## ABSTRACT

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.

## HIGHLIGHTS

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.

## INTRODUCTION

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).

## MATHEMATICAL MODELING

### 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: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.

. | Substance . | ||||||||
---|---|---|---|---|---|---|---|---|---|

NaCl . | HCOONa . | CH_{3}COONa
. | CaCl_{2}
. | MgCl_{2}
. | Na_{2}SO_{4}
. | MgSO_{4}
. | KCl . | HCl . | |

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 . | ||||||||
---|---|---|---|---|---|---|---|---|---|

NaCl . | HCOONa . | CH_{3}COONa
. | CaCl_{2}
. | MgCl_{2}
. | Na_{2}SO_{4}
. | MgSO_{4}
. | KCl . | HCl . | |

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 |

### Modeling flux for osmotic process (Model-1)

*et al.*2006):where

*J*

_{w}is water flux,

*A*is the water permeability constant for the membrane,

*σ*is the reflection coefficient, Δ

*π*is the osmotic pressure difference, and Δ

*P*is the hydraulic pressure, which is zero for FO, and with respect to RO, Δ

*P*> Δ

*π*, and for PRO, Δ

*π*> Δ

*P*as shown in Figure 1.

*J*

_{w}, of the FO process is dependent on the flux over the selective layer of the membrane and is given as follows:where

*π*

_{d,w}−

*π*

_{f,w}is the difference of the osmotic pressure through the FO membrane selective layer, which is also called effective pressure (Δ

*π*

_{eff}). 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

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:

*L*is the channel length, and

*d*

_{h}is the hydraulic diameter. The relation describing hydraulic diameter for rectangular duct iswhere

*a*and

*b*are the dimensions of the duct. The coefficient of mass transfer,

*k*, is associated with Sh to give

*J*

_{w}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,

*C*

_{p}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) =

*C*

_{f,w}and

*C*(

*δ*) =

*C*

_{f},

_{b}, where

*C*

_{f,b}is the bulk feed concentration and

*C*

_{f,w}is the concentration at the membrane wall, lead to the following equations:

*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:where

*J*

_{w}is the water permeate, and

*π*

_{f,w}and

*π*

_{f,b}are the feed solution osmotic pressures at the membrane wall and the bulk, respectively. The positive sign of the exponent indicates that

*π*

_{f,w}>

*π*

_{f,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.

*et al.*2010):

*π*

_{d,w}and

*π*

_{d,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: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:

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

*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.where

*K*is the solute resistivity for diffusion within the porous support layer, which is given aswhere

*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: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:

*π*

_{f,i}is the osmotic pressure of the feed inside the active layer. The positive sign of the exponent denotes that

*π*

_{f,i}>

*π*

_{f,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:

*et al.*(1997) described this phenomenon for FO as follows:

*B*) is neglected. Equation (22) can be rearranged to get the following water flux equation:

*π*

_{d,b}is now modified by the DICP modulus, given bywhere

*π*

_{d,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.

*π*

_{d,i}<

*π*

_{d,b}by substituting Equation (12) into Equation (23), we get

### Modified model for ICP layer (Model-2)

*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,

*J*

_{s}, over the dense layer for DICP is given as

The term *x* is the vertical distance from the membrane selective layer that is determined inside the porous support layer and the coefficients *E _{i}* represent constants accompanied by the mathematical relation of diffusion coefficient and their values depend on the type of salt used. The

*E*values for NaCl salt are listed in Table 2.

_{i}E
. _{i} | E_{1}
. | E_{2}
. | E_{3}
. | E_{4}
. |
---|---|---|---|---|

Value | 14,900 × 10^{−13} | −398 × 10^{−13} | 418 × 10^{−13} | −77.6 × 10^{−13} |

E
. _{i} | E_{1}
. | E_{2}
. | E_{3}
. | E_{4}
. |
---|---|---|---|---|

Value | 14,900 × 10^{−13} | −398 × 10^{−13} | 418 × 10^{−13} | −77.6 × 10^{−13} |

*et al.*2023; Chong

*et al.*2024) the following:

*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.

### Recovery percentage

*V*

_{p}) by the feed volume (

*V*

_{f}). The recovery is defined as

## RESULTS AND DISCUSSION

### Mathematical modeling of flux behavior in FO process and biological process

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.

Membrane . | TFC . | CA . | CTA . |
---|---|---|---|

Water permeability, slope y | 6.606x | 4.69x | 0.727x |

Correlation factor, R^{2} | 0.9921 | 0.9937 | 0.9849 |

Membrane . | TFC . | CA . | CTA . |
---|---|---|---|

Water permeability, slope y | 6.606x | 4.69x | 0.727x |

Correlation factor, R^{2} | 0.9921 | 0.9937 | 0.9849 |

*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:

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.

*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.

Parameter . | Model-1 . | Model-2 . |
---|---|---|

Correlation factor, R^{2} | 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 |

Parameter . | Model-1 . | Model-2 . |
---|---|---|

Correlation factor, R^{2} | 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 |

## CONCLUSIONS

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.

## DATA AVAILABILITY STATEMENT

All relevant data are available on request from the authors.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Analysis and Improvement Proposal of a Wastewater Treatment Plant in a Mexican Refinery*

*Concentration Poisonous Metallic Radicals From Industrial Water by Forward and/or Reverse Osmosis*

*MSc Thesis*