## 1. Introduction

Separation of dissolved and suspended matter from a solvent constitutes a major unit operation, and is important in virtually every industry, including water treatment, environmental remediation, resource extraction, food processing, and effluent treatment. Membrane based separation processes have become extremely popular owing primarily to their lower operating expenses and lower energy consumption compared to other processes, such as distillation. Other advantages of membrane processes include (1) easy integration with other types of separation processes; (2) availability of a variety of membrane materials with different properties that enables tailored separation for targeted components; (3) high rejection of dissolved solutes and ions; and (4) compact design.

Among different membrane processes, pressure driven filtration processes, classified as microfiltration (MF), ultrafiltration (UF), nanofiltration (NF), and reverse osmosis (RO), are widely used for separating constituents from the liquid phase. MF and UF processes are used for separation of particles and macromolecules via physical retention (sieving) of the suspended matter by microporous membranes based on particle size. Typically, MF membranes retain particles >100 nm, and have large pore sizes, whereas UF membranes retain macromolecules, colloids, and proteins in size range of 5–100 nm. These two processes are not suitable for rejection of salts and divalent ions, as the pore size of the membranes used is larger than these entities. RO has been widely used in desalination of water for the past four decades due to its high rejection of monovalent salts [

1] which are removed via a solution-diffusion mechanism. Because of higher hydraulic resistance and osmotic pressure development, RO plants must operate at very high pressures, making the process energy intensive.

NF processes act as a bridge between UF and RO. NF membranes provide a higher permeability and a lower rejection of monovalent ions (<70%) compared to RO processes, but offer a reasonable rejection of multivalent ions (>99%) and organic matter (>90%) [

2,

3]. NF membranes are generally classified according to the size of contaminants they remove. “Tight” NF membranes can approach the salt rejection of RO membranes, whereas “loose” NF membranes can be similar in performance to UF membranes [

4]. NF processes provide a facile method for softening, and even for partial desalination of brackish waters, while employing considerably lower operating pressures than RO.

The major challenge of the pressure driven membrane processes is reduced separation performance due to fouling. Fouling may be defined as the irreversible deposition of retained particles, colloids, macromolecules, salt, etc. on the surface or within the pores of membranes [

5]. This includes adsorption [

6,

7,

8], pore blocking [

9,

10], precipitation or scaling [

11,

12] and cake formation [

13,

14]. Fouling mechanisms during MF and UF processes typically include pore blocking, solute adsorption, and cake/gel formation. In the salt rejecting NF and RO membranes, however, fouling is mainly governed by the adsorption of contaminants on the surface and scaling by divalent ions like Ca

^{2+} and Mg

^{2+}. Fouling reduces membrane performance and longevity as well as flux and permeate quality, and subsequently increases operating costs [

3].

A particular problem of interest in the context of NF process is the combined fouling due to cake formation by charged colloids retained by the membrane, and concentration polarization (CP) due to the retained ions. Such fouling mechanisms are evident in numerous NF processes like desalination, water treatment, softening, produced water treatment in petroleum extraction and refining, etc. [

14,

15,

16]. The cake formation and the CP phenomena are not additive, but manifest in a more complex manner, depending on the particle charge, particle size, ion concentration in the feed, membrane characteristics, and the influence of operating and hydrodynamic conditions [

17,

18,

19,

20,

21].

Although many studies have been conducted to provide insight into the effect of physicochemical parameters that synergistically influence colloidal fouling, the importance and methodology of membrane preconditioning (e.g., membrane compaction) and the determination of substantial parameters (e.g., critical flux) before fouling experiment have not been reported. The aim of this paper is to present a standard experimental and data analysis protocol for NF colloidal fouling experiments. The developed methodology covers preparation and characterization of water samples and colloidal particles, pre-test membrane compaction and critical flux determination, measurement of experimental data during the fouling test, and the analysis of that data to determine the relative importance of various fouling mechanisms. The standard protocol is illustrated with data from a series of flat sheet, bench-scale experiments.

## 2. Theoretical Background and Development of Data Analysis Model

In salt rejecting NF/RO membranes, the CP by salt ions and fouling by colloidal particles, organic matter, and microorganisms are two interconnected phenomena that reduce the water flux through the membranes. Hoek and Elimelech have postulated the first model, namely, cake enhanced concentration polarization (CECP), to elucidate the mechanism of flux decline by the combined effect of CP and colloidal fouling [

14]. They explained the fouling mechanism as arising from hindered back-diffusion of salt ions within the colloidal deposit layers, resulting in an increase of CP as well as the transmembrane osmotic pressure (TMOP). In this section, the engineering basis and the mathematical equations used in the data analysis for the quantification of fouling based on CECP model are presented. These equations are used in the subsequent sections to analyze experimental data from a series of bench-scale experiments to determine the fouling mechanisms present. The NF experiments were conducted using 65 nm radius silica particles with 2300 kg/m

^{3} density and ~−30 mV surface charge (at pH 7.0 in 10 mM NaCl solution) and NF membranes with the effective surface area of 140 cm

^{2} and ~18 mV surface charge (at pH 7.0 in 10 mM NaCl solution).

Resistance through a NF membrane is made up of three major components: the hydrodynamic resistance of the membrane in the absence of foulants, the resistance due to the accumulation of ions at the membrane surface (concentration polarization), and the resistance due to the accumulation of colloids at the membrane surface (cake layer fouling).

Membrane Resistance: The hydrodynamic resistance of the membrane itself in the absence of foulants is determined by measuring flow and pressure over time through the membrane using pure water, and calculating the pure water flux (

${\nu}_{\mathrm{w}}^{0}$, m

^{3}/m

^{2}s). Membrane resistance (

R_{m}, 1/m) is then calculated using the following equation:

where

μ is the dynamic viscosity of the pure water (Pa·s) and Δ

P is the transmembrane pressure (Pa).

Concentration Polarization: The resistance due to the accumulation of ions at the membrane surface (CP), in the absence of colloids, is captured by the generated TMOP. The TMOP reduces the effective pressure driving force for solvent transport and is determined by measuring flow, pressure, and permeate salt concentration over time through the membrane using a salt water solution with a specific NaCl concentration. From this experimental data, the salt water flux (

${\nu}_{w}^{s}$, m

^{3}/m

^{2}s) and the observed salt rejection (

R_{o}) can be calculated. The resistance due to the effects of the accumulated ions, the TMOP (Δ

π, Pa), can then be calculated using the following equations:

where

C_{i,f} and

C_{i,p} (mol/m

^{3}) are the salt concentration in the feed and permeate, respectively. Using evaluated

R_{o} and Δ

π, the initial electrolyte mass transfer coefficient in the salt CP layer (

k_{i}, m/s) is then calculated combining the van’t Hoff equation and film theory (see

Appendix A for the derivation of this equation):

where

R is the universal gas constant (J/mol K) and

T is the absolute temperature of water (K).

Cake Layer Fouling: The cake layer hydrodynamic resistance (

R_{c}, 1/m) due to the accumulation of colloidal particles at the membrane surface can be evaluated by calculating the mass of colloids deposited on the membrane (

M_{c}, kg) using a simple mass balance around the membrane and then estimating the hydrodynamic drag exerted by that mass of spherical colloids within the cake layer. The Kuwabara cell model [

22] can be used to estimate the hydraulic resistance through the cake layer:

where

ε_{c} is the average cake layer porosity,

${\delta}_{\mathrm{c}}$ is the thickness of cake layer (m),

a is the particle diameter (m),

A_{m} is the effective membrane area (m

^{2}),

A_{K} is the Kuwabara correction factor accounting for the effect of neighboring particles in the cake layer, and

g* accounts for electroosmotic effect in swarm of charged colloidal particles [

23]. The parameter

g* quantifies this effect in terms of an electroviscous resistance, which is additional to the hydrodynamic resistance of the cake layer. It also directly relates the cake volume fraction (

${\varphi}_{\mathrm{c}}$) and zeta potential of particles (

ψ_{p}) to the electroviscous resistance. Our previous studies showed that the electroosmotic effect becomes significant for cake layers with higher cake volume fraction and zeta potential [

23]. For

${\varphi}_{\mathrm{c}}$ ~ 0.5 and

ψ_{p} ~ −30 mV in the present study, there would be no electroosmotic backflow and thus

g* = 1 [

23]. The expressions of

A_{K} and

M_{c} are given as follows:

In these equations, ${\varphi}_{\mathrm{c}}$ is the cake volume fraction (${\varphi}_{\mathrm{c}}=1-{\epsilon}_{\mathrm{c}}$), and ${\rho}_{\mathrm{p}}$ is the density of colloidal particles (2300 kg/m^{3} in this study).

Cake Enhanced Osmotic Pressure: Knowing the membrane resistance from the pure water flux experiment and the TMOP due to concentration polarization from the salt water flux experiment, the overall cake enhanced osmotic pressure (CEOP) resulting from the combined effects of the retained colloids and the retained ions can be determined from the actual membrane experimental data. According to film theory, the permeate flux in the fouling experiment can be written as:

where

$\Delta {P}_{t}$,

$\Delta {P}_{\mathrm{c}}$, and

$\Delta {P}_{\mathrm{m}}$ (Pa) are the total, trans-cake, and trans-membrane hydraulic pressures, respectively. The total applied pressure (

$\Delta {P}_{t}$ as the driving force of transport through the membrane) is the summation of the trans-cake hydraulic pressure (

$\Delta {P}_{\mathrm{c}}$), the trans-membrane pressure (

$\Delta {P}_{\mathrm{m}}$), and CEOP (

$\Delta {\pi}_{\mathrm{m}}$). By knowing the membrane resistance (

R_{m}), cake resistance (

R_{c}), and permeate flux, the CEOP is calculated as follows:

The CEOP can also be calculated based on the modified van’t Hoff equation,

where

${k}_{\mathrm{i}}^{\ast}$ (m/s) is the hindered mass transfer coefficient (

${k}_{\mathrm{i}}^{\ast}$) which consists of two parts: one describes the mass transfer within the cake layer (

δ_{c}), which can be considered as internal mass transfer coefficient, and the other one is related to the mass transfer within the CP layer (between the surface of cake layer and the bulk,

δ_{s} =

δ −

δ_{c}), which is considered as external mass transfer coefficient [

14,

17,

24]:

where

δ (m) is the thickness of the mass boundary layer,

${D}_{i}$ (m

^{2}/s) is the bulk diffusivity,

δ/

D_{i} is the inverse of mass transfer coefficient within the CP layer (1/

k_{i}), and

${D}_{\mathrm{i}}^{\ast}$ (m

^{2}/s) is the hindered diffusivity. Hence, the following equation is derived for the hindered mass transfer coefficient:

The hindered diffusivity (

${D}_{\mathrm{i}}^{\ast}$) in Equation (12) is related to bulk diffusivity (

${D}_{i}$), tortuosity (

ζ), and cake porosity (

${\epsilon}_{\mathrm{c}}$) as

${D}_{\mathrm{i}}^{\ast}={D}_{i}{\epsilon}_{\mathrm{c}}{\varsigma}^{-1}$. Tortuosity in the present work is also related to porosity by

$\varsigma =1-\mathrm{ln}{\epsilon}_{\mathrm{c}}^{2}$ [

17,

24]. Hence, Equations (9) and (10) are both related to cake porosity (or cake volume fraction) which is calculated by setting these two equations equal [

23]. After finding the cake porosity, CEOP is calculated using either of these equations.

The pressure drops are non-dimensionalized by dividing them by the applied transmembrane pressure (

$\Delta {P}_{t}$). Hence, the summation of non-dimensional trans-cake pressure (

$\Delta {P}_{\mathrm{c}}^{\ast}$), trans-membrane pressure (

$\Delta {P}_{\mathrm{m}}^{\ast}$), and CEOP (

$\Delta {\pi}_{\mathrm{m}}^{\ast}$) can be expressed as follows: