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To evaluate the disposal effluent from the Al-Daura refinery in Iraq, which comprises oily wastewater, a mathematical model has been developed for both forward osmosis (FO) and osmotic membrane bioreactor (OsMBR). The procedure is explained mathematically, accounting for both the concentration and polarization aspects. As a result of mathematical modeling, the water flux was determined by the osmotic pressure, the concentration, and the polarization of the feed and draw solutions. Based on traditional methods of predicting water flux using external and internal concentration polarizations, it is determined that water flux will occur in the first model (Model-1). To increase the accuracy of Model-1, the resistivity (K) of the solute has been modified to be independent of the diffusivity of the solute. The old model (Model-1) and the updated model (Model-2) overestimated water flux by 17 and 25%, respectively. It was possible to make a valid comparison between the experiment and theory based on the results of both experiments.

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  • An osmotic membrane bioreactor (OsMBR) is an excellent choice for treating oily wastewater discharged from the Al-Daura refinery. The OsMBR process converts oily wastewater into high-quality water that can be reused in a variety of applications.

  • Increasing the feed temperature, draw solution concentration, and feed flow rate increased the water flux from the forward osmosis process.

external concentration polarization, internal concentration polarization, modeling, wastewater, water flux
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The scarcity of fresh water is a fundamental issue in many parts of the world, and it affects many sectors of society. It has become increasingly evident in the 21st century that access to fresh water is one of the biggest obstacles, both in terms of consuming it and using it for other purposes (Al-Alawy & Salih 2016; Cairone et al. 2024; Guo et al. 2024). Globally, there are 2.2 billion people in the world who do not have access to clean water due to poor sanitation conditions. There is therefore no doubt that meeting the growing demands for clean water in the 21st century is going to be one of the greatest challenges of the century (Al-Alawy et al. 2017; Tortajada & Biswas 2018; United Nations 2023; Choque Campero et al., 2024; WWAP 2024). As a general rule, wastewater is generated by two types of sources: industrial and human wastes. Industrial wastes come from companies such as oil refineries, where they generate industrial wastes. A refinery may discharge effluents that contain oil, grease, and hydrocarbons. These three contaminants are the most common ones that can be found in the effluents of refineries. Among the many biotreatment techniques that are currently being developed, there are membrane-based biotreatment technologies that show great promise, especially when it comes to the production of high-quality water that is free from contaminants and known to be unharmful to living organisms (Escobar 2010; Chung et al. 2012; Zhao et al. 2012; Im et al. 2021; Andrianov et al. 2023; Boubakri et al. 2024). Among the various ways to treat oily wastewater are solvent extractions, adsorptions, chemical oxidations, and biological treatments. Among the various ways of treating oily wastewater are membrane bioreactors (MBRs), osmotic membrane bioreactors (OsMBRs), and conventional treatments. Regardless of whether the wastewater is organic or inorganic, the methods of treating it are the same. In terms of wastewater treatment processes, MBRs and OsMBRs stand out as the most effective ones. It is important to consider several factors when selecting the best water treatment strategy, such as water quality requirements, energy consumption, and operating costs (Al-Saffar & Al-Alawy 2002; Abass O et al. 2011; Abdul Wahab et al. 2015; Al-Asheh et al. 2021; Chang et al. 2022).

It has been noted that a variety of membrane separation techniques have been developed and are being applied for the treatment of industrial and municipal wastewaters in an attempt to make them drinkable. In order to achieve separation, there are two kinds of membranes that can be used. The first are those driven osmotically, such as forward osmosis (FO) membranes, and the second are those driven by pressure, such as nanofiltration, ultrafiltration, microfiltration, and reverse osmosis (RO) membranes. As the name implies, FO is a method of transporting water via natural osmosis from an aqueous solution through a membrane to an aqueous solution through a highly selective layer that has been created via nature (Ana Isabella Navarrete Pérez 2015; Mamah et al. 2022; Salamanca et al. 2023; Anh-Vu et al. 2024; Takabi et al. 2024). In contrast to pressure-driven membrane processes, FO is a naturally occurring, osmosis-driven process that involves a semi-permeable membrane. As an ideal barrier, the semi-permeable membrane allows water to pass while rejecting salts and other undesirable substances (Ma et al. 2013; Eyvaz et al. 2016; Iorhemen et al. 2016; Damirchi & Koyuncu 2021; Ozcan et al. 2023). An osmotic gradient is responsible for keeping the solute on both sides of a selectively permeable membrane when it transports water from the low-solute concentration feed solution (FS) to the concentrated draw solution (DS) (Al-Alawy et al. 2016; Damirchi & Koyuncu 2021; Kharraz et al. 2022; Han et al. 2023). In other words, FO deals with a physical phenomenon. FO processes utilize the osmotic pressure difference between feed and draw solutions to generate driving power. This result reduces energy costs and reduces membrane fouling, among other benefits. In addition to its low membrane fouling potential, the method operates with minimal hydraulic pressure and retains a wide variety of undesirable compounds (Holloway et al. 2007; Cornelissen et al. 2008; Kadhima et al. 2018; Schneider et al. 2021; Salih & Al-Alawy 2022a, 2022b; Chen et al. 2024; Liu et al. 2024).

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Osmotic pressure

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According to the relation of van't Hoff, the osmotic pressure (π) of a solution measured in bar units depends not only on the concentration of dissolved ions but also on the temperature of the solution (van't Hoff 1995; Alenezi & Merdaw 2021), as follows:
(1)
where i represents the van̕ t Hoff factor (also known as dissociation factor), Φ is the osmotic coefficient, M is the molarity, R is the ideal gas constant (L bar/K mol), and T represents the absolute temperature (K). The van't Hoff factor is used to compensate deviations from ideality behavior of solutions where particles of a solute take a finite volume and are attracted to each other by van der Waals forces (Johnson et al. 2021). Generally, Φ depends on the solute type and concentration, where with a diluted concentration of the solute it will be equal to one. In the ideal solution, Φ is equal to one (Qasem et al. 2023). Table 1 illustrates values of osmotic coefficients and the van̕ t Hoff factor for a bunch of solutes with respect to physiological importance.
Table 1

Osmotic coefficients and van't Hoff factors for a number of solutes (Khudair 2011; Bowden et al. 2012)

Substance
NaClHCOONaCH3COONaCaCl2MgCl2Na2SO4MgSO4KClHCl
Φ 0.93 0.96 0.94 0.86 0.89 0.74 0.58 0.92 0.95 
i 2 2 2 3 3 3 2 2 2 
Substance
NaClHCOONaCH3COONaCaCl2MgCl2Na2SO4MgSO4KClHCl
Φ 0.93 0.96 0.94 0.86 0.89 0.74 0.58 0.92 0.95 
i 2 2 2 3 3 3 2 2 2 
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Modeling flux for osmotic process (Model-1)

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Here's how FO, RO, and pressure-retarded osmosis (PRO) transport water (Cath et al. 2006):
(2)
where Jw is water flux, A is the water permeability constant for the membrane, σ is the reflection coefficient, Δπ is the osmotic pressure difference, and ΔP is the hydraulic pressure, which is zero for FO, and with respect to RO, ΔP > Δπ, and for PRO, Δπ > ΔP as shown in Figure 1.
Figure 1
Directions and values of water flux as a function of ΔP (Cath et al. 2006).
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Directions and values of water flux as a function of ΔP (Cath et al. 2006).

Figure 1
Directions and values of water flux as a function of ΔP (Cath et al. 2006).
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Directions and values of water flux as a function of ΔP (Cath et al. 2006).

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The water flux, Jw, of the FO process is dependent on the flux over the selective layer of the membrane and is given as follows:
(3)
where πd,w − πf,w is the difference of the osmotic pressure through the FO membrane selective layer, which is also called effective pressure (Δπeff). The osmotic reflection coefficient (σ) is assumed to be equal to one.

Concentration polarization (CP) is an important issue in water treatment with the usage of membrane technology and this problem has been investigated by many researchers. CP affects the permeate negatively via increasing osmotic pressure at the wall of the membrane active side. Usually, CP can take place on the membrane sides. At feed side, the solute will be concentrated on the membrane wall. While at the permeate side, the solute will be diluted at the membrane wall. These two phenomena are known as concentrative external concentration polarization (CECP) and dilutive external concentration polarization (DECP), respectively. In case of using an asymmetric membrane, one of these boundary layers take place inside the porous support layer of the membrane leading to protecting it from turbulence associated with cross flow as well as shear over the membrane surface. This state is referred to as either concentrative internal concentration polarization (CICP) or dilutive internal concentration polarization (DICP). In general, concentration polarization occurs in electrochemical systems when the bulk solution is significantly different from the electrode surface in terms of concentration. There can be a decrease in electrochemical reactions due to this difference. There are two types of this phenomenon, internal (ICP) and external concentration polarization (ECP). It is important to note that ICP occurs within the porous structure of an electrode, and ECP occurs at the interface between the electrode and the bulk solution (McCutcheon & Elimelech 2006; Tan & Ng 2008; Chae et al. 2024).

External concentration polarization

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Concentrative ECP takes place in the FO process when the feed solution is cited toward the membrane active layer. It's important to know the overall effective osmotic driving force to account for the flux in FO. Therefore, it's necessary to calculate the concentration of the feed at the surface of the active layer. The surface concentration can be calculated experimentally through utilizing boundary layer (BL) film theory (McCutcheon & Elimelech 2006). Based on the Sherwood number (Sh) determination in a rectangular channel, the following flow system can be formed:

For laminar flow:
(4)
And for turbulent flow:
(5)
where Re represents Reynolds number, Sc is Schmidt number, L is the channel length, and dh is the hydraulic diameter. The relation describing hydraulic diameter for rectangular duct is
(6)
where a and b are the dimensions of the duct. The coefficient of mass transfer, k, is associated with Sh to give
(7)
Here, D is the diffusivity of the solute. A mass balance can be presented at the membrane wall as follows:
(8)
With steady state, no change of accumulation of mass and the flux of influent solute to the membrane surface should be equated by fluxes of effluent solute passing from the membrane (via diffusion) and across the membrane (into the permeate) as follows:
(9)
where Jw is the water flux, M is the mass of solute, t is the time, x is the perpendicular distance to membrane wall, C is the solute concentration inside the BL, Cp is the concentration of the solute in the permeate side, and α is the membrane surface area. There is no limit to the application of Equation (9) at any plane in the BL and not only at the membrane surface, since the steady flow of solutes over the boundary layer prevents accumulation of solute within it. The last part in Equation (9) represents the solute that should transfer through the BL and the membrane to the permeate side. Rearranging and solving Equation (9) over the thickness of BL (δ) with the BL conditions: C(0) = Cf,w and C(δ) = Cf,b, where Cf,b is the bulk feed concentration and Cf,w is the concentration at the membrane wall, lead to the following equations:
(10)
Equation (10) can be integrated to give (assuming concentration of solute in permeate is zero)
(11)
where δ is the thickness of the BL. Rearranging Equation (11) using the van't Hoff relation gives the model of the CECP for each of the permeate fluxes. The mass transfer coefficient is utilized to determine the CECP mode, as follows:
(12)
where Jw is the water permeate, and πf,w and πf,b are the feed solution osmotic pressures at the membrane wall and the bulk, respectively. The positive sign of the exponent indicates that πf,w > πf,b. CECP take place solely at the membrane feed side. Besides, it's assumed in this relation that the ratio of feed solute concentration to the bulk concentration at the membrane wall balances the ratios of osmotic pressures. This is acceptable hypothesis for diluted solutions where osmotic pressure is related to salt concentration.
A diluted ECP has a lower membrane wall concentration than the bulk solute, as indicated by the following equation. This state decreases the draw solution's effective driving force. Dilutive ECP characterized with that the concentration of membrane wall of the draw solute is less than that of the bulk (Tang et al. 2010):
(13)
Here, πd,w and πd,b are the draw solution osmotic pressures at the membrane wall and at the bulk, respectively. It is assumed, as revealed in Equation (13), that the ratio of the draw solute concentration of the membrane wall to the concentration of bulk balances the ratio of osmotic pressures at the same side. To represent the FO process flux in the existence of ECP, we begin with the general FO equation of flux, given as follows:
(14)
where A is the permeability coefficient of pure water. The coefficient of osmotic reflection is assumed to have a magnitude of 1, because no reverse salt transfer occurs across the membrane. Equation (14) predicts flux that is proportional to driving force without the presence of CECP or DECP, which might be valid when the permeate rate is very low. When permeate flux becomes relatively high, this equation must be modified for both the CECP and DECP to be included:
(15)

The utilization of this flux model is limited because the dense symmetric membranes for osmotic processes are unused. Due to this, it must take into account the state where the membrane is asymmetric, making ICP effects the most important.

Internal concentration polarization

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Using the convection–diffusion relation, Lee et al. (1981) developed a model for the determination of the CICP layer in PRO mode. The simplified model, however, was applicable for low water fluxes. Loeb et al. (1997) utilized the upcoming relation for DICP (FO mode) and CICP, respectively, which was used by McCutcheon & Elimelech (2006) to describe the FO process.
(16)
where K is the solute resistivity for diffusion within the porous support layer, which is given as
(17)
where S is the membrane structural parameter, t is membrane thickness, ε is the porosity of the support layer, τ is the tortuosity of the membrane, and D is the solute diffusion coefficient. In this equation, K is dependent on diffusivity. B in Equation (16) is the salt permeability coefficient within active layer of the membrane, which can be calculated using the following relation:
(18)
where R is the solute rejection, which is related to the membrane characteristic. Salt permeability coefficient (B) is nearly negligible when compared with the other terms in Equation (16). Thus, salt flux in the direction of water flux is ignored as well as any movement of salt from the permeate (draw solution) side to the feed side (Gray et al. 2006). Therefore, Equation (16) can be simplified for water flux as follows:
(19)
The exponential term in Equation (19) represents the correction factor, which can be considered as the CECP modulus and defined as
(20)
where πf,i is the osmotic pressure of the feed inside the active layer. The positive sign of the exponent denotes that πf,i > πf,b, which indicates concentrative influence. Substitute Equation (12) into Equation (19) to get an analytical relation that gathers the influence of ICP and ECP on water flux, as follows:
(21)
All the elements in Equation (21) can be determined analytically or by experiments. Therefore, water flux can be determined for the PRO mode as shown in Figure 2.
Figure 2
Specify ICP and ECP profiles for osmotic driving forces. (a) A plot depicting concentrative and dilutive external CP. (b) The plot shows a concentration of internal CP and a dilution of external CP (PRO mode). (c) The plot displays dilutive and concentrated internal concentrations of CO (FO mode) (McCutcheon & Elimelech 2006).
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Specify ICP and ECP profiles for osmotic driving forces. (a) A plot depicting concentrative and dilutive external CP. (b) The plot shows a concentration of internal CP and a dilution of external CP (PRO mode). (c) The plot displays dilutive and concentrated internal concentrations of CO (FO mode) (McCutcheon & Elimelech 2006).

Figure 2
Specify ICP and ECP profiles for osmotic driving forces. (a) A plot depicting concentrative and dilutive external CP. (b) The plot shows a concentration of internal CP and a dilution of external CP (PRO mode). (c) The plot displays dilutive and concentrated internal concentrations of CO (FO mode) (McCutcheon & Elimelech 2006).
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Specify ICP and ECP profiles for osmotic driving forces. (a) A plot depicting concentrative and dilutive external CP. (b) The plot shows a concentration of internal CP and a dilution of external CP (PRO mode). (c) The plot displays dilutive and concentrated internal concentrations of CO (FO mode) (McCutcheon & Elimelech 2006).

Close modal
The dilutive ICP state is characterized for FO, where the membrane active layer is located against the feed side and the porous support layer is against the draw side (Kessler & Moody 1976). When water penetrates the active layer, the draw solution inside the porous structure is diluted. This is called dilutive ICP (Figure 2). Loeb et al. (1997) described this phenomenon for FO as follows:
(22)
When considering that the salt permeability (B) is neglected. Equation (22) can be rearranged to get the following water flux equation:
(23)
Here πd,b is now modified by the DICP modulus, given by
(24)
where πd,i represents the draw solution concentration inside the active layer. The negative sign refers to the direction of water flux, which is away from the membrane active layer, which means the CP effect is dilutive, i.e. πd,i < πd,b by substituting Equation (12) into Equation (23), we get
(25)
The parameters in Equation (25) are system conditions and membrane parameters that are measurable. The iteration steps required to obtain water flux using Model-1 for FO mode are shown in Figure 3. In this relation, DICP is combined with CECP. Operating conditions and concentrations of feed and draw solutions are given.
Figure 3
Iteration process using software solved mathematically for water flux, Jw, (using Model-1) for FO mode.
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Iteration process using software solved mathematically for water flux, Jw, (using Model-1) for FO mode.

Figure 3
Iteration process using software solved mathematically for water flux, Jw, (using Model-1) for FO mode.
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Iteration process using software solved mathematically for water flux, Jw, (using Model-1) for FO mode.

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Modified model for ICP layer (Model-2)

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Model-2 focuses on developing the correlations between solute resistivity coefficients (K) and their relations hips with diffusivity, while the equations used in Model-1 that relate mass transfer coefficients (film model) are used in Model-2 as well. There is a possibility that the constant solute resistance coefficient (K) may not be true, as it is attributed to the fact that the diffusivity coefficient may not be constant, especially in the case of a large concentration difference between the solute and the membrane. In order to analyze the influence of K on water flux, modeling by using a constant value of K would be under-researched. A relation for ICP layer modeling has been developed by Loeb et al. (1997), which has been modified from the governed equations used in this study. The solute flux, Js, over the dense layer for DICP is given as
(26)
The flux of the solute over the porous layer of the FO membrane can be written using the equation of convection and diffusion as follows:
(27)
The solute diffusion coefficient can be represented as a function of the solute concentration over Distance x as follows:
(28)

The term x is the vertical distance from the membrane selective layer that is determined inside the porous support layer and the coefficients Ei represent constants accompanied by the mathematical relation of diffusion coefficient and their values depend on the type of salt used. The Ei values for NaCl salt are listed in Table 2.

Table 2

Ei values for NaCl salt

EiE1E2E3E4
Value 14,900 × 10−13 −398 × 10−13 418 × 10−13 −77.6 × 10−13 
EiE1E2E3E4
Value 14,900 × 10−13 −398 × 10−13 418 × 10−13 −77.6 × 10−13 
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Combining Equations (26) and (27) yields
(29)
The relevant boundary conditions are defined in Figure 2 for the FO mode and represented as
By applying the boundary conditions, Equation (29), which is a separable differential equation, can be solved by MATLAB software for dilutive ICP (FO mode) to give (Tan & Ng 2013; Majid et al. 2023; Chong et al. 2024) the following:
(30)
In a similar manner, the relation for CICP (PRO mode) for the FO process can also be modified and written as
(31)
When solving Equation (29), with respect to NaCl as a draw salt, Equations (30) and (31) become, respectively:
(32)
(33)
The modified solute resistivity (K*) will be written as
(34)
The solute resistivity (K*) used in this study has to be calculated for the FO membrane with respect to membrane structure and it is constant for each membrane and not influenced by other conditions of the process. The iteration steps required to obtain water flux using Model-2 for FO mode are shown in Figure 4. The operating conditions and concentrations of feed and draw solutions are given.
Figure 4
Iteration process using software solved mathematically for water flux, Jw, (using Model-2) for FO mode.
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Iteration process using software solved mathematically for water flux, Jw, (using Model-2) for FO mode.

Figure 4
Iteration process using software solved mathematically for water flux, Jw, (using Model-2) for FO mode.
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Iteration process using software solved mathematically for water flux, Jw, (using Model-2) for FO mode.

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Recovery percentage

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The recovery refers to the quantity of feed that's recovered as permeate and it is written as a percentage (Kessler & Moody 1976). The membrane recovery was calculated by dividing the permeate volume (Vp) by the feed volume (Vf). The recovery is defined as
(35)
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Mathematical modeling of flux behavior in FO process and biological process

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Three types of membranes, cellulose triacetate (CTA), cellulose acetate (CA), and thin-film composite (TFC), have been selected based on previous studies and practical experience in this field. Several studies have demonstrated that CTA and CA membranes are efficient for FO, as well as being cost-effective. The TFC membrane is widely used in RO, and was compared with CTA and CA membranes in this study. The FO water flux was calculated theoretically using Equation (14) without consideration of concentration polarization or fouling influences. Moreover, two models were used in this study, the conventional model referred to as Model-1 and the modified model referred to as Model-2. There are a number of factors that affect the equations used for determining water flux in the two models, including the temperature of the feed and draw solutions, their concentrations, their flow rates, the solute diffusion coefficient, permeability coefficient, mass transfer coefficient, the solute resistance, modified solute resistance, membrane thickness, tortuosity, and porosity.

Figure 5 represents the relation between pure water flux vs. applied pressure for TFC, CA, and CTA membranes, and Table 3 represents the results of water permeability for these three membranes, respectively. From the figure and table, we can see that the permeability of TFC and CA membranes is much superior to that of the CTA membrane, where the value of the TFC membrane was nine times greater than that of the CA membrane, and one and a half times greater than that of the CA membrane. These results prove that TFC and CA RO membranes give high fluxes under hydraulic pressure.
Table 3

Water permeability of TFC, CA, and CTA membranes

MembraneTFCCACTA
Water permeability, slope y 6.606x 4.69x 0.727x 
Correlation factor, R2 0.9921 0.9937 0.9849 
MembraneTFCCACTA
Water permeability, slope y 6.606x 4.69x 0.727x 
Correlation factor, R2 0.9921 0.9937 0.9849 
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Figure 5
Pure water flux against applied pressure for (a) TFC, (b) CA, and (c) CTA membranes at 30 °C.
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Pure water flux against applied pressure for (a) TFC, (b) CA, and (c) CTA membranes at 30 °C.

Figure 5
Pure water flux against applied pressure for (a) TFC, (b) CA, and (c) CTA membranes at 30 °C.
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Pure water flux against applied pressure for (a) TFC, (b) CA, and (c) CTA membranes at 30 °C.

Close modal
Figure 6 illustrates the theoretical and experimental effects of concentration on water flux for the three draw solutions, sodium chloride, sodium formate, and sodium acetate, as well as the models that are applied to each draw solution. Based on the flux data, the three draw solutions perform approximately the same for each model, demonstrating that the modified model, which represents ICP development, is more predictive and realistic than the traditional model (Model-1). The conventional model (model-1) over-predicted the water flux by more than 25% while the modified model (Model-2) over-predicted the water flux by 17%.
Figure 6
Theoretical and experimental water flux for abiotic processes using (a) sodium chloride, (b) sodium formate, and (c) sodium acetate (T = 30 °C, QFS = QDS = 3 l/min, CTA membrane).
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Theoretical and experimental water flux for abiotic processes using (a) sodium chloride, (b) sodium formate, and (c) sodium acetate (T = 30 °C, QFS = QDS = 3 l/min, CTA membrane).

Figure 6
Theoretical and experimental water flux for abiotic processes using (a) sodium chloride, (b) sodium formate, and (c) sodium acetate (T = 30 °C, QFS = QDS = 3 l/min, CTA membrane).
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Theoretical and experimental water flux for abiotic processes using (a) sodium chloride, (b) sodium formate, and (c) sodium acetate (T = 30 °C, QFS = QDS = 3 l/min, CTA membrane).

Close modal
Figure 7(a) shows the theoretical and experimental water flux for an abiotic process done on CTA membranes. Therefore, it is evident that both models exhibit approximately the same response to changes in feed flow rate, based on the comparison between them. Figure 7(b) and 7(c) shows the deviation of the theoretical models from the experimental results for CA and TFC membranes, respectively. The same response was observed that Model-2 is more predictive than Model-1, proving the modifications carried out on ICP were satisfied.
Figure 7
Theoretical and experimental water flux for abiotic processes, (a) CTA, (b) CA, and (c) TFC membrane (T = 30 °C, QFS = QDS = 3 l/min, 0.6 M NaCl).
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Theoretical and experimental water flux for abiotic processes, (a) CTA, (b) CA, and (c) TFC membrane (T = 30 °C, QFS = QDS = 3 l/min, 0.6 M NaCl).

Figure 7
Theoretical and experimental water flux for abiotic processes, (a) CTA, (b) CA, and (c) TFC membrane (T = 30 °C, QFS = QDS = 3 l/min, 0.6 M NaCl).
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Theoretical and experimental water flux for abiotic processes, (a) CTA, (b) CA, and (c) TFC membrane (T = 30 °C, QFS = QDS = 3 l/min, 0.6 M NaCl).

Close modal
Figure 8 illustrates the relationship between the mass transfer coefficient and the feed flow rate of a CTA membrane. According to this figure, we can see an increase in the mass transfer coefficient (k) for CTA membranes as a function of the flow rate. Based on the following mathematical equation, we know that the mass coefficient is directly proportional to the velocity through its direct relationship with Sherwood's number, which can be expressed as follows:
(36)
Figure 8
Mass transfer coefficient vs. feed flow rate for CTA membrane (T = 30 °C, QDS = 3 l/min, 0.6 M NaCl).
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Mass transfer coefficient vs. feed flow rate for CTA membrane (T = 30 °C, QDS = 3 l/min, 0.6 M NaCl).

Figure 8
Mass transfer coefficient vs. feed flow rate for CTA membrane (T = 30 °C, QDS = 3 l/min, 0.6 M NaCl).
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Mass transfer coefficient vs. feed flow rate for CTA membrane (T = 30 °C, QDS = 3 l/min, 0.6 M NaCl).

Close modal

Furthermore, it is directly proportional to the Reynolds number, which is used in order to measure the flow speed. As a result of the experiments (Al-Alawy et al. 2016), it has been demonstrated that the feed flow rate increases with an increase in water flux. Accordingly, as the flow rate increases, the mass transfer coefficient increases, as does the water flux, both of which increase simultaneously as the flow rate increases.

An illustration of a continuous biological process is shown in Figure 9(a) and 9(b), where the two models have been applied to a single CTA membrane in both side-stream and submerged modes, respectively, for the purpose of illustrating a biological continuous process. Due to the modification of the model (Equation (32)), it has been found that the water flux has been overpredicted by approximately 37%, while the traditional model (Equation (12)) has overpredicted by nearly 60%, therefore the change in the model has yielded a significantly higher prediction. The efficiency of the side-stream mode is better than the submerged mode in terms of productivity when comparing their obtained results with the practical results.
Figure 9
Theoretical and experimental water flux for biological processes (a) side-stream mode and (b) submerged mode (T = 30 °C, QFS = QDS = 3 l/min).
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Theoretical and experimental water flux for biological processes (a) side-stream mode and (b) submerged mode (T = 30 °C, QFS = QDS = 3 l/min).

Figure 9
Theoretical and experimental water flux for biological processes (a) side-stream mode and (b) submerged mode (T = 30 °C, QFS = QDS = 3 l/min).
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Theoretical and experimental water flux for biological processes (a) side-stream mode and (b) submerged mode (T = 30 °C, QFS = QDS = 3 l/min).

Close modal
Based on the investigation, the large deviation in forecasted water flow was attributed to fouling parameters that were affected, resulting in a decline in water flow. With respect to the modified model, the diffusivity coefficient equation (Equation (28)) is incorporated into the integration term and yields K*, which is independent of the diffusivity coefficient. According to Figure 10 and Table 4, this will improve the accuracy of the prediction of water flux in the FO process.
Table 4

Statistical analysis of the models for the side-stream mode

ParameterModel-1Model-2
Correlation factor, R2 0.9635 0.970 
Variance 0.6626 0.93125 
Standard of deviation 0.814 0.965 
Confidence level 95% 95% 
No. of observation 4 4 
ParameterModel-1Model-2
Correlation factor, R2 0.9635 0.970 
Variance 0.6626 0.93125 
Standard of deviation 0.814 0.965 
Confidence level 95% 95% 
No. of observation 4 4 
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Figure 10
Deviation of predicted flux for Model-1 and Model-2 from experimental flux for the side-stream mode.
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Deviation of predicted flux for Model-1 and Model-2 from experimental flux for the side-stream mode.

Figure 10
Deviation of predicted flux for Model-1 and Model-2 from experimental flux for the side-stream mode.
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Deviation of predicted flux for Model-1 and Model-2 from experimental flux for the side-stream mode.

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  • 1. OsMBRs are an effective means of treating oily wastewater discharged from Al-Daura refineries. With OsMBR, high-quality water can be produced from oily wastewater, which can be reused in a wide range of applications.

  • 2. CTA membranes have a higher flux permeability than TFC and CA membranes. Consequently, CTA membranes produced two times more water flux than CA membranes and six times more than TFC membranes. These membranes were arranged according to the water flux order, including CTA, CA, and TFC. In comparison to CA and TFC membranes, CTA membranes have a higher reverse salt flux.

  • 3. In response to higher feed temperatures, higher draw solution concentrations, and higher feed flow rates, a greater amount of water was reclaimed through FO. An increase in run time and draw flow rate is accompanied by a decrease in water flux.

  • 4. When the concentration of the draw solution increases over time, reverse salt flux decreases.

  • 5. There was approximately a 25% difference between the numerical model (Model-1) and the experimental value of water flux. Compared with the experimental results, the updated model (Model-2) showed a deviation of almost 17%, indicating a more realistic estimate.

  • 6. In both methods, side-stream and submerged, the withdrawal solution flow rate and experiment time increase, resulting in a decrease in water flux. However, the side-stream mode produced the best productivity results. In the same way, reverse salt flux is also applicable.

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All relevant data are available on request from the authors.

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The authors declare there is no conflict.

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