A combined network model for membrane fouling

doi:10.1016/j.jcis.2014.06.021

Journal of Colloid and Interface Science

Volume 432, 15 October 2014, Pages 10-18

https://doi.org/10.1016/j.jcis.2014.06.021 Get rights and content

Highlights

•
Membrane fouling leads to a decline in the flux with time.
•
The rate of flux decline also commonly decreases with time.
•
However, in many experiments the rate of flux decline can increase with time.
•
This is observed only when multiple fouling mechanisms interact with one another.
•
A network model accounts for these interactions to capture the system behaviour.

Abstract

Membrane fouling during particle filtration occurs through a variety of mechanisms, including internal pore clogging by contaminants, coverage of pore entrances, and deposition on the membrane surface. Each of these fouling mechanisms results in a decline in the observed flow rate over time, and the decrease in filtration efficiency can be characterized by a unique signature formed by plotting the volumetric flux, $\hat{Q}$ , as a function of the total volume of fluid processed, $\hat{V}$ . When membrane fouling takes place via any one of these mechanisms independently the $\hat{Q} \hat{V}$ signature is always convex downwards for filtration under a constant transmembrane pressure. However, in many such filtration scenarios, the fouling mechanisms are inherently coupled and the resulting signature is more difficult to interpret. For instance, blocking of a pore entrance will be exacerbated by the internal clogging of a pore, while the deposition of a layer of contaminants is more likely once the pores have been covered by particulates. As a result, the experimentally observed $\hat{Q} \hat{V}$ signature can vary dramatically from the canonical convex-downwards graph, revealing features that are not captured by existing continuum models. In a range of industrially relevant cases we observe a concave-downwards $\hat{Q} \hat{V}$ signature, indicative of a fouling rate that becomes more severe with time. We derive a network model for membrane fouling that accounts for the inter-relation between fouling mechanisms and demonstrate the impact on the $\hat{Q} \hat{V}$ signature. Our formulation recovers the behaviour of existing models when the mechanisms are treated independently, but also elucidates the concave-downward $\hat{Q} \hat{V}$ signature for multiple interactive fouling mechanisms. The resulting model enables post-experiment analysis to identify the dominant fouling modality at each stage, and is able to provide insight into selecting appropriate operating regimes.

Graphical abstract

Introduction

Understanding membrane fouling is a key goal in separation science, and is an area in which detailed mathematical modelling can provide key insight for membrane design optimization. Membrane fouling is a complex process due to interaction between membrane properties, solution composition, and process conditions [1]. Historically, in a typical filtration set-up there are four key membrane fouling mechanisms:

(i)
Standard blocking – small particles pass into the membrane pores and a finite number adhere to the walls causing pore constriction.
(ii)
Intermediate blocking – larger particles land on the membrane surface and partially cover a pore.
(iii)
Complete blocking – larger particles land on the membrane surface and cover a pore entirely.
(iv)
Caking – a layer of particles builds up on the membrane surface, which provides a resistance in the form of an additional porous medium through which the feed must also permeate [2].

There are various ways in which fouling behaviour may be characterized. One approach is direct monitoring of foulant deposition on the surface of a membrane, which may be achieved through a variety of techniques, including microscopy, laser sensoring, reflectometry, spectroscopy, and 3D imaging [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. A comprehensive review of various experimental techniques, capabilities and sensitivity specific to membrane protein fouling is provided in [13]. Other more recent approaches include computational fluid mechanics [14], genetic programming (GP) models [15], artificial neural network (ANN) tools for modelling highly complex and nonlinear membrane systems [16], diffusion-limited-aggregation simulation techniques [17], and multiscale modelling of protein fouling in ultrafiltration [18].

While direct measurement is a useful tool in understanding membrane fouling, a more common and widely accepted approach involves indirect measurement. For example, examining the temporal variations in volumetric flux, defined as volume per unit membrane area per unit time, defined here as $\hat{Q}$ , provides key insight. When filtering at constant transmembrane pressure the flux through the membrane will reduce with time as a result of fouling (Fig. 1a). A second indirect measurement that is often made is the total throughput, defined as the volume processed per unit membrane area, $\hat{V}$ , given by $\hat{V} (\hat{t}) = \int_{0}^{\hat{t}} \hat{Q} (s) d s,$ where $\hat{t}$ denotes time. As a result of fouling we generally find that $\hat{V}$ approaches a constant value as $\hat{t} \to \infty$ when a constant transmembrane pressure is applied (Fig. 1b). There are several articles confirming that the mode of fouling changes during the course of membrane plugging (see, for example, [19], [20]) with the initial stages dominated by pore blocking, followed by complete blocking, while in later stages cake formation is predominantly responsible for the continued reduction in flux.

Mathematical modelling of membrane fouling dates back to 1935 [21], focusing initially on models for individual fouling mechanisms. It emerges that these mechanisms can be expressed by $\frac{d^{2} \hat{t}}{d {\hat{V}}^{2}} = c {(\frac{d \hat{t}}{d \hat{V}})}^{n},$ where c is the generalized filtration constant and n may take the values 2, 3/2, 1 or 0, which correspond respectively to complete blocking, standard pore blocking, intermediate pore blocking, and cake filtration. This model effectively describes distinct fouling mechanisms in a definite period of filtration [22], [23], [24], [25]. However, Iritani et al. [26] reported that these individual models were unable to fit their experimental observations and indicated that a combination of effects would probably lead to better results.

Two-stage models that combine two individual fouling mechanisms sequentially are able to improve on the agreement with experiment. For example, a combination of pore blocking followed by cake-layer build-up was shown to describe the fouling of track-etched membranes by BSA [27] and proteins [28]. This approach can be generalized to other combinations of two fouling mechanisms [29] and has been further extended to capture three sequential fouling mechanisms, such as pore constriction followed by pore blocking and finally transitioning to caking [30].

In all of these cases, the indirect measurement of flux and throughput with time provide information on the overall fouling process. However, they are unable to reveal information on the dominant active fouling mechanism at any given time. Combining both of these visualizations in a graph of $\hat{Q}$ versus $\hat{V}$ provides a much richer understanding of the system behaviour, from which the resulting $\hat{Q} \hat{V}$ signature allows the dominant mode of fouling at any instant to be inferred.

Generally, when the feed is such that fouling takes place via an individual blocking mechanism such as complete blocking or caking, the $\hat{Q} \hat{V}$ signature is convex-downwards, that is, ${\hat{Q}}^{'} (\hat{V}) < 0$ and ${\hat{Q}}^{″} (\hat{V}) > 0$ , where primes denote differentiation (see solid curve in Fig. 2a). In this case, the graph depicts a membrane fouling modality where the rate of flux decline reduces as more fluid volume is processed. For example, the reduction of flux of fluid due to fouling reduces the rate of arrival of foulants (or particles), and hence reduces the subsequent rate of fouling.

However, recent experiments have revealed the possibility of an alternative, concave downwards $\hat{Q} \hat{V}$ signature, i.e., ${\hat{Q}}^{'} (\hat{V}) < 0$ and ${\hat{Q}}^{″} (\hat{V}) < 0$ , as depicted by the dashed curve in Fig. 2a. Such a graph may be identified with a regime in which the rate of fouling increases with time, in contrast to the more conventional observations. One scenario that may give rise to such a fouling behaviour is where contaminants are able to adsorb and desorb to the internal pore structure in such a way that the reduction in pore radius leads to an exacerbated flux decline (see, for example, Fig. 2b and [31], [32]). However, experiments suggest that such a fouling behaviour may also be observed in the absence of any such adsorption mechanisms (see, for example, [19], [29], [30], [33]). Furthermore, it is clear that such fouling behaviour is not constrained to the filtration of specific contaminants by a specific membrane, since significantly changing both the membrane type and contaminants within the feed still reveals qualitatively similar results (Fig. 2c and d).

In these cases, it is the transition between multiple fouling mechanisms during the filtration process that is thought to be responsible for the observed flux decline. Since current theories assume either that each fouling mechanism operates at a uniform rate, simultaneously and independently of the others, or that fouling mechanisms are sequential in operation, these are unable to reproduce the concave-downward shapes observed experimentally through the effects of standard, intermediate, complete blocking, and caking alone. To capture such behaviour and ultimately replicate the experimentally observed graphs requires a mathematical model that caters for the inter-relation between each fouling mechanism.

In this paper we develop for the first time the foundation of a discrete network model that allows multiple inter-related blocking mechanisms to be active at any given time, including fouling caused by particulate adsorption as well as size-based pore-plugging events. Specifically, the model caters for the interplay between membrane fouling effects, which may change with system state and hence time. For instance, particle adhesion within pores may lead to a reduction in pore size that eventually enables complete pore blocking. The fouling behaviour may be characterized through a series of parameters and we explore in Section 3 the specific effect that each of these has on the resulting $\hat{Q} \hat{V}$ signature. We explore the required model ingredients that lead to the physical observations exhibited in Fig. 2. The results are used in Section 4 to interpret how the dominant fouling mechanism varies during a given experiment, and enables us to determine the regimes in which continuum models appropriately describe the fouling behaviour, and those in which our new discrete modelling framework is essential. We conclude by generalizing our model to include more complex effects such as the ability to partially cover a pore or to land on the membrane material, and we demonstrate how this may be simply built onto our framework and the implications such features have on the physical observations.

Section snippets

Mathematical modelling

We consider a membrane, of uniform initial thickness $\hat{h}$ , that comprises a two-dimensional m × n array of regularly spaced uniform pores each of radii ${\hat{r}}_{0}$ , as illustrated in Fig. 3. The theory we present in this paper may easily be generalized to irregular membranes however here we choose to model a spatially homogeneous membrane to elucidate an accurate characterization of the fouling process. Such regularity is exhibited, for example, in track-etched membranes [34].

The flow rate through pore $(i, j$

Standard blocking

If a particle of size a arrives at pore $(i, j)$ and $a < r_{i, j}$ then the particle enters into the internal pore structure. Once within the pore we allow for a finite probability of particle adhesion to the pore wall, $p_{a}$ . This probability will in practice be a complex function that depends on many features, such as membrane and contaminant composition and geometry. Here we suppose that the probability of adhesion within the pore, $p_{a}$ obeys the following law: $p_{a} = p_{a 0} (1 - e^{- A / k}),$ where $p_{a 0}$ is a constant, $A = 2 π rh$

Results

We first present results for a regime in which all four fouling mechanisms introduced in Section 3 are possible. In this case we require the feed to be composed of contaminants with radius $a < 1$ , so that their size is smaller than that of the membrane pores and the particles may enter the pores. We also set $Q^{*} \neq 0$ so that the effect of caking is present. We make two observations in this case. Firstly, the $QV$ signature is able to distinguish each dominant active blocking mechanism via the concavity

Conclusions

We have presented for the first time a discrete network model that is able to capture the inherent inter-relation among the various mechanisms via which a membrane may foul. In particular, the model captures pore constriction via the entry of particles into the pores and adhesion onto the pore walls, which allows entry of a finite number of particles before the pore becomes completely blocked. Following this, a layer of particles will be deposited on the surface that contribute further to

Acknowledgments

The authors gratefully acknowledge helpful discussions from the Oxford–Pall 2013 In-House Study Group participants.

References (35)

C.Y. Tang et al.
Adv. Colloid Interface Sci.
(2011)
J.C. Chen et al.
Adv. Colloid Interface Sci.
(2004)
H. Li et al.
J. Membr. Sci.
(1998)
J. Altmann et al.
J. Membr. Sci.
(1997)
M. Hamachi et al.
Chem. Eng. Sci.
(1999)
A.P. Mairal et al.
J. Membr. Sci.
(1999)
L.-H. Cheng et al.
J. Membr. Sci.
(2011)
T. Chilcott et al.
J. Membr. Sci.
(2002)
L. Gaedt et al.
J. Membr. Sci.
(2002)
C.M. Bondy et al.
J. Membr. Sci.
(2010)

M. Ferrando et al.

J. Membr. Sci.

(2005)

R. Chan et al.

J. Membr. Sci.

(2004)

M. Rahimi et al.

Chem. Eng. Process.: Process Intensif.

(2009)

A. Fouladitajar et al.

Desalination

(2013)

Q.-F. Liu et al.

J. Membr. Sci.

(2008)

M. Wessling

Sep. Purif. Technol.

(2001)

G. De Luca et al.

J. Membr. Sci.

(2014)

Cited by (59)

Comparing operating strategies for unsteady crossflow membrane filtration with pore blocking
2024, Chemical Engineering Science
Crossflow filtration with pore blocking is an unsteady system affected by operating conditions. We model pore blocking and solve the governing partial differential equations using the method of lines. The loss in permeability is neither spatially nor temporally uniform. Operating at constant flux results in a greater permeate of clean water but at a higher operating cost than operating at constant pressure.
Forecasting multicycle hollow fiber ultrafiltration fouling using time series analysis
2023, Journal of Water Process Engineering
This study proposes the application of time series analysis and exponential smoothing (ETS) to predict multicycle membrane fouling with high accuracy and great interpretability. First, lab-scale filtration tests were performed with foulant surrogates to assess the potential of ETS models in predicting multicycle membrane fouling. Two models, ETS (A,A,A) and ETS (A,A,M), were found to be most suitable for describing the variation of transmembrane pressure. The ETS (A,A,A) model showed a slight edge over the ETS (A,A,M) model when reversible fouling was dominant. In contrast, the ETS (A,A,M) performed better when irreversible fouling was prevalent. Then, the effectiveness of ETS models in multicycle forecasting during the filtration of natural water resources was evaluated. Residual mean squared errors between 0.02 bar and 0.15 bar were obtained for horizons going from 2 up to 10 filtration cycles. Finally, ETS models were applied to predict the variation in permeability of drinking water treatment plant membrane skid. A three-month dataset was utilized to predict the next 41 days of permeability variation, yielding an average error of 3.2 %. Along with that, a web application designed to assist membrane users in utilizing ETS was developed and presented at the end. The web application can be accessed through the following link: https://fouling-ets.shinyapps.io/shiny-ets-forecasting/.
Connected mechanistic process modeling to predict a commercial biopharmaceutical downstream process
2023, Computers and Chemical Engineering
Mechanistic modeling has shown to contribute greatly to the process understanding of chromatography and filtration processes. However, these are mostly considered individually and not connected for an entire downstream process. In this study, mechanistic models were connected to describe an entire downstream process of a Fab fragment. For the capture step, a transport-dispersion model (TDM) combined with an extended Langmuir isotherm was applied. Depth filtration was modeled with a combined pore blocking model. The polishing ion exchange chromatography steps were described by a TDM combined with the colloidal particle adsorption model. The tangential flow filtration model accounts for both the Donnan effects and flow limitations. The presented downstream process model could predict online and offline data recorded at 12,000 L manufacturing scale. Process variations of 23 manufacturing batches were adequately reproduced by the model based on the consideration of input process parameter variations.
A review on state-of-the-art applications of data-driven methods in desalination systems
2022, Desalination
The substitution of conventional mathematical models with fast and accurate modeling tools can result in the further development of desalination technologies and tackling the need for freshwater. Due to the great capability of data-driven methods in analyzing complex systems, several attempts have been made to study various desalination systems using data-driven approaches. In this state-of-the-art review, the application of various artificial intelligence and design of experiment data-driven methods for analyzing different desalination technologies have been thoroughly investigated. According to the applications of data-driven methods in the field of desalination, the reviewed investigations are classified into five categories namely performance prediction using operational parameters, performance prediction using design parameters, optimization and correlation development, maintenance, and control of desalination systems. For each category, valuable information about the data-driven methods such as inputs, outputs, hyper-parameter tuning methods, and size of datasets have been provided and the main remarks are reported. The findings showed that data-driven methods can play a vital role in each aforementioned application for both thermal and membrane-based desalination technologies. Eventually, the research gaps are highlighted and a roadmap is also provided for future data-driven analysis of various desalination systems and their further advancement.
Artificial intelligence-incorporated membrane fouling prediction for membrane-based processes in the past 20 years: A critical review
2022, Water Research
Citation Excerpt :
A prediction of membrane fouling is of great significance for the selection of pretreatment measures or the development of antifouling techniques (Schmitt and Do 2017; Shim et al., 2021), which has attracted increasing interests in the past decades (Bagheri et al., 2019). Mechanistic models based on membrane flux, transmembrane pressure (TMP), pollutant (salt) rejection, and other operating parameters have been developed to understand and predict membrane fouling behaviours in membrane-based processes for water and wastewater treatment (Griffiths et al., 2014; Pimentel et al., 2015; Zuthi et al., 2017). However, due to the complexity of membrane fouling processes, it is rather challenging to accurately predict the occurrence or evolution of membrane fouling using these classical mathematical models (Nandi et al., 2010; Wu et al., 2021; Zhao et al., 2020a).
Membrane fouling is one of major obstacles in the application of membrane technologies. Accurately predicting or simulating membrane fouling behaviours is of great significance to elucidate the fouling mechanisms and develop effective measures to control fouling. Although mechanistic/mathematical models have been widely used for predicting membrane fouling, they still suffer from low accuracy and poor sensitivity. To overcome the limitations of conventional mathematical models, artificial intelligence (AI)-based techniques have been proposed as powerful approaches to predict membrane filtration performance and fouling behaviour. This work aims to present a state-of-the-art review on the advances in AI algorithms (e.g., artificial neural networks, fuzzy logic, genetic programming, support vector machines and search algorithms) for prediction of membrane fouling. The working principles of different AI techniques and their applications for prediction of membrane fouling in different membrane-based processes are discussed in detail. Furthermore, comparisons of the inputs, outputs, and accuracy of different AI approaches for membrane fouling prediction have been conducted based on the literature database. Future research efforts are further highlighted for AI-based techniques aiming for a more accurate prediction of membrane fouling and the optimization of the operation in membrane-based processes.
A hybrid discrete–continuum framework for modelling filtration
2022, Journal of Membrane Science
Typical mathematical frameworks for modelling the blocking behaviour of a filter due to particle deposition fall into one of two categories: a continuum approximation, whereby particle deposition is assumed to occur in such a way that all pores in the material are in the same state of blocking at any given time; or a discrete model, where blocking is treated as individual events in both space and time. While the former is computationally inexpensive, the latter allows for variation from pore to pore. This pore-to-pore variation has been shown to provide a qualitative change in the observed filtration behaviour that is essential to reproduce experimental observations. We present a hybrid model that describes the location of particle depositions in a continuum manner while retaining a discrete, stochastic component to capture the time at which a blocking event occurs. The model is able to grade between the aforementioned extreme continuum and discrete cases through a parameter that controls the spatial extent of a blocking event. This enables us to uncover the way in which the nature of the blocking process changes between these two pre-existing models. The model also captures the key ingredients of a fully discrete stochastic model at a fraction of the computational cost, making it ready to use to describe other complex filtration scenarios.

View all citing articles on Scopus

View full text

A combined network model for membrane fouling

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Mathematical modelling

Standard blocking

Results

Conclusions

Acknowledgments

Adv. Colloid Interface Sci.

Adv. Colloid Interface Sci.

J. Membr. Sci.

J. Membr. Sci.

Chem. Eng. Sci.

J. Membr. Sci.

J. Membr. Sci.

J. Membr. Sci.

J. Membr. Sci.

J. Membr. Sci.

J. Membr. Sci.

J. Membr. Sci.

Chem. Eng. Process.: Process Intensif.

Desalination

J. Membr. Sci.

Sep. Purif. Technol.

J. Membr. Sci.