. 2021 May 18;11(5):368. doi: 10.3390/membranes11050368

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)	$1 / J^{2} = 1 / J_{o}^{2} + k_{c} k_{c f} t; n = 0$ $1 / J = 1 / J_{o} + k_{i} t; n = 1$ $\frac{1}{\sqrt{J}} = \frac{1}{\sqrt{J_{o}}} + K_{s} t; n = 1.5$ $l n J = l n J_{o} - k_{c} t; n = 2$	Hermia (1982)	[60]	-	Convective	Dead-end	-	-	[13,14,164,165]
(4.2)	$\ln [\frac{{1 - R}_{obs}}{R_{obs}}] = \ln [\frac{{1 - R}_{m}}{R_{m}}] + \frac{J}{k}$	Nakao and Kinura (1981)	[114]	PEG	Convective	Dead-end	Tubular	32	[166]
(4.3)	$J_{p} {= J}_{pss} + (J_{o} {- J}_{pss}) e^{{- k}_{c} J_{o} t}$ ; Complete blocking $J_{p} = \frac{J_{o} J_{pss} (e^{k_{i} J_{pss} t})}{J_{pss} {+ J}_{o} (e^{k_{i} J_{pss} t} - 1)}$ ; Intermediate blocking $J_{p} = \frac{J_{o}}{{(J_{o} {+ J}_{o}^{\frac{1}{2}} k_{s} t)}^{2}}$ ; Standard blocking $t = \frac{1}{k_{gl} J_{pss}^{2}} \ln [(\frac{J_{p}}{J_{o}} \frac{J_{o} {- J}_{pss}}{J_{p} {- J}_{pss}}) {- J}_{pss (\frac{1}{J_{p}} - \frac{1}{J_{o}})}]$ ; 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)	$S h = \frac{1}{(W_{d} - 1) e^{- P_{e}} + \frac{1}{P_{e}} (1 - e^{- P_{e}})}$	Bacchin et al. (1996)	[24]	Clay suspensions	Diffusive	Cross-flow	Hollow fiber	88	[6]
(4.5)	$J = \frac{1}{L} [\int_{0}^{x (t)} v_{e q} (x) d x + (L - X (t)) v (t)]$ When t < t_ss $J = 1.31 {(D^{2} γ / L)}^{1 / 3} {(c_{g} / c_{0} - 1)}^{1 / 3}$ When t > t_ss	Dynamic model Song (1998)	[108]	-	Diffusive-Convective	Cross-flow	-	253	[45,55]
(4.5)		Wang and Song (1999)	[170]	Silica colloids	Diffusive-Convective	Cross-flow	Tubular	62	-
(4.6)	$J = J_{o} [e^{(- \frac{α Δ P c_{b}}{μ R_{m}} t) + \frac{R_{m}}{R_{m} + R_{p}} (1 - e^{(- \frac{α Δ P c_{b}}{μ R_{m}} t)})}]$	Ho and Zydney (2000)	[66]	BSA	Convective	Cross-flow	Stirred cell	434	[10,171,172,173]
(4.7)	$J = D f (\partial) S_{D} (\partial) \frac{A k}{Δ x} Δ C_{s} + \bar{J C_{s}} g (\partial) S_{F} (\partial)$ 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)	$V = \frac{J_{O}}{k_{b}} ({1 - e}^{(\frac{{- k}_{b}}{k_{c} J_{o}^{2}} (\sqrt{{1 + 2 k}_{c} J_{o}^{2} t} - 1))})$ Cake-complete $V = \frac{1}{k_{i}} \ln (1 + \frac{k_{i}}{k_{c} J_{o}} ({({1 + 2 k}_{c} J_{o}^{2} t)}^{1 / 2} - 1))$ Cake-intermediate $V = \frac{J_{o}}{k_{b}} ({1 - e}^{(\frac{{- 2 k}_{b} t}{{2 + k}_{s} J_{o} t})})$ Complete-standard $V = \frac{1}{k_{i}} \ln (1 + \frac{{2 k}_{i} J_{o} t}{{2 + k}_{s} J_{o} t})$ Intermediate-standard $V = \frac{2}{k_{s}} (β \cos (\frac{2 π}{3} - \frac{1}{3} a r c c o s (α)) + \frac{1}{3})$ 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)	$\frac{Q}{Q_{o}} = \frac{1}{{({1 + β Q}_{o} C_{b} t)}^{2}} e^{(- \frac{{α C}_{b} J_{o} t}{{1 + β Q}_{o} C_{b} t})} + \int_{0}^{t} \frac{({α C}_{b} J_{o} / {({1 + β Q}_{o} C_{b} t_{p})}^{2}) e^{(- ({α C}_{b} J_{o} t_{p} / ({1 + β Q}_{o} C_{b} t_{p})))}}{\sqrt{{[(R_{p o} {/ R}_{m}) + {({1 + β Q}_{o} C_{b} t_{p})}^{2}]}^{2} + 2 (f^{'} R^{'} Δ p C_{b} / μ R_{m}^{2} (t - t_{p})}} {d t}_{p}$ Model parameters: α, β, tp, f′, R′	Duclos-Orsello et al. (2006)	[174]	BSA	Convective	Dead-end	Stirred cell	152	-
(4.10)	$\frac{dJ}{dt} = - a \frac{C_{b}}{C_{bo}} ({J - J}^{}) J^{2}$ $J^{} {= J}_{(C_{bo})}^{*} {(\frac{C_{b}}{C_{bo}})}^{- n}$	Furukawa et al. (2008)	[67]	Soy less	Diffusive-Convective	Dead-end Cross-flow	Tubular	27	-
(4.11)	$J (t) = J (t \to \infty) {+ ke}^{(- bt)}$ Model parameter: b	Lin et al. (2008)	[175]	BSA Hemoglobin	Diffusive-Convective	Dead-end	Stirred glass cell	21	-
(4.12)	$J = \frac{J_{0}}{{1 + R}_{CPB}^{}}$ $J_{t_{1}} = \frac{J}{{1 + R}_{CPB}^{} t_{1}}$ Model parameter: $R_{CPB}^{*}$	Mondal and De (2009)	[176]	Pineapple juice	Convective	Cross-flow	Hollow fiber	29	[177]
(4.13)	$Δ P = \frac{\bar{J}}{k}$ $\frac{k}{k_{o}} = \frac{{(1 - \frac{σ}{ε_{o}})}^{3}}{{[1 + σ / ({1 - ε}_{o})]}^{2}}$ Model parameter: σ	Wang et al. (2017)	[178]	Aqueous solutions	Diffusive-Convective	Cross-flow	Hollow fiber	1	-

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