Computational fluid dynamic modeling of a pervaporation process for removal of styrene from petrochemical wastewater

Abstract

In this study, a predictive model was developed to describe the process of separation of volatile organic compounds. The model was based on solution–diffusion theory and computational fluid dynamics (CFD) was applied to model the pervaporation process for the separation of styrene–water in a PDMS membrane. Florry–Huggins-FV, Florry–Huggins and UNIFAC models were adopted to predict the equilibrium adsorption of styrene and water into the PDMS membrane and the results were compared. For the penetration, Duda's and Fujita's free volume theories were compared to estimate the diffusivity of penetration through the membrane. The conservation equations including the continuity and the momentum balance equations in membrane modulus were solved simultaneously using a finite element scheme. The studied system includes styrene–water solution which is adsorbed on a polydimethylsiloxane (PDMS, polydimethylsiloxane) nonporous polymer membrane. In this mass transfer model, diffusion coefficients through the membrane were taken into account as a function of concentration and they were estimated based on free volume theory. The proposed model was validated using experimental data obtained from pervaporation tests. The simulation results for various feed concentrations, feed temperatures and membrane thickness were in satisfactory agreement with experimental data. The CFD results showed that both the permeation flux and the styrene enrichment factor increased with the increase in concentration of styrene in the feed. By increasing membrane thickness the permeation flux decreased while the enrichment factor increased. The developed model can predict well the mass transfer process occurring in the feed and the membrane.

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Introduction

The pervaporation (PV) process is a separation process based on selective mass transfer through a compact membrane which is followed by a phase change of the penetrates from liquid to vapor. This process is well-known as an effective membrane process for the separation of azeotropic mixtures, close boiling point mixtures, and isomers.¹ Pervaporation has been considered as an appropriate alternative for a number of separation processes like distillation and liquid–liquid extraction. Among the major applications of pervaporation are dehydration of organic/inorganic solutions, the removal of organic species from water and the separation of organic/organic solutions.²

In this area, the knowledge of adsorption characteristics and the mass transfer through membrane layers are important. In order to predict liquid transport in binary systems through a pervaporation membrane, a number of models have been developed and presented such as the solution–diffusion model,³ residence-in-series model,^4,5 pore flow mechanism,⁶ pseudo phase-change solution–diffusion model,⁷ dynamic heat from irreversible process,⁸ Stephan–Maxwell theory,⁹ and molecular dynamic simulation.^10,11 Among these models, the solution–diffusion model is widely used to describe mass transfer in a pervaporation process. According to this model, dissolution and permeation occur in two steps: selective adsorption of liquid molecules by membrane surface in feed side and their selective permeation through membrane due to difference in solubility and permeation of the resulted compounds.¹² The PV model is not only important for understanding of the dependency of flux to process parameters, but also is helpful for design calculation of PV module.¹³

Liu et al.¹⁴ applied CFD for simulation and optimization of flow filed distribution in models with different packaging density and cross-section layout. Rezakazemi et al.¹⁵ developed comprehensive model to investigate pervaporation performance. This model was based on solving the conservation equations for water in the membrane module. Computational fluid dynamics method was used to solve the equations. Keshavarz Moraveji et al.¹⁶ applied CFD for modeling pervaporation process for the separation of ethanol from aqueous solutions with polydimethylsiloxane membrane. The proposed model was used to describe the penetration and the effect of flow rate, temperature and concentration of ethanol feed on the pervaporation process performance. Mafi et al.¹⁷ investigated mass transfer in the recovery of aromatic compounds using pervaporation based on the solution–diffusion theory. They adopted CFD to solve their mass transfer equations and determined distribution of penetrates within the membrane as well as penetration flux and concentration through the membrane. Shokouhi et al.¹⁸ focused on the absorption stage in modeling pervaporation process. They applied four thermodynamic models including Flory–Huggins, UNIQUAC, modified NRTL and modified Wilson to describe equilibrium adsorption of ethanol/water mixture into polydimethylsiloxane membrane.

The main aim of this study is to develop a comprehensive two-dimensional CFD model in order to predict the behavior of styrene removal from an aqueous solution using PV process. The membrane considered is a commercial composite membrane, polydimethylsiloxane (PDMS)/PVDF/PP. The model is based on the solution of the conservation equations for styrene in membrane module. Conservation equations of continuity and momentum equations are obtained and solved numerically using the finite element method. In this study, a mass transfer model will be developed based on the solution–diffusion theory to predict membrane-side concentration profile. The Flory–Huggins, Flory–Huggins-FV, and UNIFAC theories are adopted and compared to explain adsorption in membrane. The Lee's equation and two free-volume theories are used to model the transport step. A set of experimental data from pervaporation of binary styrene/water solution in PDMS commercial composite membrane are used to validate the presented model. In this study, the effect of operating parameters such as concentration and temperature of feed as well as membrane thickness on mass transfer flux and selectivity of membrane are evaluated in details and the model's results are compared with available experimental data.¹⁹

The modified mass transfer model

The solution–diffusion model is an appropriate model to describe permeation in a polymer membrane.³ In this model, adsorption and diffusion of species in membrane are two important steps for determining the mass transfer rates and the membrane selectivity. The major assumptions made for developing the model are as follow:

• The solution–diffusion mechanism is hold.

• No reaction occurred between species across the membrane.

• The membrane is in thermodynamic equilibrium with the feed mixture.

• The membrane temperature is constant across membrane thickness.

• The system is considered steady-state and one-dimensional with permeation through membrane thickness only.

• The feed-side boundary layer resistance and that of the membrane protection layer are neglected.

• Because of the vacuum generated, the concentration of species in permeate side is zero (PV pressure is 1 mm Hg).

Membrane adsorption

Adsorption is specified mainly by chemical nature of polymer membrane and permeable molecules. Interactions between penetrate species and polymer determines the amount of adsorption occurred in the membrane. Physical-chemical forces, such as dispersion forces, hydrogen bonding and polar forces influence on the interaction between passing molecules and polymer.²⁰ According to the dissolution–permeation model, it has been assumed that phase equilibrium exists between feed mixture and membrane, therefore, chemical potential of species in these two phases are equal.

The volume fraction of species in the membrane can be achieved by equating the chemical potentials of different species:

μ^F_i = μ^M_i (1)

In isothermal process, chemical potential is calculated from:

dμ_i = RTd ln a_i + V_idP (2)

where, V_i is molar volume, μ_i is chemical and a_i is activity of species i, and P is the pressure of the system. Unlike other chemical processes such as reverse osmosis, in pervaporation process one may neglect the pressure change compared to the activity change, because the pressure difference between the feed and permeate sides is approximately 1 bar. Therefore, eqn (2) is simplified as follows:

μ_i = μ^°_i + RT ln a_i (3)

where, μ^°_i is chemical potential of species i at the reference conditions. Therefore, the equity of chemical potential can be written as the equity of activates of species in two phases. The feed phase consists of styrene and water, so two equilibrium relationships exist for the two species in which styrene is considered as species 1, waster as species 2, and PDMS as species 3.

a^F₁ = a^M₁, a^F₂ = a^M₂ (4)

As the sum of species in each phase is one, the third equation is written as follows:

φ^M₁ + φ^M₂ + φ^M₃ = 1 (5)

These three equations were solved using fsolve function in MATLAB and volume fractions of species were used as boundary conditions.

UNIFAC model

In UNIFAC model, the calculation of activity coefficient consists of two parts: the first part relates to the difference in the shape and size of the molecules in the mixture that called combination term, and the second part relates to the inter-molecular interactions that is called residual term. The equations of this model to calculate the activity coefficient of species i are as follow:²¹

ln γ_i = ln γ^C_i + ln γ^R_i (6)

Flory–Huggins-FV model

Flory in 1953 presented the following relationship between the activity and the volume fraction of systems comprising a polymer and a species:where, s and p stand for species and polymer, respectively. This relationship is used mainly to describe the adsorption equilibrium. It is extended for ternary mixtures that include a polymer and a liquid species, as follows:

In these equations, subscripts 1, 2 and 3 represent the two liquid species (styrene and water) and polymer (PDMS), respectively, χ₂₃ and χ₁₃ represent the interaction between the soluble and membrane and χ₁₂ shows interaction of species with each other.²²

Free-volume theory is defined for polymer and solvent systems. This model takes into account the effect of free volumes of polymer and is more important for systems involving large molecules. Small molecular compounds normally contain high percentage of free volumes. However, the percentage of free volume in polymers is significantly lesser than in most of solvents.²³ The effect of free volumes is considerable on elastomers. In this study, attempts have been made to modify Flory–Huggins theory through considering the effects of free volume on elastomers for adsorption of species.

To consider the effect of free volumes, a number of parameters should be defined, as follow:

where, the free volume (v^FV_i) is determined from:²⁴

v^FV_i = (0.3 + 4.5 × 10⁻⁴T)v^vdw_i (11)

v^vdw_i is van der Waals molar volume of species i which is determined from eqn (12):²⁴

v^vdw_i = 15.17r_i (12)

where, r_i is molecular volume parameter.

To determine χ₂₃ and χ₁₃ which represent interaction of species with polymer, the Flory–Rehner equation is used, namely:²⁵

and to determine χ₁₂ which represents interaction of species with each other, the Hansen Solubility Parameters (HSP) theory is applied. χ₁₂ is obtained using eqn (14) which is based on HSP method:where, V is the molar volume of the solvent, δ is solubility parameter of species, R is universal gas constant and T is absolute temperature. HSP parameters for the various species are listed in Table 1.

Table 1 HSP parameters considered for various species^26,61

Property	Styrene	Water	PDMS
Dispersion solubility (δd)	18.6	15.5	15.9
Polar solubility (δp)	1.0	16.0	0.1
Hydrogen bonding solubility (δh)	4.1	42.3	4.7

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The empirical constant β is usually taken into account between 0.3 to 0.4, whereas it is 0.34 in the most ref. 26.

Permeation through membrane

The continuity equation for species i in differential from can be expressed as the following equation:²⁷where, ρ is density, J is flux of mass transfer, and r is the production source term. It is assumed that, mass transfer in membrane is diffusion dependent.²⁸ Therefore, the conductive mass transfer term is negligible in comparison with the diffusion term. According to these assumptions, eqn (15) is simplified to the following equation:where z is direction of mass transfer. According to the Lee's theory, for a stagnant flat membrane, mass transfer flux of species i is expressed as:²⁹where, D is diffusivity, μ is chemical potential, and c_i is concentration of species i that is related to the volume fraction according to the following relation:

c_i = ρ_iφ_i, for i = 1, 2 (18)

In isothermal, isobar and activity contribute with chemical potential of i, the following relation is hold:

dμ_i = RTdln a_i + V_idP, for i = 1, 2 (19)

where, P is pressure. By substituting eqn (18) and (19) in eqn (17), it yields:

In the above equation, pressure gradient could be neglected compared to the first term in the parenthesis, due to low pressure difference between feed and permeate water phases (about 1 bar), therefore:

The activity of permeates can be determined from Flory–Huggins-FV theory for ternary systems.³⁰ Considering the strengthening effect to calculate flux, eqn (21) is rearranged to give:

By differentiating from eqn (8) and (9) against φ^FV₁ and φ^FV₂ respectively, partial derivatives of ∂ln a₁/∂φ^FV₁, ∂ln a₁/∂φ^FV₂, ∂ln a₂/∂φ^FV₁ and ∂ln a₂/∂φ^FV₂ are resulted as follow:³¹

By taking ρ₁ and ρ₂ into account independent of z, eqn (22) and (23) are substituted in eqn (16) and the following differential equations are resulted:

Calculation of diffusion coefficient in membrane

Determination of the diffusion coefficient of different species in a membrane is typically a very important step in modeling. Diffusion is the controlling step in pervaporation process. Diffusion phenomenon in membrane is often complex due to the influence of membrane-penetrating components on each other as well as the effects of the concentration on diffusion coefficient. Various empirical and semi-empirical models with different assumptions have been presented to describe this step. A number of the most popular models are given in ref. 32 and 33.

According to Yeom and Huang study, a good agreement exists between actual data and the values calculated from free volume theory.³³ Wesseling and Bollen extended free volume theory for diffusion into multi-component mixtures. They found that the free volume for each composition can be determined from its surface component.³⁴ In this study, the latter model was used to calculate the diffusion coefficient. Two free-volume models have been considered and their results have been compared with referenced experimental data.¹⁹

Fujita's free-volume theory. Fujita's free volume theory was developed to describe the mass transfer through polymer membranes based on the movement of molecules from one place to another. The main idea of this theory is that an external molecule can enter the polymer even if free space exists.³² The basic assumption of this model is dependency of diffusion coefficient on the temperature and molecular characteristics, including its size and shape. The thermodynamic diffusivity of species in polymer is expressed in eqn (30):where, A_d and B_d are constant parameters related to the shape and size of penetrating molecule, f is free volume fraction of polymer that is function of volume fraction of component i and temperature. For the calculation of the thermodynamic diffusivity, first, free volume fraction of polymer is calculated from eqn (31):

f(φ_i,T) = f(0,T) + β_i(T)φ_i (31)

In eqn (30), φ_c polymer crystallinity, f(0,T) is the volume fraction of alone polymer and β_i(T) is a proportionality constant that represents the increase in polymer free volume due to diffusion of species i. The latter is obtained using the free volume fraction of pure liquid.³⁵

f(0,T) = 0.025 + a(T − T_g) (32)

where,

a = 4.8 × 10⁻⁵k⁻¹ T < T_g

a = 4.8 × 10⁻⁴k⁻¹ T > T_g

In the above equation, T_g is glass transition temperature of the polymer. The free volume fraction in the liquid can be calculated by the formula given by Doollittle.³⁶

Taking the volume fraction zero in eqn (30), the diffusion coefficient at infinite dilution based on the free volume theory is expressed by eqn (33).

Diffusion coefficient at infinite dilution in engineering calculations represents a diffusion coefficient at concentrations of less than 5% (molar) of soluble component. One of the most common methods for calculating this coefficient is Wilke–Chang's equation, namely:²¹

In order to estimate the diffusion coefficient values, first, by putting components properties in the eqn (34) the diffusion coefficient at infinite dilution is calculated. Then, by combining eqn (30)–(32) and using a nonlinear fitting procedure in MATLAB, constants A_d and B are obtained. Yeom et al. used Fujita's free volume theory in their study for modeling pervaporation process of ethanol–water mixture and validated their model through comparing the simulated and experimental data of permeability.³⁷

Duda's free-volume theory. The molecular permeation into polymer membrane occurs by passage of molecules through free spaces and intermolecular distance between the polymer chains.^34,38–42 So far, a number of models have been developed based on the free-volume theory to predict diffusivity of penetrants within membrane. Raisi et al.⁴³ applied the Wesslingh's free-volume theory to predict diffusivity of an odor compound within PDMS membrane. The basic of the Duda's free-volume theory for determining diffusivity in permanents/membrane ternary systems is as follows:^44,45where, E₁ and E₂ are critical energies of molecules to overcome the attractive forces keeping them next to each other, Γ is the overlap factor, V̂^∗₁, V̂^∗₂, and V̂^∗₃ are specific free volumes of pore for species 1, 2, and 3, respectively, w₁, w₂, and w₃ are mass fraction of species, ξ₁₃ and ξ₂₃ stands for critical molar volume ratios of a single jump for species 1 and 2, respectively, to the critical molar volume of a single jump for membrane, V̂^FH/Γ is mean free volume of pore per unit mass of the mixture, k_II,2 − T_g,2 and k_II,1 − T_g,1 are free volume parameters for permanents and k_II,3 − T_g,3 and k_I,3 are free volume parameters for polymer. D₀₁ and D₀₂ are zero-concentration diffusivity of penetrants in membrane in a binary system which are calculated with the following equation:where, D⁰₁ and D⁰₂ are diffusivities in infinite dilution liquid bulk condition of a penetrant/penetrant binary system that are calculated from Wilke–Chang theory.⁴⁶ The used parameters in eqn (35)–(39) for a number of solutions and polymers are given in literature.^45,47 The parameters required in the free-volume theory for styrene/water/PDMS membrane are summarized in Table 2.

Table 2 The free volume parameters of styrene/water/PDMS membrane used in this study

Parameter	Styrene (1)^47,62	Water (2)^47	PDMS (3)^44
V̂^∗ (cm^3 g^−1)	0.899	1.072	0.905
	1.20 × 10^3	2.18 × 10^−3	9.32 × 10^−4
kII − Tg (K)	−68.92	−152.29	−81.00
ξip	0.634	0.232	—
E (cal per mol)	386.86	250.01	—

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Referenced experimental description

The PV setup considered in this study has been described in.^5,48 A PDMS/PVDF/PP composite membrane with a PDMS functional layer (thickness 1, 18 μm) was used. A 20 cm × 15 cm cut of this membrane was placed in a flat form membrane module. Styrene (reagent grade, purity 99%) and water (HPLC grade, purity 99.9%) were supplied from Merck Co., Ltd (Darmstadt, Germany). Distilled water was used to prepare aqueous solutions.¹⁹ A schematic diagram of the setup is represented in Fig. 1.

Fig. 1 (a) Feed channel, (b) module used in PV experiments.

The calculation of flux and concentration profile of species

In this study, the finite element scheme is applied to solve the set of non-linear equations. To obtain the concentration profiles, the mass transfer boundary conditions for eqn (28) and (29) are as follow:

at z = 0; φ_i = φ^f_i, at z = δ; φ_i = 0 (40)

The differential eqn (28) and (29) were solved using a Galerkin finite element algorithm.⁴⁹

Mathematical modeling

Assuming Newtonian fluid for styrene/water mixture and the momentum and continuity equations can be written for feed stream.

The following assumptions have been made:

• Constant feed temperature.

• Steady feed stream for feed (the changes with time is zero).

• Flow is only in y-direction.

• Laminar stream.

• Incompressible Newtonian fluid (constant density).

• Velocity change only in the z-direction.

By solving the continuity equation and using the above assumptions, it is concluded that flow is fully-developed. By developing the equation of motion in the y-direction and some simplification, the following equation is obtained:

In the above equation, the term on the right side is constant and is determined from the pressure difference created by the pump through the module length. To solve the above equation two boundary conditions are required. Considering the available solid wall, the following two conditions are hold:

z = 0; v_y = 0, z = z₁; v_y = 0 (42)

Fig. 2 shows the domain of the present model. This model is proposed for the steady-state transportation of styrene through the membrane. As it can be seen in this figure, at zone 1 which is shown with width z₁ the feed entered from bottom and exited at the top of zone Z₁. As it flows upward in this zone, the flow of the feed is in contact with zone 2 and it absorbed by the membrane. After passing the membrane, it will goes toward zone 3 which is the vacuum zone. Because of the existence of vacuum in this part, the penetrate will desorb quickly from the membrane and will exit from the top of zone 3.

Fig. 2 Different zones and boundaries of the model.

Mass transfer equations in the feed side

To simplify mass transfer equation the following assumptions have been considered:

• No chemical reaction occurs in the feed stream.

• In the direction of flow, the mass transfer due to diffusion can be neglected compared to the convective mass transfer.

• The feed temperature is constant.

• Flow streams only in y-direction.

• The change of water concentration with time is negligible.

Applying the above assumptions in mass transfer equation is simplified to:

By substituting the Fick's law in eqn (43), it gives:

To solve the above equation, three boundary conditions are required which are as follow:

c(z,y = 0) = c₀, (45-a)

(z = z₁,y) = c_eq (45-c)

where, c₀ is initial concentration. According to eqn (45-b), no diffusional mass transfer occurs at the wall, and in eqn (45-c), it is assumed that water concentration at the membrane wall is equal to its thermodynamic equilibrium concentration.

The flow sheet/algorithm for the established solution–diffusion model is shown in Fig. 3.

Fig. 3 The flow sheet and algorithm for the established solution–diffusion model.

Solution procedure

Exact solution for solving modelling equations simultaneously is not applicable. As a result the numerical solution is used in order to solve the equations. The basic of solution of the governing and the supplementary equations is the finite element scheme. The numerical solution was performed using CFD based on finite element scheme using an appropriate mesh generator.^50,51 Based on network independence, a total number of 5885 meshes were used in this simulation. Fig. 4 shows a part of concentration profile obtained in the membrane.

Fig. 4 A part of concentration profile obtained.

In this study, the obtained equations for the feed and membrane were discretized using finite element method. PARDISO was used as a direct solver. Various types of grids were considered and compared with each other. Finally, the triangular grid with uniform size was chosen for the calculation (Fig. 5).

Fig. 5 A part of mesh created in this study.

As it is seen, in critical areas that the changes are significant finer meshes have been considered. Table 3 shows the mesh sensitivity at temperature of 30 °C and concentration of 50 ppm. The mesh shown in 8^th row has been selected.

Table 3 Details on mesh and the parameters used in simulation

	Number of elements	Number of boundary elements	Error (%)
1	1384	211	16.52
2	1585	237	15.19
3	1977	257	14.06
4	2527	291	12.34
5	3267	333	11.23
6	3972	380	9.36
7	4676	410	8.9
8	5885	490	8.69
9	6535	543	8.69
10	8160	601	8.69

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Table 4 summarizes the details on mesh and the parameters used in simulation.

Table 4 Details on mesh and the parameters used in

Mesh detail		Solver parameter
No. of degrees of freedom	12952	Solver	Stationary
Number of boundary elements	490	Linear system solver	PARDISO
No. of elements	5885	Relative tolerance	1 × 10^6
Mesh shape	Triangular	No. of iteration	8

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Results and discussions

Various models related to adsorption phenomenon and the two free volume theories presented for the penetration step have been compared. The governing equations in the feed-side with the membrane boundary conditions have been solved for styrene/water mixture in various operating conditions. The obtained results were validated against available experimental data.¹⁹

The results of Flory–Huggins-FV, Flory–Huggins, and UNIFAC models

The experimental data of styrene flux versus concentration at three temperatures, 30, 45 and 60 °C, have been extracted from ref. 19 and used. In Fig. 6–8, styrene flux is estimated using the results of Flory–Huggins, Flory–Huggins-FV and UNIFAC models.

Fig. 6 The effect of styrene concentration on styrene flux at temperature of 30 °C based on Flory–Huggins-FV, Flory–Huggin, and UNIFAC models.

Fig. 7 The effect of styrene concentration on styrene flux at temperature of 45 °C based on Flory–Huggins-FV, Flory–Huggin, and UNIFAC models.

Fig. 8 The effect of styrene concentration on styrene flux at temperature of 60 °C based on Flory–Huggins-FV, Flory–Huggin, and UNIFAC models.

As it is seen in these figures, Flory–Huggins-FV model in comparison with the two other models, UNIFAC and Flory–Huggins, predicted the experimental data more accurately. This shows that the free-volume theory had a positive impact on the modification and accuracy of Flory–Huggins model. Therefore, considering the free volume of the polymer and its effect on styrene flux is important. As can be seen, at higher temperature the accuracy of Flory–Huggins model decreased while that of Flory–Huggins-FV model is still high. It is satisfactory to consider the effects of free volume for the present system and improve the model accuracy. So, we applied Flory–Huggins-FV model for the adsorption process in this study. The average absolute errors of Flory–Huggins, Flory–Huggins-FV and UNIFAC models in all concentrations for three different temperatures (30, 45 and 60 °C) is calculated using eqn (46):

The calculated errors are presented in Fig. 9. As shown in this figure, the average absolute error of Flory–Huggins-FV is 12.71% which is less than that of Flory–Huggins and UNIFAC models which are 34.9 and 22.86 respectively. As a results, Flory–Huggins-FV can predict experimental data more accurately than the others in all temperatures and in different styrene concentrations specially in high concentration of styrene, Flory–Huggins-FV can predict the experimental data better than the others.

Fig. 9 Average absolute error for UNIFAC, Flory–Huggins and Flory–Huggins-FV in three different temperatures (30, 45 and 60 °C).

The results of Fujita's and Duda's free volume theories

Determining the diffusion coefficient of each of the species in the membrane is one of the key steps for modeling pervaporation process. Therefore, in this section, the two most widely used models, namely Fujita's and Duda's free volume theories are applied to obtain diffusion coefficients and their results are compared. These theories have been adopted to describe the effect of styrene concentration on styrene flux at a temperature of 60 °C. The results are shown in Fig. 10.

Fig. 10 Styrene flux vs. feed concentration at temperature of 60 °C based on Fujita's and Duda's free-volume theories.

As seen in Fig. 10, the both theories have similar accurately to estimate styrene flux. Hence, in future calculations of the model, the Duda's free volume theory will be used. According to Fig. 10, the accuracy of this free volume theory decreases when feed concentration increases. Thus, it is appropriate to predict experimental results when the feed concentration is low.

The effect of feed concentration

One of the important parameters affecting the mass transfer flux and selectivity of pervaporation is feed concentration. A change in feed composition directly affects the adsorption process. The permeation of species into membrane depends on their concentration, too.

To study the effect of feed concentration on the system performance, the concentration was varied from 20 to 300 ppm for styrene–water mixture in three different temperature (30, 45 and 60 °C) and the results are shown in Fig. 6–8. According to Fig. 6–8, Styrene flux in PDMS membrane increased linearly with increasing styrene concentration in feed. For instance, as shown in Fig. 6, at temperature of 30 °C, with increasing styrene feed concentration from 20 to 300 ppm, styrene flux increased from 0.52 to 10.51 g m⁻² h⁻¹. The increase in flux resulted from an increase in styrene concentration which can be assigned to the driving force for permeation as well as the mass transfer characteristics of membrane. Therefore, chemical potential is directly affected by feed concentration which means increasing styrene concentration in feed results in an increase in chemical potential throughout the membrane which in turn increases the permeation flux.

During adsorption step, the high styrene concentration in the feed intensifies its adsorption. Therefore, the penetrate concentration in membrane is increased. The latter causes membrane swelling, which increases the free volume for styrene penetration into membrane and thus increases styrene flux. In addition, the concentration of the penetrate influences on mass transfer behavior because penetration is dependent on the penetrate concentration in the membrane. In the absorption step, higher the concentration of styrene in the feed leads to higher styrene adsorption into membrane. As a results, styrene concentration within the membrane increases.

Moreover, it can be seen in Fig. 6–8 that the difference between the predicted and experimental values of styrene flux increases in higher styrene concentrations in feed. This deviation may be related to the dependency of styrene diffusivity to the concentration. In fact, the Duda's free-volume theory could not predict styrene diffusivity well in higher concentrations; however, its prediction is reasonable in low concentrations. As it has been observed, the permeation flux predicted by the model has a good agreement with experimental data at low feed concentrations.

It can be concluded that, the observed deviation in higher concentrations is assigned to the assumption of activity equilibrium at membrane-feed interface that may not be occurred in experiments. In other words, in the higher concentration, the absence of equilibrium is more important. As it is seen, the changes in flux in the model's results and experimental data are reasonably similar; therefore, the predicted and experimental data are in good agreement with each other. In higher concentrations, the deviation from equilibrium conditions is higher which results in higher deviations of the model's results. In addition, the concentration polarization in pervaporation, which could lead to further diversion, has not been considered in the present model.

The enrichment factor of styrene increases with feed concentration which should be considered in more detail. Here, the key point is the change in water flux with styrene concentration. Fig. 11 indicated water flux variation with styrene feed concentration at 30 °C. As it can be seen in Fig. 11, with increasing styrene feed concentration from 20 to 300 ppm water flux decreased from 97.58 to 82.26 g m⁻² h⁻¹. This observation is rather different from those reported by several researchers for many other organic compounds in pervaporation process.^52,53

Fig. 11 The effect of feed concentration on water flux in pervaporation process at temperature of 30 °C.

Furthermore, as can be seen in Fig. 11, the predicted mass fluxes of water are always higher than the experimental values. This result could be assigned to water penetration behavior into hydrophobic membranes. Penetration coefficient of water inside the membrane was determined based on free-volume theory, in which it is assumed that water molecules penetrate into polymer membrane without the formation of clusters. However, Nguyen et al.⁵⁴ demonstrated that water molecules easily form clusters even though the amount of water absorbed by the membrane is low. It has been shown that, water penetration into polymer membrane may be delayed by the formation of clusters of water molecules.^52,55 Clustering of water molecules occurs due to desorption forces between the adsorbed organic compound and water molecules.⁵⁶ So in practice, by forming the clusters, the size of permeate species increases and this effect leads to reduced permeability of clusters with regard to normal molecules. The latter effect has not been considered in the study, and this may be the reasoning for a higher flux predicted by the model than that observed in the experimental. Water diffusion coefficient through the membrane has been obtained based on the free-volume theory in which it has been assumed that the penetration of water molecules occurs without any accumulation in clustering inside the polymer membrane. On the other hand, the non-polar hydrophobic styrene molecules adsorbed by the membrane affect polymer matrix characteristics; they increase the hydrophobicity of membrane and inhibit partly water molecules from passing through membrane. It should be noted that, this phenomenon has been observed in organic compounds with high hydrophobicity nature which causes plasticization of membrane. This phenomenon has not been considered in this study and may be the reasoning for greater flux values predicted by the model in comparison with the experimental results.

According to Fig. 11, the proposed model can predict the permeation flux and selectivity of styrene reasonably without the need for adjustable parameters. The model could predict the effect of feed concentration on the performance of the hydrophobic pervaporation process reasonably for the separation of styrene from aqueous solutions.

The effect of feed temperature

Feed temperature is an important parameter on pervaporation process of styrene–water, as it affects the both adsorption and the permeation steps. Permeability, viscosity of species in feed mixture as well as penetration of species inside the membrane are changed with food temperature.

The effect of feed temperature on flux is shown in Fig. 12 and 13. As it is observed, by increasing temperature, flux of various species increased. As an illustration, with increasing temperature from 30 to 60 °C, styrene flux increased from 1.42 to 1.62 g m⁻² h⁻¹ and water flux increased from 95.81 to 417.8 g m⁻² h⁻¹. According to the membrane free volume theory, in polymer membranes, penetrating molecules pass through the membrane's free volumes. Free volume of membrane is formed from local movements of polymer chains. By a temperature increase, the frequency, domain and local motions of polymer chains increase which causes an increase in free volumes of membrane.⁵⁷ Hence, the penetration rate of species and their mass transfer flux are increased. Besides, equilibrium vapor pressure of penetrating molecules changes with temperature; feed temperature affects the mass transfer driving force. The latter is increased due to increase of vapor pressure of pure species, therefore, the chemical potential gradient and the mass transfer rate through membrane increases. On the other hand, the temperature increase of feed decreased viscosity of feed solution which increases diffusivity of various species in the feed and increases the penetration rates.

Fig. 12 The effect of feed temperature on styrene flux at styrene concentration of 50 ppm.

Fig. 13 The effect of feed temperature on water flux at styrene concentration of 50 ppm.

The effect of membrane thickness

The effect of membrane thickness is considered to study membrane performance in pervaporation process. In general, in pervaporation process, membrane thickness is inversely related to flux of species passing from membrane. The greater membrane thickness the smaller flux of species.

A similar trend in the performance of pervaporation have been reported in studies.^58–60 This phenomenon can be explained as follows: in larger membrane thickness, feed species should penetrate through a long tortuous route to reach the other side of the membrane, thus the resistance against penetration increases. Moreover, when the membrane thickness is increased, the uniformity of membrane structure may be changed. The results have been presented for two thicknesses of 1 and 18 micrometers in Fig. 14. As shown in Fig. 14, by increasing the thickness of the membrane, styrene flux reduces fairly while the water flow is decreased significantly. Decreasing membrane thickness from 18 to 1 μm, styrene flux increased from 1.42 to 1.55 g m⁻² h⁻¹ and water flux increased from 95.81 to 418.51 g m⁻² h⁻¹. Hence, by increasing the thickness of the membrane the enrichment factor of the process increases. Similar behavior is observed by other researches.⁵⁹

Fig. 14 The effect of membrane thickness on water and styrene fluxes at styrene concentration of 50 ppm and temperature of 30 °C.

Conclusion

In this study, the removal of styrene from petrochemical wastewater was investigated via pervaporation process. A 2D CFD model was developed and presented for the prediction of mass transfer in the pervaporation process of styrene in aqueous solutions using hydrophobic PDMS membrane based on solution–diffusion theory. In the model, the Flory–Huggins-FV, the Flory–Huggins and the UNIFAC models were applied in order to describe the adsorption and permeation process in membrane. It was found that the Flory–Huggins-FV equation results in more accurate predictions of experimental data in comparison with the Flory–Huggins and UNIFAC models. To model the adsorption phenomenon, the free-volume theory was adopted. In this regard, two common theories, namely Duda's and Fujita's free-volume theories were applied and their results were compared with experimental data of ref. 19. The both theories resulted in almost similar predictions. In low feed concentrations, this theory is more accurate in predicting the experimental results. The CFD model is capable of predicting the effect of operating parameters like concentration, velocity and temperature of feed as well as membrane thickness on the performance of the pervaporation process. The mass transfer flux is higher in thinner membrane whereas the selectivity of thicker membranes is more. By increasing styrene concentration in feed, the partial flux and enrichment factor are both increased. The results predicted by the model were compared with experimental data available in ref. 19 and a good agreement was revealed in various operating conditions. Therefore, the model successfully predicts the performance of hydrophobic pervaporation process for removal of styrene from aqueous solutions without the need for adjustable parameters.

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