Mathematical modelling of the membrane fouling process can assist the engineer in improving the membrane operation and performance,
i.e., better prediction of the fouling rate and optimisation of the corresponding retaliatory actions needed to reduce it. For a MBR system treating wastewater, capturing membrane fouling phenomena in the form of mathematical models has been a task of many different global research teams over the past two decades. Most researchers model the membrane fouling process using a phenomenological mechanistic approach. In Le-Clech
et al. [
1], it was suggested that a “Three-Stage-Fouling” model should be considered for the total membrane fouling and clogging process for the entire flux range from sub-critical up to supra-critical fluxes. Based on experimental observation, the three stages of fouling are initial passive conditioning; slow and steady fouling (usually at sub-critical fluxes); and a rapid trans-membrane pressure (TMP) jump (usually at supra-critical fluxes, although not exclusively).
Under normal plant operation, supra-critical fluxes should not usually occur for the plant as a whole, although they will occur at a localised level for individual membrane modules or portions of modules due to localised uneven membrane blocking and clogging events. So any useful fouling modelling should be capable of catering for this. Cho and Fane’s area loss model [
2] and the subsequent Ye
et al. pore loss model [
3] both consider that the sub-critical flux is exceeded at a local or regional level for certain portions of the membrane. These localised supra-critical fluxes then further reduce the area/pores contributing to sub-critical fluxes, and when they reach a critical tipping point, the dramatic TMP increase occurs. Chang
et al. [
4] developed an alternative fouling model based upon the critical suction pressure, where the model describes the sudden collapse of the cake layer under a critical TMP. This cake material can be highly compressible depending on the hydrodynamic regime employed and the cake composition (
i.e., floc size and structure). This sudden cake compression causes a rapid build up in resistance in a very short space of time, and would explain the TMP jump experienced in practice. Hermanowicz [
5] suggested and modelled an alternative clogging and fouling mechanism based on percolation theory in which the developed model uses the critical loss of diffusivity of the cake layer to explain the TMP jump stage. In this model, the cake layer is taken as being highly porous with smaller particles, aggregates and colloidal material travelling towards the concentration polarisation zone by natural convective transport, and taking up the void spaces in the cake layer which would be currently taken up by fluid. Hermanowicz [
5] theorised that when this mechanism reaches a critical point, the cake resistance increases dramatically, causing the apparent TMP jump.
Over that time many approaches of various complexities and usefulness have been adopted to explain foulability [
6], with some research groups developing models that describe the entire MBR plant configuration and operation [
7]. For instance Broeckmann
et al. [
8] developed a fouling model for a MBR system that also considered pore size and floc size distributions which are an important aspect in contributing to overall fouling potential. Busch
et al. [
9] developed a fully comprehensive model which consisted of several sub-models that modelled different aspects of the MBR process. These models could then account, for example, for membrane configuration and geometry (e.g., hollow fibre, flat sheet, multi-tubular,
etc.), the hydrodynamics of the feed flow, the hydrodynamics of the permeate flow, the cleaning mechanisms employed (e.g., backflushing, air sparging, relaxation,
etc.), and the various filtration resistance components. Gehlert
et al. [
10] created a rigorous mathematical model of an open channel cassette module for use in a MBR. In contrast, Zarragoitia-Gonzalez
et al. [
11] created a mathematical model to simulate the filtration process, and aeration influence on a submerged MBR operated under aerobic conditions. In contrast Lee
et al. [
12] developed a very simple cake build up model which just considered the mixed liquor suspended solids (MLSS) in very simple terms. Similarly Wintgens
et al. [
13] produced an easy-to-use cake build up and pore blocking model based upon a simple resistance-in-series expression, which however didn’t reflect reality that well as it depicted total membrane resistance as reaching a saturation value at supra-critical fluxes when in fact in practice it is an exponential increase when the TMP jump occurs. This does not mean to say that a simple but relevant model can never be developed, since Ognier
et al. [
14] did just this by introducing the concept of local critical flux to explain the TMP jump conditions experienced at long filtration periods when no cleaning-in-place (CiP) procedures are applied. Ye
et al. [
15] developed a very similar but slightly more complex model for predicting TMP evolution under long term sub-critical filtration of synthetic extra-polymeric substance (EPS) solutions. In an alternative vein, Psoch and Schiewer [
16] constructed a model based on pore constriction resistance, cake resistance and clean membrane resistance with air sparging and backflushing mechanisms also included.
Several classical fouling studies [
17] use a three mechanism model for the bio-fouling process made up of pore constriction, pore blockage, and cake filtration. These mechanisms can be directly related to the main bio-fouling processes observed in a MBR system. This set of models uses the classical blocking laws developed by Hermia [
17] as the start point for developing a model description. Ho and Zydney [
18] developed the first real combined fouling model which accounted for the classical complete pore blockage equation, intermediate pore blockage equation and cake filtration mechanisms. The model was validated and was in good agreement with flux decline data obtained from bovin serum albumen (BSA) filtration experiments. This model was further extended by the extensive and comprehensive work of Duclos-Orsello
et al. [
19] in which an internal pore constriction mechanism was modelled as a reduction of pore diameters by the Hagen-Poiseuille Law.