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

Table 3.

Summary of resistance-in-series models developed for permeate flux prediction.

No.	Model	Authors	Ref.	Validation Matrix	Main Transport Mechanism	Configuration	Module Type	Number of Citations	Model Validation in Publications
(3.1)	$J = \frac{\| Δ P \|}{μ R_{t}}$	Resistance Darcy’s law	-	-	Convective	Dead-end Cross-flow	Tubular	-	[12,23,128,129,130,131]
(3.2)	$J = \frac{ε d_{p}^{2} Δ P}{32 Δ x μ}$	Hagen-Poiseuille	-	Solvent	Convective	Dead-end Cross-flow	Tubular	-	[71,132,133,134]
(3.3)	$\frac{1}{A} \frac{d V}{d t} = \frac{Δ P}{[R_{m} + (V / A - J_{s s} t) α C_{b}] η}$ $J_{s s} = k l n (C_{g} / C_{b})$ 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)	$J = \frac{Δ P - Δ Π}{μ (R_{m} + R_{g})}$ $R_{m} = Δ P / J_{o}$ $R_{a c}^{*} = (J_{o} / J_{f}) - 1$ $R_{m} + R_{a d} = Δ P / J_{f}$	Adsorption resistance Gekas et al. (1993)	[136]	BSA	Convective	Cross-flow	Plate type	44	-
(3.6)	$J = \frac{Δ P - Δ Π}{μ (R_{m} + R_{g})}$ $R_{g} = α (1 - ϵ_{g}) ρ_{g} L$ $α = 180 \frac{(1 - ϵ_{g})}{ϵ^{3} d_{p}^{2} ρ_{g}}$	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)	$J = \frac{1}{μ (R_{M} + R_{c p} (z))} (P (z) - P_{p})$ $P (z) = \frac{P_{i} - P (z)}{z} = \frac{16}{R e} \frac{ρ u_{0}^{2}}{R}$	Paris et al. (2002)	[142]	Dextran T500	Diffusive-Convective	Cross-flow	Tubular	45	[143]
(3.8)	$\frac{1}{J} = \frac{μ R_{m}}{Δ P} + \frac{μ}{P_{m} Δ P} [\frac{V}{A} (\frac{c_{b} - c_{p}}{c_{g} - c_{b}}) - \frac{k_{b}}{A} \frac{c_{g}}{c_{g} - c_{b}} ω t]$ Model parameters: P_m, k_b, ω	Bhattacharjee and Datta (2003)	[144]	PEG-6000	Diffusive-Convective	Dead-end	Stirred cell	31	-
(3.9)	$J_{m f} (z, t) = \frac{Δ P (z, t)}{μ (R_{m f} + R_{c} (z, t))} = \frac{- p_{i} (z, t)}{μ (R_{m f} + R_{c} (z, t))}$	Chang et al. (2005)	[145]	Polystyrene latex	Convective	Dead-end	Hollow fiber	54	-
(3.10)	$J = \frac{Δ P A β}{μ} \frac{1}{{(V - V_{c})}^{\frac{1}{α} - 1}} + γ$ Model parameters: A, β, $γ$	Mohammadi et al. (2005)	[146]	Emulsion of oil and gelatin	Diffusive-Convective	Cross-flow	Plate and frame	26	-
(3.11)	$J (z) = \frac{Δ P (z)}{R_{m} + R_{f} + ϕ Δ P (z)}$	Yeh and Chen (2005)	[147]	Dextran T500	Convective	Cross-flow	Tubular	6	-
(3.12)	$1 - (\frac{\bar{J}}{J_{l i m}}) = e^{- {(\bar{Δ P})}_{e x p} / (R J_{l i m})}$	Yeh (2008)	[148]	Dextran T500	Convective	Cross-flow	Hollow fiber	8	-
(3.13)	$J = \frac{P_{T M}}{η_{p e r m} (R_{m} + R_{c})}$ $R_{c} = α \frac{m_{X}^{c}}{A}$ $α = α_{0} {(\frac{P_{T M}}{P_{T M, 0}})}^{n}$ Model parameters: P_TM, α, $α_{0}$	Cuellar et al. (2009)	[149]	E. coli cells	Convective	Cross-flow	Hollow fiber	7	-
(3.14)	$\bar{J} = \int_{0}^{1} \frac{- Δ P_{i} d ε}{A ε_{2} + B ε + C} + \int_{0}^{1} \frac{(m Q_{i} - n \bar{J}) ε d ε}{A ε_{2} + B ε + C}$ Model parameters: $ℰ$ , A, B, C, n	Yeh et al. (2010)	[150]	Dextran T500	Convective	Cross-flow	Tubular	1	-
(3.15)	$J_{D} = \frac{1}{(R_{s} + \frac{R_{m} R_{p}}{R_{m} + R_{p}})}$ $R_{s} = k_{s} \frac{r_{s}}{r_{p}} e x p (1 - β)$ $R_{m} = k_{m} (\frac{μ}{α})$ $R_{p} = k_{p} {(\frac{r_{s}}{r_{p}})}^{2} μ$	Marchetti et al. (2012)	[133]	Water Ethanol Acetone DMF	Convective	Cross-flow	Tubular	37	-
(3.16)	$J = \frac{Δ P}{μ [R_{m} + (R_{a d, s s} + R_{c p, s s}) (1 - e^{- b t}) + (\frac{m_{p}}{A}) α]}$ Model parameters: σ, b	Corbatón-Báguena et al. (2018)	[151]	Whey model solution	Diffusive-Convective	Cross-flow	Tubular Flat sheet	6	-

DNA: Deoxyribonucleic acid; DMF: N, N-dimethylformamide; BSA: Bovine serum albumin; PEG: Polyethylene glycol.