# A Concise Review of Theoretical Models and Numerical Simulations of Membrane Fouling

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Concentration Polarization

_{m}) and the foulant concentration at x = δ (i.e., at the boundary layer near the bulk flow) is fixed at lower limit C

_{b}[10]. Thus, the CP model enhances the integration of Equation (1) as:

_{m}and C

_{b}are the foulant concentration at the membrane surface and bulk flow, respectively. Equation (2) shows that the flux through the membrane J is dependent on the solute characteristics of D and C

_{m}/C

_{b}, and the boundary layer thickness $\delta $. Operational management thus should be directed towards increasing D or decreasing δ. Here the diffusion coefficient D can be obtained according to the Stokes-Einstein equation [18]:

_{B}is the Boltzmann’s constant; T represents the absolute temperature; μ represents the solution viscosity; and d

_{p}is the particle size. In addition, the term D/δ in the right part of Equation (2) is recognized as the mass transfer coefficient k [10]:

_{m}/C

_{b}is the CP modulus M, which represents the degree of CP [4]. It is worthwhile to note that the concentration of foulant in permeate C

_{p}is not included in the mass balance (Equation (1)). Therefore, Equation (5) can be applied to model the CP behavior where the concentration of foulant in permeate C

_{p}is negligible compared to that at the membrane surface C

_{m}and bulk flow C

_{b}(i.e., in the NF/RO process). When further considering the role of C

_{p}, the convection transport of solute towards the membrane surface (JC) should be balanced by the sum of permeate JC

_{p}and the back diffusion transport D(dC ⁄dx) away from the membrane. Thus, Equation (5) can be expressed as a common form [19]:

_{p}, and the foulant loss from the solution phase because of their attachment onto the membrane αJC

_{m}. Accordingly, the CP model can be given by [6]:

_{b}inducing from the membrane–colloid interaction [21]. At the same time, a potential energy ΔE

_{d}is provided to promote its attachment by the hydrodynamic drag force acting on the particle [18]. Therefore, the attachment coefficient α can be given by:

#### 2.2. Cake Formation

_{f}). The membrane permeate flux is thus decreased under constant applied pressure due to this cake resistance. Based on Darcy’s law [22], the permeate flux can be expressed as:

_{f}is decided by the foulant mass deposition m

_{f}and specific cake resistance α

_{f}as shown in Equation (10):

_{f}exerts important roles in membrane fouling [23,24,25]. The value of α

_{f}can be calculated by the Carmen–Kozeny equation [26]:

_{f}can also be experimentally determined according to the foulant mass deposition m

_{f}and flux variation [27]:

_{0}and J

_{f}is the initial flux and final flux, respectively, for any given filtration period.

_{f}cannot change with respect to ΔP when s = 0 for incompressible foulant cake layers, and the value of s is larger for a more compressible cake layer. It was reported that the value of s ranges from 0.4–1.0 for UF and MF membranes fouled by bentonite and kaolin particles [32]. In addition, the value of s is highly affected by the solution chemistry. For silica colloids fouling UF membranes, s is ~0.82 in distilled water, and it decreases to ~0.36 when ionic strength increases to 100 mM [33].

#### 2.3. Pore Blocking

_{c}represents the complete pore blocking coefficient.

_{s}represents the standard pore blocking coefficient.

_{i}is the intermediate pore blocking coefficient.

_{cf}is the complete pore blocking coefficient.

## 3. Numerical Simulation

#### 3.1. Computational Fluid Dynamics (CFD)

_{e}are energy, temperature, and the source term of energy, respectively.

**Table 1.**Application of CFD for fouling simulation in different membrane systems from the literature.

Membranes | Aims | Highlights | References |
---|---|---|---|

RO/NF | - Description of the effect of biofilm development on RO performance.
| - The model explained the loss of permeate flux and the increase of salt passage in time due to biofilm formation.
- Three mechanisms by which the biofilm contributed to the flux decline were identified.
| [69] |

- Investigation of flow transition behavior of commercial spacer geometries.
| - The flow transition was mainly ascribed to the interaction of the vortices’ attachment onto the filaments and the screw-vortex derived from the spacer cells.
- The minimization of CP was not effectively obtained by using this non-woven spacer design.
| [66] | |

- Simulating and addressing the effectiveness of implementing directional freezing to seawater desalination.
| - The validation of the energy consumption model showed good agreement with those reported in the literature.
- Better partition and removal efficiency were achieved under lower salinity brine, top freezing instead of bottom or radial freezing.
| [67] | |

- Studying the compression by evaluating CP extent over time.
| - The suppression effect on CP by transformed morphologies was due to the larger shear stress introduced by the flow field.
| [70] | |

Electro-osmosis | - Analyzing the effect of membrane permeance on the resonant frequency and the mass transfer.
| - The resonant frequency of the unsteady forced slip velocity was not affected by the membrane permeance.
- The permeate flux enhanced for greater membrane permeances (up to 23%) at the cost of a 5–7% higher pumping energy.
| [55] |

UF | - Analysis of shear and elongational strain rates and associated hydrodynamic influences on mobile genetic elements (MGEs) retention.
| - Significant deformation of MGEs occurred at the distance of dozens of nm away from the membrane.
- The existence of the PAC particles presented a negligible impact on the permeation of MGEs through UF membrane pores.
| [60] |

MF | - Examination of the fluid velocity distribution.
| - The velocity profile at different points was obtained by CFD models.
- The experimental data and CFD results suggested the cake resistance and mass of circular-type promoters decreased at high crossflow velocity.
| [68] |

- Predicting the permeate flux through a microfiltration membrane.
| - The results show that the predicted CFD flux values are more accurate compared with a simple calculation using the feed-side pressure in the Darcy equation.
| [61] | |

MBR | - A 3D CFD model developed the cost-effective optimization of a lab-scale AEC-MBR.
| - The height of gas–liquid dispersion for membrane scouring was optimized.
- Simultaneous cost-effective membrane scouring, and nitrogen removal was achieved.
| [65] |

- Investigating the effect of the dynamics of fouling on effective backwashing length.
| - The degree of fouling can change the backwashing velocity inside fiber lumen and have a further influence on effective backwash length.
- The signal variations of LBS correspond to the simulation data.
| [56] | |

Distillation | - A 2D CFD model was developed to study the coupled effects of temperature and concentration polarization in the direct contact membrane distillation treatment of hypersaline brines.
| - Dramatic increases were observed in solute concentration at the membrane surface, exceeding 1.6 times the feed value.
- The temperatures, concentration, and vapor flux vary considerably in the downstream direction.
| [62] |

- Description of the thermal and hydrodynamic conditions in a hollow fiber membrane module.
| - Thermal efficiency varied with feed temperature and feed velocity.
- Temperature polarization became more significant at higher feed temperature and lower feed velocity.
- A short module was better utilized for high efficiency of VMD.
| [58] | |

- CFD Simulation of the MD process for desalination of high salinity feedwater.
| - The membrane conductivity and thickness had an important influence on the DCMD performance.
- Optimal membrane thickness was found to increase with salinity.
- Better desalination of low salinity feedwater was observed with a thinner membrane, whereas a thicker one resulted in higher separation performance.
| [63] | |

Forward Osmosis (FO) | - Modelling a FO system with an asymmetric membrane using a CFD model.
| - The process of FO was well simulated through CFD software.
- The volume fraction of NaCl in the sea water chamber reduced with each time step.
| [71] |

#### 3.2. Monte Carlo Simulation

_{B}T. A uniform random number $\xi $ (0–1) is generated and compared with $exp\left(-{\beta}^{\prime}{\delta}^{\prime}{V}_{n,m}\right).$ When$\xi $ < $exp\left(-{\beta}^{\prime}{\delta}^{\prime}{V}_{n,m}\right)$, the uphill move is accepted. Otherwise ($\xi $ ≥ $exp\left(-{\beta}^{\prime}{\delta}^{\prime}{V}_{n,m}\right)$, the proposed move is rejected.

^{−2}h

^{−1}bar

^{−1}, even with ideal interlayer spacing [81]. Liu et al. [82] also dissected the different effects of local defects on fouling due to their local energy barrier towards the membrane surface. Through a collision–attachment-based MC approach, Liu and co-workers [5] systematically investigated the transitional fouling behavior from foulant–clean membrane interaction (F-M) to foulant–deposited-membrane interaction (F-F) for colloidal fouling of NF/RO membranes. They further applied the MC approach to dissect the effect of local defects of the local energy barrier of the membrane surface and found that when F-F repulsion (E

_{f}) was above a critical value (E

_{c}), flux stability was independent of defects; when E

_{f}< E

_{c}and E

_{m}(F-M energy barrier) ≥ E

_{c}, large coverage or a low energy barrier of defects enhanced fouling; for E

_{f}< E

_{c}and E

_{m}< E

_{c}, serious fouling occurred with/without defects [82].

**Table 2.**Application of MC method for fouling simulation in different membrane systems from the literature.

Membrane | Aims | Highlights | References |
---|---|---|---|

MF | - Examining the pore blocking phenomena in cross-flow microfiltration
| - Reduced pore blocking is observed in membranes with a high porosity and narrowed pore distribution.
- Particle concentration and size distribution have significant influences on particles passing through, being adsorbed, and sticking on the membrane pore.
| [74] |

- Investigating the particle capture at membrane surfaces in cross-flow MF.
| - Abrupt decline of permeability happens at short times, while the stationary flux is observed at long times.
- The quick reduction of permeability occurs with increased foulant concentration.
- The permeability is more affected in a thinner pore membrane.
| [85] | |

- Simulation of the flux behavior during the MF of multi-dispersed iron oxide particles.
| - The major flux dominating parameter was cake resistance rather than pore blocking for non-flocculating situations.
- A porosity of 0.39 was obtained for mono-dispersed particles, while a porosity of 0.33 was acquired for multi-dispersity.
| [86] | |

UF | - Calculation of the osmotic pressure and the diffusion coefficient and characterizing the cake layer development.
| - Hypernetted Chain theory allowed the MC code verification.
- The simulated osmotic pressure agreed well with and the experimental data.
| [76] |

UF/MF | - Investigating the influence of various operation conditions on cake structure.
| - A simple force balance model was developed for a rational explanation of the calculated cake volume fraction.
- The effects of operation parameters on the cake reversibility were evaluated.
- A force accumulation and transfer model was developed for the real filtration process.
| [72] |

- Simulating the pore blocking and cake layer formation of interfacial interactions.
| - The capture probability is dependent on the energy barrier and flow modes.
- The packing density reduces with the increased capture probability.
- The resistance of the cake layer elevated with higher capture probability.
| [73] | |

RO | - Understanding water and solute transport in thin film nanocomposite (TFN) membranes.
| - The addition of a small amount of porous or even non-porous NPs leads to an obvious flux elevation.
- Thicker intermediate layer with minimized NPs agglomeration is favorable to gain high flux.
| [78] |

RO/NF | - Modeling the particle transport and membrane fouling.
| - Particle transport governed by the interplay of hydrodynamic and electric effects can achieve a more stable state than hydrodynamic effects considered alone.
- Larger Reynolds numbers originated from higher shear rates could keep more particles in the bulk flow.
| [87] |

RO/NF | - Assessing the influences of framework defects on the performance of 2D nanomaterial laminate membranes.
| - 2D nanomaterial frameworks are extremely tortuous (tortuosity ≈10
^{3}) with water permeability decreasing from 20 to <1 Lm^{−2}h^{−1}bar^{−1}when thickness increased from 8 to 167 nm. - Framework defects allow salt to percolate through the framework, hindering water–salt selectivity.
- 2D nanomaterial frameworks with a packing density of 75% are projected to achieve <92% NaCl rejection at a water permeability of <1 Lm
^{−2}h^{−1}bar^{−1}, even with ideal interlayer spacing.
| [81] |

- Revealing the origin of the defects in the stacked nanosheets and their impact on the overall water-solute selectivity of the lamellar 2D membrane.
| - Small imperfections in the stacking of MoS
_{2}nanosheets lead to the emergence of microporous defects. - These defects negated the interlayer sieving effect and thus impaired the selectivity of the lamellar structure.
| [80] | |

- Simulating the fouling transition from foulant–clean-membrane interaction (F-M) to foulant-fouled-membrane interaction (F-F).
| - The long-term membrane flux maintained stability for high F-F energy barrier (E
_{f}). - Severe flux declines happened as both F-M energy barrier (E
_{m}) and E_{f}are weak. - A metastable flux behavior presented at the combination of large E
_{m}but small E_{f}.
| [5] | |

- Dissecting the role of membrane defects with a low-energy barrier on fouling development.
| - When E
_{f}was above a critical value (E_{c}), flux stability was independent of defects. - At E
_{f}< E_{c}& E_{m}≥ E_{c}, large coverage or low energy barrier of defects enhanced fouling. - For E
_{f}< E_{c}& E_{m}< E_{c}, serious fouling occurred w/o defects.
| [82] | |

MD | - Studying heat and mass transfer through hydrophobic membranes applying direct contact MD process.
| - The proposed model can simultaneously predict the vapor flux and membrane surface temperatures.
| [83] |

- Describing the vapor flux across the membrane in the membrane distillation process.
| - The higher feed solution temperature and higher pore size do not necessarily increase the vapor flux.
| [84] |

#### 3.3. Artificial Neural Networks (ANN)

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 3.**Application of ANNs for fouling simulation in different membrane system from the literature.

Membranes | Aims | Highlights | References |
---|---|---|---|

RO | - An MLP-ANN with back-propagation approach was performed to predict the dynamic K
_{w}values for a RO desalination plant.
| - The K
_{w}values were very closely to those acquired by the existing correlations for the operating pressure range and feed salinity. - The effect of feed salinity on K
_{w}value was more significant at low pressure.
| [91] |

- MLP and RBF neural networks were trained and developed to predict total TDS concentrations and permeate flow rates.
| - ANN models showed a better simulation of permeate flow rate.
- The accuracy of MLP (R = 0.9904) was slightly better than that of the RBF model (R = 0.9853).
| [96] | |

- ANNs were used to dissect the crucial reasons for fouling in a full-scale RO plant.
| - Total chlorine, electrical conductance, and TDS were essential parameters for early fouling.
- Turbidity, nitrate, organic nitrogen, and nitrite were important indicators for later fouling.
| [97] | |

- DNN was first time applied to simulate cake growth and flux decline based on real-time OCT images.
| - DNN could improve the prediction accuracy of fouling layer growth of NF/RO membranes with R
^{2}> 0.99.
| [98] | |

NF | - A bootstrap-based ANN was developed to simulate the rejections of organic matters.
| - Good agreement was obtained between the predicted and experimental rejections for the bootstrap-based ANN model (R
^{2}= 0.9862).
| [99] |

- ANN, SVM, and RF was employed to forecast the separation performance of an organic solvent NF process.
| - ANN obtained high accuracy in predicting the permeance (R
^{2}= 0.90) and rejection (R^{2}= 0.91).
| [100] | |

UF | - Short-term fouling fluctuations were predicted during UF process by the ANN incorporate continuous fluorescence characterization data.
| - The trained neural networks offered a potentially powerful modeling approach in predicting the development of fouling resistance (mean absolute percentage error <5%).
| [101] |

- ANN combined Darcy’s law was to predict the specific cake resistance and total suspended solids in a UF pilot plant.
| - The predicted specific cake resistance and TSS data provided early indications of membrane fouling propensity.
- Model provides an easy implementation to in an industrial-scale UF plant.
| [102] | |

MF | - An ANN was developed to predict the characteristics of a MF system.
| - The prediction accuracy of permeation flux using ANN (R
^{2}= 0.996) was better than that of complete blocking (R^{2}= 0.186), intermediate blocking (R^{2}= 0.988), standard blocking (R^{2}= 0.866), and cake filtration (R^{2}= 0.858).
| [103,104] |

MBR | - A feed-forward ANN model was developed for the early prediction of OMBR system performance
| - The most effective input variables for predicting flux and fouling behavior were pH, conductivity, ammonia nitrogen, and total nitrogen concentrations.
| [105] |

- RBF-ANN was introduced to predict the interactions of sludge flocs—membranes.
| - The trained RBF-ANN saved up to 98% of computation time for quantification of the interfacial interactions in comparation with the XDLVO theory.
| [106] | |

MD | - A feed-forward ANN was used to model permeate flux in an air gap MD system.
| - The overall agreement between the ANN predictions and experimental data was good (R
^{2}= 0.992).
| [77] |

- A combined ANN—GA was developed to evaluate the effects of operating parameters on membrane fouling in MD.
| - The model simulations were experimentally validated with R
^{2}> 0.98.
| [107] |

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**MDPI and ACS Style**

Yang, H.; Yu, X.; Liu, J.; Tang, Z.; Huang, T.; Wang, Z.; Zhong, Y.; Long, Z.; Wang, L. A Concise Review of Theoretical Models and Numerical Simulations of Membrane Fouling. *Water* **2022**, *14*, 3537.
https://doi.org/10.3390/w14213537

**AMA Style**

Yang H, Yu X, Liu J, Tang Z, Huang T, Wang Z, Zhong Y, Long Z, Wang L. A Concise Review of Theoretical Models and Numerical Simulations of Membrane Fouling. *Water*. 2022; 14(21):3537.
https://doi.org/10.3390/w14213537

**Chicago/Turabian Style**

Yang, Haiyan, Xuri Yu, Junxia Liu, Zhiwei Tang, Tianyi Huang, Zhihong Wang, Yiyun Zhong, Zhihong Long, and Lin Wang. 2022. "A Concise Review of Theoretical Models and Numerical Simulations of Membrane Fouling" *Water* 14, no. 21: 3537.
https://doi.org/10.3390/w14213537