2.1 Dimensionless equations

We scale the variables via $\tilde{\boldsymbol{x}}=l\boldsymbol{x}$ , $\tilde{\boldsymbol{u}}={\mathcal{U}}\boldsymbol{u}$ , $\tilde{t}=(\unicode[STIX]{x1D6FF}l/(c_{\infty }V_{m}\unicode[STIX]{x1D6FE}))t$ , $\tilde{R}=\unicode[STIX]{x1D6FF}lR$ , $\tilde{c}=c_{\infty }c$ and $\tilde{p}=(\unicode[STIX]{x1D707}{\mathcal{U}}/(\unicode[STIX]{x1D6FF}^{2}l))p$ , where ${\mathcal{U}}$ and $c_{\infty }$ are characteristic velocity and concentration scales respectively. The time scale is chosen to balance obstacle growth with contaminant concentration in (2.6), and the pressure scale is chosen to balance the pressure gradient over the macroscale with viscous forces over the obstacle microscale. The flow problem (2.1) then transforms to

where $\unicode[STIX]{x1D6FC}=c_{\infty }V_{m}\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D6FF}{\mathcal{U}})$ .

For the concentration problem given by (2.2), (2.5) and (2.6), we obtain the dimensionless solute-transport equation

where $Pe={\mathcal{U}}l/D$ is the Péclet number and $k=\unicode[STIX]{x1D6FE}/(\unicode[STIX]{x1D6FF}{\mathcal{U}})$ . The boundary condition (2.8b ) expresses the flux across the moving boundary relative to the boundary, and as such does not contain any velocity component. This system reduces to that considered in Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015) when $R$ is independent of $t$ , i.e. when $V_{m}=0$ , which can be seen by scaling $t\sim \unicode[STIX]{x1D6FC}$ and then taking $\unicode[STIX]{x1D6FC}\rightarrow 0$ . We assume that $\unicode[STIX]{x1D6FC}$ , $Pe$ and $k$ are all $O(1)$ parameters, which corresponds to the richest asymptotic limit. That is, all mechanisms contribute at leading order, from which all asymptotic sublimits may be distilled. In practice, the contaminant accumulates on the obstacles over a much longer timescale than the fluid takes to travel through the porous medium. This manifests in the smallness of the parameter $\unicode[STIX]{x1D6FC}$ , and we exploit this feature in § 4.2 to consider the physically relevant effect of slow contaminant accumulation.

In dimensionless units, the obstacles now form a two-dimensional hexagonal lattice of circles whose centres are a distance of $\unicode[STIX]{x1D6FF}$ apart, and a circle with centre at $\boldsymbol{x}$ has radius $\unicode[STIX]{x1D6FF}R(\boldsymbol{x},t)$ (figure 1).

3.1 Flow problem

Under the spatial transforms (3.1) and (3.2), the flow equations (2.7) become

Expanding the flow velocity and pressure in powers of $\unicode[STIX]{x1D6FF}$ as usual in the multiple-scales method (see Dalwadi et al. Reference Dalwadi, Griffiths and Bruna2015 for details), we find that the leading-order pressure, $p_{0}$ , is independent of the microscale, i.e. $p_{0}=p_{0}(\boldsymbol{x},t)$ . Considering the next order in (3.5) allows us to write where $\overline{p}$ is an arbitrary function of the macroscale and time only, and the matrix function $\unicode[STIX]{x1D646}$ and the vector function $\unicode[STIX]{x1D72B}$ satisfy the so-called cell problem: Here, $\unicode[STIX]{x1D644}$ is the two-dimensional identity matrix and the dependence of $\unicode[STIX]{x1D646}$ and $\unicode[STIX]{x1D72B}$ on $\boldsymbol{x}$ and $t$ is due to the dependence of the microscale boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{f}(\boldsymbol{x},t)$ on the macroscale variable and time.

Integrating (3.6a ) over $\unicode[STIX]{x1D714}_{f}(\boldsymbol{x},t)$ and using (3.4b ), we obtain the homogenized Darcy relation

at leading order, where ${\mathcal{K}}(\unicode[STIX]{x1D719})$ is a scalar function that contains the necessary microscale structure information, and is defined by

We note that the integral of $\unicode[STIX]{x1D646}$ in (3.8b ) is a multiple of the identity matrix due to the symmetry of the cell problem described by (3.7). This would not be the case if, instead of circles, we had considered obstacles with fewer than two orthogonal planes of symmetry. Finally, to obtain a closed homogenized flow problem, we consider the $O(\unicode[STIX]{x1D6FF})$ terms in (3.5b–d ), given by

Integrating (3.9a ) over $\unicode[STIX]{x1D714}_{f}(\boldsymbol{x},t)$ and using Reynolds transport theorem in conjunction with the periodic and no-slip boundary conditions (3.9b,c ) yields a continuity equation for the macroscale fluid velocity

The system (3.8) and (3.10) determines the flow problem given the obstacle structure, which manifests itself through the source term on the right-hand side of (3.10). This reflects the fact that reducing the available pore space within a cell for an incompressible fluid will push the fluid out of that cell. If $\unicode[STIX]{x2202}R/\unicode[STIX]{x2202}t\equiv 0$ , we recover the system derived in Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015).

3.2 Solute-transport problem

Under the spatial transforms (3.1) and (3.2), the solute-transport problem (2.8) becomes

Expanding $c(\boldsymbol{x},\boldsymbol{y},t)$ in powers of $\unicode[STIX]{x1D6FF}$ , we find that the leading-order concentration, $c_{0}$ , is independent of the microscale, i.e.  $c_{0}=c_{0}(\boldsymbol{x},t)$ . The $O(\unicode[STIX]{x1D6FF})$ terms in (3.11) allow us to write $c_{1}$ as

where $\bar{c}$ is an arbitrary function of the macroscale and time only, and the components of the function $\unicode[STIX]{x1D71E}$ satisfy the cell problem

and $n_{y,i}$ is the $i$ th component of the unit vector $\boldsymbol{n}_{\boldsymbol{y}}$ .

The effect of the obstacle growth only appears at the next order. Namely, the $O(\unicode[STIX]{x1D6FF}^{2})$ terms in (3.11) are

Integrating (3.14a ) over $\unicode[STIX]{x1D714}_{f}$ and applying the boundary conditions (3.5c ), (3.9b ), and (3.14b–c ) gives

where $\text{d}s$ denotes the differential element of the obstacle surface $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{f}(\boldsymbol{x},t)$ . Using Reynolds transport theorem, equation (3.15) reduces to

noting that $c_{0}$ is independent of $\boldsymbol{y}$ . From now on, we simply write $\unicode[STIX]{x1D719}$ but it should be understood that the porosity will be, in general, a function of $\boldsymbol{x}$ and $t$ . In a similar manner, henceforth we use $\unicode[STIX]{x1D714}_{f}(\unicode[STIX]{x1D719})$ instead of $\unicode[STIX]{x1D714}_{f}(\boldsymbol{x},t)$ and likewise for $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{f}$ . Finally, using (3.12), (3.16) reduces to

where $(\unicode[STIX]{x1D645}_{\unicode[STIX]{x1D71E}}^{T})_{ij}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{j}/\unicode[STIX]{x2202}y_{i}$ is the transpose of the Jacobian matrix of $\unicode[STIX]{x1D71E}$ , the solution to the cell problem (3.13).

To express (3.17) in terms of the volumetric average concentration $C(\boldsymbol{x},t)$ defined in (3.4a ), we note that $C(\boldsymbol{x},t)\sim |\unicode[STIX]{x1D714}|^{-1}\int _{\unicode[STIX]{x1D714}_{f}(\unicode[STIX]{x1D719})}c_{0}\,\text{d}\boldsymbol{y}=\unicode[STIX]{x1D719}c_{0}(\boldsymbol{x},t)$ at leading order in $\unicode[STIX]{x1D6FF}$ . Using this relation and (3.8a ), we find that

at leading order in $\unicode[STIX]{x1D6FF}$ , where the effective diffusion coefficient is

and the effective adsorption coefficient is

using the fact that $\unicode[STIX]{x1D719}=1-2\unicode[STIX]{x03C0}R^{2}/\sqrt{3}$ , and $|\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{f}(\unicode[STIX]{x1D719})|=4\unicode[STIX]{x03C0}R$ . As is the case for the permeability ${\mathcal{K}}$ , we find that the diffusion tensor is a multiple of the identity due to the symmetry of the cell problem (3.13). Finally, the condition to couple contaminant concentration with obstacle growth (2.8c ) becomes

The effective permeability ${\mathcal{K}}(\unicode[STIX]{x1D719})$ (from (3.8b )) and the effective diffusion ${\mathcal{D}}(\unicode[STIX]{x1D719})$ (from (3.18b )) that encapsulate the microscale physics can be computed by solving the respective cell problems (3.7) and (3.13) for a given cell porosity $\unicode[STIX]{x1D719}$ , determined by the size of the obstacles. We calculate this numerically using the finite-element software Comsol Multiphysics and find that ${\mathcal{K}}$ and ${\mathcal{D}}$ are both monotonically increasing with the porosity, as expected, and that there is a sharp decrease in both functions as the porosity tends down towards the critical porosity where blocking occurs (see figure 2).

Figure 2.

The functions (a) ${\mathcal{K}}(\unicode[STIX]{x1D719})$ and (b) $Pe{\mathcal{D}}(\unicode[STIX]{x1D719})$ (defined in (3.8b ) and (3.18b ), respectively) calculated using Comsol Multiphysics, for circular obstacles whose centres lie on a hexagonal lattice.

The homogenization procedure has significantly reduced the mathematical complexity of the growing multiply-connected domain in the full problem (2.8), with only a marginal increase in the complexity of the coefficients in the resulting governing equations (3.18), which capture the microscale structure of the problem in a systematic manner.

4.1 Model set-up

We are now in a position to use the homogenized equations (3.8), (3.10) and (3.18) to quantify the effect of blocking and porosity gradients on filter efficiency. We consider a canonical industrial process of interest whereby a three-dimensional filter separates two reservoirs, and a unidirectional flow is induced through the filter. As is common in industrial set-ups, we assume that the filter is graded in the same direction as the fluid flow. This reduces the problem to a one-dimensional description.

We define the direction of porosity variation as $x$ , where $x\in (0,1)$ within the filter (since the dimensional characteristic length $l$ is chosen to be the depth of the filter). Thus, the set-up is similar to that illustrated in figure 1. The upstream is defined for $x\in (-\infty ,0)$ and the downstream for $x\in (1,\infty )$ .

Provided the boundary conditions allow for unidirectionality, equations (3.8) and (3.10) yield a unidirectional macroscale flux $\boldsymbol{U}=U(x,t)\boldsymbol{e}_{x}$ (where $\boldsymbol{e}_{x}$ is the unit vector in the $x$ -direction). When filtering at constant (dimensionless) pressure drop $\unicode[STIX]{x0394}P$ , we find that

where

Thus, the governing equation (3.18a ) becomes

and the relationship between obstacle growth and concentration (3.18d ) is

Considering the limit of (3.18b ) and (4.3) as $\unicode[STIX]{x1D719}\rightarrow 1$ , and using continuity of fluid flux, we obtain the following governing equations for the upstream $C^{-}$ and downstream $C^{+}$ concentrations

We impose continuity of concentration and concentration flux at the boundaries between the filter and the reservoirs. In the far-field of the reservoirs, the concentration tends to a constant value. We may take the upstream concentration $C^{-}\rightarrow 1$ (by choice of our non-dimensionalization), while the downstream concentration $C^{+}$ tends to a constant value that must be determined as part of the solution. Mathematically, this corresponds to

Thus, with appropriate initial conditions, our homogenized system is given by (4.1)–(4.6). We solve this system numerically using the method of lines, discretizing in space with a second-order finite difference scheme and using the MATLAB program ode15s to solve in time.

A suitable measure of filter efficiency is the cumulative contaminant removal, defined by

where the second equality can be deduced from using the relationship $\unicode[STIX]{x1D719}=1-2\unicode[STIX]{x03C0}R^{2}/\sqrt{3}$ to obtain $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}/\unicode[STIX]{x2202}t=-(|\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{f}|/|\unicode[STIX]{x1D714}|)\unicode[STIX]{x2202}R/\unicode[STIX]{x2202}t$ , in conjunction with (3.18c ) and (4.4). A measure of the corresponding structural changes within the filter is captured by the quantity

which expresses the ratio of the distance between obstacles and the obstacle centres, or an effective pore size in the filter.

Figure 3.

Filtration through an initially uniform porosity filter, $\unicode[STIX]{x1D719}(x,0)=0.8$ , with $Pe=3$ , $k=1$ . (a) The pore size, $P$ , defined by (4.8), throughout the filter at four snapshots in time (shown by asterisks in b–d). We start with $t=0$ and end with the first time at which a pore size in the filter reaches $0.01$ , defined as $t_{0.01}$ . The two intermediate snapshots are at time $t_{0.01}/3$ and $2t_{0.01}/3$ , where we use $t_{0.01}$ calculated from the solution to the full homogenized equations for all cases. (b) The cumulative contaminant adsorption ${\mathcal{T}}$ , defined by (4.7), as a function of time. (c) The time taken until the pore size reaches a minimum value. (d) The cumulative contaminant adsorption ${\mathcal{T}}$ , defined by (4.7), as a function of minimum pore size. The solid black curves denote the solutions to the full homogenized equations (4.1)–(4.6), the dashed red curves denote the asymptotic approximation when obstacle growth is small (4.12) and the dotted yellow curves denote the asymptotic solution when we further assume a weak porosity gradient (4.13). We see excellent agreement throughout, with only small deviations that occur when the pore size becomes small.

We begin by considering the case of a filter whose initial porosity is uniform. Solving (4.6) we find that as filtration proceeds the pore size $P(x,t)$ decreases in time throughout the filter (figure 3 a) and the rate of contaminant removal ${\mathcal{T}}^{\prime }(t)$ reduces (figure 3 b). The reduction in pore size is more rapid close to the filter entrance than towards the filter exit (figure 3 a), which causes the filter to block at the entrance while the rest of the filter is still fit for purpose. This has significant practical disadvantages, and so we now use our model to explore strategies that minimize this wastefulness by varying the initial filter porosity. To do so, we first note that the homogenized system (4.6) can be further simplified by exploiting two relevant asymptotic regimes. We discuss these in the following section.

4.2 Asymptotic reductions of the homogenized equations

In the majority of filtration applications the growth of the obstacles, and thus the blocking of the filter, takes place over a timescale that is much longer than the time taken for the fluid to flow through the filter. Mathematically, this corresponds to the quasi-steady limit in which $\unicode[STIX]{x1D6FC}\rightarrow 0$ . At leading order in this limit, returning to the three-dimensional equations for generality, the flow equations (3.8) and (3.10) become

where ${\mathcal{K}}(\unicode[STIX]{x1D719})$ is defined in (3.8b ); and the transport equations (3.18) become

where ${\mathcal{D}}(\unicode[STIX]{x1D719})$ and $f(\unicode[STIX]{x1D719})$ are defined in (3.18b,c ), respectively. In this regime, the growth in $R$ decouples from the flow and transport problems, which are now quasi-steady.

For a unidirectional porosity variation, the flow velocity becomes independent of $x$ and so (4.9a ) simplifies to

where $\overline{K}$ is given in (4.2). Moreover, the upstream and downstream concentrations can be solved to obtain

where $A(t)$ and $B(t)$ are arbitrary functions of time. Substituting (4.11) into (4.6), the system for $C$ becomes

The governing equation for the porosity (4.4) is unchanged:

This problem is similar to that considered in Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015), where the absence of microstructural changes due to filter clogging meant that $U(t)\equiv 1$ in (4.12), and $\unicode[STIX]{x2202}R/\unicode[STIX]{x2202}t\equiv 0$ instead of (4.12d ). We can solve the system (4.12) numerically using the method of lines, discretizing in space with a second-order finite difference scheme and using the MATLAB program ode15s to solve in time.

We can make further analytic progress by considering the case where, in addition to the slow obstacle growth, the deviation of the porosity $\unicode[STIX]{x1D719}(x,t)$ from its average value in space $\bar{\unicode[STIX]{x1D719}}(t)=\int _{0}^{1}\unicode[STIX]{x1D719}(x,t)\,\text{d}x$ is small. In this case, we express the porosity as its average in space plus the deviation, as follows $\unicode[STIX]{x1D719}(x,t)=\bar{\unicode[STIX]{x1D719}}(t)+\unicode[STIX]{x1D6F7}(x,t)$ , and assume that $\max _{x\in (0,1)}|\unicode[STIX]{x1D6F7}|\ll \bar{\unicode[STIX]{x1D719}}$ for any given time. Thus, the leading-order coefficients in the linear governing equation (4.12a ) are constant in $x$ , and the solution can be written in terms of hyperbolic functions. The $O(\unicode[STIX]{x1D6F7})$ correction terms in (4.12a ) can then be solved using the method of variation of parameters (a similar calculation is carried out in Dalwadi et al. Reference Dalwadi, Griffiths and Bruna2015). From this, we may express the concentration $C$ as a sum of the concentration distribution in a filter with constant porosity, $C_{0}(x,t)$ , and the adjustment due to the porosity gradient, $C_{1}(x,t)$ , i.e.

where $|C_{1}|\ll C_{0}$ ; $C_{0}$ and $C_{1}$ are functions of $\unicode[STIX]{x1D719}$ that can be solved for explicitly and are given in appendix A. Using this analytic solution, we are able to reduce the entire problem to solving (4.12d ). We solve this system numerically using the method of lines, discretizing in space with a second-order finite difference scheme and using the MATLAB program ode15s to solve in time.

In figure 3, we compare the results from these two asymptotic results, (4.12) and (4.13), with the full numerical solution, (4.1)–(4.6) (solid black lines), as dashed red and dotted yellow lines, respectively. Both asymptotic solutions offer very good agreement in the pore size $P$ with the full numerical solution (figure 3 a), and in ${\mathcal{T}}$ until around $t=0.45$ (figure 3 b).

The discrepancy between numeric and asymptotic solutions appears when considering the time taken to reach a minimum pore size rather than in the cumulative contaminant removal metric ${\mathcal{T}}$ . We are able to contextualize this metric by introducing relative timings in a filter lifespan. That is, we define the minimum pore size

and the time taken until the minimum pore size reaches a given value,

Blocking occurs at $t_{0}$ , but we must consider small finite values of $P^{\ast }$ due to the singularity in (4.12) when $\unicode[STIX]{x1D719}\rightarrow 0$ , and hence when $P^{\ast }\rightarrow 0$ , at the point of blocking. For example, in figure 3 the simulations are stopped at $t_{0.01}$ .

From figure 3(b), we see that although the maximum time to removal is different between the asymptotic and numerical solutions, the total cumulative removals are very close. We show the former in more detail in figure 3(c), where we see that the numerical and asymptotic solutions agree well in $t_{P^{\ast }}$ until $P^{\ast }$ falls below around $P^{\ast }=0.05$ . We show the latter in more detail in figure 3(d), where we use $P^{\ast }$ as the abscissa instead of time, against cumulative contaminant removal. We see that there is excellent agreement between numeric and asymptotic solutions throughout, and henceforth use the full asymptotic approximation (4.13), which offers a very accurate representation of the solution to the full homogenized system (4.1)–(4.6) while enabling further analytic study of the system. Moreover, we note that we halt simulations when $P^{\ast }$ reaches $0.01$ , and refer to this as ‘blocking’, and we see from figure 3(d) that this is unlikely to yield major differences in values of total contaminant removal.

4.3 Linearly graded filters

As shown in figure 3, in a depth filter with a uniform porosity the early portion of the media will block before the later portion has been used to its full effect. This can be counteracted by using a filter whose porosity decreases with depth.

In this section we consider linearly graded filters at $t=0$ , characterized by $\unicode[STIX]{x1D719}(0,x)=\unicode[STIX]{x1D719}_{0}+m(x-0.5)$ , where we vary the initial average porosity, $\unicode[STIX]{x1D719}_{0}$ , and the initial porosity gradient, $m$ . In a similar manner to Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015), we assume that the pressure drop is kept constant across each filter considered and, noting that the system is invariant to the scaling

for any constant $\unicode[STIX]{x1D714}$ , we impose $U(0)=1$ and vary $Pe$ and $k$ accordingly. For short times, a filter with spatially uniform porosity (i.e.  $m=0$ ) provides the largest cumulative removal for a given average porosity, and there is an optimum value for this average porosity (figure 4 a). This optimum exists because a highly porous filter has less surface area with which to remove contaminant, but a less porous filter will admit a lower flow rate through the filter, relying more on diffusion than advection for contaminant transport and thus lowering the contaminant removal. The result that there is no apparent difference in the total contaminant removal of a filter with a positive or negative porosity gradient for small time replicates the results for the initial instantaneous adsorption obtained in Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015). However, in contrast to Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015) we are now able to capture the asymmetry in removal that arises over longer times as a result of blockages within the filter. In particular, negative initial porosity gradients do provide an inherently larger cumulative removal at a given time than positive initial porosity gradients (figure 4 b,c), and their final cumulative removal is larger (figure 4 d). We note that whilst negative initial porosity gradients do last for a longer time before blocking, the filters that take the longest time to block do not automatically remove the most contaminant (figure 5).

In Dalwadi et al. (Reference Dalwadi, Griffiths and Bruna2015), a further metric was introduced to measure the uniformity of uptake, thus allowing us to account for the superior performance of filters with a negative porosity gradient without explicitly accounting for filter evolution. As we are able to account for filter evolution in this work, ${\mathcal{T}}$ is now the better measure of filtration. We show in appendix B that this alternative metric can only provide qualitative insights into filtration, and should not be used when the filter structure evolves in time.

Figure 4.

Cumulative contaminant removal, ${\mathcal{T}}$ , defined by (4.7), for varying $\unicode[STIX]{x1D719}_{0}$ and $m$ in initial porosity distributions of the form $\unicode[STIX]{x1D719}(x)=\unicode[STIX]{x1D719}_{0}+m(x-0.5)$ , given by the solution to (4.1)–(4.6): (a) $t=10^{-4}$ ; (b) $t=0.1$ ; (c) $t=0.7$ ; (d) $t=2$ . Each curve represents a different value of $\unicode[STIX]{x1D719}_{0}$ , which increments in steps of $0.1$ from $\unicode[STIX]{x1D719}_{0}=0.4$ to $\unicode[STIX]{x1D719}_{0}=0.8$ and the arrow in (d) denotes curves of increasing $\unicode[STIX]{x1D719}_{0}$ . We vary $m$ for a given $\unicode[STIX]{x1D719}_{0}$ such that $\unicode[STIX]{x1D719}\in [0.2,0.95]$ . Therefore, the available range of $m$ varies with $\unicode[STIX]{x1D719}_{0}$ . We use the reference values $\unicode[STIX]{x1D719}=0.6$ , $Pe=5$ , $k=0.1$ from which to modify appropriate parameters. The dashed curves denote that the minimum pore size has reached $0.01$ and the filtration has stopped.

Figure 5.

(a) The time until the minimum pore size reaches 0.01 for varying $\unicode[STIX]{x1D719}_{0}$ and $m$ in initial porosity distributions of the form $\unicode[STIX]{x1D719}(x)=\unicode[STIX]{x1D719}_{0}+m(x-0.5)$ , determined by solving (4.1)–(4.6). The dashed black lines denote the times at which snapshots are taken in figures 4 and 7. (b) The total cumulative adsorption over the lifespan of a filter versus the time taken to block. Each line represents a different initial average porosity, and the unlabelled arrows denote the direction of increasing $m$ for each line. In both figures, each line represents a different value of $\unicode[STIX]{x1D719}_{0}$ , which increments in steps of $0.1$ from $\unicode[STIX]{x1D719}_{0}=0.4$ to $\unicode[STIX]{x1D719}_{0}=0.8$ and the labelled arrow denotes lines of increasing $\unicode[STIX]{x1D719}_{0}$ . We vary $m$ for a given $\unicode[STIX]{x1D719}_{0}$ such that $\unicode[STIX]{x1D719}\in [0.2,0.95]$ . Therefore, the available range of $m$ varies with $\unicode[STIX]{x1D719}_{0}$ . We use the reference values $\unicode[STIX]{x1D719}=0.6$ , $Pe=5$ , $k=0.1$ from which to modify appropriate parameters.

For a given initial average porosity, the largest cumulative removal does not necessarily correspond to a filter with the most negative initial porosity gradient (figure 4 d). If the porosity gradient is too negative, the removal is skewed towards the filter exit, and thus blocking occurs at the filter exit too quickly. This effect causes the sharp drop-off in filter performance seen in figure 4(c,d) for large negative values of $m$ . Thus, an optimum initial porosity gradient exists for a given initial average porosity. Whilst this ‘optimal’ initial porosity gradient maximizes contaminant removal over the space of initially linear graded filters with a given initial average porosity, it will not necessarily maximize contaminant removal over the space of all filters with a given initial average porosity. We explore this idea in more detail in the next section.

4.4 Filters that block everywhere at once

Using the framework we have set up in this paper, we can address the question of which initial porosity distributions maximize the cumulative removal for given operating conditions. We are able to bypass much of the difficulty involved in the full optimal control problem by noting that, for a given initial average filter porosity, the cumulative adsorption (4.7) is maximized when a filter blocks all at once, rather than in just one place. Most filters will only block in one place, and this is the case for all of the filters represented in figure 4, where the filters block at either the filter entrance or the filter exit.

By starting with a filter that is blocked everywhere and running our simulations backwards in time, we can obtain initial porosity distributions whose lifespans end with blocking occurring everywhere in the filter at once. To avoid the computational issues arising when blocking occurs, we approximate the concept of a filter that blocks everywhere at once by a filter that reaches a pore size of $0.01$ everywhere at once. An issue with this procedure is that the full homogenized system for the concentration (4.1)–(4.6) is parabolic in time, and thus running the simulations backwards in time results in an ill-posed problem. However, the evolution equation for the filter porosity (4.12d ) is well-posed if run backwards in time, and only relies on knowing the concentration for a given porosity. Thus, if we use the asymptotic solutions for the concentration derived in § 4.2, we do not have to run the concentration evolution equation backwards in time and we are able to side-step the issue with the ill-posedness of this problem.

A related issue, due to solving the system of equations backwards in time, is that only the final flow velocity can be imposed, and the initial flow velocity is now an output to the calculation. Thus, a general final flow velocity $U(t_{0.01})$ will not necessarily result in $U(0)=1$ , the condition we prescribe in the forward problem. Moreover, we vary $Pe$ and $k$ according to the initial porosity in the forward problem, using the invariant scaling (4.16), and thus it is not immediately clear what values of $Pe$ and $k$ we should impose for the inverse problem. We overcome these problems by exploiting the fact that we impose a constant pressure drop across the filter, and thus $\overline{K}(0)=U(0)\overline{K}(t_{0.01})/U(t_{0.01})=\overline{K}(t_{0.01})/U(t_{0.01})$ , where the first equality holds from (4.10), and the second equality holds by using $U(0)=1$ (the filter for which we are aiming). This allows us to move between the initial and final average permeabilities as a function of the final flow velocity, as well as imposing values of $Pe$ and $k$ using the final porosity. Hence, we are able to use a shooting method for just one parameter, the final flow velocity $U(t_{0.01})$ , shooting to achieve $U(0)=1$ .

To summarize, we impose a final constant pore size with $P(x,t_{0.01})\equiv 0.01$ and a final flow velocity and run our simulation in reverse, halting the process when the porosity at any point reaches a set maximum. We iterate this process, varying the final flow velocity, until the initial flow velocity $U(0)=1$ to within an accuracy of $2\times 10^{-3}$ . We repeat this process for different maximum porosity constraints to obtain a family of initial porosity distributions that block everywhere at once when contaminant flows through, each with a different maximum (and average) porosity. The maximum porosities we choose are given by $1-\unicode[STIX]{x03C0}R^{2}$ , where $R$ takes the values $0.1174$ , $0.15$ , $0.2$ , $0.25$ and $0.3$ . We produce this range of initial porosity distributions as there may be physical constraints on how large/small the porosity/fibres can be.

Each resulting filter has a decreasing porosity with depth and, compared with the region near the filter entrance, the porosity gradient in each filter is more negative towards the centre of the filter and smaller towards the end of the filter (figure 6 a). This effect is more pronounced for larger maximum porosities. We compare how these filters remove contaminant against the linear filters we considered in § 4.3 and see that the ideal filters do remove around $20\,\%$ more contaminant for a given initial average porosity (figure 6 b). However, these filters cannot have arbitrarily large initial average porosity, as a filter whose initial porosity is everywhere close to $1$ will not block everywhere at once, and thus a linearly graded filter with a large enough initial average porosity can remove more contaminant than a filter with a lower initial average porosity that blocks everywhere at once. As a result, we may term these pseudo-optimal filters. As a filter may require a maximum porosity to maintain structural integrity, the choice of initial porosity distribution will depend on the operating conditions.

Figure 6.

(a) The initial porosity distributions for filters that block everywhere at once with different initial average porosities, $\unicode[STIX]{x1D719}_{0}$ . The initial average porosities are: 0.320, 0.390, 0.465, 0.548 and 0.607 (all to three significant figures). (b) The cumulative contaminant removal ${\mathcal{T}}$ for the pseudo-optimal filters (red asterisks) compared with the linear filters considered in § 4.3 (black crosses). The data for the black crosses is from figure 4. We use dimensionless parameters corresponding to the reference values $\unicode[STIX]{x1D719}(x,0)\equiv 0.6$ , $Pe=5$ , $k=0.1$ in the manner described in the text. In (b) we see that the pseudo-optimal filters do provide increased contaminant removal for a given initial average porosity, but the average porosity of the pseudo-optimal filters cannot be increased to an arbitrarily large value.