Intensification of Multiphase Reactions in Petroleum Processing: A Simulation Study of SK Static Mixer Using NaClO for H₂S Removal

Abstract

During crude oil exploration and extraction, the presence of H₂S not only poses a threat to operational safety but also accelerates equipment corrosion, highlighting the urgent need for efficient and cost-effective processing solutions. This study employs a coupled numerical simulation approach that integrates computational fluid dynamics (CFD) and population balance models (PBM) to systematically investigate the multiphase flow characteristics within SK static mixers. By embedding mass transfer rates and reaction kinetics equations for hydrogen sulfide and sodium hypochlorite into the Euler-Euler multiphase flow model using user-defined functions (UDFs), the effects of equipment structure on the efficiency of the crude oil desulfurization process are examined. The results indicate that the optimized SK static mixer (with 15 elements, an aspect ratio of 1, and a twist angle of 90°) achieves an H₂S removal efficiency of 72.02%, which is 18.84 times greater than that of conventional empty tube reactors. Additionally, the micro-mixing time is reduced to 0.001 s, and the coefficient of variation (CoV) decreases to 0.21, while maintaining acceptable pressure drop levels. Using the CFD-PBM model, the dispersion behavior of droplets within the static mixer is investigated. The results show that the diameter of the inlet pipe significantly affects droplet dispersion; smaller diameters (0.1 and 1 mm) enhance droplet breakup through increased shear force and turbulence effects. The findings of this study provide theoretical support for optimizing crude oil desulfurization processes and are of significant importance for enhancing the economic efficiency and safety of crude oil extraction operations.

1. Introduction

Hydrogen sulfide (H₂S) is a natural impurity found in crude oil [1], originating from biological degradation, microbial sulfate reduction, thermochemical reduction of organic matter, and the thermochemical decomposition of sulfur-containing minerals [2]. The removal of H₂S from crude oil is critical for several reasons. Firstly, H₂S is a highly toxic gas that poses a significant threat to human safety during the processing and transportation of crude oil. Secondly, it is corrosive and can cause substantial damage to equipment. Thirdly, H₂S negatively impacts the quality of refined products [3,4]. To address these challenges, both physical and chemical methods are commonly employed to remove H₂S from crude oil. While conventional physical desulfurization [5] techniques—such as steam stripping and distillation—are well suited for large-scale operations, they necessitate extensive utility systems, rendering them economically unfeasible for low-flow applications, particularly in marginal oil fields or during the early stages of oil production development. In contrast, the chemical desulfurization technologies [6]—especially sodium hypochlorite solution absorption (NaClO)—present distinct advantages [7]. This technology facilitates the efficient removal of H₂S from crude oil through the on-site electrolytic generation of a NaClO oxidant from a sodium chloride solution, requiring only sodium chloride (NaCl) as feedstock. Consequently, it offers excellent economic viability and operational safety, making it a more attractive option for applications where conventional methods may not be cost-effective.

NaClO is an efficient oxidizing agent [8] capable of rapidly oxidizing H₂S present in crude oil at the oil-water interface within the liquid film [6]. Leveraging this reaction characteristic, the efficiency of H₂S removal can be significantly enhanced by employing a continuous stirred tank reactor (CSTR). However, CSTRs are associated with high costs and energy consumption. Therefore, using a static mixer instead of CSTRs offers several advantages in chemical production processes. Static mixers provide higher mixing efficiency compared to CSTRs and are particularly well suited for fast reaction systems [9]. Currently, static mixers have been successfully applied in a variety of chemical processes, including H₂S removal [10], CO oxidation [11], hydrogenation of nitrile butadiene rubber [12], photocatalysis [13], diesel desulphurization [14], arsenic removal from industrial phosphoric acid [15], ozone decomposition [16], biodiesel synthesis [17] and glycerolysis of fatty acid methyl ester [18]. In summary, static mixers demonstrate significant potential for a wide range of industrial applications, outperforming traditional CSTRs in terms of mixing efficiency, processing capacity, and energy consumption.

Compared to experimental methods, computational fluid dynamics (CFD) has emerged as a primary tool for studying static mixers due to its advantages of low cost, high efficiency, and the capability to simulate multiple scenarios [19,20]. Haddadi et al. utilized a CFD-PBM model to simulate the turbulent dispersion of water-silicone oil and water-benzene systems in a SK static mixer, demonstrating that an increase in the number of mixing elements leads to a reduction in the average droplet diameter [21]. Deng and Zhou applied CFD-PBM to investigate the mass transfer behavior in liquid-liquid phases and the impact of fluid parameters on mass transfer rates, focusing on an SK static mixer and a water-diesel fuel system. [22]. In another study, Cao et al. employed a coupled CFD-PBM model using a water-methyl isobutyl ketone (MIBK) system as a case study to examine how fluid velocity affects interfacial mass transfer behavior [23]. Meng et al. investigated the turbulent flow within a static mixer across a Reynolds number range of 2640 to 17,600 through both experimental and numerical simulations to evaluate its performance in enhancing mass transfer [24]. When simulating the photochemical degradation process, Albertazzi et al. found that the degradation efficiency of the static mixer was twice that of an empty reactor [25]. Moon et al. compared CO₂ conversion rates and found that the performance of the static mixer (83.07%) significantly outperformed that of the empty tube reactor (25.27%) [26]. Gong et al. optimized the structure of the static mixer, showing that under optimal conditions, biodiesel production efficiency could be effectively improved [17]. Furthermore, Xie et al. confirmed that improved mixing in static mixers resulted in high Ce³⁺ conversion in a shorter time frame [27]. Santana et al. demonstrated that static mixers greatly enhance oil conversion in transesterification reactions [28]. These studies collectively provide theoretical support for design improvements by illustrating the application of CFD in the analysis of mixer flow fields, enhancement of mass transfer, and optimization of reaction conditions.

Currently, research on the removal of hydrogen sulfide from crude oil using static mixers combined with the NaClO oxidation method primarily focuses on experimental explorations (Shen, 2010) [10], while relevant numerical simulation studies have not yet been reported. Nonetheless, existing research has highlighted the feasibility of using numerical simulations to model multiphase mass transfer and reaction processes involving static mixers [25,26,27,28]. In light of this, this study conducts a numerical simulation of multiphase flow behavior within static mixers based on computational fluid dynamics (CFD). By incorporating reaction kinetics and mass transfer rate equations, the effects of static mixer structure on hydrogen sulfide removal efficiency, mixing performance, and pressure drop are systematically analyzed. The above simulation efforts aim to provide a theoretical basis for the technological optimization of hydrogen sulfide removal processes.

Current studies employing the computational fluid dynamics–population balance model (CFD–PBM) coupled model for static mixers primarily focus on their internal geometries and configurations [21,22,23]. However, a notable limitation is the insufficient attention to the effects of inlet parameters of the disperse phase on liquid-liquid dispersion processes. To address this gap, the present study systematically varies the diameter of the water inlet pipe, elucidating the influence of external flow conditions on liquid-liquid dispersion characteristics. This innovative approach not only fills a significant gap in the literature but also provides new perspectives and valuable theoretical insights for analyzing multiphase flow phenomena in static mixers.

2. Materials and Methods

2.1. Physical Model and Boundary Conditions

The physical model of the SK static mixer is illustrated in Figure 1 and is divided into four distinct zones: A (crude oil inlet), B (transition), C (mixing), and D (outlet). The relevant structural parameters are provided in

Table 1. Structure of SK static mixer.

Parameters	Value
Length of Zone A	15 mm
Length of Zone B	30 mm
Length of Zone D	30 mm
Diameter of crude oil inlet D1	15 mm
Diameter of solvation inlet D2	10 mm

. Zone C features a mixing element composed of blades that are rotated through 180°, with adjacent blades rotating in opposite directions and offset by 90°.

The physical parameters of the crude oil and sodium hypochlorite solution utilized in this study are presented in

Table 2. Physical properties.

Properties	Crude Oil	Sodium Hypochlorite Solution
Density (293 K)	810 kg/m3	998.2 kg/m3
Viscosity (293 K)	0.01 kg/(m·s)	0.001 kg/(m·s)
Concentration of H2S	100 mg/kg	0 mg/kg
Concentration of NaClO	0 g/kg	10 g/kg
Oil–water interfacial tension	0.01 N/m

.

The initial H₂S concentration for this simulation was predominantly based on field investigation data reported by Townsend-Small et al. [29]. Their study indicated that H₂S concentrations commonly reach hazardous levels (above 100 ppm) in most oil well environments. To ensure the practical relevance of the simulation results while maintaining computational efficiency, an initial H₂S concentration of 100 mg/kg in crude oil was established.

The selection of NaClO solution concentration was informed by recent research conducted by Ghalwa’s group, which demonstrated that electrolysis of sodium chloride solution using Pb/PbO₂ electrodes can yield NaClO solutions with effective chlorine concentrations of up to 20 g/kg [30]. Given the relatively low initial H₂S concentration in our simulation system, a NaClO concentration of 10 g/kg was chosen for the reaction simulation following thorough evaluation. This concentration not only optimizes reaction efficiency but also minimizes reagent waste, ensuring sufficient reaction completion and economic viability for practical industrial applications.

In the numerical simulation setup, the inlet boundary conditions for both the crude oil and sodium hypochlorite solution were designated as velocity inlets, while the outlet boundary conditions were configured as outflow. The inlet flow rate of the crude oil was set at 1 m/s, and the volume flow ratio between the crude oil and the sodium hypochlorite solution was maintained at 20:1 throughout the simulation.

2.2. Computational Method and Assumptions

This study utilizes Fluent 14.5 software for numerical simulations, employing the Eulerian-Eulerian multiphase flow model. The realizable k-ϵ model is chosen as the turbulence model, while the Standard Wall Function (SWF) [31] is applied for boundary definition. The pressure-based SIMPLE algorithm is used to compute the flow field, and the second-order upwind scheme is employed to solve the pressure, momentum, and energy equations. In this simulation, the mixing cells and walls are treated as fixed walls.

The continuity and momentum equations [31] are as follows:

$$\nabla \cdot ({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q}})=0$$

(1)

$$\nabla \cdot ({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q}})={\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}\mathrm{g}-{\mathsf{\alpha }}_{\mathrm{q}}\nabla \mathrm{p}+\nabla \cdot ({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\mu }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{q}}(\nabla {\mathrm{u}}_{\mathrm{q}}+{(\nabla {\mathrm{u}}_{\mathrm{q}})}^{\mathrm{T}})+{\mathrm{F}}_{\mathrm{q}}$$

(2)

$${\displaystyle \sum {\mathsf{\alpha }}_{\mathrm{q}}=1}$$

(3)

where ${\mathsf{\alpha }}_{\mathrm{q}}$ is the volume fraction of the phase q. ${\mathsf{\rho }}_{\mathrm{q}}$ is the density of the qth-phase, kg/m³. ${\mathrm{u}}_{\mathrm{q}}$ is the velocity of the phase q, m/s. $\mathrm{g}$ is the gravitational acceleration, m/s². ${\mathrm{F}}_{\mathrm{q}}$ is the sum of all the forces on the qth-phase, N. $\mathrm{p}$ is the pressure, Pa. ${\mathsf{\mu }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{q}}$ is the effective viscosity, Pa·s.

$${\mathsf{\mu }}_{\mathrm{e}\mathrm{f}\mathrm{f},\mathrm{q}}={\mathsf{\mu }}_{\mathrm{q}}+{\mathsf{\mu }}_{\mathrm{t}}$$

(4)

where ${\mathsf{\mu }}_{\mathrm{q}}$ is the continuous-phase molecular viscosity, Pa·s. ${\mathsf{\mu }}_{\mathrm{t}}$ is the turbulent viscosity, Pa·s.

The turbulence kinetic energy equation [21] is as follows:

$$\nabla \cdot ({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{k}}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q}})=\nabla \left[\left({\mathsf{\mu }}_{\mathrm{q}}+{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathrm{k},\mathrm{q}}}}\right)\nabla {\mathrm{k}}_{\mathrm{q}}\right]+{\mathrm{G}}_{\mathrm{k}}+{\mathrm{G}}_{\mathrm{b}}-\mathsf{\rho }\mathsf{\epsilon }-{\mathrm{Y}}_{\mathrm{M}}$$

(5)

$$\nabla \cdot ({\mathsf{\alpha }}_{\mathrm{q}}{\mathsf{\rho }}_{\mathrm{q}}{\mathrm{k}}_{\mathrm{q}}{\mathrm{u}}_{\mathrm{q}})=\nabla \left[\left(\mathsf{\mu }+{\displaystyle \frac{{\mathsf{\mu }}_{\mathrm{t}}}{{\mathsf{\sigma }}_{\mathsf{\epsilon }}}}\right)\nabla \mathsf{\epsilon }\right]+\mathsf{\rho }{\mathrm{C}}_{1\mathsf{\epsilon }}{\mathrm{S}}_{\mathsf{\epsilon }}-\mathsf{\rho }{\mathrm{c}}_2{\displaystyle \frac{{\mathsf{\epsilon }}^2}{\mathrm{k}+\sqrt{\mathrm{v}\mathsf{\epsilon }}}}+{\mathrm{C}}_{1\mathsf{\epsilon }}{\displaystyle \frac{\mathsf{\epsilon }}{\mathrm{k}}}{\mathrm{C}}_{2\mathsf{\epsilon }}{\mathrm{G}}_{\mathrm{b}}$$

(6)

where ${\mathrm{G}}_{\mathrm{k}}$ is the turbulent kinetic energy generation term due to the mean velocity gradient, W/m³. ${\mathrm{G}}_{\mathrm{b}}$ is the turbulent kinetic energy generation term due to buoyancy, W/m³. $\mathsf{\epsilon }$ is the turbulence dissipation rate. ${\mathrm{Y}}_{\mathrm{M}}$ is the dissipation term of the turbulent kinetic energy, W/m³. ${\mathsf{\sigma }}_{\mathrm{k},\mathrm{q}}$ is the Planck numbers corresponding to the turbulent kinetic energy k. ${\mathsf{\sigma }}_{\mathsf{\epsilon }}$ is the Planck numbers corresponding to the dissipation rate $\mathsf{\epsilon }$. ${\mathrm{C}}_{1\mathsf{\epsilon }}$ and ${\mathrm{C}}_{2\mathsf{\epsilon }}$ are the empirical constants. ${\mathrm{S}}_{\mathsf{\epsilon }}$ is the user-defined source term, W/m³.

The Euler-Euler model offers interphase interaction models to simulate mass transfer and chemical reaction processes. Within this framework, users can define mass transfer rates, reaction kinetics, source terms, and other critical parameters through User Defined Functions (UDFs), facilitating a more accurate simulation of mass transfer and reaction behaviors in real-world industrial processes. The mass transfer rate equations and reaction kinetics employed in this study are derived from the research findings of Vilmain et al. [6].

The reaction equation is as follows:

$${\mathrm{H}}_2\mathrm{S}+\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}\mathrm{O}=\mathrm{S}\downarrow +{\mathrm{H}}_2\mathrm{O}+\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}$$

(7)

The reaction kinetic equation is as follows:

$$-{\mathrm{r}}_{{\mathrm{H}}_2\mathrm{S}}={\mathrm{k}}_{{\mathrm{H}}_2\mathrm{S}/\mathrm{C}\mathrm{l}{\mathrm{O}}^{-}}\cdot {\mathrm{C}}_{{\mathrm{H}}_2\mathrm{S}}\cdot {\mathrm{C}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}\mathrm{O}}$$

(8)

where ${\mathrm{r}}_{{\mathrm{H}}_2\mathrm{S}}$ is the reaction rate, mol/(L∙s). ${\mathrm{k}}_{{\mathrm{H}}_2\mathrm{S}/\mathrm{C}\mathrm{l}{\mathrm{O}}^{-}}$ is the reaction rate constant, ${\mathrm{k}}_{{\mathrm{H}}_2\mathrm{S}/\mathrm{C}\mathrm{l}{\mathrm{O}}^{-}}$ = 6.75 × 10⁶ L/(s∙mol) (The reaction temperature is 293 K). ${\mathrm{C}}_{{\mathrm{H}}_2\mathrm{S}}$ is the concentration of hydrogen sulfide in crude oil, mol/L. ${\mathrm{C}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}\mathrm{O}}$ is the concentration of sodium hypochlorite in the solvation, mol/L.

The mass transfer rate equation is as follows:

$$\mathrm{d}\mathrm{J}={\mathrm{K}}_{\mathrm{L}}({{\mathrm{C}}_{\mathrm{A}}}^{\mathrm{E}\mathrm{q}}-{\mathrm{C}}_{\mathrm{A}})$$

(9)

$${\mathrm{K}}_{\mathrm{L}}={\displaystyle \frac{\sqrt{{\mathrm{k}}_{{\mathrm{H}}_2\mathrm{S}/\mathrm{C}\mathrm{l}{\mathrm{O}}^{-}}\cdot {\mathrm{D}}_{{\mathrm{H}}_2\mathrm{S}}\cdot {\mathrm{C}}_{\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}\mathrm{O}}}}{2}}$$

(10)

where ${\mathrm{C}}_{\mathrm{A}}$ is the solute concentration in the bulk, mol/m³. ${{\mathrm{C}}_{\mathrm{A}}}^{\mathrm{E}\mathrm{q}}$ is the solute concentration in equilibrium, mol/m³. ${\mathrm{D}}_{{\mathrm{H}}_2\mathrm{S}}$ is the aqueous phase diffusion coefficient of hydrogen sulfide, ${\mathrm{D}}_{{\mathrm{H}}_2\mathrm{S}}$ = 1.75 × 10⁻⁹ m²/s (The mass transfer temperature is 293 K).

The Population Balance Model (PBM) establishes population balance equation (PBE) for droplet number density based on the microscopic dynamics of dispersed-phase droplets, including processes such as breakup, agglomeration, and growth [32]. This model has found widespread applications in chemical engineering, food processing, bioengineering, and emulsion research. The primary objective of employing the PBM in this study is to analyze the behavior of droplet breakup and agglomeration within the static mixer. In the PBM framework, the Luo model [29] is adopted for the breakup process, while a turbulent model [30] is utilized for the agglomeration process.

In Fluent 14.5, the PBE can be solved using the finite volume method. The transport equation for the number density function is as follows [33,34]:

$${\displaystyle \frac{\partial \mathrm{n}(\mathrm{V},\mathrm{t})}{\partial \mathrm{t}}}+\nabla \cdot [\mathrm{u}\times \mathrm{n}(\mathrm{V},\mathrm{t})]={\mathrm{B}}_{\mathrm{C}}(\mathrm{V},\mathrm{t})-{\mathrm{D}}_{\mathrm{C}}(\mathrm{V},\mathrm{t})+{\mathrm{B}}_{\mathrm{B}}(\mathrm{V},\mathrm{t})-{\mathrm{D}}_{\mathrm{B}}(\mathrm{V},\mathrm{t})$$

(11)

where V is the droplet breakage volume fraction, m³. $\mathrm{n}(\mathrm{V},\mathrm{t})$ is the number density distribution function. u is the solvent velocity, m/s. ${\mathrm{B}}_{\mathrm{C}}(\mathrm{V},\mathrm{t})$ represents the droplet aggregation and birth function of volume V. ${\mathrm{D}}_{\mathrm{C}}(\mathrm{V},\mathrm{t})$ represents the droplet aggregation and death function of size L. ${\mathrm{B}}_{\mathrm{B}}(\mathrm{V},\mathrm{t})$ represents the droplet breakage birth function of volume V. ${\mathrm{D}}_{\mathrm{B}}(\mathrm{V},\mathrm{t})$ represents the droplet fragmentation death function of volume V.

The droplet breakup rate equation is as follows [35,36]:

$$\mathsf{\Omega }(\mathrm{V},{\mathrm{V}}^{\prime })=0.923(1-{\mathsf{\alpha }}_{\mathrm{d}})\mathrm{n}{\displaystyle \frac{{\mathsf{\epsilon }}^{1/3}}{{\mathrm{d}}^{2/3}}}{\displaystyle {\int }_{{\Im }_{\min }}^1\frac{{(1+\Im )}^2}{{\Im }^{11/3}}}\exp \left\{-{\displaystyle \frac{12\left[{\mathrm{f}}_{\mathrm{B}\mathrm{V}}^{2/3}+{(1-{\mathrm{f}}_{\mathrm{B}\mathrm{V}})}^{2/3}-1\right]\mathsf{\sigma }}{\mathsf{\beta }{\mathsf{\rho }}_{\mathrm{c}}{\mathsf{\epsilon }}^{2/3}{\mathrm{d}}^{5/3}{\Im }^{11/3}}}\right\}\mathrm{d}\Im$$

(12)

where $\Im$ is the ratio of the turbulent vortex diameter to the diameter of the broken droplet. $\mathsf{\sigma }$ is the interfacial tension, N/m². ${\mathrm{f}}_{\mathrm{B}\mathrm{V}}$ is the volume ratio of the crushed sub-droplet to the mother droplet. ${\mathsf{\rho }}_{\mathrm{c}}$ is the continuous-phase density, kg/m³.

The droplet aggregation rate equation is as follows [37,38]:

$${\mathrm{a}}_1({\mathrm{d}}_{\mathrm{i}},{\mathrm{d}}_{\mathrm{j}})={\mathsf{\zeta }}_{\mathrm{T}}\sqrt{{\displaystyle \frac{8\mathsf{\pi }}{15}}}\mathsf{\gamma }{\displaystyle \frac{{({\mathrm{d}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{j}})}^3}{8}}$$

(13)

$${\mathrm{a}}_2({\mathrm{d}}_{\mathrm{i}},{\mathrm{d}}_{\mathrm{j}})={\mathsf{\zeta }}_{\mathrm{T}}{2}^{3/2}\sqrt{\mathsf{\pi }}{\displaystyle \frac{{({\mathrm{d}}_{\mathrm{i}}+{\mathrm{d}}_{\mathrm{j}})}^2}{4}}\sqrt{{\mathrm{U}}_{\mathrm{i}}^2+{\mathrm{U}}_{\mathrm{j}}^2}$$

(14)

where ${\mathrm{a}}_1({\mathrm{d}}_{\mathrm{i}},{\mathrm{d}}_{\mathrm{j}})$ is rate the viscous subrange. ${\mathrm{a}}_2({\mathrm{d}}_{\mathrm{i}},{\mathrm{d}}_{\mathrm{j}})$ is rate the inertial subrange. ${\mathsf{\zeta }}_{\mathrm{T}}$ is the pre-factor that considers the collision capture efficiency coefficient due to turbulence. $\mathsf{\gamma }$ is the shear rate. ${\mathrm{U}}_{\mathrm{i}}^2$ is the mean square velocity of droplet, m/s.

The computational process of the CFD-PBM coupled model is outlined as follows: First, local flow field parameters, including phase volume fractions and turbulence dissipation rates, are obtained through CFD simulations. These parameters are subsequently input into the PBM model, which incorporates droplet breakup and coalescence kinetic models to calculate the non-uniform droplet size distribution. Based on the resulting distribution, the interaction forces between the liquid phases in the CFD model and the turbulence kinetic energy are dynamically corrected. This forms a closed-loop iterative computational mechanism that enables the synergistic optimization of flow field information and particle size distribution. The specific schematic diagram of the CFD-PBM coupling model is shown in Figure 2.

The main assumptions underlying this study are as follows:

• The acceleration of gravity is 9.81 m/s² in the direction of the z-axis.
• Each phase of the multiphase flow is assumed to be an incompressible fluid [39,40].
• The surfaces of the mixing elements are assumed to be smooth, and their turbulent effects arise only from fluid shear [41,42].
• In this study, the volumetric flow rate of crude oil significantly exceeds that of the aqueous phase. Therefore, crude oil is set as the continuous phase and water is set as the dispersed phase during oil-water mixing.
• Mass transfer occurs exclusively at the oil-water interface.

2.3. Characterization Methods

2.3.1. Hydrogen Sulfide Removal Efficiency

The hydrogen sulfide removal efficiency (HSRE) was utilized to assess the effectiveness of the H₂S removal process. A higher HSRE value indicates a more effective process for removing hydrogen sulfide from crude oil within the static mixer.

$$\mathrm{C}\mathrm{O}\mathrm{H}\mathrm{S}={\displaystyle \frac{{\displaystyle \int {\mathsf{\alpha }}_{\mathrm{o}\mathrm{i}\mathrm{l}\mathrm{c}}\mathrm{d}\mathrm{s}}}{{\displaystyle \int \mathrm{d}\mathrm{s}}}}$$

(15)

$$\mathrm{H}\mathrm{S}\mathrm{R}\mathrm{E}=1-{\displaystyle \frac{\mathrm{C}\mathrm{O}\mathrm{H}\mathrm{S}}{{\mathsf{\alpha }}_{\mathrm{o}\mathrm{i}\mathrm{l}1}}}$$

(16)

where $\mathrm{C}\mathrm{O}\mathrm{H}\mathrm{S}$ is the average mass fraction of H₂S in the crude oil within the cross-section grid, mg/kg. ${\mathsf{\alpha }}_{\mathrm{o}\mathrm{i}\mathrm{l}1}$ imports the mass fraction of H₂S in the crude oil within the inlet cross-section grid.

2.3.2. Mixing Effect

• The coefficient of variation (CoV) is used to evaluate the mixing effectiveness, representing the deviation of the crude oil volume fraction at the sampling point from the average volume fraction of crude oil across the entire cross-section. A lower CoV indicates a better mixing effect [43].

  $$\mathrm{C}\mathrm{o}\mathrm{V}={\displaystyle \frac{\mathsf{\sigma }}{\overline{\mathrm{x}}}}$$

  (17)

  $$\mathsf{\sigma }=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{n}=1}^{\mathrm{N}}{({\mathrm{x}}_{\mathrm{i}}-\overline{\mathrm{x}})}^2}}{\mathrm{N}-1}}}$$

  (18)

  where N is the number of grids in the static mixer cross-section, and, in this paper, we take N to be 2000. ${\mathrm{x}}_{\mathrm{i}}$ is the volume fraction of oil-phase components in each grid. $\overline{\mathrm{x}}$ is the average volume fraction of crude oil at the entire outlet plane.
• Mixing efficiency was further assessed using the micro-mixing time (tm, in seconds). This parameter indicates the time required to achieve homogeneous mixing of crude oil and water at a microscopic level. A lower tm value reflects a higher mixing efficiency of the static mixer [44].

  $${\mathrm{t}}_{\mathrm{m}}=17.24\sqrt{{\displaystyle \frac{{\mathsf{\mu }}_0}{{\mathsf{\epsilon }}_{{\mathsf{\mu }}_0}}}}$$

  (19)

  where ${\mathsf{\mu }}_0$ represents the kinematic viscosity of the continuous phase fluid, m²/s. ${\mathsf{\epsilon }}_{{\mathsf{\mu }}_0}$ denotes the energy dissipation rate per unit time in the mixer (calculated by simulation software, m²/s³).

2.3.3. Droplet Dispersion Effect

To evaluate the dispersion performance of the static mixer, this study introduces the Sauter mean diameter (SMD, d₃₂) as a quantitative indicator. The d₃₂ is directly related to the specific surface area of the dispersed phase (the surface area per unit volume ∝ 1/d₃₂); a larger specific surface area leads to higher rates of mass transfer, heat transfer, or reaction. Additionally, d₃₂ also reflects the uniformity of the dispersion; a smaller value indicates a higher proportion of small droplets in the system, signifying superior dispersion performance [45].

$${\mathrm{d}}_{32}={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}{{\mathrm{d}}_{\mathrm{i}}}^3}}{{\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{n}}_{\mathrm{i}}{{\mathrm{d}}_{\mathrm{i}}}^2}}}$$

(20)

where ${\mathrm{d}}_{\mathrm{i}}$ is the droplet diameter, mm. ${\mathrm{n}}_{\mathrm{i}}$ is the number of droplets with diameter ${\mathrm{d}}_{\mathrm{i}}$.

2.3.4. Differential Pressure

Differential pressure refers to the pressure drop resulting from frictional resistance, turbulent dissipation, and interphase interactions as the fluid flows through the mixing equipment. A lower pressure drop is preferable, as long as the process requirements are satisfied.

$$\mathrm{P}={\mathrm{P}}_1-{\mathrm{P}}_2$$

(21)

where ${\mathrm{P}}_1$ is the pressure at the inlet, kPa. ${\mathrm{P}}_2$ is the pressure at the outlet, kPa.

2.4. Grid Irrelevance Test

In this study, Fluent Meshing was employed to construct the hexahedral mesh for the SK static mixer. The mixer parameters included 10 elements, an aspect ratio (L/D₁) of 1, and a twist angle of 90°. Five mesh sizes were utilized: 1 mm, 0.1 mm, 0.01 mm, 0.001 mm, and 0.0001 mm, corresponding to grid numbers of 213,983; 246,897; 379,328; 423,896; and 513,215, respectively. The simulations were conducted under the following conditions: an H₂S concentration in crude oil of 100 mg/kg, a NaClO concentration in water of 10 g/kg, an oil flow velocity of 1 m/s, and a water flow velocity of 0.1125 m/s.

The calculation results are shown in Figure 3. When the mesh size of the SK-type static mixer is set to 0.0001 mm, the HSRE values are essentially consistent with those obtained at mesh sizes of 0.001 mm (with a variation of HSRE less than 0.9%). Therefore, it can be concluded that the use of a mesh size of 0.001 mm meets the grid independence verification criteria.

2.5. Model Validation

Based on the hydrogen sulfide removal experimental results from crude oil reported by Shen [10], the reliability of the UDFs used in this simulation has been verified. The physical model and boundary conditions of the static mixer in the simulation are consistent with those of the aforementioned experiment: the static mixer has a length of 250 mm and a diameter of 15 mm, equipped with 7 SK-type elements. The flow rate of the aqueous solution is set at 30 mL/min, while the oil-water ratio is increased from 25:1 to 45:1 by varying the crude oil flow rate.

The simulated values of H₂S content in the crude oil at the outlet of the static mixer were compared with the experimental values, as shown in

Table 3. Comparative analysis of simulation and experimental results.

Oil-to- Water Ratios	Oil Phase H2S Content at Inlet	Oil Phase H2S Content at Outlet (Experiment)	Oil Phase H2S Content at Outlet (Simulation)	Error (%)
25:1	15.48	5.29	6.25	18.22
30:1	15.46	5.86	6.99	19.23
35:1	16.10	7.21	7.52	4.29
40:1	16.95	9.19	9.11	0.84
45:1	25.40	18.52	14.89	19.60

. There is a discrepancy between the simulation results and the experimental results, primarily due to the mass transfer and reaction equations used in the simulation being investigated at a temperature of 293 K, while the oil temperature during the experiment was 338 K and the NaClO solution temperature was 298 K. However, the absolute difference between the simulation and experimental values is less than 4 mg/kg, and the overall trends are consistent, indicating that the numerical model is accurate, reasonable, and applicable.

A reliability check of the PBM method employed in this paper is based on the experimental results of toluene-water two-phase mixing in an SK-type static mixer, as investigated by Wang et al. [46]. The physical model of the static mixer used in the simulation is consistent with the experimental setup: the tube length is 500 mm, the inner diameter is 24 mm, 10 SK-type elements are installed, and the aspect ratio of the mixing elements is 1.5. Distilled water was set as the continuous phase, while toluene served as the dispersed phase. Simulation was conducted using the experimental conditions from groups 2, 3, 7, and 10 of the response surface experiments described by Wang et al. The simulated value of d₃₂ at the outlet of the static mixer was compared with the experimental values, and the results are presented in

Table 4. Comparative analysis of simulation and experimental results.

No.	Water Velocity (m/s)	Toluene Velocity (m/s)	Experimental Results (mm)	Simulation Results (mm)	Error (%)
2	0.27	0.21	1.99	2.08	4.42
5	0.30	0.71	1.69	1.85	9.42
7	0.15	0.36	3.1	3.48	1.21
10	0.18	0.85	2.35	2.27	3.32

. Notably, the relative errors between the simulated and experimental values are within 10%.

3. Results and Discussion

3.1. Structural Optimization of SK Static Mixer

Using a static mixer with a diameter of 15 mm, the effects of the number of elements (A), aspect ratio (B), and twist angle (C) on the hydrogen sulfide removal efficiency (HSRE) were investigated through response surface methodology. The study was conducted under the following conditions: 100 mg/kg H₂S in crude oil, 10 g/kg NaClO in water, a crude oil velocity of 1 m/s, and a water velocity of 0.1125 m/s. The simulated data are presented in

Table 5. Factors and levels of Box-Behnken experiment.

Code	Factors	Levels
A	Number of elements	10	14	18
B	Aspect ratio	1	2	3
C	Twist angle (°)	45	60	90

,

Table 6. Results of Box-Behnken experiment.

No.	A	B	C	HSRE
1	10	1	60	0.4462
2	18	1	60	0.6504
3	10	3	60	0.3077
4	18	3	60	0.4674
5	10	2	45	0.3811
6	18	2	45	0.5062
7	10	2	90	0.4012
8	18	2	90	0.5161
9	14	1	45	0.6141
10	14	3	45	0.4504
11	14	1	90	0.6511
12	14	3	90	0.4711
13	14	2	60	0.5321

and

Table 7. Analysis of Variance.

Source	Sum of Squares (SS)	Degrees of Freedom (df)	Mean Square (MS)	F	p	Significance
Model	1252	9	139.2	22.58	0.01321	Yes
A	419.2	1	419.2	68.03	0.003732	Yes
B	535.5	1	535.5	86.90	0.002614	Yes
C	9.614	1	9.614	1.560	0.3002	No
AB	4.951	1	4.951	0.8034	0.4361	No
AC	2.254	1	2.254	0.3658	0.5880	No
BC	0.9474	1	0.9474	0.1537	0.7212	No
A2	145.7	1	145.7	23.65	0.01661	Yes
B2	5.616	1	5.616	0.9114	0.4102	No
C2	0.5007	1	0.5007	0.08126	0.7942	No

. Table 5 details the factors and levels of the Box-Behnken experiment.

Figure 4 illustrates the crude oil path-lines for experiments No. 1 and 2. The geometry of the mixing elements generates shear and turbulent disturbances, which accelerate fluid flow within the channel. Increasing the number of elements enhances multi-scale fluid cutting, turbulent pulsation, and mixing uniformity, thereby disrupting the crude oil–water phase boundary. This disruption promotes the diffusion of NaClO, facilitating mass transfer and its reaction with H₂S, ultimately resulting in improved hydrogen sulfide removal efficiency.

Figure 5 presents the path-lines for experiments No. 1 and 3. A smaller aspect ratio (which corresponds to more compact mixing units) enhances fluid cutting, folding, and redistribution within a shorter flow path. This configuration reduces residence time, induces local acceleration through narrow passages, increases turbulence and shear, disrupts liquid boundary layers, and enlarges the liquid-liquid contact area. Furthermore, it minimizes flow resistance, enabling higher flow rates for the same pressure drop. These combined effects all contribute to facilitating the hydrogen sulfide removal efficiency.

Figure 6 illustrates the path-lines for experiments No. 9 and 11. The mixing elements configured at 45° and 90° generate shear and turbulence by altering the flow direction. The 90° configuration leads to sharp deflections and localized high-pressure zones, while the 45° configuration induces oblique shear and facilitates kinetic energy transfer through successive splitting and reconvergence. Despite their differing mechanisms, both configurations result in a comparable increase in flow velocity, with no significant differences observed between them.

A second-order polynomial model was constructed to predict hydrogen sulfide removal efficiency (HSRE) based on the design matrix. The model was fitted using MATLAB 24.1 software, resulting in the following Equation (22):

$$\begin{array}{c}\mathrm{Y}=-0.7352+0.1693*\mathrm{A}-0.09349*\mathrm{B}+0.003471*\mathrm{C}-0.002781*\mathrm{A}*\mathrm{B}-0.00008118*\mathrm{A}*\mathrm{C}-\\ 0.0002105*\mathrm{B}*\mathrm{C}-0.004991*{\mathrm{A}}^2+0.01568*{\mathrm{B}}^2-0.00001056{\mathrm{C}}^2\end{array}$$

(22)

In the equation, Y represents the predicted hydrogen sulfide removal efficiency (HSRE). The model indicates that the linear terms A (number of elements) and C (twist angle), as well as the quadratic term B² (aspect ratio), have a positive effect on HSRE. Conversely, the linear term B and the interaction terms AB, AC and BC, along with the quadratic terms A² and C², exert a negative effect on HSRE. The analysis of variance (ANOVA) presented in Table 7 indicates a significant model (F = 22.58, p < 0.05) with a high coefficient of determination (R² = 0.9855), confirming the reliability and predictive capability of the model.

Three-dimensional response surface analysis (Figure 7, Figure 8 and Figure 9) revealed a positive correlation between hydrogen sulfide removal efficiency (HSRE) and the number of elements, while a negative correlation was observed with the aspect ratio. The twist angle demonstrated a negligible influence (p > 0.05), and no significant interactions were found between the variables. The optimal conditions determined through stepwise regression and response surface analysis were identified as 15 elements, an aspect ratio of 1 and a twist angle of 90°. Under these optimal conditions, the simulation yielded an HSRE of 0.7202 (corresponding to an outlet H₂S concentration of 27.98 mg/kg), with a model-predicted value of 0.6601, resulting in an error of 8.34%.

3.2. Static Mixer Performance

Simulation was employed to compare the fluid flow characteristics in the optimized static mixer with those in an isometric empty pipe, focusing on their effects on mixing, hydrogen sulfide removal efficiency (HSRE), and pressure drop (P). The inlet conditions for both systems were maintained consistent with those specified in Section 3.1.

3.2.1. Hydrogen Sulfide Removal Efficiency

The static mixer demonstrated a substantial reduction in H₂S concentration, with the outlet hydrogen sulfide removal efficiency (HSRE) increasing from 0.0363 in the empty pipe to 0.7202 in the optimized static mixer (as shown in Figure 10). Figure 11 illustrates a positive correlation between the number of elements and the HSRE, which increased from 0.1815 with 3 elements to 0.7184 with 15 elements, thereby confirming the efficacy of the SK mixer in enhancing hydrogen sulfide removal efficiency.

3.2.2. Mixing Effect

Figure 12 compares the oil volume fraction distributions in the static mixer and the empty pipe. The empty pipe exhibits smooth laminar flow with a distinct oil-water interface, while the static mixer induces alternating rotational motions that enhance radial fluid exchange and promote oil-water mixing. The mixing efficiency was quantified using the coefficient of variation (CoV) and micro-mixing time (t_m). The CoV decreased from 0.8798 to 0.2143, indicating improved mixing homogeneity. Additionally, t_m was reduced from 0.04593 s to 0.001 s, signifying an increase in mass exchange between fluid micro-clusters per unit time and enhanced mixing efficiency. The data presented in Figure 13 show that by increasing the number of elements from 3 to 15, the CoV was reduced by a factor of 3.21, and t_m decreased by a factor of 45.51, demonstrating the superior mixing efficiency of the static mixer.

In summary, the static mixer significantly enhances oil-water contact efficiency by increasing turbulent mixing effects. This improvement accelerates both the mass transfer and reaction rates of H₂S, ultimately leading to highly efficient hydrogen sulfide removal.

3.2.3. Differential Pressure

The radial pressure drop cloud diagrams for the static mixer and empty pipe are illustrated in Figure 14. The radial pressure drop increased from 2.796 kPa in the empty pipe to 19.28 kPa in the static mixer. While mixing efficiency improved with the addition of more elements, the pressure drop also increased proportionally. This rise in pressure drop indicates enhanced turbulent intensity, characterized by stronger vortices and shear forces during fluid passage through the mixer. Such turbulence reduces the thickness of the oil-water mass transfer boundary layer, thereby accelerating the mass transfer and reaction rates of H₂S, ultimately leading to improved hydrogen sulfide removal efficiency. Consequently, it can be concluded that reducing the number of mixing elements should be prioritized, provided that the process requirements are satisfactorily met.

3.3. Water Inlet Diameter

The maximum size of water droplets is influenced by the diameter of the water inlet diameter. To systematically analyze the effect of maximum droplet size, the impact on droplet size in a static mixer was studied by varying the diameter of the water inlet diameter. Droplet dispersion was assessed using the Sauter Mean Diameter (d₃₂), where a smaller d₃₂ indicates more effective fluid breakup.

Figure 15 illustrates that the water inlet diameter ranged from 0.1 mm to 10 mm. The d₃₂ does not decrease significantly after 12 mixing elements. Initially, as the number of elements increases, the d₃₂ reduces due to the high shear forces that break up large droplets. However, as more elements are added, the aggregation of small droplets becomes more pronounced. When the number of mixing elements exceeds 12, the processes of droplet fragmentation and aggregation reach a dynamic equilibrium, causing the d₃₂ to stabilize. Consequently, further increases in the number of elements have a diminishing effect on the d₃₂. Notably, the d₃₂ at the static mixer outlet is slightly higher than that observed with 15 mixing elements. After exiting the static mixer, the droplets primarily coalesce, a phenomenon attributed to the reduced shear forces and turbulent dissipation, which increase droplet collision frequency and promote aggregation. Additionally, surface tension drives the coalescence of smaller droplets to minimize interfacial energy. It was also observed that for configurations with 3, 6, 9, and 12 mixing elements, when the water inlet diameter exceeded 1 mm, its influence on the final d₃₂ was significantly reduced. During the early mixing stage (e.g., within the first 12 elements), although large droplets (>1 mm) were not fully fragmented, their low proportion meant that the d₃₂ was mainly determined by smaller droplets. As the number of mixing elements increased beyond 12, droplet fragmentation approached its limit, leading to a diminished effect of the initial maximum droplet size on the d₃₂.

Figure 16 presents droplet size distributions (DSD) for various water inlet diameters (0.1 mm, 1 mm, 5 mm, and 10 mm). The distributions reveal a dominance of droplets within the size range of 0.001 mm to 0.1 mm. As the number of mixing elements increases, the DSDs exhibit narrower normal distributions and a higher proportion of smaller droplets. This trend reflects the continuous shear-induced fragmentation process and the dynamic equilibrium established between fragmentation and aggregation within the static mixer.

The comprehensive results of the above analysis demonstrate that the static mixer significantly increases the oil-water contact interface area by facilitating the uniform dispersion of droplets. This enhanced dispersion, in turn, improves the mass transfer and reaction processes of H₂S, ultimately leading to a substantial increase in removal efficiency.

4. Conclusions

This study demonstrates the effectiveness of SK static mixers for H₂S removal from crude oil by employing a combined CFD-PBM approach. Structural optimization using response surface methodology revealed that increasing the number of elements to 15 and reducing the aspect ratio to 1:1 maximized turbulent shear and contact between reactant interfaces, achieving an H₂S removal efficiency of 72.02%. This represents a significant improvement over conventional empty tubes, which exhibited an efficiency of only 3.63%. The optimized geometry induced multi-scale fluid folding and radial momentum exchange, resulting in a reduction of micro-mixing time (t_m) to 0.001 s and a 75.6% decrease in phase separation, as indicated by a coefficient of variation (CoV) of 0.21. While the pressure drop increased linearly with the number of elements (recording 19.28 kPa compared to 2.80 kPa for the empty tube), the trade-off between energy consumption and process efficiency favors the use of static mixers for fast oxidation reactions, such as those occurring in NaClO-H₂S systems. Furthermore, droplet dispersion analysis revealed that the Sauter Mean Diameter (d₃₂) stabilized after 12 mixing elements, with 95% of the droplets measuring below 0.1 mm. This small droplet size provides sufficient interfacial area to facilitate effective mass transfer.

The numerical simulations in this study incorporate necessary simplifying assumptions; however, actual industrial flow conditions are inherently more complex. This discrepancy represents a primary limitation of our current approach. Future studies should concentrate on developing correction models through systematic experimental validation to enhance the industrial applicability of the optimized mixer design.