# Prediction of Permeate Flux in Ultrafiltration Processes: A Review of Modeling Approaches

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## Abstract

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## 1. Introduction

## 2. Theory

_{w}is the solvent permeate flux, ΔP is the transmembrane pressure, L

_{p}is the membrane permeability, µ

_{w}is the solvent viscosity, and R

_{m}is the intrinsic membrane resistance. However, when solutes are added to the solvent, the behavior is entirely different [78]. This means that the flux would increase up to a certain limit. In this regard, Field et al. [65] introduced the concept of critical flux for microfiltration, stating that there is a permeate flux below which fouling is not observed. For operational curves of permeate flux versus transmembrane pressure, three areas or zones related with membrane fouling were described: a subcritical zone (Zone 1), where the transmembrane pressure is low, in which only the concentration polarization phenomenon exists and the permeate flux is lower than the critical flux; a Zone 2, characterized by the formation and consolidation of the cake layer, where pore blocking or particle adsorption can also occur; a Zone 3 due to the compaction of the cake, which is undesirable because it represents irreversible fouling, which is difficult to remove even using chemical membrane cleaning. The critical point, or critical transmembrane pressure, and the limiting point, which is the maximum permeate flux where the increase of permeate flux is not possible after a certain point, can be distinguished in the critical flux theory [18]. The limiting flux is affected by shear stress applied to the system as well as by the feed and module characteristics.

#### 2.1. Concentration Polarization Models

_{p}is the permeate concentration, C

_{b}is the bulk stream concentration, C

_{g}is the gel concentration at the membrane surface, D is the diffusivity coefficient, δ is the boundary layer thickness, and k is the mass transfer coefficient. This relationship includes phenomena occurring in UF processes, where solutes as macromolecules or colloids are conveyed by permeate flux to the membrane surface, and a portion of them is rejected by the membrane and diffused back into the bulk. In this regard, Aimar and Sanchez (1986) [85] have shown that the subsequent decrease in mass transfer coefficient can explain a limiting flux. They used the theories developed and quantified for heat transfer to membrane processes. In particular, the heat transfer work on transfer coefficient variations, theoretically established by Field (1990) [86], was combined with a mass transfer film theory in order to examine the limiting-flux phenomenon. In this context, the rejected solutes tend to form a gel layer on the membrane surface, which acts as an additional resistance [83]. This model assumes that C

_{g}is constant, and the flux of solvent is dependent only on the characteristics of D, C

_{g}, and δ. Fane et al. [87] have applied a correction based on the effective free area correction modifying the assumption in the conventional model for concentration polarization that implies a homogeneously permeable membrane surface. These authors described the membrane surface as a mosaic of regions of different solvent permeabilities depending on the manufacturing process and the structural changes caused by usage, damage to the membrane surface, and plugging of pores, among others. Blatt et al. [2], said that the hydraulic permeability of a gel or concentrated dispersion of submicroscopic particles is a complex function of the solid’s concentration and such variables as the size, shape, resistance, and state of aggregation of particles or molecules comprising the solid phase. In turn, Jonsson [88] indicated that even though polarization phenomena at the membrane–solute interface are usually characterized by the film-theory where the longitudinal mass transport within the boundary layer is assumed negligible, the effect of pressure impacts the permeate flux. This author established that it had been observed that as pressure is increased, permeate flux first increases and then remains more or less pressure independent (phenomena first explained by Blatt et al. [2]). It should be pointed out that the effect of ΔP was not considered by film theory; therefore, models including it can improve the capacity of prediction. Bacchin et al. [24] proposed a model which combined a contribution of both cake filtration and deposition kinetics on fouling. In this regard, the cake filtration law describes the fouling resistance as the sum of the membrane hydraulic resistance (R

_{m}) and cake resistance (R

_{c}). The latter resistance is assumed proportional to the amount deposited on the membrane (M

_{d}) and the cake-specific resistance (α). The deposition rate to the interface is expressed as the amount brought by convection (J

_{c}), minus a back flux (n). Considering the complexity of phenomena involved in membrane filtration, as well as the drawbacks reported in the literature for film theory, a series of modifications to the original model have been developed to date. Table 1 depicts a compilation of the classical and most-used models (considering the number of citations and validations), including the concentration polarization phenomena in the equation.

#### 2.2. Osmotic Pressure Models

_{1}, B

_{2}, and B

_{3}are the osmotic virial coefficients, and C

_{p}is the concentration of the macromolecular solution (g L

^{−1}). The B

_{1}coefficient describes the so-called van’t Hoff’s limiting law for osmotic pressure, which is applicable at very dilute concentrations [50]. The osmotic pressure models consider that the flux is limited by the high osmotic pressure arising in the concentration-polarized layer in the membrane interface. Once the gel layer is formed on the membrane surface, the osmotic pressure plays a key role in permeate flux decay [83]. In this regard, Kedem and Katchalsky [112] were the first authors to develop a model including the osmotic pressure for the permeate flux prediction (shown in Equation (4)).

_{m}is the membrane resistance. From the model developed by Kedem and Katchalsky [112], several publications and models have been developed in which osmotic pressure is integrated directly in the equation or is related to other parameters included in the model. Table 2 summarizes the models with major number of citations and validated, which includes, directly or indirectly, osmotic pressure.

#### 2.3. Resistance-in-Series Models

_{t}is the total resistance given by:

_{m}the membrane resistance, R

_{cp}the concentration polarization resistance, R

_{f}is the irreversible resistance, and R

_{g}is the gel layer resistance. Viscosity is explicitly presented in Darcy’s law. Here, it increases with solute concentration and decreases with temperature. At the same time, if the membrane is sensitive to temperature changes, this must be considered in Darcy’s law’s membrane resistance term [127]. According to our literature review, these models are the most used and reported. The main differences among them are the form in which the different resistances are analyzed and quantified for the prediction of permeate flux [1,46,83]. Table 3 shows the most relevant predictive models (based on number of citations and validations) published in the literature based on the resistance in series.

#### 2.4. Fouling Models

#### 2.5. Non-Phenomenological Models

## 3. Analysis of Model Goodness-of-Fit

- (i)
- Type of configuration: models tested or developed for cross-flow filtration of fruit juices were selected.
- (ii)
- Validation: models with more than one validation were considered.
- (iii)
- The number of citations: models with a high number of citations were selected in order to take into account the scientific impact of each model.
- (iv)
- Membrane module: models tested or developed in fruit juice processing with hollow fiber and tubular membranes were selected.
- (v)
- Mathematical complexity: Considering the easy application of the models, the most straightforward models were preferred.

## 4. Results and Discussion of Selected Models’ Performance

#### 4.1. Models’ Performance in Bergamot Juice Clarification

^{2}h in the first 10 min of operation, followed by a long period of gradual flux decrease (until 190 min of operation) that ended with a steady-state flux of about 4 kg/m

^{2}h. Table 7 shows the results obtained for the statistical validation, in which more than one model was validated with p-values higher than 0.05 in at least one of the statistics used. According to Conidi et al. [27], the rapid decline of permeate flux in the UF of bergamot juice was attributed to gel-layer formation phenomena. This is confirmed by the results obtained in this work where the shear-induced model [57] showed 91.08% in the R-squared and lowered RMSE and MAPE (Table 7). Thus, it is evident that for bergamot juice, the gel-layer formation is confirmed. Based on the shear-induced model results, it can be deduced that the hydrodynamic diffusion of these particles occurs because they, individually, move randomly thanks to the current caused by the cut-off of the flow. Thus, the shear-induced model [57] assumes that after some time, the tangential flow compensates the convective transport of solutes on the surface of the membrane, preventing the growth of the gel-layer, and leading to a stationary value, as illustrated in Figure 2a. It should be mentioned that the shear-induced model assumes that the predominant mechanism is gel-layer formation, neglecting the existence of pore blockage. This may explain the goodness-of-fit obtained for this model: 91.08% (R-square), but not higher. Vincent Vela et al. [70] reported that the shear-induced model loses capacity of prediction at high cross-flow velocity (>2 m/s) and ΔP higher than 1 bar, a fact that was not corroborated in this work since the operating conditions were lower than the limits reported by Vincent Vela et al.

^{2}of 97.92 and lower values of RMSE and MAPE.

#### 4.2. Models’ Performance in Kiwifruit Juice Clarification

^{2}≤ 52.86). Concentration polarization models assume that the cake layer, once it is formed, remains constant. In this case, this is not appreciated since the drop in permeate flux was observed along the whole process.

^{2}(97.43%) and lower values of RMSE and MAPE, as shown in Table 8. This model is based on a convective transport with respect to the driving force in a porous medium in which separation occurs by size exclusion [4,46,197,198]. Despite the high capacity of fit for this model, R-square could be higher if some changes are made in the equation, such as some modifications in the Leveque relationship (used for the determination of mass-transfer coefficient) [100,199,200,201], which includes the gel-layer growing [202,203] or the viscous effect, and modification in the Sherwood correlation to improve the capacity to determine the mass-transfer and the thickness of the gel layer with a higher precision [61,204].

^{2}and lower values of RMSE and MAPE (Table 8). This model assumes that an initial blocking step exists on the membrane surface, followed by the development of a gel layer in a steady-state [66,176]. The model has also been validated with a BSA solution [66], protein solution [205], wastewater containing oil [206], and concentrated protein solution [207] with good goodness-of-fit.

^{2}(98.98%) and lower values of RMSE and MAPE. Thus, non-phenomenological models appear to be the most adequate for permeate flux curves with high variability.

#### 4.3. Models’ Performance in Pomegranate Juice Clarification

^{−2}h

^{−1}to 1.81 kg m

^{−2}h

^{−1}after 600 min of operation can be observed. The absence of a stationary point implies difficulty for modeling. In addition, the filtration of pomegranate juice was carried out at low ΔP and a high cross-flow velocity. According to several authors, these conditions lead to a loss in the capacity of prediction of some models [58,61,168,203]. The validation of the selected models for pomegranate juice in terms of permeate flux prediction is shown in Table 9.

^{2}99.2%). It is particularly interesting that the ARIMA model and the one reported by Yee et al. [191] have been validated for more than 10 h of operation with a great capacity of forecast [171,191,209,210]. Thus, both models appear to be adequate for the long-term prediction of permeate flux.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Membrane area: m^{2} | Sh | Sherwood number, dimensionless |

A_{d} | Membrane area in cell, m^{2} | Sc | Schmidt number, dimensionless |

C_{b} | Bulk concentration, kgm^{−3} | Pe | Peclet number, dimensionless |

C_{CL} | Boundary layer concentration, kgm^{−3} | W_{d} | Stability factor with respect to deposition (parameter in No. 4.4, Table 4) |

C_{g} | Gel concentration, kgm^{−3} | s | Sedimentation coefficient |

C_{gv} | Concentration gel layer in volume, m^{3} | t | Time, s |

C_{m} | Intrinsic concentration, kgm^{−3} | t_{R} | Fouling phase time, s |

C_{p} | Permeated concentration, kgm^{−3} | τ | Tortuosity, dimensionless |

C_{0v} | Feed concentration, kgm^{−3} | T | Temperature, °C |

D_{m} | Equivalent diffusion diameter of macromolecules (parameter in No. 1.11), m | U | Flow velocity, ms^{−1} |

D | Diffusivity, m^{2}s−^{1} | U_{0} | Initial flow velocity, ms^{−1} |

d | Diameter of the module, m | v_{(L)} | Average permeate velocity on the length of the filter channel, ms^{−1} |

d_{h} | Hydraulic diameter, m | v_{f} | Local filtrate velocity (parameter No. 1.8), ms^{−1} |

d_{p} | Pore diameter, m | X(t) | Position change of the equilibrium zone (parameter in No. 4.5) |

F | Intermolecular interactions | b | Radius of the stirred cell (parameter of No. 1.8) |

f_{c} | Marchetti correction factor | V | Permeate volume (parameter in No. 3.10), L |

f_{cp} | Capilar effect | V_{c} | Permeate volume at the reference time point (parameter in No. 3.10), L |

b | Inverse of the solute density (parameter of No. 1.9) | V | Permeate volume, m^{3} |

H | Height of liquid over membrane (parameter of No. 1.9), m | V_{0} | Initial permeate volume, m^{3} |

η | Non-dimensional distance (parameter of No. 1.9) = x/H | v | Specific partial volume, kgm^{3} |

ϕ | Non-dimensional concentration (parameter of No. 1.9) = c/co | v_{0} | Specific partial initial volume, kgm^{3} |

f_{d} | Dipole effect | VRF | Volume reduction factor, dimensionless |

f_{e} | Steric effect | v | Kinematic viscosity, m^{2}s^{−1} |

H | Thickness of the gel layer, m | x_{i} | Proportional parameter of permeability in No. 1.3 |

J | Permeate flux, ms^{−1} | ∆x | Membrane thickness, m |

J_{f} | Final permeate flux, ms^{−1} | X_{12} | Flory-Huggins parameter |

J_{lim} | Limit permeate flux, ms^{−1} | z* | Axial position for osmotic pressure, m |

J_{∞} | Saturation (equilibrium) volumetric permeate flux, m^{3}m^{−2}s^{−1} | ||

J_{ss} | Steady-state permeate flux, ms^{−1} | Greek symbols | |

J_{W} | Flux of pure water permeate, ms^{−1} | δ | Thickness of the boundary layer, m |

J* | Hydraulic lifting speed, ms^{−1} | ∆π | Osmotic pressure, Pa |

J^{o} | Balance between solute input and output, | γ | Shear rate, ms^{−1} |

k | Mass transfer coefficient, ms^{−1} | γm | Shear rate at the wall (parameter in No. 1.11), s^{−1} |

k_{o} | Ideal mass transfer coefficient, ms^{−1} | Ɛ | Porosity of the membrane, dimensionless |

K | Boltzmann constant (parameter in No. 1.11), Jmol^{−1}K^{−1} | α | Specific resistance of the deposit on membrane, kgm^{2} |

L | Length of the module, m | ϵ | Specific gel resistance, m^{−2} |

L_{p} | Membrane permeability, mPa^{−1}s^{−1} | Ɛ_{CL} | Boundary layer porosity, dimensionless |

L_{ph} | Effective permeability reverse flow, | β | Parameter of No. 1.5 |

m_{p} | Deposited cake weight, kg | σ | Reflection coefficient |

MW | Membrane cut-off limit, gmol^{−1} | Ɛ_{g} | Solidity of the gel layer, % |

∆P | Transmembrane pressure, Pa | ε_{st} | Steady-state value of the average solidity, % |

Q | Flow rate, m^{3}s^{−1} | μ_{0} | Initial viscosity, Pa s |

R_{m} | Membrane resistance, m^{−1} | μ_{b} | Viscosity in the bulk, Pa s |

R_{M} | Fouled membrane resistance, m^{−1} | μ | Viscosity, Pa s |

R_{ad,ss} | Adsorption resistance, m^{−1} | ρ | Feed density, kgm^{−3} |

R_{cp} | Concentration polarization resistance, m^{−1} | ρ_{pol} | Membrane polymer density, kgm^{−3} |

R_{cp,ss} | Steady-state concentration polarization resistance, m^{−1} | γ_{a} | Axial speed, ms^{−1} |

Re | Reynolds number, dimensionless | υ_{p}^{o} | Osmotic pressure limiting flux (m^{3} m^{−2} s^{−1}) |

R_{f} | Irreversible resistance, m^{−1} | X_{12} | Flory–Huggins interaction parameter |

R_{g} | Gel resistance, m^{−1} | ϕ | Volume fraction of particles at the distance x from the membrane surface |

R_{m} | Hydraulic resistance, m^{−1} | ϕ | 1/Jlim (parameter of No. 3.11), m^{2}sm^{−3} |

${\mathrm{R}}_{m}^{*}$ | Experimental resistance (constant ΔP), m^{−1} | ||

R_{m i−1} | Accumulated resistance at t_{i−1}, | ||

${\mathrm{R}}_{\mathrm{ps}}^{*}$ | Concentration polarization differential resistance, dimensionless | ||

R_{t} | Total resistance, m^{−1} | ||

r_{CL} | Specific resistance of the boundary layer, m^{−1} | ||

r_{i} | Cell radius, m | ||

r_{o} | Initial cell radius, m | ||

r_{p} | Particle radius, m | ||

r_{pp} | Membrane pore radius, m |

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**Figure 2.**Permeate flux evolution obtained experimentally and predicted values by selected models for fruit juice clarification. (

**a**) Bergamot juice; (

**b**) kiwifruit juice; (

**c**) pomegranate juice.

**Table 1.**Summary of the relevant models for the concentration polarization category, including the Boundary layer, polarized concentration, and gel models.

No. | Model | Authors | Ref. | Validation Matrix | Main Transport Mechanism | Configuration | Module Type | Number of Citations | Model Validation in Publications |
---|---|---|---|---|---|---|---|---|---|

(1.1) | $J=\frac{D}{\delta}\mathrm{ln}\frac{{C}_{g}}{{C}_{b}}=k\mathrm{ln}\frac{{C}_{g}}{{C}_{b}}$ | Film theory | [46] | - | Diffusive | Cross-flow | - | - | [14,89] |

(1.2) | $J={\left(D/\pi t\right)}^{1/2}\mathrm{ln}\left[\frac{{c}_{g}-{c}_{p}}{{c}_{o}-{c}_{p}}\right]$ | Trettin and Doshi (1980) | [62] | BSA | Diffusive | Dead-end | Unstirred cell | 76 | - |

(1.3) | ${J=x}_{1}k\mathrm{ln}\left(\frac{{C}_{g}}{{C}_{b}}\right)+\left({1-x}_{1}\right)k\mathrm{ln}\left(\frac{{C}_{g}}{{C}_{b}}\right)$ | Modified gel-polarization Fane et al. (1981) | [87] | Gamma Globulin BSA | Diffusive-Convective | - | - | 164 | - |

(1.4) | $J=0.078{\left({\frac{rp}{L}}^{4}\right)}^{1/3}\gamma \mathrm{ln}\left(\frac{{C}_{g}}{{C}_{b}}\right)$ | Zydney and Colton (1986) | [63] | Blood | Diffusive | Cross-flow | - | 274 | - |

(1.5) | $J=\frac{\frac{\Delta P}{{\mu R}_{m}}}{1+2\beta \frac{\left(t-\frac{D\left({C}_{gv}{-C}_{ov}{-C}_{ov}\mathrm{ln}\left(\raisebox{1ex}{${C}_{g}$}\!\left/ \!\raisebox{-1ex}{${C}_{o}$}\right.\right)\right)}{{\left(\frac{\Delta P}{{\mu R}_{m}}\right)}^{2}}\right)}{\frac{{C}_{g}{-C}_{o}}{{C}_{o}}\frac{\raisebox{1ex}{${d}_{h}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\frac{\Delta P}{{\mu R}_{m}}}}}$ | Shear-induced diffusion Davis (1992) | [57] | PEG | Diffusive-Convective | Cross-flow | Tubular | 158 | [70] |

(1.6) | $J=\frac{\Delta P}{L\beta v{\left(L\right)}^{2}}\left[1-\frac{{L}_{p}}{\Delta P}v\left(L\right)\right]$ | Song and Elimelech (1995) | [90] | - | Diffusive-Convective | Cross-flow | Rectangular channel | 246 | [91] |

(1.7) | $J\xb76\pi {\mu}_{0}\xb7{\displaystyle \sum _{i=0}^{\infty}}{f}_{i}{\varnothing}^{i}=-kT\frac{{\left(1-\varnothing \right)}^{3}}{\left(\varnothing -{\varnothing}_{p}\right)}(1+{\displaystyle {\displaystyle \sum}_{i=2}^{\infty}}{A}_{i}{\varnothing}^{i-1})\frac{d\varnothing}{dx}$ | Jonsson and Jonsson (1996) | [92] | Silica sol | Diffusive-Convective | Cross-flow | - | 71 | - |

(1.8) | $J=\frac{2\pi {{\displaystyle \int}}_{0}^{b}{V}_{f}\left(r\right)r\mathrm{dr}}{{\pi b}^{2}}$ | Saksena and Zydney (1997) | [93] | BSA and IgG | Diffusive-Convective | Dead-end | Stirred cell | 51 | - |

(1.9) | $JH/{D}_{o}=-{\left[\frac{{\left(1-b{c}_{o}\varnothing \right)}^{6.5}}{\varnothing -{\varnothing}_{p}}\frac{\partial \varnothing}{\varnothing \eta}\right]}_{\eta =0}$ | Bhattacharjee and Datta (1991) | [94] | PEG-6000 | Diffusive-Convective | Dead-end | Unstirred cell | 9 | - |

(1.10) | ${J}_{t}\left(t\right)=\left({J}_{0}-{J}_{\infty}\right){e}^{-\frac{t}{{t}_{0}}}+{J}_{\infty}$ | The relaxation model Konieczny (2002) | [95] | Water potable | Diffusive-Convective | Cross-flow | Tubular | 22 | [96] |

(1.11) | $\overline{J}=0.807{\left(\frac{{\gamma}_{w}}{L}{\left(\frac{KT}{3\pi \mu {D}_{m}}\right)}^{2}\right)}^{1/3}ln\frac{{c}_{g}}{{c}_{o}}$ Model parameter: ${\mathsf{\gamma}}_{w}$ | Neggaz et al. (2007) | [97] | Pectin Albumin | Diffusive-Convective | Cross-flow | Hollow fiber | 6 | - |

(1.12) | $J=\frac{\Delta {P-2502C}_{o}}{{1.79\times 10}^{13}\mu +{9.327x10}^{14}{C}_{o}{r}_{p}^{2/3}}$; Brownian diffusion $J=\frac{\Delta {P-2502C}_{o}}{{1.79\times 10}^{13}\mu +62.73\frac{{C}_{o}}{{r}_{p}{}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}$; Shear induced diffusion $J=\frac{\Delta {P-2502C}_{o}}{{1.79\times 10}^{13}\mu +\frac{{340C}_{o}}{{\left[12.639{r}_{p}^{2}+\frac{{2.204\times 10}^{-19}}{a}\right]}^{2}}}$; Combined diffusion | Singh et al. (2013) | [64] | Synthetic Fruit juice | Diffusive-Convective | Cross-flow | Spiral-wound | 10 | - |

**Table 2.**Summary of developed models for permeate flux prediction in which osmotic pressure is considered.

No. | Model | Authors | Ref. | Validation Matrix | Main Transport Mechanism | Configuration | Module Type | Number of Citations | Model Validation in Publications |
---|---|---|---|---|---|---|---|---|---|

(2.1) | $J=\frac{\left|\Delta P\right|-\left|\Delta \pi \right|}{{\mu R}_{m}}$ | Osmotic pressure Keden and Katchalsky (1958) | [112] | Water | Convective | Dead-end | - | 442 | [113,114,115,116,117] |

(2.2) | $J=A(\Delta P-\Delta \pi )$ | Goldsmith (1971) | [118] | Dextran fractions (polysaccharides) | - | Cross-flow Dead-end | Tubular Stirred cell | 138 | - |

(2.3) | $J=\frac{\Delta P-a{c}_{b}^{n}\mathrm{exp}(nJ/k)}{{R}_{m}}$ Model parameters: a, n | Wijmans et al. (1984) | [111] | - | - | - | - | 201 | [80,119] |

(2.4) | ${R}_{P}\left(t\right)=\frac{{R}_{ma}}{1-\frac{\sigma \Delta {\pi}_{m}}{\Delta P}}-{R}_{ma}={R}_{ma}\left(\frac{{J}_{w}}{{\upsilon}_{w}}-1\right)$ | Bhattacharjee and Bhattacharya (1992) | [58] | BSA | Convective | Dead-end | Unstirred cell | 36 | [17] |

(2.5) | $J=\frac{\Delta P-\Delta \pi}{\left[{R}_{m}+\left(V/A-{J}^{*}t\right)\alpha \left({c}_{b}-{c}_{p}\right)\right]\mu}$ Model parameter: α | Bhattacharjee and Bhattacharya (1992) | [25] | PEG | Convective | Dead-end | Unstirred cell | 50 | - |

(2.6) | $J=\frac{{\upsilon}_{p}^{o}}{{1+R}_{\mathrm{ps}}^{*}\left[{1-e}^{\left({-K}_{1}t\right)}\right]}$ Model parameters: ${R}_{\mathrm{ps}}^{*}$, K _{1} | Bhattacharya et al. (2001) | [21] | Sugar cane | Convective | Dead-end | Stirred cell | 42 | [120] |

(2.7) | $J=\frac{\Delta P-\Delta {\pi}_{i-1}}{{R}_{mi-1}+\left(\frac{{R}_{m}^{*}\Delta t}{{t}_{R}}\right)+\left(\frac{{m}_{i}-\Delta t}{{J}_{i-1}}\right)}$ Model parameters: t _{R}, m_{i} | Kanani and Ghosh (2007) | [121] | HSA | Convective | Dead-end | Stirred cell | 28 | - |

(2.8) | $J=\frac{\Delta P+\frac{\sigma RT}{{V}_{1}}\left[\mathrm{ln}\left(\frac{{\rho}_{pol}-{C}_{m}}{{\rho}_{pol}-{C}_{p}}\right)+\left\{\left(1-\frac{1}{n}\right)+{X}_{12}\frac{{C}_{m}+{C}_{p}}{{\rho}_{pol}}\right\}\frac{{C}_{m}-{C}_{p}}{{\rho}_{pol}}\right]}{\mu {R}_{m}}$ Model parameters: α, n, X _{12} | Sarkar et al. (2010) | [122] | PEG-6000 | Diffusive-Convective | Dead-end | Stirred cell | 2 | - |

(2.9) | ${J=k}_{o}{\left(\frac{{\mu}_{b}}{{\mu}_{o}}\right)}^{1/3}{{\displaystyle \int}}_{{C}_{b}}^{{C}_{w}}\left(\frac{{\mu}_{o}}{\mu}\right)\left(\frac{{M}_{p}}{RT}\right)\left(\frac{d\mathsf{\Pi}}{dC}\right)\frac{dC}{C}$ | Binabaji et al. (2015) | [123] | Protein solution | Diffusive | Cross-flow | Tangential flow filtration (TFF) Cassette | 6 | - |

No. | Model | Authors | Ref. | Validation Matrix | Main Transport Mechanism | Configuration | Module Type | Number of Citations | Model Validation in Publications |
---|---|---|---|---|---|---|---|---|---|

(3.1) | $J=\frac{\left|\Delta P\right|}{\mu {R}_{t}}$ | Resistance Darcy’s law | - | - | Convective | Dead-end Cross-flow | Tubular | - | [12,23,128,129,130,131] |

(3.2) | $J=\frac{\epsilon {d}_{p}^{2}\Delta P}{32\Delta x\mu}$ | Hagen-Poiseuille | - | Solvent | Convective | Dead-end Cross-flow | Tubular | - | [71,132,133,134] |

(3.3) | $\frac{1}{A}\frac{dV}{dt}=\frac{\Delta P}{\left[{R}_{m}+\left(V/A-{J}_{ss}t\right)\alpha {C}_{b}\right]\eta}$ ${J}_{ss}=kln\left({C}_{g}/{C}_{b}\right)$ Model parameters: J _{ss}, α | Agitation resistance Chudacek and Fane (1984) | [135] | Silica sol Albumin Dextran | Convective | Dead-end Cross-flow | Unstirred cell | 167 | - |

(3.5) | $J=\frac{\Delta P-\Delta \mathsf{\Pi}}{\mu \left({R}_{m}+{R}_{g}\right)}$ ${R}_{m}=\Delta P/{J}_{o}$ ${R}_{ac}^{*}=\left({J}_{o}/{J}_{f}\right)-1$ ${R}_{m}+{R}_{ad}=\Delta P/{J}_{f}$ | Adsorption resistance Gekas et al. (1993) | [136] | BSA | Convective | Cross-flow | Plate type | 44 | - |

(3.6) | $J=\frac{\Delta P-\Delta \mathsf{\Pi}}{\mu \left({R}_{m}+{R}_{g}\right)}$ ${R}_{g}=\alpha \left(1-{\u03f5}_{g}\right){\rho}_{g}L$ $\alpha =180\frac{\left(1-{\u03f5}_{g}\right)}{{\u03f5}^{3}{d}_{p}^{2}{\rho}_{g}}$ | De and Bhattacharya (1997) | [137] | Mixture of sucrose and poly(vinyl alcohol) | Diffusive-Convective | Cross-flow | Stirred cell | 66 | [61,131,138,139,140,141] |

(3.7) | $J=\frac{1}{\mu \left({R}_{M}+{R}_{cp}\left(z\right)\right)}\left(P\left(z\right)-{P}_{p}\right)$ $P\left(z\right)=\frac{{P}_{i}-P\left(z\right)}{z}=\frac{16}{Re}\frac{\rho {u}_{0}^{2}}{R}$ | Paris et al. (2002) | [142] | Dextran T500 | Diffusive-Convective | Cross-flow | Tubular | 45 | [143] |

(3.8) | $\frac{1}{J}=\frac{\mu {R}_{m}}{\Delta P}+\frac{\mu}{{P}_{m}\Delta P}\left[\frac{V}{A}\left(\frac{{c}_{b}-{c}_{p}}{{c}_{g}-{c}_{b}}\right)-\frac{{k}_{b}}{A}\frac{{c}_{g}}{{c}_{g}-{c}_{b}}\omega t\right]$ Model parameters: P _{m}, k_{b}, ω | Bhattacharjee and Datta (2003) | [144] | PEG-6000 | Diffusive-Convective | Dead-end | Stirred cell | 31 | - |

(3.9) | ${J}_{mf}\left(z,t\right)=\frac{\Delta P\left(z,t\right)}{\mu \left({R}_{mf}+{R}_{c}\left(z,t\right)\right)}=\frac{-{p}_{i}\left(z,t\right)}{\mu \left({R}_{mf}+{R}_{c}\left(z,t\right)\right)}$ | Chang et al. (2005) | [145] | Polystyrene latex | Convective | Dead-end | Hollow fiber | 54 | - |

(3.10) | $J=\frac{\Delta PA\beta}{\mu}\frac{1}{{\left(V-{V}_{c}\right)}^{\frac{1}{\alpha}-1}}+\gamma $ Model parameters: A, β, $\gamma $ | Mohammadi et al. (2005) | [146] | Emulsion of oil and gelatin | Diffusive-Convective | Cross-flow | Plate and frame | 26 | - |

(3.11) | $J\left(z\right)=\frac{\Delta P\left(z\right)}{{R}_{m}+{R}_{f}+\varphi \Delta P\left(z\right)}$ | Yeh and Chen (2005) | [147] | Dextran T500 | Convective | Cross-flow | Tubular | 6 | - |

(3.12) | $1-\left(\frac{\overline{J}}{{J}_{lim}}\right)={e}^{-{\left(\overline{\Delta P}\right)}_{exp}/\left(R{J}_{lim}\right)}$ | Yeh (2008) | [148] | Dextran T500 | Convective | Cross-flow | Hollow fiber | 8 | - |

(3.13) | $J=\frac{{P}_{TM}}{{\eta}_{perm}\left({R}_{m}+{R}_{c}\right)}$ ${R}_{c}=\alpha \frac{{m}_{X}^{c}}{A}$ $\alpha ={\alpha}_{0}{\left(\frac{{P}_{TM}}{{P}_{TM,0}}\right)}^{n}$ Model parameters: P _{TM}, α, ${\alpha}_{0}$ | Cuellar et al. (2009) | [149] | E. coli cells | Convective | Cross-flow | Hollow fiber | 7 | - |

(3.14) | $\overline{J}={\displaystyle {{\displaystyle \int}}_{0}^{1}}\frac{-\Delta {P}_{i}d\epsilon}{A{\epsilon}_{2}+B\epsilon +C}+{\displaystyle {{\displaystyle \int}}_{0}^{1}}\frac{(m{Q}_{i}-n\overline{J})\epsilon d\epsilon}{A{\epsilon}_{2}+B\epsilon +C}$ Model parameters: $\mathcal{E}$, A, B, C, n | Yeh et al. (2010) | [150] | Dextran T500 | Convective | Cross-flow | Tubular | 1 | - |

(3.15) | ${J}_{D}=\frac{1}{\left({R}_{s}+\frac{{R}_{m}{R}_{p}}{{R}_{m}+{R}_{p}}\right)}$ ${R}_{s}={k}_{s}\frac{{r}_{s}}{{r}_{p}}exp\left(1-\beta \right)$ ${R}_{m}={k}_{m}\left(\frac{\mu}{\alpha}\right)$ ${R}_{p}={k}_{p}{\left(\frac{{r}_{s}}{{r}_{p}}\right)}^{2}\mu $ | Marchetti et al. (2012) | [133] | Water Ethanol Acetone DMF | Convective | Cross-flow | Tubular | 37 | - |

(3.16) | $J=\frac{\Delta P}{\mu \left[{R}_{m}+\left({R}_{ad,ss}+{R}_{cp,ss}\right)\left(1-{e}^{-bt}\right)+\left(\frac{{m}_{p}}{A}\right)\alpha \right]}$ Model parameters: σ, b | Corbatón-Báguena et al. (2018) | [151] | Whey model solution | Diffusive-Convective | Cross-flow | Tubular Flat sheet | 6 | - |

**Table 4.**Summary of models developed for permeate flux prediction based on fouling and adsorption mechanisms.

No. | Model | Authors | Ref. | Validation Matrix | Main Transport Mechanism | Configuration | Module Type | Number of Citations | Model Validation in Publications |
---|---|---|---|---|---|---|---|---|---|

(4.1) | $1/{J}^{2}=1/{J}_{o}^{2}+{k}_{c}{k}_{cf}t;n=0$ $1/J=1/{J}_{o}+{k}_{i}t;n=1$ $\frac{1}{\sqrt{J}}=\frac{1}{\sqrt{{J}_{o}}}+{K}_{s}t;n=1.5$ $lnJ=ln{J}_{o}-{k}_{c}t;n=2$ | Hermia (1982) | [60] | - | Convective | Dead-end | - | - | [13,14,164,165] |

(4.2) | $\mathrm{ln}\left[\frac{{1-R}_{\mathrm{obs}}}{{R}_{\mathrm{obs}}}\right]=\mathrm{ln}\left[\frac{{1-R}_{m}}{{R}_{m}}\right]+\frac{J}{k}$ | Nakao and Kinura (1981) | [114] | PEG | Convective | Dead-end | Tubular | 32 | [166] |

(4.3) | ${J}_{p}{=J}_{\mathrm{pss}}+\left({J}_{o}{-J}_{\mathrm{pss}}\right){e}^{{-k}_{c}{J}_{o}t}$; Complete blocking ${J}_{p}=\frac{{J}_{o}{J}_{\mathrm{pss}}\left({e}^{{k}_{i}{J}_{\mathrm{pss}}t}\right)}{{J}_{\mathrm{pss}}{+J}_{o}\left({e}^{{k}_{i}{J}_{\mathrm{pss}}t}-1\right)}$; Intermediate blocking ${J}_{p}=\frac{{J}_{o}}{{\left({J}_{o}{+J}_{o}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{k}_{s}t\right)}^{2}}$; Standard blocking $t=\frac{1}{{k}_{\mathrm{gl}}{J}_{\mathrm{pss}}^{2}}\mathrm{ln}\left[\left(\frac{{J}_{p}}{{J}_{o}}\frac{{J}_{o}{-J}_{\mathrm{pss}}}{{J}_{p}{-J}_{\mathrm{pss}}}\right){-J}_{\mathrm{pss}\left(\frac{1}{{J}_{p}}-\frac{1}{{J}_{o}}\right)}\right]$; Gel layer formation | Cros-flow HermianField et al. (1995) | [65] | Dodecane-water emulsion | Convective | Cross-flow | Flat-sheet | 945 | [9,167,168,169] |

(4.4) | $Sh=\frac{1}{\left({W}_{d}-1\right){e}^{-{P}_{e}}+\frac{1}{{P}_{e}}\left(1-{e}^{-{P}_{e}}\right)}$ | Bacchin et al. (1996) | [24] | Clay suspensions | Diffusive | Cross-flow | Hollow fiber | 88 | [6] |

(4.5) | $J=\frac{1}{L}\left[{\displaystyle {{\displaystyle \int}}_{0}^{x\left(t\right)}}{v}_{eq}\left(x\right)dx+\left(L-X\left(t\right)\right)v\left(t\right)\right]$ When t < t _{ss}$J=1.31{\left({D}^{2}\gamma /L\right)}^{1/3}{\left({c}_{g}/{c}_{0}-1\right)}^{1/3}$ When t > t _{ss} | Dynamic model Song (1998) | [108] | - | Diffusive-Convective | Cross-flow | - | 253 | [45,55] |

Wang and Song (1999) | [170] | Silica colloids | Diffusive-Convective | Cross-flow | Tubular | 62 | - | ||

(4.6) | $J={J}_{o}\left[{e}^{\left(-\frac{\alpha \Delta P{c}_{b}}{\mu {R}_{m}}t\right)+\frac{{R}_{m}}{{R}_{m}+{R}_{p}}\left(1-{e}^{\left(-\frac{\alpha \Delta P{c}_{b}}{\mu {R}_{m}}t\right)}\right)}\right]$ | Ho and Zydney (2000) | [66] | BSA | Convective | Cross-flow | Stirred cell | 434 | [10,171,172,173] |

(4.7) | $J=Df(\partial ){S}_{D}(\partial )\frac{Ak}{\Delta x}\Delta {C}_{s}+\overline{J{C}_{s}}g(\partial ){S}_{F}(\partial )$ Model parameters: $\partial $, S _{D}, S_{F} | Darnon et al. (2002) | [172] | Β-Lactoglobulin and yeast extract | Diffusive-Convective | Cross-flow | Tubular | 12 | - |

(4.8) | $V=\frac{{J}_{O}}{{k}_{b}}\left({1-e}^{\left(\frac{{-k}_{b}}{{k}_{c}{J}_{o}^{2}}\left(\sqrt{{1+2k}_{c}{J}_{o}^{2}t}-1\right)\right)}\right)$ Cake-complete $V=\frac{1}{{k}_{i}}\mathrm{ln}\left(1+\frac{{k}_{i}}{{k}_{c}{J}_{o}}\left({\left({1+2k}_{c}{J}_{o}^{2}t\right)}^{1/2}-1\right)\right)$ Cake-intermediate $V=\frac{{J}_{o}}{{k}_{b}}\left({1-e}^{\left(\frac{{-2k}_{b}t}{{2+k}_{s}{J}_{o}t}\right)}\right)$ Complete-standard $V=\frac{1}{{k}_{i}}\mathrm{ln}\left(1+\frac{{2k}_{i}{J}_{o}t}{{2+k}_{s}{J}_{o}t}\right)$ Intermediate-standard $V=\frac{2}{{k}_{s}}\left(\beta \mathrm{cos}\left(\frac{2\pi}{3}-\frac{1}{3}arccos\left(\alpha \right)\right)+\frac{1}{3}\right)$ Cake-standard Model parameters: K _{b}, k_{c}, k_{i}, k_{s}, α, β | Bolton et al. (2004) | [173] | IgG BSA | Convective | Cross-flow | Tubular | 201 | - |

(4.9) | $\frac{Q}{{Q}_{o}}=\frac{1}{{\left({1+\beta Q}_{o}{C}_{b}t\right)}^{2}}{e}^{\left(-\frac{{\alpha C}_{b}{J}_{o}t}{{1+\beta Q}_{o}{C}_{b}t}\right)}+\phantom{\rule{0ex}{0ex}}{\displaystyle {{\displaystyle \int}}_{0}^{t}}\frac{\left({\alpha C}_{b}{J}_{o}/{\left({1+\beta Q}_{o}{C}_{b}{t}_{p}\right)}^{2}\right){e}^{\left(-\left({\alpha C}_{b}{J}_{o}{t}_{p}/\left({1+\beta Q}_{o}{C}_{b}{t}_{p}\right)\right)\right)}}{\sqrt{{\left[\left({R}_{po}{/R}_{m}\right)+{\left({1+\beta Q}_{o}{C}_{b}{t}_{p}\right)}^{2}\right]}^{2}+2({f}^{\prime}{R}^{\prime}\Delta p{C}_{b}/\mu {R}_{m}^{2}(t-{t}_{p})}}{dt}_{p}$ Model parameters: α, β, tp, f′, R′ | Duclos-Orsello et al. (2006) | [174] | BSA | Convective | Dead-end | Stirred cell | 152 | - |

(4.10) | $\frac{\mathrm{dJ}}{\mathrm{dt}}=-a\frac{{C}_{b}}{{C}_{\mathrm{bo}}}\left({J-J}^{*}\right){J}^{2}$ ${J}^{*}{=J}_{\left({C}_{\mathrm{bo}}\right)}^{*}{\left(\frac{{C}_{b}}{{C}_{\mathrm{bo}}}\right)}^{-n}$ | Furukawa et al. (2008) | [67] | Soy less | Diffusive-Convective | Dead-end Cross-flow | Tubular | 27 | - |

(4.11) | $J\left(t\right)=J\left(t\to \infty \right){+\mathrm{ke}}^{(-\mathrm{bt})}$ Model parameter: b | Lin et al. (2008) | [175] | BSA Hemoglobin | Diffusive-Convective | Dead-end | Stirred glass cell | 21 | - |

(4.12) | $J=\frac{{J}_{0}}{{1+R}_{\mathrm{CPB}}^{*}}$ ${J}_{{t}_{1}}=\frac{J}{{1+R}_{\mathrm{CPB}}^{*}{t}_{1}}$ Model parameter: ${R}_{\mathrm{CPB}}^{*}$ | Mondal and De (2009) | [176] | Pineapple juice | Convective | Cross-flow | Hollow fiber | 29 | [177] |

(4.13) | $\Delta P=\frac{\overline{J}}{k}$$\frac{k}{{k}_{o}}=\frac{{\left(1-\frac{\sigma}{{\epsilon}_{o}}\right)}^{3}}{{\left[1+\mathsf{\sigma}/\left({1-\mathsf{\epsilon}}_{o}\right)\right]}^{2}}$ Model parameter: σ | Wang et al. (2017) | [178] | Aqueous solutions | Diffusive-Convective | Cross-flow | Hollow fiber | 1 | - |

No. | Model | Authors | Ref. | Validation Matrix | Main Transport Mechanism | Configuration | Module Type | Number of Citations | Model Validation in Publications | |
---|---|---|---|---|---|---|---|---|---|---|

(5.1) | $\overline{J}=\frac{({J}_{o}-{J}^{o})({J}_{\mathrm{lim}}-{J}^{o})}{{J}^{o}+{J}_{\mathrm{lim}}-{J}^{o}}$ Model parameter: J ^{0} | Surface renovation theory Koltuniewicz (1992) | [179] | BSA | Diffusive-Convective | Cross-flow | - | 44 | - | |

(5.2) | $J\left(t\right){=J}_{p,o}-\mathrm{at}\hspace{1em}\hspace{1em}J\left(t\right)\le {J}_{\mathrm{th}}$ $J\left(t\right)=\left({J}_{p,o}{-J}_{\mathrm{th}}\right){e}^{-b\prime t}{+J}_{\mathrm{th}}-\mathrm{at}J\left(t\right)\ge {J}_{\mathrm{th}}$ | Threshold model Ochando-Pulido et al. (2015) | [6] | - | Diffusive-Convective | Cross-flow | - | 192 | [59,189] | |

(5.3) | $J=\frac{{e}^{s}}{{1-e}^{-\mathrm{tp}*}}\sqrt{{\pi S}^{*}}\left[\mathrm{erf}\left(\sqrt{{S}^{*}{+t}_{p}*}\right)-\mathrm{erf}\left(\sqrt{{S}^{*}}\right)\right]$ Model parameters: S*, t _{p} | Surface renovation theory Hasan et al. (2013) | [190] | Fermentation broths | Diffusive-Convective | Cross-flow | Unstirred cell | 16 | [120] | |

(5.4) | $\mathrm{ln}\left({J-J}_{\infty}\right){=\mathrm{ln}k}_{f}{+b}_{f}t$ ${J=J}_{\mathrm{ss}}{k}_{f}{e}^{{-b}_{f}t}$ | Yee et al. (2009) | [191] | PEG | Diffusive-Convective | Cross-flow | Tubular | 30 | [171] | |

(5.5) | $J\left(f\right){=J}_{o}\mathrm{exp}\left\{\frac{-t}{f\left(t\right)}\right\}$ $f\left(t\right){=A}_{1}{+A}_{2}t$ Model parameter: A _{1}, A_{2} | Empirical model Mallubhotla and Belfort (1996) | [59] | Yeast | - | Dead-end | Unstirred cell | 29 | [68] | |

(5.6) | $J\left(f\right){=J}_{o}\mathrm{exp}\left\{\frac{-t}{f\left(t\right)}\right\}$ $f\left(t\right){=B}_{1}{+B}_{2}{t+B}_{3}{t}^{2}$ | Modified Mallubhotla and Belfort | Modification empirical model Soler-Cabezas et al. (2015) | [68] | Waster water | - | Cross-flow | Hollow fiber | 11 | - |

$J\left(t\right){=C}_{1}+\frac{{C}_{2}}{\mathrm{tan}\left({C}_{3}{t+C}_{4}\right)}$ | Inverse Tangential | |||||||||

$J\left(t\right){=J}_{\mathrm{pss}}{+(J}_{o}{-J}_{\mathrm{pss}}{)e}^{-\left({D}_{1}{t+D}_{2}{t}^{2}\right)}$ | Exponential quadratic | |||||||||

$J\left(t\right){=E}_{1}+\frac{{E}_{2}}{\mathrm{ln}\left({E}_{3}{t+E}_{4}\right)}$ | Inverse logarithmic | |||||||||

$J\left(t\right){=F}_{1}\frac{\left({F}_{2}{+e}^{{F}_{3}t}\right)}{\left({F}_{4}{+e}^{{F}_{5}t}\right)}$ | Exponential double | |||||||||

Model parameters: B, C, D, E, F | ||||||||||

(5.7) | Computational model of system dynamics (SD) | Zhu et al. (2016) | [8] | Raw water | - | Cross-flow | Stirred cell | 0 | - | |

(5.8) | Adaptive neuro-diffusive inference system model (ANFIS) | Salahi et al. (2015) | [7] | Wastewater | - | Cross-flow | Hollow fiber | - | - | |

(5.9) | PCA model of simultaneous multilevel analysis of components with invariant patterns (MSCA-P) | Modeling for Data Mining Klimkiewicz et al. (2016) | [15] | Enzymes | - | - | - | 1 | - | |

(5.10) | Neural network (ANN’s) per layer | Corbatón-Báguena et al. (2016) | [72] | PEG | - | Cross-flow | Tubular | 6 | - | |

(5.11) | Neural network (ANN’s) per layer | Díaz et al. (2017) | [12] | Water | - | Cross-flow | Tubular | 0 | - | |

(5.12) | ${Y}_{t}={\varphi}_{1}{Y}_{t-1}+{\varphi}_{2}{Y}_{t-2}+\dots +{\varphi}_{p}{Y}_{t-p}{+\epsilon}_{t}$ | AR | ARIMA Ruby-Figueroa et al. (2017) | [69] | Fruit juices | - | Cross-flow | Tubular Hollow fiber | 6 | - |

$\Delta {Y}_{t}{=Y}_{t}{-Y}_{t-1}$ | I | |||||||||

${Y}_{t}{=\mathsf{\epsilon}}_{t}{+\theta}_{1}{\epsilon}_{t-1}{+\theta}_{2}{\epsilon}_{t-2}{+\dots +\theta}_{q}{\epsilon}_{t-q}$ | MA |

**Table 6.**Description of the UF membrane, operating conditions, and physicochemical characteristics of the fruit juices analyzed in this work *.

Bergamot | Kiwi Fruit | Pomegranate | Reference | |
---|---|---|---|---|

DCQ II-006C | Koch Series-Cor TM HFM 251 | FUC 1582 | ||

Membrane characteristics and operation | ||||

Membrane material | Polysulfone (PS) | Polyvinylidene fluoride (PVDF) | Triacetate cellulose (CTA) | - |

Configuration | Hollow Fiber | Tubular | Hollow Fiber | - |

Area (m^{2}) | 0.16 | 0.23 | 0.26 | - |

MWCO (kDa) | 100 | 100 | 150 | - |

ΔP (bar) | 1 | 0.85 | 0.6 | - |

Temperature (°C) | 20 | 25 | 25 | - |

Flow (Lh^{−1}) | 114 | 800 | 400 | - |

Porosity (dimensionless) | 0.0057 | 1.1 | 0.0007 | |

Tortuosity (dimensionless) | 3 | 3 | 0.03 | - |

Membrane thickness (m) | 4.7 × 10^{−7} | 2.0 × 10^{−6} | 0.00023 | [34] |

Pore density, N (number of pores m ^{−1}) | 6.0 × 10^{12} | 4.0 × 10^{16} | 1.0 × 10^{13} | [46] |

Module length, L (mm) | 330 | 406 | 136 | [61] |

Module diameter (m) | 0.0021 | 0.025 | 0.0008 | [30,46,192] |

Hydraulic resistance (m^{−1}) | 3.6 × 10^{12} | 1.6 × 10^{12} | 2.1 × 10^{12} | - |

Hydraulic permeability (mPa^{−1}s^{−1}) | 2.7 × 10^{−10} | 5.9 × 10^{−10} | 4.6 × 10^{−10} | - |

Fruit juices characteristics | ||||

Total soluble solids (°Brix) | 9.4 | 12.6 | 18.7 | [30,38,43,193] |

Titratable Acidity | 53.86 (gL^{−1}) | - | 1.04 (% citric acid) | [30,38,43,193] |

pH | 2.40 | 3.19 | 3.61 | [30,38,43,193] |

Total phenolic compounds | 660 (mg/L) | 421.6 (mg/L) | 1930 (mg GAE/100 L) | [30,38,43,193] |

Turbidity (%) | 33.67 | - | [30,38,43,193] | |

Feed density, ρ (kgm^{−3}) | 1091 | 1070 | 1131 | [194,195] |

Feed viscosity, μ (Pa s) | 0.0019 | 0.0014 | 0.0017 | [31,196] |

Concentration in food (%) | 12 | 10.08 | 4.9 | [27,33,36] |

**Table 7.**Results of the simulation for selected models of bergamot juice clarification in terms of RMSE, MAPE, R

^{2}, and Shapiro–Wilk (S-W) and Kruskal–Wallis (K-W) residual analysis tests. Statistically validated models are in bold.

Models | RMSE | MAPE | R^{2} | S-W | K-W | |
---|---|---|---|---|---|---|

Concentration polarization model | Davis (1992)/Shear-Induced Diffusion | 0.80 | 11.76 | 91.08 | 0.00 | 0.10365 |

Osmotic pressure models | Keden & Katchalsky (1958) | 0.25 | 5.70 | 99.17 | 0.0117 | 0.05 |

Wijmanset al. (1984) | 0.49 | 11.70 | 99.22 | 0.6855 | 0.0004 | |

Resistance in series models | Hagen-Poiseuille (1839) | 0.22 | 3.99 | 99.78 | 0.00034 | 0.8364 |

De et al. (1997) | 0.36 | 4.81 | 97.47 | 0.00 | 0.8692 | |

Fouling and adsorption models | Ho and Zydney (2000) | 1.64 | 31.52 | 90.25 | 1.554 × 10^{−15} | 0.00 |

Song (1998)/Dynamic model | 1.51 | 35.90 | 97.56 | 0.00 | 0.00 | |

Mondal et al (2009) | 1.76 | 18.23 | 87.01 | 0.0 | 0.00002 | |

Non-Phenomenological models | Yee et al. (2009) | 2.03 | 28.91 | 84.91 | 0.000088 | 0.1038 |

Ruby-Figueroa et al. (2017)/ARIMA models | 0.40 | 8.24 | 97.92 | 2.99 × 10^{−15} | 0.056 |

**Table 8.**Results of the simulation for selected models of kiwi juice clarification in terms of RMSE, MAPE, R

^{2}, and residual analysis tests Shapiro–Wilk (S-W) and Kruskal–Wallis (K-W). Statistically validated models are in bold.

Models | RMSE | MAPE | R^{2} | S-W | K-W | |
---|---|---|---|---|---|---|

Concentration polarization models | Davis (1992)/Shear-Induced Diffusion | 2.91 | 22.35 | 52.86 | 0.00 | 1.213 × 10^{−10} |

Osmotic pressure models | Keden and Katchalsky (1958) | 9.51 | 115.03 | 97.76 | 0.002 | 0.00 |

Wijmanset al. (1984) | 0.33 | 3.14 | 97.98 | 0.075 | 0.45 | |

Resistance in series models | Hagen-Poiseuille (1839) | 0.64 | 8.21 | 98.45 | 0.00 | 0.0032 |

De et al. (1997) | 0.48 | 5.46 | 97.43 | 0.0012 | 0.2238 | |

Fouling and adsorption models | Ho and Zydney (2000) | 1.07 | 8.92 | 95.95 | 4.152 × 10^{−12} | 0.1015 |

Song (1998)/Dynamic model | 3.94 | 43.51 | 67.94 | 0.00 | 0.00 | |

Mondal et al (2009) | 0.96 | 11.17 | 93.18 | 0.0 | 0.058 | |

Non-Phenomenological models | Yee et al. (2009) | 0.64 | 7.16 | 97.67 | 1.438 × 10^{−13} | 0.2047 |

Ruby-Figueroa et al. (2017)/ARIMA models | 0.33 | 3.74 | 98.98 | 0.0250 | 0.3801 |

**Table 9.**Results of the simulation for selected models of pomegranate juice clarification in terms of RMSE, MAPE, R

^{2}, and residual analysis tests Shapiro–Wilk (S-W) and Kruskal–Wallis (K-W). Statistically validated models are in bold.

Models | RMSE | MAPE | R^{2} | S-W | K-W | |
---|---|---|---|---|---|---|

Concentration polarization models | Davis (1992)/Shear-Induced Diffusion | 1.64 | 27.56 | 85.58 | 2.22 × 10^{−}^{9} | 0.8234 |

Osmotic pressure models | Keden and Katchalsky (1958) | 4.89 | 67.03 | 98.92 | 0.0001 | 3.581 × 10^{−9} |

Wijmanset al. (1984) | 0.49 | 7.85 | 98.91 | 0.00 | 0.964 | |

Resistance in series models | Hagen-Poiseuille (1839) | 0.81 | 21.00 | 98.28 | 0.00 | 0.1974 |

De et al. (1997) | 0.72 | 16.64 | 96.73 | 2.33× 10^{−13} | 0.37255 | |

Fouling and adsorption models | Ho & Zydney (2000) | 2.01 | 51.69 | 75.91 | 2.93× 10^{−12} | 0.088 |

Song (1998)/Dynamic model | 3.41 | 50.78 | 80.64 | 1.154 × 10^{−14} | 0.00 | |

Mondal et al (2009) | 1.60 | 17.45 | 92.40 | 0.0 | 0.3804 | |

Non-Phenomenological models | Yee et al. (2009) | 0.46 | 11.09 | 99.20 | 2.991× 10^{−12} | 0.2262 |

Ruby-Figueroa et al. (2017)/ARIMA models | 0.25 | 4.08 | 99.70 | 0.00 | 0.6320 |

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Quezada, C.; Estay, H.; Cassano, A.; Troncoso, E.; Ruby-Figueroa, R.
Prediction of Permeate Flux in Ultrafiltration Processes: A Review of Modeling Approaches. *Membranes* **2021**, *11*, 368.
https://doi.org/10.3390/membranes11050368

**AMA Style**

Quezada C, Estay H, Cassano A, Troncoso E, Ruby-Figueroa R.
Prediction of Permeate Flux in Ultrafiltration Processes: A Review of Modeling Approaches. *Membranes*. 2021; 11(5):368.
https://doi.org/10.3390/membranes11050368

**Chicago/Turabian Style**

Quezada, Carolina, Humberto Estay, Alfredo Cassano, Elizabeth Troncoso, and René Ruby-Figueroa.
2021. "Prediction of Permeate Flux in Ultrafiltration Processes: A Review of Modeling Approaches" *Membranes* 11, no. 5: 368.
https://doi.org/10.3390/membranes11050368