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)	$J=\frac{D}{\delta }\ln \frac{C_g}{C_b}=k\ln \frac{C_g}{C_b}$	Film theory	[46]	-	Diffusive	Cross-flow	-	-	[14,89]
(1.2)	$J={\left(D/\pi t\right)}^{1/2} \ln \left[\frac{c_g-c_p}{c_o-c_p}\right]$	Trettin and Doshi (1980)	[62]	BSA	Diffusive	Dead-end	Unstirred cell	76	-
(1.3)	${J=x}_{1}k\ln \left(\frac{C_g}{C_b}\right)+\left({1-x}_1\right) k\ln \left(\frac{C_g}{C_b}\right)$	Modified gel-polarization Fane et al. (1981)	[87]	Gamma Globulin BSA	Diffusive-Convective	-	-	164	-
(1.4)	$J=0.078{\left({\frac{rp}{L}}^4\right)}^{1/3}\gamma \ln \left(\frac{C_g}{C_b}\right)$	Zydney and Colton (1986)	[63]	Blood	Diffusive	Cross-flow	-	274	-
(1.5)	$J=\frac{\frac{\Delta P}{{\mu R}_m}}{1+2\beta \frac{\left(t-\frac{D\left(C_{gv}{-C}_{ov}{-C}_{ov}\ln \left(\raisebox{1ex}{$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)	$J·6\pi {\mu }_0·{\displaystyle \sum _{i=0}^{\infty }}{f}_i{\varnothing }^i=-kT\frac{{\left(1-\varnothing \right)}^3}{\left(\varnothing -{\varnothing }_p\right)}(1+{\displaystyle {\displaystyle \sum }_{i=2}^{\infty }}{A}_i{\varnothing }^{i-1})\frac{d\varnothing }{dx}$	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)	$J=\frac{\Delta {P-2502 C}_o}{{1.79 \times 10}^{13}\mu +{9.327x10}^{14}{C}_o{r}_p^{2/3}}$; Brownian diffusion $J=\frac{\Delta {P-2502 C}_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 $J=\frac{\Delta {P-2502 C}_o}{{1.79 \times 10}^{13}\mu +\frac{{340C}_o}{{\left[12.639r_p^2+\frac{{2.204\times 10}^{-19}}{a}\right]}^2}}$; Combined diffusion	Singh et al. (2013)	[64]	Synthetic Fruit juice	Diffusive-Convective	Cross-flow	Spiral-wound	10	-

BSA: Bovine serum albumin; PEG: Polyethylene glycol; IgG: Immunoglobulin G.