A combined network model for membrane fouling
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 , 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, , given bywhere denotes time. As a result of fouling we generally find that approaches a constant value as 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 bywhere 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 versus provides a much richer understanding of the system behaviour, from which the resulting 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 signature is convex-downwards, that is, and , 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 signature, i.e., and , 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 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 , that comprises a two-dimensional m × n array of regularly spaced uniform pores each of radii , 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
Standard blocking
If a particle of size a arrives at pore and 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, . 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, obeys the following law:where is a constant,
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 , so that their size is smaller than that of the membrane pores and the particles may enter the pores. We also set so that the effect of caking is present. We make two observations in this case. Firstly, the 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.
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