2.1 Modelling assumptions: outline

The geometry of a cylindrical pleated membrane filter cartridge is sketched in figures 1 and 2 and described in the Introduction. Figure 2(b) also introduces the coordinates that will be used in this paper: the $Z$ -direction is along the cylinder’s axis, the $X$ -direction is radially inwards measured from the cartridge outer wall, and $Y$ is arc length around the outer cylinder boundary, measured as indicated. We idealize the rather complicated flow scenario depicted in figure 2 in several ways. We assume there is no variation in the $Z$ -direction, and that all pleats are identical. This justifies our considering flow confined to a cross-section at constant $Z$ , within a single pleat, which we assume to be part of a periodic array (periodic in arc length $Y$ ). We simplify further by neglecting the curvature of the cylindrical cartridge, considering instead one section of a linear periodic array in rectangular Cartesian $(X,Y)$ -coordinates. We restrict our attention to the case of tightly packed pleats, as shown in figure 3(a). In this situation, the length $L$ of the pleat (from outer to inner cartridge boundary; $X$ -direction in figures 2 b and 3 b) is much greater than the pleat thickness (the thickness of support layers plus membrane in the $Y$ -direction), so that the vast majority of the flow through the pleat is expected to pass through the pleat length rather than its ends. This observation suggests neglecting the flow through the ends of the pleats (the pleat tips and valleys) as being negligible relative to the flow through the straight section of the membrane (the section parallel to the $X$ -axis). With this in mind, we make our final simplification, idealizing the pleat geometry to be rectangular and imposing no-flux conditions at pleat tips and valleys where shown in figure 3(b). This simplification is justified further: (i) by noting that the high membrane curvature in the pleat tips and valleys is likely to lead locally to very low membrane permeability and high resistance to flow (particularly on the inside of the tight curve); and (ii) by limited experimental data (Fotovati et al. Reference Fotovati, Hosseini, Vahedi Tafreshi and Pourdeyhimi2011) on filters subjected to dust-laden air and then analysed, which indicate little or no dust particle deposition at the actual fold locations.

The sketches in figures 3(b) and 3(c) summarize the simplified flow problem to be solved. Figure 3(b) clarifies how the pleat geometry of 3(a) is idealized (the same colour coding is used to distinguish between inflow and outflow sides of the membrane), while figure 3(c) shows the solution domain, with the boundary conditions to be applied on the pressure. The flow region considered is from the lower to the upper periodicity boundary (the dashed symmetry lines in figure 3 b): $-H\leqslant Y\leqslant H$ , and from $X=0$ to $X=L$ along the membrane, with the membrane itself occupying region $-D/2\leqslant Y\leqslant D/2$ , $0\leqslant X\leqslant L$ (hatched region in figure 3 c). Our small aspect ratio assumptions, to be discussed further below, are represented by the dimensionless parameters ${\it\epsilon}=H/L\ll 1$ , ${\it\delta}=D/H\ll 1$ . Figures 3(b) and 3(c) also indicate the inflow and outflow, the no-flux boundaries at the pleat valley ( $X=L$ , $0\leqslant Y\leqslant H$ ) and tip ( $X=0$ , $-H\leqslant Y\leqslant 0$ ), and the symmetry conditions at the support layer mid-surfaces ( $Y=\pm H$ ). In general throughout this paper we use upper-case characters to denote dimensional variables, while the lower-case equivalent will be dimensionless.

Table 1.

Approximate dimensional parameter values (A. Kumar, 2014, Private communication).

The tight packing means that the whole flow domain considered is occupied by porous medium (support layer or membrane), within which Darcy flow of an incompressible Newtonian feed solution, viscosity ${\it\mu}$ , is assumed, with velocity $\boldsymbol{U}=(U,V)$ and pressure $P$ . We assume the feed solution to be a dilute suspension of particles, which are advected passively through the support layers. The permeabilities of support layers and membrane are $K$ , $K_{m}$ , respectively, with $K_{m}/K\ll 1$ in accordance with data for real filter cartridges (see table 1). We allow support layer permeability $K$ to vary along the pleat, but assume it is symmetric above and below the membrane. For the most part we assume that $K$ varies only in the coordinate $X$ along the pleat, $K(X)$ ; the case of permeability that can vary also in the $Y$ -direction, $K(X,Y)$ , is discussed briefly in § 2.3.1. (Note that, due to the annular configuration of the cartridge, the valley ends of pleats will be more compressed than the tip ends, leading to lower permeability at the valleys; see figures 1 and 2.) In general $K_{m}$ will vary in both space and time as membrane fouling occurs, with the variation in time being quasi-static: $K_{m}(X,T)$ . (We do not account for any fouling of the support layers, which are not designed to capture particles: the pores of these layers are very much larger than those of the membrane filter.) The time evolution of the membrane permeability similarly induces time variation into the solution for the pressure and fluid velocity within the pleat. However, since no explicit time-dependence appears in the Darcy flow model, we will mostly suppress the time dependence to simplify notation, writing $K_{m}(X)$ , $P(X,Y)$ , etc. This (quasi-static) assumption that only the dynamics of membrane fouling need explicit consideration amounts to an assumption (borne out by data) that fouling occurs on a time scale long compared with that of fluid transit time across the pleated cartridge.

To arrive at a tractable fouling model we consider a membrane composed of an identical array of uniformly distributed cylindrical pores of radius $A(T)$ . Within our model, membrane resistance is assumed to increase in time due to fouling by two mechanisms: (i) $A(T)$ decreases in time due to adsorption of tiny particles within the pores; and (ii) pores become blocked from above by particles too large to pass through pores. In order to model (ii) we monitor $N(X,T)$ , the number of unblocked pores per unit area of membrane. Again, we will mostly suppress the time-dependence here to simplify notation, writing just $A$ and $N(X)$ . Membrane permeability will be expressed as a function of both $A$ and $N$ .

2.2 Governing equations

The feed is assumed to be a dilute suspension of particles, which do not affect the fluid dynamics directly (though they do have an indirect effect via the fouling, which results in increased system resistance). We therefore use a single-phase model, in which Darcy velocity $\boldsymbol{U}=(U,V)$ within the support layers is given in terms of the pressure $P(X,Y)$ by

(2.1) $$\begin{eqnarray}\displaystyle \boldsymbol{U}=(U,V)=-\frac{K}{{\it\mu}}\boldsymbol{{\rm\nabla}}P,\quad \boldsymbol{{\rm\nabla}}=(\partial _{X},\partial _{Y}). & & \displaystyle\end{eqnarray}$$

Incompressibility of the feed solution requires

(2.2) $$\begin{eqnarray}\displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}=0\quad \Rightarrow \quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(K\boldsymbol{{\rm\nabla}}P)=0, & & \displaystyle\end{eqnarray}$$

within the support layers. As discussed in § 2.1 above, we assume that the flow is driven by an imposed pressure difference, $P_{0}$ , across the pleated membrane and that there is no flux through the pleat valley on the inflow side and the pleat tip on the outflow side, and we impose symmetry across the support layer centrelines $Y=\pm H$ (see figure 3). Hence, we impose boundary conditions

(2.3a-c ) $$\begin{eqnarray}\displaystyle P^{+}(0,Y)=P_{0},\quad P_{X}^{+}(L,Y)=0,\quad P_{Y}^{+}(X,H)=0, & & \displaystyle\end{eqnarray}$$

(2.4a-c ) $$\begin{eqnarray}\displaystyle P_{X}^{-}(0,Y)=0,\quad P^{-}(L,Y)=0,\quad P_{Y}^{-}(X,-H)=0, & & \displaystyle\end{eqnarray}$$

where we use $\pm$ superscripts to distinguish between quantities evaluated in $Y\gtrless 0$ respectively, on either side of the membrane. For most of our simulations we take $P_{0}$ to be a specified constant, reflecting flow driven by a fixed pressure drop between inlet and outlet; but we will also present some results for fixed-flux scenarios, where $P_{0}$ increases in time in order to maintain the same flux as the system resistance increases due to fouling (see § 2.3.4).

Similar to (2.1) we also assume Darcy flow across the membrane, which is itself a porous medium, though much less permeable than the support layers. We are primarily concerned with the flux, $V_{m}$ , per unit area across the membrane in the $Y$ -direction. Anticipating in advance the fact that the pressure within the membrane will be independent of the coordinate $Y$ perpendicular to the membrane (due to the small aspect ratio $D/L={\it\epsilon}{\it\delta}\ll 1$ ), and given that the pressure is continuous across the interface between the membrane and support layers, the vertical pressure gradient within the membrane may be written, correct to leading order in ${\it\epsilon}{\it\delta}$ , as $(P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2})/D$ . The Darcy law for the flow through the membrane then gives

(2.5) $$\begin{eqnarray}\displaystyle |V_{m}|=\frac{K_{m}}{{\it\mu}D}[P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2}],\quad 0\leqslant X\leqslant L, & & \displaystyle\end{eqnarray}$$

where, by continuity of flux, $|V_{m}(X)|=|V(X,D/2)|=|V(X,-D/2)|$ (with $V$ as defined in (2.1)),

(2.6) $$\begin{eqnarray}\displaystyle |V_{m}|=\frac{K}{{\it\mu}}\frac{\partial P^{+}}{\partial Y}\bigg|_{Y=D/2}=\frac{K}{{\it\mu}}\frac{\partial P^{-}}{\partial Y}\bigg|_{Y=-D/2},\quad 0\leqslant X\leqslant L. & & \displaystyle\end{eqnarray}$$

The total flux through the membrane, $Q$ (per unit length along the axis of the cartridge), which will be an important performance characteristic in our simulations, is defined by

(2.7) $$\begin{eqnarray}\displaystyle Q=\int _{0}^{L}|V_{m}|\,\text{d}X=\int _{0}^{L}\frac{K}{{\it\mu}}\left.\frac{\partial P^{+}}{\partial Y}\right|_{Y=D/2}\,\text{d}X. & & \displaystyle\end{eqnarray}$$

Membrane permeability $K_{m}$ changes, over time scales long compared to the fluid transit time, due to pore shrinkage (arising from particle adsorption) and to blocking of pores by large particles. We now consider these fouling phenomena in more detail. We assume that membrane pores are long thin cylindrical tubes, of length $D$ and radius $A(T)$ , spanning the membrane, which initially all have the same radius, $A(0)=A_{0}$ . A more sophisticated model would allow for non-uniform shrinkage of pores due to the adsorption, but in our simple model we assume uniform adsorption, so that the pore radius does not vary spatially. Where an individual pore (at position $X$ and time $T$ ) is unblocked the total flux through it $Q_{u,pore}(X,T)$ is given (approximately) by the Hagen–Poiseuille formula

(2.8) $$\begin{eqnarray}\displaystyle Q_{u,pore}=\frac{1}{R_{u}}(P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2})\quad \text{where }R_{u}=\frac{8{\it\mu}D}{{\rm\pi}A^{4}}, & & \displaystyle\end{eqnarray}$$

and $R_{u}$ is the pore resistance. Blocking occurs when a large particle becomes trapped at the entrance to a pore, obstructing the flow. We model this effect by adding an extra resistance of magnitude $8{\it\mu}D{\it\rho}_{b}/({\rm\pi}A_{0}^{4})$ , where ${\it\rho}_{b}$ is a dimensionless number, in series with the Hagen–Poiseuille resistance $R_{u}$ . The flux through a blocked pore, $Q_{b,pore}(X,T)$ , is thus given by

(2.9) $$\begin{eqnarray}\displaystyle Q_{b,pore}=\frac{1}{R_{b}}(P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2})\quad \text{where }R_{b}=\frac{8{\it\mu}D}{{\rm\pi}A_{0}^{4}}\left(\left(\frac{A_{0}}{A}\right)^{4}+{\it\rho}_{b}\right). & & \displaystyle\end{eqnarray}$$

The dimensionless parameter ${\it\rho}_{b}$ characterizes blocking strength: for large values of ${\it\rho}_{b}$ pore resistance increases dramatically after blocking, while for small values resistance is almost unchanged. In the limit ${\it\rho}_{b}\rightarrow \infty$ our model captures the simplest blocking assumption: that deposition of a large particle over a pore blocks it completely.

We can now relate the number densities of unblocked and blocked pores, $N(X,T)$ and $N_{0}-N(X,T)$ respectively (where $N_{0}=N(X,0)$ ), to the membrane permeability $K_{m}$ by noting that the flux $|V_{m}|$ of fluid (per unit area of membrane) is

(2.10) $$\begin{eqnarray}\displaystyle |V_{m}|=N(X,T)Q_{u,pore}+(N_{0}-N(X,T))Q_{b,pore} & & \displaystyle\end{eqnarray}$$

so that, on substituting for $Q_{u,pore}$ from (2.8) and for $Q_{b,pore}$ from (2.9) in the above and comparing to (2.5), we obtain

(2.11) $$\begin{eqnarray}\displaystyle K_{m}=\frac{{\rm\pi}A_{0}^{4}}{8}\left(\frac{N}{(A_{0}/A)^{4}}+\frac{N_{0}-N}{(A_{0}/A)^{4}+{\it\rho}_{b}}\right). & & \displaystyle\end{eqnarray}$$

To complete the model we need equations describing the evolution of $N(X,T)$ , the local number density of unblocked pores, and $A(T)$ the pore radius. We assume a pore is blocked whenever a particle with radius $S>A(T)$ is advected to the pore entrance. If we assume a cumulative particle size distribution function $G(S)$ , giving the number of particles per unit volume of fluid with radius smaller than $S$ , then the concentration of particles of size $S>A(T)$ is $G_{\infty }-G(A)$ (where $G_{\infty }=\lim _{S\rightarrow \infty }G(S)$ is the total particle concentration). The probability that a particular pore is blocked (per unit time) is thus $(G_{\infty }-G(A))$ multipled by the flux through the pore, $Q_{u,pore}$ (given by (2.8)):

(2.12) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt}\left[\begin{array}{@{}c@{}}\text{Probability per unit time that}\\ \text{pore of radius }A\text{ is blocked}\end{array}\right]=\frac{{\rm\pi}A^{4}}{8{\it\mu}D}(G_{\infty }-G(A))(P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2}). & & \displaystyle\end{eqnarray}$$

Given that we assume unblocked pores all have radius $A(T)$ , the number of pores blocked per unit time, per unit area, is equal to $N(X,T)$ times the probability, per unit time, that a pore of radius $A(T)$ is blocked. It follows that the rate of change of the number density (per unit area) of unblocked pores is given by

(2.13) $$\begin{eqnarray}\displaystyle \frac{\partial N}{\partial T}=-N\frac{{\rm\pi}A^{4}}{8{\it\mu}D}(G_{\infty }-G(A))(P^{+}|_{Y=D/2}-P^{-}|_{Y=-D/2}). & & \displaystyle\end{eqnarray}$$

In order to describe fouling we make the simplest possible assumption, namely uniform adsorption within the pores, so that particle radius decreases uniformly according to

(2.14) $$\begin{eqnarray}\displaystyle \frac{\partial A}{\partial T}=-E,\quad A|_{T=0}=A_{0}, & & \displaystyle\end{eqnarray}$$

for some constant $E$ . Deposition within pores in reality will be controlled by a complex interplay between suspended particles in the feed, and the membrane material, the details of which will vary from one system to another; in the absence of detailed experimental data our model reflects an assumption that the rate of loss of pore area is proportional to the pore circumference only, other effects being largely the same from one pore to another.

All our simulations will be conducted with an exponential cumulative particle distribution of the form

(2.15) $$\begin{eqnarray}\displaystyle G(S)=G_{\infty }(1-\text{e}^{-BS}), & & \displaystyle\end{eqnarray}$$

where $B^{-1}$ is a characteristic particle size. This functional form was not chosen to fit to any specific dataset, but has the general features required. While some particle size distributions characterized in the literature (see e.g. Giglia & Straeffer Reference Giglia and Straeffer2012, with two distinct characteristic particle sizes) are more complicated, we recall that our model implicitly assumes two separate particle size distributions: the ‘macroscopic’ particles modelled by $G(S)$ with pore-blocking potential; and the ‘microscopic’ particles implicit in our pore adsorption model.

Table 2.

Approximate dimensionless parameter values.

2.3 Scaling, non-dimensionalization and asymptotics

Even with the simplifications introduced, the model above is complicated, and we therefore exploit asymptotic analysis based on the small aspect ratio of the pleat ( ${\it\epsilon}=H/L\ll 1$ ), and on the thinness of the membrane relative to the support layer ( ${\it\delta}=D/H\ll 1$ ); see table 2 below, which summarizes the model parameters and gives estimates, where available. For the most part we consider filtration driven by a fixed pressure drop, $P_{0}$ , between inlet and outlet, and this is the basis on which we non-dimensionalize below. We will also present some simulations for a prescribed flux scenario, for which the scalings are a little different; we comment briefly on this in § 2.3.4 below.

2.3.1 Fluid dynamics

In order to exploit the asymptotic simplifications, we introduce dimensionless variables as follows:

(2.16) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}r@{}}P^{\pm }(X,Y)=P_{0}p^{\pm }(x,y),\quad (X,Y)=(Lx,Hy),\\ K(X)=K_{av}k(x),\quad K_{m}(X)=K_{m0}k_{m}(x),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

where $K_{av}=(1/L)\int _{0}^{L}K(X)\,\text{d}X$ is the average support layer permeability and $K_{m0}$ is a typical initial membrane permeability. For definiteness, we can take it to be the initial membrane permeability in the expression (2.11), with $A=A_{0}$ and $N=N_{0}$ . For conciseness in the following we will indicate dependence on variables only where necessary, but it should be understood that all functions except the support layer permeability $k$ vary in both space and time. In the dimensionless coordinates our idealized problem for the pressure $p^{\pm }(x,y)$ within the support layers (2.2)–(2.4) becomes

(2.17) $$\begin{eqnarray}\displaystyle {\it\epsilon}^{2}(k(x)p_{x}^{+})_{x}+(k(x)p_{y}^{+})_{y}=0,\quad {\it\delta}/2\leqslant y\leqslant 1, & & \displaystyle\end{eqnarray}$$

(2.18a-c ) $$\begin{eqnarray}\displaystyle p^{+}(0,y)=1,\quad p_{x}^{+}(1,y)=0,\quad p_{y}^{+}(x,1)=0, & & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle {\it\epsilon}^{2}(k(x)p_{x}^{-})_{x}+(k(x)p_{y}^{-})_{y}=0,\quad -1\leqslant y\leqslant -{\it\delta}/2, & & \displaystyle\end{eqnarray}$$

(2.20a-c ) $$\begin{eqnarray}\displaystyle p_{x}^{-}(0,y)=0,\quad p^{-}(1,y)=0,\quad p_{y}^{-}(x,-1)=0. & & \displaystyle\end{eqnarray}$$

This system is closed by enforcing flux continuity across the membrane, (2.5) and (2.6), which gives

(2.21) $$\begin{eqnarray}\displaystyle p_{y}^{+}|_{y={\it\delta}/2}=p_{y}^{-}|_{y=-{\it\delta}/2}={\it\epsilon}^{2}{\it\Gamma}\frac{k_{m}(x)}{k(x)}[p^{+}|_{y={\it\delta}/2}-p^{-}|_{y=-{\it\delta}/2}], & & \displaystyle\end{eqnarray}$$

where the key dimensionless parameter ${\it\Gamma}$ is defined by

(2.22) $$\begin{eqnarray}\displaystyle {\it\Gamma}=\frac{K_{m0}L^{2}}{K_{av}HD} & & \displaystyle\end{eqnarray}$$

and gives a scaled measure of the relative importance of the resistance of the packing material to that of the membrane, such that if ${\it\Gamma}\gg 1$ the packing material provides most of the resistance whereas if ${\it\Gamma}\ll 1$ the membrane provides most of the resistance. We now seek asymptotic solutions for $p^{\pm }$ in the distinguished limit ${\it\Gamma}=O(1)$ , ${\it\epsilon}\ll 1$ and ${\it\delta}\ll 1$ (note that our solution is asymptotically valid for all ${\it\Gamma}\ll 1/{\it\epsilon}^{2}$ ) by expanding $p^{\pm }$ in powers of ${\it\epsilon}$ as follows:

(2.23a,b ) $$\begin{eqnarray}\displaystyle \hspace{-25.00003pt}p^{+}(x,y)=p_{0}^{+}(x)+{\it\epsilon}^{2}p_{1}^{+}(x,y)+\cdots \,,\quad p^{-}(x,y)=p_{0}^{-}(x)+{\it\epsilon}^{2}p_{1}^{-}(x,y)+\cdots \,. & & \displaystyle\end{eqnarray}$$

We will determine coupled equations for the as yet unknown functions $p_{0}^{\pm }(x)$ by seeking a solvability condition on the first-order solutions $p_{1}^{\pm }(x,y)$ . This is effected by substituting the expansions (2.23) into (2.17)–(2.20) and taking the $O({\it\epsilon}^{2})$ terms,

(2.24) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left(k(x)\frac{\partial p_{1}^{+}}{\partial y}\right)+\frac{\partial }{\partial x}\left(k(x)\frac{\partial p_{0}^{+}}{\partial x}\right)=0, & & \displaystyle\end{eqnarray}$$

(2.25a,b ) $$\begin{eqnarray}\displaystyle \frac{\partial p_{1}^{+}}{\partial y}\bigg|_{y=1}=0,\quad k(x)\frac{\partial p_{1}^{+}}{\partial y}\bigg|_{y=0^{+}}={\it\Gamma}k_{m}(x)(p_{0}^{+}-p_{0}^{-}), & & \displaystyle\end{eqnarray}$$

(2.26) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left(k(x)\frac{\partial p_{1}^{-}}{\partial y}\right)+\frac{\partial }{\partial x}\left(k(x)\frac{\partial p_{0}^{-}}{\partial x}\right)=0, & & \displaystyle\end{eqnarray}$$

(2.27a,b ) $$\begin{eqnarray}\displaystyle \frac{\partial p_{1}^{-}}{\partial y}\bigg|_{y=-1}=0,\quad k(x)\frac{\partial p_{1}^{-}}{\partial y}\bigg|_{y=0^{-}}={\it\Gamma}k_{m}(x)(p_{0}^{+}-p_{0}^{-}). & & \displaystyle\end{eqnarray}$$

Integrating (2.24) between $y=0$ and $y=1$ and applying the boundary conditions (2.25) leads to a solvability condition in the form of an ordinary differential equation (ODE) (in $x$ ) for $p_{0}^{+}(x)$ and $p_{0}^{-}(x)$ . Similarly, integration of (2.26) between $y=-1$ and $y=0$ and application of the boundary conditions (2.27) leads to a second ODE. Boundary conditions on these two ODEs come from the leading-order terms of (2.18) and (2.20). The resulting simplified model for $p_{0}^{+}$ and $p_{0}^{-}$ is

(2.28) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial x}\left(k(x)\frac{\partial p_{0}^{+}}{\partial x}\right)={\it\Gamma}k_{m}(x)(p_{0}^{+}-p_{0}^{-}), & & \displaystyle\end{eqnarray}$$

(2.29) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial x}\left(k(x)\frac{\partial p_{0}^{-}}{\partial x}\right)=-{\it\Gamma}k_{m}(x)(p_{0}^{+}-p_{0}^{-}), & & \displaystyle\end{eqnarray}$$

(2.30a,b ) $$\begin{eqnarray}\displaystyle p_{0}^{+}|_{x=0}=1,\quad \left.\frac{\partial p_{0}^{+}}{\partial x}\right|_{x=1}=0, & & \displaystyle\end{eqnarray}$$

(2.31a,b ) $$\begin{eqnarray}\displaystyle \left.\frac{\partial p_{0}^{-}}{\partial x}\right|_{x=0}=0,\quad p_{0}^{-}|_{x=1}=0. & & \displaystyle\end{eqnarray}$$

In addition $p_{1}^{+}$ and $p_{1}^{-}$ can be found by solving the differential equations (2.24) and (2.26) subject to boundary conditions (2.25) and (2.27) respectively, giving

(2.32) $$\begin{eqnarray}\displaystyle p_{1}^{+}(x,y)=-\frac{(k(x)p_{0x}^{+})_{x}}{k(x)}\left(\frac{y^{2}}{2}-y\right)+h^{+}(x), & & \displaystyle\end{eqnarray}$$

(2.33) $$\begin{eqnarray}\displaystyle p_{1}^{-}(x,y)=-\frac{(k(x)p_{0x}^{-})_{x}}{k(x)}\left(\frac{y^{2}}{2}+y\right)+h^{-}(x), & & \displaystyle\end{eqnarray}$$

for some functions $h^{\pm }(x)$ .

This model, at leading order, describes two porous medium flows (with pressures $p_{0}^{+}(x)$ and $p_{0}^{-}(x)$ ) separated by a membrane through which fluid is driven (from one side to the other) by the local pressure difference.

Generalization to support layer with $y$ -dependent permeability $k(x,y)$

It is straightforward to generalize this treatment to a support layer with permeability $k(x,y)$ . Dependence on $y$ could be introduced by (for example) choice of a support material with a layered structure. The result obtained is identical to (2.28)–(2.31) except that $k(x)$ is replaced by $\hat{k}^{+}(x)$ in (2.28) and $\hat{k}^{-}(x)$ in (2.29), where $\hat{k}^{+}(x)$ and $\hat{k}^{-}(x)$ are the $y$ -averages of the permeability, above and below the membrane, respectively,

(2.34a,b ) $$\begin{eqnarray}\displaystyle \hat{k}^{+}(x)=\int _{0}^{1}k(x,y)\,\text{d}y\quad \text{and}\quad \hat{k}^{-}(x)=\int _{-1}^{0}k(x,y)\,\text{d}y. & & \displaystyle\end{eqnarray}$$

When we come to suggest possible improvements to the pleated filter system in § 4 we will return to this generalized formulation.

The small- ${\it\Gamma}$ limit

In this limit the dominant resistance to flow is that of the membrane (as opposed to that of the porous support layers) and, to leading order in ${\it\Gamma}$ , the solution to (2.28)–(2.31) is just

(2.35a,b ) $$\begin{eqnarray}\displaystyle p_{0}^{+}=1\quad \text{and}\quad p_{0}^{-}=0. & & \displaystyle\end{eqnarray}$$

It is apparent that the situation here is identical to that of a flat membrane across which a constant pressure difference is applied. We would thus expect membrane fouling to occur uniformly along the length of the membrane, leading to optimal membrane performance (see appendix A and the simulations of § 3).

Method of solution

It is apparent from (2.28) and (2.29) that $(k(x)(p_{0}^{+}+p_{0}^{-})_{x})_{x}=0$ . This statement is readily integrated twice to obtain an expression for $p_{0}^{-}$ in terms of $p_{0}^{+}$ ,

(2.36) $$\begin{eqnarray}\displaystyle p_{0}^{-}(x)=-p_{0}^{+}(x)-c_{1}(t)\int _{0}^{x}\frac{\text{d}x^{\prime }}{k(x^{\prime })}-c_{2}(t), & & \displaystyle\end{eqnarray}$$

for some $c_{1}(t)$ and $c_{2}(t)$ (which are independent of $x$ , but will vary in time as fouling occurs). By substituting (2.36) into (2.28) we obtain a single equation for $p_{0}^{+}$ containing two arbitrary functions of time,

(2.37) $$\begin{eqnarray}\displaystyle (k(x)p_{0x}^{+}(x))_{x}-2{\it\Gamma}k_{m}(x)p_{0}^{+}(x)={\it\Gamma}k_{m}(x)\left(c_{1}(t)\int _{0}^{x}\frac{\text{d}x^{\prime }}{k(x^{\prime })}+c_{2}(t)\right), & & \displaystyle\end{eqnarray}$$

which must be solved subject to the four boundary conditions

(2.38) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle p_{0}^{+}|_{x=0}=1,\quad p_{0x}^{+}|_{x=0}=-\frac{c_{1}(t)}{k(0)},\\ \displaystyle p_{0}^{+}|_{x=1}=-\left(c_{1}(t)\int _{0}^{1}\frac{\text{d}x}{k(x)}+c_{2}(t)\right),\quad p_{0x}^{+}|_{x=1}=0.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Hence, with $p_{0}^{+}$ determined, we have the leading-order solution for the pressure within the support layers, from (2.23) and (2.36).

The flux of fluid through the pleat

The total dimensionless flux $q$ , which we will use to characterize membrane performance later, is defined in terms of total dimensional flux $Q$ (2.7) by $Q=Q_{0}q$ , where $Q_{0}=K_{m0}P_{0}L/({\it\mu}D)$ . By mass conservation the total flux of fluid flowing across the membrane is equal to that flowing across the inlet boundary and so

(2.39) $$\begin{eqnarray}\displaystyle q=-\frac{k}{{\it\Gamma}}\frac{\partial p_{0}^{+}}{\partial x}\bigg|_{x=0}. & & \displaystyle\end{eqnarray}$$

Another useful quantity for understanding the progress of fouling is the flux $|V_{m}|$ (as defined in (2.6)), per unit area, through the membrane (from top to bottom) as a function of position $X$ along the membrane. When we define the dimensionless analogue, $|v_{m}|$ , of this quantity by $|V_{m}|=|v_{m}|K_{m0}P_{0}/({\it\mu}D)$ , this satisfies the relation

(2.40) $$\begin{eqnarray}\displaystyle |v_{m}(x,t)|=p_{0}^{+}(x,t)-p_{0}^{-}(x,t). & & \displaystyle\end{eqnarray}$$

2.3.2 Membrane fouling

The membrane permeability $k_{m}$ and the flow vary in time due only to the fouling by deposited particles, which occurs in a spatially non-uniform manner. In (2.11) we expressed membrane permeability in terms of the pore radius $A$ , and the number of unblocked pores per unit area, $N$ . We scale each of these quantities with their initial values,

(2.41a,b ) $$\begin{eqnarray}\displaystyle A=A_{0}a,\quad N(X)=N_{0}n(x). & & \displaystyle\end{eqnarray}$$

We note further that (2.13) defines a natural time scale for the problem (that of blocking), while $G_{\infty }$ gives a natural scale for the particle size distribution, motivating us to rescale as follows:

(2.42a-e ) $$\begin{eqnarray}\displaystyle \hspace{-20.00003pt}T=\frac{8{\it\mu}D}{{\rm\pi}P_{0}G_{\infty }A_{0}^{4}}t,\quad G=G_{\infty }g(s),\quad S=A_{0}s,\quad B=\frac{b}{A_{0}},\quad g(s)=1-\text{e}^{-bs} & & \displaystyle\end{eqnarray}$$

while (2.11) gives a natural choice of $K_{m0}$ and leads to the definition

(2.43) $$\begin{eqnarray}\displaystyle K_{m0}=\frac{{\rm\pi}A_{0}^{4}N_{0}}{8}. & & \displaystyle\end{eqnarray}$$

Applying these rescalings together with our original non-dimensionalization (2.16) to (2.11)–(2.14), we obtain the remaining dimensionless equations in the model:

(2.44) $$\begin{eqnarray}\displaystyle & \displaystyle k_{m}(x,t)=a^{4}\left(n+\frac{1-n}{1+{\it\rho}_{b}a^{4}}\right), & \displaystyle\end{eqnarray}$$

(2.45) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial n}{\partial t}=-na^{4}\text{e}^{-ba}(p^{+}|_{y={\it\delta}/2}-p^{-}|_{y=-{\it\delta}/2}),\quad n|_{t=0}=1, & \displaystyle\end{eqnarray}$$

(2.46) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial a}{\partial t}=-{\it\beta},\quad a|_{t=0}=1, & \displaystyle\end{eqnarray}$$

where the dimensionless parameter ${\it\beta}$ is given by

(2.47) $$\begin{eqnarray}\displaystyle {\it\beta}=\frac{8{\it\mu}ED}{{\rm\pi}A_{0}^{5}P_{0}G_{\infty }}. & & \displaystyle\end{eqnarray}$$

2.3.3 Fluid velocity and streamfunction in support layers

It will be convenient in our simulations to be able to visualize the fluid flow through the support layers. Since the flow within these layers is quasi-static and two-dimensional, a streamfunction ${\it\psi}$ may be defined. From the asymptotic solution for the pressure, (2.23), and using the dimensionless form of the Darcy equation, we have dimensionless velocity in the upper and lower support layers given by

(2.48) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{\pm }(x,y)=(-k(x)p_{0x}^{\pm }(x),(k(x)p_{0x}^{\pm }(x))_{x}(y\mp 1)). & & \displaystyle\end{eqnarray}$$

From the streamfunction definition

(2.49) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{\pm }(x,y)=({\it\psi}^{\pm }(x,y)_{y},-{\it\psi}^{\pm }(x,y)_{x}) & & \displaystyle\end{eqnarray}$$

we find

(2.50) $$\begin{eqnarray}\displaystyle {\it\psi}^{+}(x,y)=-k(x)p_{0x}^{+}(x)(y-1),\quad {\it\psi}^{-}(x,y)=-k(x)p_{0x}^{-}(x)(y+1)-c_{1}, & & \displaystyle\end{eqnarray}$$

where the integration constant in ${\it\psi}^{-}$ (the same $c_{1}$ that was introduced in (2.36)) was chosen to match streamlines on the $x$ -axis (the filter membrane location).

2.3.4 Modification for the constant flux case

In the alternative scenario where the total flux $Q_{0}$ through the membrane is fixed instead of the pressure drop, we define the inlet pressure by $P_{0}{\it\zeta}(t)$ , where ${\it\zeta}$ increases monotonically as the membrane is fouled, in such a way as to sustain constant total flux $Q=Q_{0}$ (as defined by (2.7)). By modifying (2.39),

(2.51) $$\begin{eqnarray}\displaystyle {\it\zeta}\left[\frac{k}{{\it\Gamma}}\frac{\partial p_{0}^{+}}{\partial x}\right]_{x=0}^{x=1}=q_{0},\text{constant},\quad \text{for all }t, & & \displaystyle\end{eqnarray}$$

with dimensionless flux as defined there.

2.4 Model summary

Our final model is represented by (2.23), (2.36)–(2.38), (2.44)–(2.46). At this stage we now emphasize each quantity that depends on space and/or time. At each instant of time we must solve the boundary value problem (2.37),

(2.52) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt}\frac{\partial }{\partial x}\left(k(x)\frac{\partial p_{0}^{+}(x,t)}{\partial x}\right)-2{\it\Gamma}k_{m}(x,t)p_{0}^{+}(x,t)={\it\Gamma}k_{m}(x,t)\left(c_{1}(t)\int _{0}^{x}\frac{\text{d}x}{k(x)}+c_{2}(t)\right), & & \displaystyle\end{eqnarray}$$

subject to conditions (2.38)

(2.53a,b ) $$\begin{eqnarray}\displaystyle p_{0}^{+}(0,t)=1,\quad p_{0x}^{+}(0,t)=-\frac{c_{1}(t)}{k(0)}, & & \displaystyle\end{eqnarray}$$

(2.54a,b ) $$\begin{eqnarray}\displaystyle p_{0}^{+}(1,t)=-\left(c_{1}(t)\int _{0}^{1}\frac{\text{d}x}{k(x)}+c_{2}(t)\right),\quad p_{0x}^{+}(1,t)=0, & & \displaystyle\end{eqnarray}$$

in terms of which the pressures in the support layers are given by (2.23), (2.36). Note that we have four boundary conditions (2.53), (2.54) for the second-order equation (2.52), which ensures that the unknown functions $c_{1}(t)$ and $c_{2}(t)$ are fixed also. The membrane permeability $k_{m}(x,t)$ varies quasi-statically in (2.52) due to the fouling; it satisfies (2.44),

(2.55) $$\begin{eqnarray}\displaystyle k_{m}(x,t)=a(t)^{4}\left[n(x,t)+\frac{(1-n(x,t))}{(1+{\it\rho}_{b}a(t)^{4})}\right],\quad \text{where }a(t)=1-{\it\beta}t. & & \displaystyle\end{eqnarray}$$

The number density of unblocked pores, $n(x,t)$ , varies according to (2.45),

(2.56) $$\begin{eqnarray}\displaystyle \frac{\partial n(x,t)}{\partial t}=-n(x,t)a(t)^{4}\text{e}^{-ba(t)}\left(2p_{0}^{+}(x,t)+c_{1}(t)\int _{0}^{x}\frac{\text{d}x}{k(x)}+c_{2}(t)\right),\quad n(x,0)=1. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The solution scheme for this system is straightforward. At time $t=0$ assign $k_{m}(x,0)=k_{m0}=1$ . Then: (i) solve the boundary value problem (2.52)–(2.54) for $p_{0}^{+}(x,t)$ ; (ii) use this solution, and the current membrane permeability $k_{m}(x,t)$ and pore radius $a(t)$ as given by (2.55) to solve (2.56) for $n(x,t)$ ; (iii) update $k_{m}(x,t)$ and $a(t)$ via (2.55) according to the new $n(x,t)$ ; and (iv) use the updated $k_{m}(x,t)$ and return to step (i); repeat.