5)

In the above formula, D_ij is the diffusion term, P_ij is the stress generation term, Φ_ij is the stress strain term, and ε_ij is the viscous dissipation term.

$$D_{ij}=\frac{\partial }{\partial x_k}\left(\frac{{\mu }_{\mathrm{t}}}{{\sigma }_k}\times \frac{\partial \overline{u_i{}^{\prime }{u}_j{}^{\prime }}}{\partial x_k}+\mu \frac{\partial \overline{u_i{}^{\prime }{u}_j{}^{\prime }}}{\partial x_k}\right)$$

(6)

$$P_{ij}=-p(\overline{u_i{}^{\prime }{u}_j{}^{\prime }}\frac{\partial u_i}{\partial x_k}+\overline{u_j{}^{\prime }{u}_k{}^{\prime }}\frac{\partial u_i}{\partial u_k})$$

(7)

$${\varphi }_{ij}=-C_1\rho \frac{\epsilon }{k}\left(\overline{u_i{}^{\prime }{u}_j{}^{\prime }}-\frac{2}{3}{\delta }_{ij}k\right)-C_2\left(P_{ij}-\frac{1}{3}{\delta }_{ij}{P}_{kk}\right)$$

(8)

$${\epsilon }_{ij}=\frac{2}{3}\rho \epsilon {\delta }_{ij}$$

(9)

In the formula, μ_t is the turbulent viscosity, C₁ and C₂ are constants with C₁ = 1.8 and C₂ = 1.8, k is the turbulent kinetic energy, ε is the dissipation rate, δ_ij is the Kronecker-delta function, P_kk = 2p, and σ_k is the Prandtl number.

Boundary Conditions and Efficiency

Figure 3 shows the fluid domain and meshing of the hydrocyclone. The multiphase flow mixed model (mixture model) was employed to simulate the interaction between oil and water. In this approach, the inlet for the mixed-phase liquid was designated as a velocity inlet (velocity-int), while the overflow outlet and the bottom outlet were configured as free outlets (out-flow). The density of the oil phase was defined as 880 kg/m³, while the dynamic viscosity of the oil phase was set at 112.2 mPa·s. Additionally, the diameter of the oil droplets was established as 0.35 mm. In this context, the oil phase was designated as the discrete phase. On the other hand, the density of the water phase was established as 998 kg/m³, and the dynamic viscosity of the water phase was set at 1.003 mPa·s. Consequently, the water phase was regarded as the continuous phase. The turbulence calculation was performed utilizing the RSM. For the coupling of velocity and pressure, the SIMPLEC algorithm, known for its robust computational stability and excellent convergence, was employed. Non-slip boundary conditions were assigned to the walls. For discretization, the momentum, turbulent kinetic energy, and turbulent dissipation rate were subjected to a second-order upwind scheme. The set convergence accuracy was 10⁻⁶. Impermeable and non-slip boundary conditions were assigned to the walls. The operational parameter range of the simulation was from 40 m³/d to 150 m³/d. The diversion ratio was defined within a range of 15% to 60%, and the moisture content was considered within a range of 92% to 99%.

The oil–water separation efficiency serves as the key parameter in assessing the simulation outcomes of the axial center-piercing hydrocyclone. The separation performance of the hydrocyclone is evaluated based on the mass efficiency. The calculation formula [28] is given as follows:

$$E_{\mathrm{Z}}=1-(1-F)\frac{{\phi }_{\mathrm{d}}}{{\phi }_{\mathrm{i}}}$$

(10)

In the above formula, E_Z is the mass efficiency of the hydrocyclone, F is the overflow shunt ratio, φ_d is the oil concentration at the bottom outlet in mg/L, and φ_i is the inlet oil concentration in mg/L.

3.2. Optimal Design of Response Surface

3.2.1. Basic Principles of Response Surface

Response surface analysis directly shows the influence of variables on the target value, and also reflects the interaction between variables. Through a reasonable and limited experimental design method, the response surface method constructs the approximate function expression between the objective function and variables, and fits the global approximation instead of the real response surface model. The original implicit objective function is explicit. On this basis, a set of optimal solutions is obtained, which can simplify the analysis process, solve the optimization problem, and improve the optimization efficiency. The basic relationship between the response dependent variable Y and the design independent variable x is as follows:

$$Y=\overline{y}(x)+\delta$$

(11)

In the above formula, $\overline{y}(x)$ represents the approximate expression of the unknown mathematical model, x stands for the design independent variable, and δ denotes the error value. The quadratic polynomial expression [29] derived from the fundamental relationship is presented as follows:

$$y(x)={\beta }_0+{\displaystyle \sum _{i=1}^n{\beta }_i{x}_i}+{\displaystyle \sum _{i=1}^n{\beta }_i{}_i}{x}_i^2+{\displaystyle \sum _{1\le i\le j}^n{\beta }_{ij}{x}_i{x}_j}+\epsilon$$

(12)

In the above formula, y(x) depends on the target value, β₀ and β_j are polynomial coefficients, x_j is the j experimental scalar, β_ii represents the second order effect of factor x_j, β_ij represents the interaction effect of factor x_i and x_j, and n is the number of experimental variables. ε represents the response observation error. The parameters of the mathematical model are determined by the least-squares method, and then the response surface analysis is carried out.

3.2.2. Response Surface Optimization Design

The BBD (Box–Behnken design) test of the response surface method was employed to establish the testing scheme for evaluating the influence of structural parameters on the efficiency of the axial center-piercing hydrocyclone. The levels and corresponding coding values for each factor are provided in

Table 2. Horizontal design of CCD (central composite design) test factors.

Factors	Symbol	Horizontal
		Lower Limit (−1)	Upper Limit (1)	Central Point (0)
Length of inverted cone L1/mm	x1	100	200	150
Angle of the large cone α1/°	x2	32	48	40
Angle of the small cone α2/°	x3	2	4	3
Length of underflow pipe L2/mm	x4	250	350	300
Principal diameter L3/mm	x5	70	86	78

. Utilizing the BBD experimental design approach, the five targeted structural parameters were treated as independent variables. The number of factors was defined as 5, and the optimal values obtained from the single-factor method served as the central values for these five factors. Given the absence of interfering factors, such as variations in the experimental environment during the simulation process, the Block factor was set to 1. Additionally, the central test was repeated six times, while the number of dependent variables remained 1.

By considering the distinct levels of structural parameters as presented in Table 2, a total of 46 BBD experimental design groups were configured. For each experimental group, a fluid domain model of the hydrocyclone was established, adhering to the specific structural parameters assigned to that group. The model was then uniformly meshed to facilitate numerical simulation analysis. Subsequently, the efficiency (E_z) of each test group was computed. The outcomes of the response surface test design and the results obtained from numerical simulations are summarized in

Table 3. BBD design and test results.

Serial Number	x1	x2	x3	x4	x5	Ez/%
1	150	20	1.5	250	35	94.36
2	100	20	1.5	250	39	93.96
3	200	24	1.5	300	39	92.49
4	150	20	1	250	39	93.56
5	150	16	1.5	300	35	96.13
6	150	20	1	350	39	94.04
7	150	16	2	300	39	96.7
8	100	20	2	300	39	96.12
9	150	16	1.5	250	39	95.69
10	150	20	1	300	43	92.72
11	150	20	1.5	250	43	92.45
12	150	24	2	300	39	95.5
13	200	20	1.5	300	35	95.2
14	150	20	1.5	300	39	94.41
15	200	20	1.5	300	43	93.02
16	200	20	1.5	250	39	94.13
17	150	20	1.5	300	39	94.42
18	150	20	1.5	300	39	94.42
19	150	20	1.5	300	39	94.42
20	150	20	1.5	350	35	94.95
21	200	20	2	300	39	96.23
22	200	20	1	300	39	94.11
23	150	20	2	350	39	96.28
24	150	20	1.5	350	43	92.99
25	150	16	1.5	350	39	95.97
26	150	16	1.5	300	43	95.24
27	150	20	2	250	39	96.01
28	100	20	1.5	300	43	92.86
29	150	24	1	300	39	91.66
30	100	20	1.5	350	39	94.42
31	150	20	2	300	35	97.55
32	150	20	2	300	43	93.3
33	150	24	1.5	300	35	93.87
34	150	20	1	300	35	94.03
35	200	20	1.5	350	39	93.33
36	150	20	1.5	300	39	94.42
37	150	24	1.5	250	39	92.1
38	150	24	1.5	350	39	91.43
39	150	24	1.5	300	43	92.63
40	100	20	1	300	39	93.9
41	150	16	1	300	39	95.88
42	100	20	1.5	300	35	92.55
43	100	24	1.5	300	39	91.49
44	200	16	1.5	300	39	94.17
45	150	20	1.5	300	39	94.42
46	100	16	1.5	300	39	94.4

.

The quadratic polyphase fitting of the data shown in Table 3 was carried out by using the second-order model. The regression equation between the optimized structure parameter x and the separation efficiency E_Z of the axial center-piercing hydrocyclone was obtained by multivariate linear regression analysis, as shown in Equation (13).

$$\begin{array}{l}{E}_Z=15.03247+0.19354x_1+0.04125x_2+0.56125x_3+0.10357x_4+2.77062x_5+1.53750\times {10}^{-3}{x}_1{x}_2\\ -1\times {10}^{-3}{x}_1{x}_3-1.26\times {10}^{-4}{x}_1{x}_4-3.1125\times {10}^{-3}{x}_1{x}_5+0.3775x_2{x}_3-1.1875\times {10}^{-3}{x}_2{x}_4-5.4688\times {10}^{-3}{x}_2{x}_5\\ +2.1\times {10}^{-3}{x}_3{x}_4+0.36750x_3{x}_5-6.25\times {10}^{-5}{x}_4{x}_5-1.995\times {10}^{-4}{x}_1{}^2+0.015703x_2{}^2+3.075x_3{}^2-8.98333\times \\ {10}^{-5}{x}_4{}^2-0.0235416x_5{}^2\end{array}$$

(13)

4. Results and Discussion

4.1. Model Optimization Result

To evaluate the accuracy of the regression equation, the least-squares method was employed to derive the mathematical relationship between the structural parameter (x) and the separation efficiency (E_Z). The structural parameter configuration leading to the maximum separation efficiency was determined as the optimization outcome of the response surface design. Upon optimization, it was found that the inverted cone length was 166.0 mm, the large cone angle was 32.4°, the small cone angle was 3.9°, the underflow tube length was 311.7 mm, and the main diameter was 70.4 mm. The comparative representation of the structural parameters of the axial center-piercing hydrocyclone before and after optimization is shown in Figure 4.

4.2. Analysis of Variance

The variance analysis method was employed to test the significance of the mathematical model involving the five structural parameters and separation efficiency constructed by the response surface method. The results of the variance analysis are presented in

Table 4. Analysis of variance results of the regression equation.

Types	Sum of Squares of Deviation	Degree of Freedom	Mean Square	F-Value	p-Value
Model	84.43	20	4.22	9.36	<0.0001
A—Inverted cone length	0.56	1	0.56	1.23	0.2778
B—Angle of the large cone	33.09	1	33.09	73.37	<0.0001
C—Angle of the small cone	19.78	1	19.78	43.86	<0.0001
D—Length of underflow pipe	0.083	1	0.083	0.18	0.6722
E—Principal diameter	11.27	1	11.27	24.99	<0.0001
AB	0.38	1	0.38	0.84	0.3685
AC	2.500 × 10−3	1	2.500 × 10−3	5.543 × 10−3	0.9412
AD	0.40	1	0.40	0.88	0.3572
AE	1.55	1	1.55	3.44	0.0756
BC	2.28	1	2.28	5.06	0.0336
BD	0.23	1	0.23	0.50	0.4859
BE	0.031	1	0.031	0.068	0.7965
CD	0.011	1	0.011	0.024	0.8770
CE	2.16	1	2.16	4.79	0.0382
DE	6.250 × 10−4	1	6.250 × 10−4	1.386 × 10−3	0.9706
A2	2.17	1	2.17	4.81	0.0377
B2	0.55	1	0.55	1.22	0.2796
C2	5.16	1	5.16	11.44	0.0024
D2	0.44	1	0.44	0.98	0.3327
E2	1.24	1	1.24	2.74	0.1104
Residual	11.28	25	0.45
Lack of Fit	11.28	20	0.56	33825.40	<0.0001
Pure Error	8.333 × 10−5	5	1.667 × 10−5
Cor Total	95.70	45

. It can be seen that the regression equation illustrating the relationship between the structural parameters and separation efficiency (E_Z) has a significance level of P < 0.0001 < 0.05. This implies that the functional relationship portrayed by the regression equation is statistically significant. Within the range of the upper and lower limit parameters specified in Table 4, the regression equation, shown in Equation (7), can be utilized to predict the inverted cone length, large cone angle, small cone angle, bottom flow tube length, main diameter, and separation efficiency of the axial center-piercing hydrocyclone under different multi-structural parameter conditions.

To verify the predictive accuracy of the regression equation, an error statistical analysis of Equation (10) was conducted. The error statistical analysis results are presented in

Table 5. Statistical analysis of regression model errors.

Statistical Item	Value	Statistical Item	Value
Std. Dev.	0.67	R-squared	0.8822
Mean	94.22	Adj R-squared	0.7879
C.V. %	0.71	Pred R-squared	0.5287
PRESS	45.10	Adeq Precision	13.380

. The closer the correlation coefficient R-squared (coefficient of determination) is to 1, the stronger the correlation. Higher and more closely aligned values of Adj-Squared (Adjusted R-squared) and Pred-Squared (Prediction Variance Explained) indicate that the regression model effectively captures the relationship between input and output variables. A coefficient of variation (C.V.) < 10% signifies higher accuracy and credibility of the test results. Adeq Precision represents the ratio of effective signal to noise, and a value exceeding 4 indicates that the model is reasonable. The statistical analysis results confirmed that the regression models adhered to the aforementioned testing principles, implying robust applicability for both models.

4.3. Comparison of Structural Shunt Performance before and after Optimization

To analyze the variations in the internal tangential velocity of the axial center-piercing hydrocyclone with treatment capacity before and after optimization, the treatment volumes of the mixture were set at 64 m³/d, 88 m³/d, and 112 m³/d, respectively. Subsequently, the tangential velocities were measured at three distinct sections, as depicted in Figure 5. These sections include section I at the entrance of the large cone segment, section II at the interface between the large cone and small cone sections, and section III at the entrance of the inverted cone. The tangential velocity distribution within the hydrocyclone is illustrated in Figure 5. Maintaining consistent operating conditions, the tangential velocity was highest at section I, followed by section II, and then section III with the lowest velocity. With the increase in processing capacity, the tangential velocities at sections I, II, and III exhibited a gradual increase both before and after optimization. Under identical operational parameters, the tangential velocities at sections I, II, and III of the optimized hydrocyclone were greater than those of the axial center-piercing hydrocyclone at the same respective positions before optimization.

Figure 6 presents a comparison of the oil-phase volume fraction within the hydrocyclone at sections I and II, as depicted in Figure 4, under the same operational parameters. Figure 6 reveals that the oil-phase volume fraction at sections I and II post-optimization was notably superior to that prior to optimization. This outcome can be attributed to the higher tangential velocities at sections I and II following optimization. The tangential velocity governs the centrifugal force within the flow field of the hydrocyclone. Within a specific range, higher tangential velocities are conducive to enrichment of the oil phase. It is evident that the optimized response surface structure can increase the oil-phase volume fraction within the separation chamber by augmenting the tangential velocity within the swirl field. This in turn enhances the separation efficiency of the mixed liquid, demonstrating the superior separation performance of the optimized hydrocyclone structure. This outcome verified the efficacy of the response surface optimization method in enhancing the separation performance of the axial center-piercing hydrocyclone.

To further validate the impact of the optimized hydrocyclone on separation efficiency, numerical simulations were conducted for various water contents using both the optimized and unoptimized models. The water content of the inlet mixture varied between 92% and 99%, with a treatment capacity of 150 m³/d and a diversion ratio of 30%. The same numerical simulation methodology and boundary conditions as described above were employed for this analysis. Through a comparison of separation efficiency and oil concentration for the unoptimized and optimized hydrocyclone models with different water content conditions, the separation efficiency curve depicted in Figure 7 was obtained. As the water content in the inlet mixture increased, the separation efficiency of the cross-axis cyclone showed a gradual improvement. As the moisture content of the inlet mixture varied from 92% to 99%, the separation efficiency of the unoptimized structure rose from 88.2% to 96.3%, while the separation efficiency of the optimized structure increased from 94.7% to 97.2%. Under identical conditions, the optimized structure consistently exhibited higher separation efficiency compared to the unoptimized structure. The peak separation performance of 97.2% was achieved when the moisture content reached 98%. This indicated that the hydrocyclone structure optimized through the response surface approach exhibited superior separation performance under varying water content conditions.

To further evaluate the applicability of the optimized hydrocyclone structure under varying mixture capacities, a comparative study was conducted by numerically simulating the optimized and unoptimized hydrocyclone across different Q_i conditions. The mixture treatment capacity was ranged from 40 m³/d to 150 m³/d, the mixture’s water content was set at 98%, and the split ratio was maintained at 30%. A comparative graph illustrating the separation efficiency and oil concentration at the bottom flow outlet of the hydrocyclone before and after optimization with different mixture capacities is depicted in Figure 8. It can be seen that as the mixture capacity increased, the separation efficiency of the hydrocyclone also exhibited a gradual enhancement. Prior to optimization, the structure’s separation efficiency rose from 48.4% to 96.6%, whereas the separation efficiency of the optimized cyclone structure increased from 56.5% to 98.6%. The optimal treatment capacity was observed at 100 m³/d. For the same mixture capacity, the optimized hydrocyclone exhibited superior separation efficiency compared to the unoptimized structure.

Figure 9 presents a comparison of the separation performance between the hydrocyclone structures before and after optimization with different shunt ratios. The split ratio, denoted as F, was varied within the range of 6% to 39%. The mixture treatment capacity remained fixed at 100 m³/d, and the water content was set at 98%. The separation efficiency of the hydrocyclone exhibited an initial increase followed by a subsequent decrease as the shunt ratio varied. The lowest separation efficiency for both the unoptimized and optimized structures was observed at a shunt ratio of F = 6%, resulting in 48% separation efficiency before optimization and 50% after optimization. However, at a shunt ratio of F = 30%, the separation efficiency reached its peak. Specifically, before optimization, the separation efficiency was 96.6%, while after optimization, it improved to 98.6%. This trend underscores the suitability of the designed hydrocyclone separator for high shunt ratio conditions. Moreover, the improved separation efficiency after optimization, compared to the unoptimized state, for varying shunt ratios validated the enhanced reliability of the optimized structure’s separation performance.

4.4. Experimental Verification

Utilizing the insights gained from the response surface analysis, an optimized prototype of the axial center-piercing hydrocyclone was fabricated. The prototype was constructed using plexiglass material for transparent observation of the internal flow patterns. Focusing on the overflow split ratio of the prototype, experimental investigations were conducted into the separation performance of the optimized rear cyclone under varying shunt ratios. The experimental procedure is illustrated in Figure 10. During the experimentation, a frequency conversion controller was used to regulate the screw pump’s frequency for managing the water intake from the water tank. The liquid supply of the metering plunger pump from the oil tank was controlled by adjusting the measuring scale. The shunt ratio was manipulated by installing cut-off valves on the inlet, overflow, and bottom outlet pipelines. Real-time monitoring of flow and pressure at the entrance was achieved using a float flowmeter and pressure gauge, respectively.

Heavy-duty gear oil with similar viscosity and simulated density was utilized as the experimental oil. The viscosity of the oil phase at 25 °C was 1.03 Pa·s, and the density was 880 kg/m³, as determined by a Malvern rheometer. In the experiment, water and oil were pumped through a screw pump and a metering plunger pump, respectively. After thorough mixing in a static mixer, the oil–water mixture was directed into the hydrocyclone prototype. Following the swirling separation of oil and water, the oil phase exited through the central axis overflow port, while the water phase exited from the sidewall bottom flow port.

To verify the accuracy of the aforementioned numerical simulation methods and results, the internal oil core state of the flow field was examined at different overflow split ratios, specifically 6%, 18%, and 30%. The distribution of the oil core under different shunt ratios is depicted in Figure 11. As the shunt ratio increased, the lower part of the oil core became shorter, and the simulated position closely resembled the experimental oil core under the same parameters. The comparison of the separation performance between the simulation and experiment under different shunt ratios is illustrated in Figure 10. The same 12 experimental points were replicated in the simulation tests, with the inlet liquid intake adjusted to the optimal treatment capacity of the optimized hydrocyclone (100 m³/d), while maintaining a stable water content of 98%. Once the inlet reached stability, the separation efficiency initially increased and then decreased as the overflow shunt ratio was raised (Figure 12). When the shunt ratio was below 30%, the separation performance of the hydrocyclone improved as the split ratio increased. However, when the split ratio F was less than 15%, the efficiency of the hydrocyclone showed significant changes. Experimentally, the efficiency increased from 48% to 82%, and the optimal separation performance occurred at a shunt ratio of F = 30%. The maximum separation efficiencies observed in the simulation and experiment were 98.6% and 97.1%, respectively.

As the shunt ratio continued to increase, the separation efficiency declined. The change pattern of the internal oil core and separation efficiency of the hydrocyclone exhibited a more consistent trend between experimental and numerical simulation results with an increase in shunt ratio. This finding confirmed the reliability of the numerical simulation method.

In a similar manner, an experiment was conducted to examine the impact of inlet liquid intake on the separation efficiency of the axial center-piercing hydrocyclone. The inlet flow parameters were adjusted within the range of 40 m³/d to 150 m³/d, covering 10 experimental points identical to those in the simulation tests of the separator. The overflow shunt ratio was maintained at a stable 30%. The comparison between the experimental and simulated separation efficiency values of the axial center-piercing hydrocyclone, as influenced by varying inlet liquid intake, is illustrated in Figure 13. The separation efficiency of the over-axis cyclone rose with the increase in capacity up to 100 m³/d. However, as the liquid intake further increased, the separation efficiency tended to decrease. When the processing capacity was below 100 m³/d, both the experimental and simulation efficiencies showed significant changes. Specifically, the experimental efficiency increased from 58% to 97.1%. The optimal processing capacity for the axial center-piercing hydrocyclone was determined to be 100 m³/d. At this capacity, the experimental separation efficiency reached 97.1%, while the simulation efficiency was 98.6%. Therefore, there was strong alignment between the experimental and numerical simulation results, affirming the validity of the proposed simulation method.

5. Conclusions

In this paper, we present an innovative design for an axial center-piercing hydrocyclone and improve its performance using the response surface method. By constructing a mathematical model that links structural parameters with separation efficiency, we pinpoint the optimal configuration for achieving efficient separation.

• According to the downhole working conditions of injection and production in the same well, a novel axial center-piercing hydrocyclone structure was proposed. The optimal structural parameters were determined through the response surface method, as follows: main diameter D = 70.4 mm, large cone angle α₁ = 32.4°, small cone angle α₂ = 3.9°, bottom flow tube length L₃ = 311.7 mm, and inverted cone length L₄ = 166.0 mm.
• The reliability of the optimized structure was validated through a comparison of internal tangential velocity cloud images and the distribution of the oil-phase volume fraction before and after optimization. Numerical simulations of the optimized structure demonstrated that the separation efficiency after optimization surpassed that before optimization under equivalent parameter conditions, confirming the enhanced efficiency of the optimized structure. The optimal operational parameters for the optimized over-axis cyclone were determined. At a treatment capacity of 100 m³/d, water content of 98%, and a shunt ratio of 30%, the separation efficiency reached its peak at 98.6%, a 2% improvement over the pre-optimized configuration.
• Reliability verification of the experiments was conducted under various shunt ratios and processing capacities. The experimental results showed close agreement with the numerical simulation results.

By optimizing the axial center-piercing hydrocyclone, we enhance its suitability for central piercing conditions, thereby offering crucial technical support and assurance for its utilization in downhole applications. This study constitutes a significant contribution to the broader field of downhole technology, particularly in advancing the application of hydrocyclones.