Table 5.

Summary of non-phenomenological models used for the prediction of permeate flux.

No.	Model		Authors	Ref.	Validation Matrix	Main Transport Mechanism	Configuration	Module Type	Number of Citations	Model Validation in Publications
(5.1)	$\overline{J}=\frac{(J_o-J^o)(J_{\lim }-J^o)}{J^o+J_{\lim }-J^o}$ Model parameter: J0		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*, tp		Surface renovation theory Hasan et al. (2013)	[190]	Fermentation broths	Diffusive-Convective	Cross-flow	Unstirred cell	16	[120]
(5.4)	$\ln \left({J-J}_{\infty }\right){=\ln k}_f{+b}_{f}t$ ${J=J}_{\mathrm{ss}}{k}_f{e}^{{-b}_{f}t}$		Yee et al. (2009)	[191]	PEG	Diffusive-Convective	Cross-flow	Tubular	30	[171]
(5.5)	$J\left(f\right){=J}_o\exp \left\{\frac{-t}{f\left(t\right)}\right\}$ $f\left(t\right){=A}_1{+A}_{2}t$ Model parameter: A1, A2		Empirical model Mallubhotla and Belfort (1996)	[59]	Yeast	-	Dead-end	Unstirred cell	29	[68]
(5.6)	$J\left(f\right){=J}_o\exp \left\{\frac{-t}{f\left(t\right)}\right\}$ $f\left(t\right){=B}_1{+B}_2{t+B}_3{t}^2$	Modified Mallubhotla and Belfort	Modification empirical model Soler-Cabezas et al. (2015)	[68]	Waster water	-	Cross-flow	Hollow fiber	11	-
	$J\left(t\right){=C}_1+\frac{C_2}{\tan \left(C_3{t+C}_4\right)}$	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
	$J\left(t\right){=E}_1+\frac{E_2}{\ln \left(E_3{t+E}_4\right)}$	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

BSA: Bovine serum albumin; PEG: Polyethylene glycol.