1. Introduction
The water produced is a mixture of different materials (fluids, dissolved and suspended solids) originating from the extraction of oil and gas in underground reservoirs. During oil production, a large amount of water is produced with a high load of organic and inorganic compounds, featuring extremely saline, oily, and toxic effluents for living beings. In general, for each barrel of oil produced, about three barrels of residual water are released [
1,
2,
3].
Due to the high risk to health and environmental imbalance [
2,
4], Brazilian and international environmental regulatory companies have established norms and regulations related to the mandatory treatment of this produced water before its disposal or reuse [
3]. In Brazil, the National Environment Council (CONAMA) is the main regulator of effluent release conditions and standards, establishing a maximum oil and wax content of 20 mg/L for disposal [
5].
Currently, there are different technologies and methods for treating water polluted with oil, which takes into account the physical, physical–chemical, chemical, and biological principles that occur in the processes [
4,
6]. Among these methods are flotation [
7,
8,
9], coagulation [
6,
10,
11,
12,
13], the use of hydrocyclones [
14,
15,
16,
17], biological treatment [
18,
19,
20,
21] and membrane separation technologies [
21,
22,
23].
In general, the higher the crude oil content in the water, the lower the removal efficiency; therefore, it is necessary to use a combination of different methods that can enhance the processes of water purification [
10]. Khan et al. [
12], for example, combined coagulation and flocculation to remove large particles of oil and inorganic matter from the produced water and, as a result, obtained a favorable environment for the biodegradation microorganisms, and finally, using microfiltration membrane removed the remaining dissolved oil microdroplets.
According to Jepsen et al. [
14], traditional technologies for the treatment of produced water (such as the use of hydrocyclones) are already working within their fundamental limit, so new filtration technologies must be proposed to enhance water purification. For example, hydrocyclone is not efficient enough to treat oil dispersed in water with oil droplets of small diameter, especially when the oil concentration is low.
In this context, ceramic membrane technologies have excelled in presenting excellent chemical, thermal and mechanical properties, favoring high performance in severe operating conditions, such as high temperatures and the presence of aggressive chemicals [
20]. Another advantage of membrane technology is related to the large volume of treated water, thereby, membrane separation has become more efficient compared to conventional techniques [
24].
Membrane separation processes can be classified into four categories according to the pore size: microfiltration, ultrafiltration, nanofiltration, and reverse osmosis [
24,
25,
26]. The pore size of the membrane determines the filtration properties of water/oil emulsions, which can also vary, depending on the concentration of the oil in the emulsion and, consequently, on the size of the oil drop [
13,
27].
The operational parameters also play an important role in the potential of membrane separation. For example, the velocity of the feed flow [
28], the transmembrane pressure [
29], the temperature, the pH, and the size of the dissolved molecules [
24] are very important during oily wastewater separation processes.
The use of ceramic membranes to treat oily effluents has grown considerably due to their high filtration efficiency, excellent hydrophilic properties, and chemical and hydrothermal stability [
30]. However, the performance is strongly affected by the concentration polarization phenomenon (formation of an oil layer and other contaminants from the effluent parallel to the membrane surface) [
31,
32,
33,
34] and, by the membrane encrustation that generally occurs as a result of adsorption of components of the feed solution on the membrane surface [
35,
36].
All potential assigned to the ceramic membranes justifies the intensification of scientific works that seek to enhance the treatment of produced water with this technology [
29,
31,
37,
38,
39,
40]. In the literature, several studies have been reported using computational fluid dynamics as a tool to investigate the process separation of oily waters [
40,
41,
42,
43,
44,
45,
46].
Cunha (2014) [
39] numerically studied the treatment of petroleum industrial effluents using a ceramic membrane. In this work, a mathematical model, later implemented in the Ansys CFX, was developed to describe the physical problem, aiming to evaluate the formation of the concentration polarization layer on the porous membrane surface. The author observed that the formation of the polarization layer by concentration is influenced by the hydrodynamic behavior of the flow and by the properties of the fluid mixture and the porous medium.
Magalhães et al. [
45] in their work on the non-isothermal treatment of oily waters, investigated the thermal influence on the oil/water separation process using a module equipped with a ceramic membrane. The authors observed that an increase in the temperature has influenced the fluid properties and the flow behavior inside the equipment, causing changes in the pressure, concentration, and speed fields.
Shirazian et al. [
41] investigate the mass transfer in wastewater treatment using membrane reactors. The authors observed that the velocity reached its full development after a short distance from the reactor inlet, and that the total flow of contaminant decreased dramatically in the region close to the membrane entrance, due to the resistance imposed by the membrane to the flow passage.
In this context, the CFD (Computational Fluid Dynamics) technique provides information and new models for the application of membranes and contributes to the understanding of the filtration mechanisms [
43,
46,
47,
48,
49,
50,
51,
52,
53,
54]. The computational fluid dynamics enables the understanding and development of new membrane separation processes, allowing researchers to study, in detail, phenomena such as fouling, concentration polarization, and pore-clogging, responsible for reducing the filtration rate and favoring contamination of the permeate.
Based on the CFD simulations, Yang et al. [
53] have studied a new configuration of a cylindrical 37-channel porous inorganic membrane tube by increasing membrane filtration area and increasing permeation efficiency of inner channels. In this very interesting paper, the authors have concluded that the permeate efficiency of the inner channels is smaller than that of the outer channels and that the difference in permeate flux of a channel in different radial rings diminishes gradually as the resistance of the skin layer increases. Tong et al. [
54] reported a similar CFD study using a 19-core tandem ceramic membrane module. The authors concluded that when the volume flow rate changes from 26 to 89 m
3/h, the resistance of each part of the membrane module system increases gradually. The increase in resistance loss in the membrane element is faster than that in the plates and the bell mouths.
Because of the above, this work aims to study the hydrodynamics of water and oil flow in a ceramic membrane module, and verify the effect of the separation process variables on the equipment’s performance via computational fluid dynamics (Ansys Fluent®).
2. Methodology
2.1. The Physical Problem and Geometry
In this study, the flow of a mixture of oil and water in a porous membrane system was analyzed. It is, therefore, a two-phase system with fluid flow in porous and non-porous media. The ceramic membrane filtration module is of the shell-and-tube type (domain understudy), consisting of the main cylinder (hull) with a tangential inlet and outlet and four internal cylindrical tubes which are the porous ceramic membranes (membranes 1, 2, 3, and 4), as shown in
Figure 1.
Table 1 summarizes all the geometric dimensions of the separation module used in the numerical simulations.
The tangential inlet of the module is a tube of a circular cross-section through which the contaminated effluent enters the system. The tangential outlet tube, also of circular cross-section, is intended for the outlet of the concentrate. Part of the filtrate (oil) is retained inside the membranes after filtration. Thus, the separation between oil and water occurs as the feed mixture, containing a certain oil concentration, enters tangentially into the separator. In this way, the fluid mixture is forced to cross the membranes, in such a way that the oil fraction is retained in the porous structure, and only the aqueous phase easily crosses the membrane, generating the permeate. This is caused, mainly due to the difference in transmembrane pressure. Finally, the four outlets coming from the membranes receive the permeate flow, while the outlet in the hull receives the concentrate (mixture with high oil concentration).
For a numerical solution to the conservation equations that govern the domain under study, it is necessary to convert the continuous domain into a discrete domain. These control volumes are three-dimensional partitions of the geometric domain, composed of edges and nodal points, in which the discretized governing equations must be solved, and represent the physical phenomenon. The numerical method causes errors of truncation and idealizations, which decrease, as elements are added to the mesh, that is, as the finite limits of the solution are reduced.
From a computational point of view, the separation module has nine structural domains, which are: the four volumes of fluids related to the four membranes (porous medium, through which water flows and the oil is retained), the four permeates (part membrane, through which the permeate flows) and the cylindrical hull, comprised between the internal surfaces, which limit the separator and the outer surfaces of the membranes (non-porous medium, through which the mixture of oil and water flows towards the outlet). From this, to adequately predict the proposed problem, three hybrid meshes (tetrahedral and hexahedral elements) with different densities of control volumes (elements) were built, using the Ansys Meshing® software. To build the hull domain, hexahedral elements were used near the outer walls of the membranes and the hull itself, and tetrahedral elements for the other regions. To describe the domain of the membranes, structured meshes were built with only hexahedral elements, which was maintained for the cylindrical domains (inner tube of the membrane through which the permeate flows). The entire domain was developed using the mesh construction technique called o-grid.
After building the meshes using the Ansys Meshing® software, it was observed that all meshes were within the recommended limits for deformation (below 0.95) and orthogonality (above 0.1) values.
Figure 2 shows mesh 2, with emphasis on the studied domains. In this figure, the mesh of the hull, membrane, and permeate domains can be seen. Besides, it is possible to see a cross-section of the entire module, focusing on the membrane mesh, as well as the mesh entrance and exit regions.
After construction, these meshes were evaluated for dependence on the numerical results obtained by the simulations with the number of elements.
2.2. Mathematical Modeling
To describe the flow of fluids in the regions of the hull and cylindrical tube (through which the permeate flows), the multiphase model with the Eulerian–Eulerian type formulation was used. This method of approach is capable of modeling multiple phases, treated as separate, but interacting with each other. The phases can be liquid, gaseous, or solid, in almost any combination. Eulerian–Eulerian solution treatment is used for each phase separately, even if one of the phases is made up of particles. The model makes no distinction between the fluid–fluid and fluid–solid (granular) flow. A granular flow is simply one that involves at least one phase designated as granular. A single pressure is shared by all phases, and the equations of moment and continuity are solved for each phase [
55].
The formulation of the model is based on the assumption that two or more phases are continuous and immiscible. For each phase added to the mathematical model, a new variable corresponding to the volumetric fraction (αq), of the respective phase “q”, is introduced. The volumetric fraction represents a relationship between the volume occupied by each phase and the total volume of the cell. In each cell, the laws of conservation of mass and linear moment are satisfied for each phase individually.
In each control volume (V), the sum of the volumetric fractions of the phases is equal to 1 (one). So, you can write:
where:
The effective density of phase q is defined as follows:
where
is the physical density of phase
.
Based on this methodology, the following conditions can be met:
: indicates that the cell volume is filled by the q phase.
: indicates that the cell volume does not contain the q phase.
: indicates that the cell volume is partially filled by the q phase. In this condition, there is an interface between the q phase and one or more phases present in the cell.
Based on the local value of , of the phases existing in the physical process, the properties and process variables are weighted in each region of the multiphase flow. In this work, it was considered that the mass conservation and linear momentum equations are solved for each of the present phases (continuous and dispersed). For each of the physical situations, the following considerations were adopted:
Flow in permanent and isothermal regime;
Newtonian fluid, incompressible and with constant thermo-physical–chemical properties;
Interfacial mass transfer, interfacial momentum, and mass source are negligible;
The non-drag interfacial forces (lift forces, wall lubrication, virtual mass, turbulent dispersion, and solid pressure) are neglected;
The walls of the separation module are rigid (not deformable) and have null roughness;
The fluid flow in the feed is considered as a mixture of immiscible water and oil (not emulsion);
The porous medium (ceramic membrane) has a uniform distribution of porosity and permeability;
There are no reaction or adsorption phenomena of the solute on the contact surface in the porous medium.
2.2.1. Formulation for the Non-Porous Domain
The mass conservation equation for multiphase flow is given by Equation (4), as follows:
where the sub-index “
” represents the phase involved in the water/oil two-phase mixture;
,
and
are the volumetric fraction, density, and velocity vector, respectively.
The linear momentum conservation equation for multiphase flow is defined by Equation (5), as follows:
where
is the stress tensor for the q phase and
is the term for the interface forces between phases p and q, P is the pressure, shared by all existing phases.
The tension related to the phase can be defined by:
where
and
are, respectively, the viscosity and shear stress of phase
, and
is the unit tensor.
The interface strength depends on friction, pressure, cohesion, and other effects, such that the following condition must be met:
The Fluent solver uses a simplified interaction term, as follows:
where
is the interfacial momentum transfer coefficient;
and
are the velocities of the phases.
For a two-phase flow, it is assumed that the second phase is in the form of drops. This has an impact on how each fluid is assigned to each phase, for example, in flows where there are unequal quantities of two fluids, the predominant fluid must be modeled as the primary fluid since the dispersed fluid is more likely to form droplets or bubbles [
56]. The exchange coefficient for bubbling, liquid–liquid, or gas–liquid mixtures can be written as follows:
where
is the interfacial area defined by
is the drag function, which is defined according to the exchange rate model used, and the term
is the “particulate relaxation time”, defined as follows:
where
is the diameter of the drop.
To determine the drag function
, the Schiller and Naumann model [
57] was used, as follows:
where
, is the drag coefficient, given by:
with
being the relative Reynolds number, defined for the primary phase
and the secondary phase
, as follows:
To describe the flow of the mixture inside the domain in the turbulent regime, we used the Shear-Stress Transport (SST
-
) model developed by Menter [
58]. This model is based on the coupling of the standard k-ω models [
59] with the k-ε model [
60], which, respectively, are characteristic for presenting good results near the walls and in the regions away from the numerical domain wall. This model was applied individually for each continuous and dispersed phases. Details about of the turbulence model used in this research can found in the literature [
52].
2.2.2. Formulation for Porous Medium
In the Eulerian–Eulerian multiphase model, the general approach of modeling porous media, physical laws, and equations is applied to the corresponding phase for the conservation of mass and linear momentum. In Fluent software, a porous medium is modeled as a region containing fluid elements, where the equation for the linear momentum is modified by the addition of a source term of dissipation. This source term is composed of two parts: one referring to the loss of viscous effects (Darcy’s Law, the first term on the right side of Equation (14)) and one referring to the loss of inertial effects (the second term on the right side of Equation (14)).
where
is the source term for the i-th (x, y, or z) equation of the linear momentum,
is the magnitude of the velocity, and M and N are prescribed matrices. This momentary sink contributes to the pressure gradient in the porous cell, creating a pressure drop that is proportional to the speed of the fluid in the cell.
Thus, homogeneous porous media are defined as follows:
where
is the permeability, and
is the inertial resistance factor, simplifying the matrices M and N as diagonal matrices, with
and
, respectively, occupying the values on the matrix diagonals. Given the low velocities developed in the volumes relative to the porous medium, the term referring to inertial resistance was neglected in this research.
2.3. Boundary Conditions
- (a)
Module input
At the input of the separator module, a prescribed mass flow condition has been established. Specifying the mass flow rate allows the total pressure to vary in response to the numerical solution. In this boundary condition, the absolute reference system, the flow direction (normal to the inlet surface), the turbulence intensity, I = 5%, and the turbulent viscosity ratio
= 10, given by Equations (16) and (17), have been established.
and
where
is the rate of velocity fluctuation, and
is the mean velocity of free flow. The values of k and ε are computed as a function of this intensity [
60].
- (b)
Concentrate and permeate outputs
For the permeate and concentrate outputs, a prescribed pressure boundary condition was applied. There was a zero gauge pressure at the outlets, that is, the ambient pressure. The pressure difference between the input of the separation module and the outputs (atmospheric pressure) drives the flow of produced water through the separation module.
- (c)
Module wall and membrane surface
The conditions of the wall were used to connect the fluid and solid regions, surfaces of the module, and membranes, in contact, externally, with the volume of the hull and, internally, with the domains related to permeate.
On the internal surfaces of the hull, surfaces of the supports (membrane ends), hull and permeate, non-slip boundary conditions, and null roughness were applied. For the internal and external surfaces of the membranes, the condition of the interior wall (open surface) was used, which allowed the flow of produced water through the membranes. Additional information can be found in
Table 2.
2.4. Numerical Procedures
For the pressure–velocity coupling, the Coupled algorithm, available in the Ansys Fluent® software, was used. The Coupled algorithm solves the continuity equation based on the results of calculating the conservation of linear momentum and pressure, in a coupled way, which gives it, concerning the segregated solution algorithms, a convergence range with a smaller number of iterations.
The relaxation factors used in the pressure–velocity coupling methods called Coupled are shown in
Table 3. The convergence criteria used in the simulations are described in
Table 4.
2.5. Thermo-Physical Properties of Fluids
The physical properties of the substances used in the numerical simulations are described in
Table 5.
2.6. Simulated Cases
In this research, different simulations were performed, varying the mass flow,
, the volumetric oil concentration at the entrance,
, the average diameter of the oil droplet particles,
, the permeability of the membranes, K, and the porosity of the membranes, ε. The idea is to evaluate the effect of these variables on the fluid dynamic behavior of the phases inside the module of porous ceramic membranes and the separation performance of this equipment. The values of the variables at entry in the initial conditions, in each case studied, are shown in
Table 6.
Case 5 was considered as a standard case. With Cases 1, 2, and 3, the influence of the mesh on the results obtained was investigated. Cases 2, 4, and 5 were simulated to verify the effect of the mass flow rate of the mixture at the entrance of the separation module, and Cases 5, 6 and 7, were used to evaluate the effect of the oil volumetric concentration at the entrance of the separation module. The effect of the average diameter of the oil droplet was verified with the results obtained with the simulations of Cases 5, 8, and 9, and the effect of the membrane permeability, with Cases 5, 10, and 11. To verify the influence of the porosity of the membranes in the process, Cases 5, 12, and 13 were simulated.