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)	$J=\frac{\Delta P-a{c}_b^n\exp (nJ/k)}{R_m}$ 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: $R_{\mathrm{ps}}^{*}$, K1	Bhattacharya et al. (2001)	[21]	Sugar cane	Convective	Dead-end	Stirred cell	42	[120]
(2.7)	$J=\frac{\Delta P-\Delta {\pi }_{i-1}}{R_{m i-1}+\left(\frac{R_m^{*}\Delta t}{t_R}\right)+\left(\frac{m_i-\Delta t}{J_{i-1}}\right)}$ Model parameters: tR, mi	Kanani and Ghosh (2007)	[121]	HSA	Convective	Dead-end	Stirred cell	28	-
(2.8)	$J=\frac{\Delta P+\frac{\sigma RT}{V_1}\left[\ln \left(\frac{{\rho }_{pol}-C_m}{{\rho }_{pol}-C_p}\right)+\left\{\left(1-\frac{1}{n}\right)+X_{12}\frac{C_m+C_p}{{\rho }_{pol}}\right\}\frac{C_m-C_p}{{\rho }_{pol}}\right]}{\mu R_m}$ Model parameters: α, n, X12	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	-

HSA: Human serum albumin solution; BSA: Bovine serum albumin; PEG: Polyethylene glycol.