2.1 Microstructure with isolated fibres

First, we consider the case when fibres are not allowed to touch at any time  $\tilde{t}$ . We recall that this was the assumption used by Dalwadi et al. (Reference Dalwadi, Bruna and Griffiths2016), but here we extend it to polydisperse fibres. We define the domain by setting the location of the fibres in $\tilde{\unicode[STIX]{x1D6FA}}(\tilde{t})$ with centres located at $\tilde{\boldsymbol{x}}^{m}$ and radii $\tilde{r}^{m}(\tilde{\boldsymbol{x}},\tilde{t})$ for $m\in M$ , where $M$ is the set of all fibres in the whole filter medium, and prescribe $\tilde{\unicode[STIX]{x1D6FA}}(\tilde{t}=0)$ . We denote the surface of each fibre as

(2.1) $$\begin{eqnarray}\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t})=\{\tilde{\boldsymbol{x}}\in \tilde{\unicode[STIX]{x1D6FA}}:\Vert \tilde{\boldsymbol{x}}-\tilde{\boldsymbol{x}}^{m}\Vert =\tilde{r}^{m}(\tilde{\boldsymbol{x}}^{m},\tilde{t})\},\quad m\in M.\end{eqnarray}$$

The interface between the pore and solid subdomains is the union of all the fibre surfaces, that is, $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}(\tilde{t})=\bigcup _{m\in M}\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t})$ . We note that the assumption that fibres are not in contact implies that $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t})\cap \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{n}(\tilde{t})=\emptyset$ for $n\neq m$ .

The contaminant particles and the fluid occupy the pore space of the filter medium, $\tilde{\unicode[STIX]{x1D6FA}}_{f}(\tilde{t})$ . We assume that the particles are sufficiently small that they do not influence the fluid flow and that the flow is incompressible and Newtonian. We also assume that the flow is sufficiently slow and thus satisfies the Stokes equations

(2.2a ) $$\begin{eqnarray}\displaystyle & -\tilde{\unicode[STIX]{x1D735}}\tilde{p}+\unicode[STIX]{x1D707}\tilde{\unicode[STIX]{x1D6FB}}^{2}\tilde{\boldsymbol{u}}=\mathbf{0},\quad \tilde{\boldsymbol{x}}\in \tilde{\unicode[STIX]{x1D6FA}}_{f}(\tilde{t}), & \displaystyle\end{eqnarray}$$

(2.2b ) $$\begin{eqnarray}\displaystyle & \hspace{36.0pt}\tilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }\tilde{\boldsymbol{u}}=0,\quad \hspace{0.2pt}\tilde{\boldsymbol{x}}\in \tilde{\unicode[STIX]{x1D6FA}}_{f}(\tilde{t}), & \displaystyle\end{eqnarray}$$

where $\tilde{p}(\tilde{\boldsymbol{x}},\tilde{t})$ is the fluid pressure (Pa), $\tilde{\boldsymbol{u}}(\tilde{\boldsymbol{x}},\tilde{t})$ is the fluid velocity $(\text{m}~\text{s}^{-1})$ , $\tilde{\unicode[STIX]{x1D735}}$ is the nabla operator with respect to the spatial coordinate  $\tilde{\boldsymbol{x}}$ , and $\unicode[STIX]{x1D707}$ is the viscosity (Pa s). The radial fibre growth results in the following no-slip boundary condition on the fibre surface:

(2.2c ) $$\begin{eqnarray}\tilde{\boldsymbol{u}}=-\frac{\unicode[STIX]{x2202}\tilde{r}^{m}}{\unicode[STIX]{x2202}\tilde{t}}\boldsymbol{n}^{m},\quad \tilde{\boldsymbol{x}}\in \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t}),~m\in M,\end{eqnarray}$$

where $\boldsymbol{n}^{m}$ is the unit normal to the fibres’ surface $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}$ pointing into the solid domain. Despite the time-dependent nature of the boundary condition (2.2c ), we use the steady-state Stokes equations (2.2a ) and (2.2b ) because the time scale of fibre growth is slow compared with that over which the fluid flow attains a steady state.

We assume that the contaminant particles are uniform in size. We also assume that they are much smaller than the typical fibre diameter and that the contaminant suspension is dilute, as is common for many filtration scenarios. This allows us to neglect particle–particle and hydrodynamic interactions and to describe the contaminant by its number concentration $\tilde{c}(\tilde{\boldsymbol{x}},\tilde{t})$ evolving according to

(2.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\tilde{c}}{\unicode[STIX]{x2202}\tilde{t}}=\tilde{\unicode[STIX]{x1D735}}\boldsymbol{\cdot }(D\tilde{\unicode[STIX]{x1D735}}\tilde{c}-\tilde{\boldsymbol{u}}\tilde{c}),\quad \tilde{\boldsymbol{x}}\in \tilde{\unicode[STIX]{x1D6FA}}_{f}(\tilde{t}),\end{eqnarray}$$

where $\tilde{c}$ is measured in particle count  $\text{m}^{-3}$ , and $D$ ( $\text{m}^{2}~\text{s}^{-1}$ ) is the diffusivity coefficient. Using a linear adsorption model without desorption (Baret Reference Baret1969) and (2.2c ), the boundary condition for the concentration reads

(2.4) $$\begin{eqnarray}-D\tilde{\unicode[STIX]{x1D735}}\tilde{c}\boldsymbol{\cdot }\boldsymbol{n}^{m}=k\tilde{c},\quad \tilde{\boldsymbol{x}}\in \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t}),~m\in M,\end{eqnarray}$$

where $k~(\text{m}~\text{s}^{-1})$ is the adsorption coefficient. For a detailed derivation of the boundary condition (2.4), we refer to Dalwadi et al. (Reference Dalwadi, Bruna and Griffiths2016). This represents a balance between the diffusive flux of particles to the fibre surface and the net adsorption as a result of contact, which, as discussed in the introduction (§ 1), may be due to a combination of mechanisms.

We assume that the contaminant particles become immobile once they adsorb onto the fibre surface and add to the fibre volume. Since we assume that the fibres grow radially, the radius of each fibre changes proportionally to the volumetric particle flux averaged over the surface of the fibre $\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}$ . This implies that

(2.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\tilde{r}^{m}}{\unicode[STIX]{x2202}\tilde{t}}=\frac{1}{|\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}|}\int _{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}}\unicode[STIX]{x1D70C}^{-1}v\,k\tilde{c}\,\text{d}s,\quad \tilde{\boldsymbol{x}}\in \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}(\tilde{t}),~m\in M,\end{eqnarray}$$

where $|\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}|=\int _{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}_{s}^{m}}\,\text{d}s$ , $v$ ( $\text{m}^{3}$ ) is the volume of a contaminant particle, and $\unicode[STIX]{x1D70C}$ is the packing density of contaminant particles on the fibre surface. If we ignore voids between the contaminant particles adsorbed onto the fibre surface, then $\unicode[STIX]{x1D70C}=1$ ; if they are perfectly packed around the fibre, then $\unicode[STIX]{x1D70C}=0.74$ . Generally, the particles do not pack so well and form so-called dendrites, or very dense tree-like structures (Brown Reference Brown1993, pp. 201–205). We consider $\unicode[STIX]{x1D70C}=0.3$ in this study.

2.1.1 Non-dimensionalization

We introduce the following non-dimensionalization:

(2.6a-f )

where $l$ is the characteristic thickness of the filter medium (m), ${\mathcal{U}}$ is the characteristic face velocity $(\text{m}~\text{s}^{-1})$ , ${\mathcal{T}}$ is the characteristic filtration time (s), and ${\mathcal{C}}$ is the inlet contaminant concentration (particle count  $\text{m}^{-3}$ ). Here $\unicode[STIX]{x1D6FF}$ is the ratio of the microscopic quasi-periodic unit-cell diameter to the (macroscopic) filter depth. The assumption to apply the method of multiple scales is that there is a separation between the microscopic and macroscopic length scales, that is, $\unicode[STIX]{x1D6FF}\ll 1$ .

We also introduce the following dimensionless groups:

(2.7a-e ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{l}{{\mathcal{U}}{\mathcal{T}}},\quad \unicode[STIX]{x1D6FD}=\frac{l\unicode[STIX]{x1D6FF}}{k{\mathcal{T}}},\quad \unicode[STIX]{x1D6FE}=\frac{D\unicode[STIX]{x1D6FF}}{lk},\quad \unicode[STIX]{x1D701}=\frac{{\mathcal{U}}\unicode[STIX]{x1D6FF}}{k},\quad \unicode[STIX]{x1D702}=\frac{{\mathcal{T}}v{\mathcal{C}}k}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}l},\end{eqnarray}$$

and the dimensionless domains $\unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x1D6FA}_{f}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{s}^{m}$ , analogous to their dimensional counterparts. Inserting this non-dimensionalization into (2.2) yields

(2.8a ) $$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u} & = & \displaystyle \mathbf{0},\quad \hspace{44.39996pt}\boldsymbol{x}\in \unicode[STIX]{x1D6FA}_{f}(t),\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u} & = & \displaystyle 0,\quad \hspace{44.39996pt}\boldsymbol{x}\in \unicode[STIX]{x1D6FA}_{f}(t),\end{eqnarray}$$

(2.8c ) $$\begin{eqnarray}\displaystyle \displaystyle \boldsymbol{u} & = & \displaystyle -\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}\boldsymbol{n}^{m},\quad \boldsymbol{x}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{s}^{m}(t),~m\in M.\end{eqnarray}$$

The mass-transport problem (2.3) and (2.4) transforms into

(2.9a ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D735}c-\unicode[STIX]{x1D701}\boldsymbol{u}c), & & \displaystyle \boldsymbol{x}\in \unicode[STIX]{x1D6FA}_{f}(t),\end{eqnarray}$$

(2.9b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D735}c\boldsymbol{\cdot }\boldsymbol{n}=-\unicode[STIX]{x1D6FF}c, & & \displaystyle \boldsymbol{x}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{s}(t).\end{eqnarray}$$

Finally, using (2.4) and (2.5), the coupling condition in dimensionless form reads

(2.10) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}=\frac{1}{|\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{s}^{m}|}\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{s}^{m}}\unicode[STIX]{x1D702}c\,\text{d}s,\quad m\in M.\end{eqnarray}$$

2.1.2 Homogenized model

At the microscale, we consider the filter medium to consist of quasi-periodic unit cells $w(\boldsymbol{x},t)$ (see figure 1). We allow for fibres of different sizes that may be randomly arranged within each periodic cell. We introduce a microscale variable $\boldsymbol{y}=\boldsymbol{x}/\unicode[STIX]{x1D6FF}$ , which is defined in the unit cell $w(\boldsymbol{x},t)$ . The macroscale variable $\boldsymbol{x}$ spans across the whole filter medium. We denote the fluid and solid subdomains as $w_{f}(\boldsymbol{x},t)$ and $w_{s}(\boldsymbol{x},t)$ , respectively. The internal fluid–solid interface is denoted as $\unicode[STIX]{x2202}w_{s}(\boldsymbol{x},t)$ , which consists of the fibre surfaces $\unicode[STIX]{x2202}w_{s}(\boldsymbol{x},t)=\bigcup _{m\in M_{w}}\unicode[STIX]{x2202}w_{s}^{m}$ , where $M_{w}$ is the set of the fibres found in the unit cell, which is a subset of all fibres $M$ in the filter medium, $M_{w}\subset M$ . The outer fluid boundary of the unit cell is denoted $\unicode[STIX]{x2202}w_{f}(\boldsymbol{x},t)=\unicode[STIX]{x2202}w\cap w_{f}$ .

Using the method of multiple scales, we seek a solution to the problem (2.8)–(2.10) as a function of $\boldsymbol{x}$ and  $\boldsymbol{y}$ , and treat these two variables as independent. The extra freedom this gives is removed by enforcing that the solution is exactly periodic in  $\boldsymbol{y}$ ; small variations from one unit cell to the next are thereby captured through the macroscale variable  $\boldsymbol{x}$ . We insert the change of variables $\boldsymbol{y}=\boldsymbol{x}/\unicode[STIX]{x1D6FF}$ and expand all dependent variables in the form $\boldsymbol{u}=\boldsymbol{u}^{(0)}+\unicode[STIX]{x1D6FF}\boldsymbol{u}^{(1)}+\cdots \,$ , and similarly for the pressure $p$ and the concentration  $c$ .

The macroscopic quantities are introduced using the following averaging:

(2.11) $$\begin{eqnarray}\overline{G}(\boldsymbol{x},t,\cdot )=\frac{1}{|w(\boldsymbol{x},t)|}\int _{w_{f}}g(\boldsymbol{x},\boldsymbol{y},t,\cdot )\,\text{d}\boldsymbol{y}=\unicode[STIX]{x1D719}G(\boldsymbol{x},t,\cdot ),\end{eqnarray}$$

where $g(\boldsymbol{x},\boldsymbol{y},t)$ is a microscopic quantity, $\overline{G}(\boldsymbol{x},t)$ is its volumetric average, $G(\boldsymbol{x},t)$ is its intrinsic average, and $\unicode[STIX]{x1D719}=|w_{f}|/|w|$ is the porosity.

For our homogenization, we use the volumetric average for the velocity $\overline{\boldsymbol{U}}$ and the intrinsic average for the pressure $P$ and concentration  $C$ . The velocity $\overline{\boldsymbol{U}}$ is the Darcy velocity and should not be confused with the actual velocity of the fluid travelling through the pores.

The derivation of the homogenized model via the method of multiple scales is analogous to that presented by Dalwadi et al. (Reference Dalwadi, Bruna and Griffiths2016) for monodisperse fibres. The same methodology can be easily extended to fibres with polydisperse sizes in the unit cell, when the fibres do not touch. For this reason, here we just present the final homogenized model and refer the reader to Dalwadi et al. (Reference Dalwadi, Bruna and Griffiths2016) for the details. In what follows, we drop the superscript (0) that refers to the leading-order quantities in $\unicode[STIX]{x1D6FF}$ and simply write $\overline{\boldsymbol{U}}^{(0)}\equiv \overline{\boldsymbol{U}}$ , and similarly for $P$ and  $C$ .

The homogenization of the flow problem (2.8) leads to Darcy’s equation, which relates $\overline{\boldsymbol{U}}$ and $P$ in terms of the permeability tensor  $\boldsymbol{{\mathcal{K}}}$ :

(2.12a ) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{U}}(\boldsymbol{x},t) & = & \displaystyle -\boldsymbol{{\mathcal{K}}}\unicode[STIX]{x1D735}_{\boldsymbol{x}}P,\end{eqnarray}$$

(2.12b ) $$\begin{eqnarray}\displaystyle \displaystyle \boldsymbol{{\mathcal{K}}}(\boldsymbol{x},t) & = & \displaystyle \frac{1}{|w|}\int _{w_{f}}\unicode[STIX]{x1D646}\,\text{d}\boldsymbol{y}.\end{eqnarray}$$

Here, $\unicode[STIX]{x1D646}$ is a matrix-valued function and together with a vector-valued function $\unicode[STIX]{x1D72B}$ they satisfy the following cell problem at each location  $\boldsymbol{x}$ :

(2.13a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D644}-\unicode[STIX]{x1D735}_{\boldsymbol{y}}\unicode[STIX]{x1D72B}+\unicode[STIX]{x1D6FB}_{\boldsymbol{y}}^{2} & & \displaystyle \,\hspace{-12.0pt}\unicode[STIX]{x1D646}=\mathbf{0},\phantom{\text{periodic}}\hspace{1.6pt}\quad \boldsymbol{y}\in w_{f}(\boldsymbol{x},t),\end{eqnarray}$$

(2.13b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{y}}\boldsymbol{\cdot } & & \displaystyle \,\hspace{-12.0pt}\unicode[STIX]{x1D646}=0,\phantom{\text{periodic}}\hspace{1.6pt}\quad \boldsymbol{y}\in w_{f}(\boldsymbol{x},t),\end{eqnarray}$$

(2.13c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \,\hspace{-12.0pt}\unicode[STIX]{x1D646}=\mathbf{0},\phantom{\text{periodic}}\hspace{1.6pt}\quad \boldsymbol{y}\in \unicode[STIX]{x2202}w_{s}(\boldsymbol{x},t),\end{eqnarray}$$

(2.13d ) $$\begin{eqnarray}\displaystyle & & \displaystyle \,\hspace{-12.0pt}\unicode[STIX]{x1D646},\unicode[STIX]{x1D72B}~\text{periodic},\quad \boldsymbol{y}\in \unicode[STIX]{x2202}w_{f}(\boldsymbol{x},t),\end{eqnarray}$$

where $\unicode[STIX]{x1D644}$ is the identity matrix. For isotropic filter media, the permeability tensor becomes a multiple of the identity, that is, $\boldsymbol{{\mathcal{K}}}={\mathcal{K}}\unicode[STIX]{x1D644}$ with ${\mathcal{K}}$ being a scalar. The macroscopic analogue to the incompressibility condition (2.8b ) is

(2.14) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\boldsymbol{x}}\boldsymbol{\cdot }\overline{\boldsymbol{U}}=\frac{\unicode[STIX]{x1D6FC}}{|w|}\mathop{\sum }_{m\in M_{w}}\int _{\unicode[STIX]{x2202}w_{s}^{m}}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}\,\text{d}s=\frac{\unicode[STIX]{x1D6FC}}{|w|}\mathop{\sum }_{m\in M_{w}}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}(2\unicode[STIX]{x03C0}r^{m})=-\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is given in (2.7).

Under the multiple-scales transformation, the contaminant-transport equation (2.9) becomes

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D719}C)}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}_{\boldsymbol{x}}\boldsymbol{\cdot }(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}\boldsymbol{{\mathcal{D}}}\unicode[STIX]{x1D735}_{\boldsymbol{x}}C-\unicode[STIX]{x1D701}\overline{\boldsymbol{U}}C)-{\mathcal{A}}C,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6FE}$ are given in (2.7) and ${\mathcal{A}}$ is the effective surface area of the fibres, defined as ${\mathcal{A}}=|\unicode[STIX]{x2202}w_{s}|/|w|$ (that is, the surface area per unit volume of the medium). The effective diffusion coefficient $\boldsymbol{{\mathcal{D}}}=\boldsymbol{{\mathcal{D}}}(\boldsymbol{x},t,q)$ is computed as

(2.16) $$\begin{eqnarray}\boldsymbol{{\mathcal{D}}}=\unicode[STIX]{x1D644}-\frac{1}{|w_{f}|}\int _{w_{f}}\unicode[STIX]{x1D645}_{\unicode[STIX]{x1D6E4}}^{\,\text{T}}\,\text{d}\boldsymbol{y},\end{eqnarray}$$

where $(\unicode[STIX]{x1D645}_{\unicode[STIX]{x1D6E4}}^{\,\text{T}})_{ij}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}_{j}/\unicode[STIX]{x2202}y_{i}$ is the transpose of the Jacobian matrix of the vector-valued function $\unicode[STIX]{x1D71E}$ . Its components $\unicode[STIX]{x1D6E4}_{i}$ satisfy the cell problem:

(2.17a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{\boldsymbol{y}}^{2}\unicode[STIX]{x1D6E4}_{i} & = & \displaystyle 0,\hspace{20.30002pt}\quad \boldsymbol{y}\in w_{f}(\boldsymbol{x},t),\end{eqnarray}$$

(2.17b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{y}}\unicode[STIX]{x1D6E4}_{i}\boldsymbol{\cdot }\boldsymbol{n}_{\boldsymbol{y}}^{m} & = & \displaystyle (\boldsymbol{n}_{\boldsymbol{y}}^{m})_{i},\hspace{5.0pt}\quad \boldsymbol{y}\in \unicode[STIX]{x2202}w_{s}^{m}(\boldsymbol{x},t),~m\in M_{w},\end{eqnarray}$$

(2.17c ) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-21.60004pt}\unicode[STIX]{x1D6E4}_{i}\text{periodic},\quad \boldsymbol{y}\in \unicode[STIX]{x2202}w_{f}(\boldsymbol{x},t),\end{eqnarray}$$

where $(\boldsymbol{n}_{\boldsymbol{y}}^{m})_{i}$ are the components of $\boldsymbol{n}_{\boldsymbol{y}}^{m}$ . We note that, in an analogous fashion to the permeability, we have $\boldsymbol{{\mathcal{D}}}={\mathcal{D}}\unicode[STIX]{x1D644}$ for the case of isotropic filter media.

Finally, we obtain the following macroscopic coupling condition from (2.10):

(2.18) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D702}C,\quad m\in M_{w},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is given in (2.7). Multiplying the left-hand side of (2.18) by $|\unicode[STIX]{x2202}w_{s}^{m}|/|w|$ and summing over all fibres in the unit cell, we obtain the following relation between the fibre radii and the macroscopic porosity:

(2.19) $$\begin{eqnarray}\mathop{\sum }_{m\in M_{w}}\frac{|\unicode[STIX]{x2202}w_{s}^{m}|}{|w|}\frac{\unicode[STIX]{x2202}r^{m}}{\unicode[STIX]{x2202}t}=\frac{1}{|w|}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\mathop{\sum }_{m\in M_{w}}\unicode[STIX]{x03C0}(r^{m})^{2}\right)=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t}.\end{eqnarray}$$

Doing the same for the right-hand side of (2.18) and noticing that $\sum _{m\in M_{w}}|\unicode[STIX]{x2202}w_{s}^{m}|/|w|={\mathcal{A}}$ , from (2.18) and (2.19) we obtain

(2.20) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D702}{\mathcal{A}}C.\end{eqnarray}$$

The diffusion of contaminant particles has a twofold impact on the filtration process. First, it acts as a bulk transport mechanism of the contaminant particles, appearing in the dimensionless parameter  $\unicode[STIX]{x1D6FE}$ . Second, it corresponds to the driving feature in the capture mechanism, expressed by (2.20). The molecular diffusion $D$ and correspondingly the dimensionless parameter $\unicode[STIX]{x1D6FE}$ scale as $q^{-1}$ , where $q$ is the size of the contaminant particles, while the diffusion component in the adsorption coefficient $k$ scales as $q^{-2/3}$ according to an empirical adsorption model from Baron & Willeke (Reference Baron and Willeke2001, pp. 205–210). This means that, as particle size increases, the molecular diffusion  $D$ converges to zero faster than the diffusion effect in the adsorption. As a result, it is possible to have a filtration regime where the diffusion term in (2.15) is negligible but particle adsorption is still mostly driven by diffusion. This will be the case in our advection-only regime.

For the sake of clarity, in this study we shall assume that $\unicode[STIX]{x1D702}$ is constant for all filtration regimes and for all fibres, but recognize that in reality it may vary for different filtration regimes and fibres and may also change with time. If we wanted to allow for different adsorption coefficients for different fibre surfaces, we would need to derive the equations in terms of an effective adsorption coefficient instead of the effective surface area  ${\mathcal{A}}$ , but we do not consider this here.

2.2 Microstructure with closely located fibres

In random microstructures, the distance between different fibres varies (see figure 2). As contaminants are being deposited and fibres are growing, fibres located close to one another come into contact and form an agglomerate, while other more distant fibres can continue growing individually. To account for this scenario, we introduce the following agglomeration algorithm. If, as fibres grow radially, two or more fibres come into contact, we replace them with one larger fibre located at the centre of mass of the original fibres and with cross-sectional area equal to the sum of the areas of the individual fibres (see scenario 1 in figure 3 for an illustrative example). When replacing two fibres with a single larger fibre, the resulting fibre may overlap with other fibres located nearby in the unit cell. In such an instance, we recursively replace the overlapping or closely located fibres until the resulting fibre becomes isolated (see scenario 2 in figure 3). Owing to the periodic boundary conditions on the unit cell, the agglomeration algorithm also accounts for the periodic images of fibres.

Figure 3.

Agglomeration algorithm of touching fibres. Scenario 1: two fibres come into contact with one another at some point and are unified into a single fibre with the same volume with the same centre of mass. Scenario 2: two fibres come into contact with one another at some point and are unified into a single fibre which then overlaps with a neighbouring fibre. The unified fibre is then joined with the third fibre to form a fibre with the same volume and centre of mass as the three fibres.

The multiscale model of (2.12)–(2.17) and (2.20) is valid for any random and polydisperse configuration of fibres with different radii in the same unit cell. When, due to the fibre growth (2.18), two or more fibres come into contact, we perform the geometry transformation described above and then reapply the same model to the new geometry.

While this algorithm is clearly an idealization of the real process, it allows us to account for the formation of fibre agglomerates while keeping our strategy simple and robust. In the appendix we investigate the effect of the fibres coming into contact on the numerical simulations, in particular to confirm that the effective diffusivity  $\boldsymbol{{\mathcal{D}}}$ converges to a limiting case.

The continuum assumption that was used to model the contaminant transport is violated as the distance between two fibres becomes comparable with the particle size. To resolve this issue, one can introduce into the agglomeration algorithm a critical distance between two fibres at which they coalesce to form an agglomerate, but we do not include this in our analysis here.

2.3 Multiscale algorithm

In this subsection, we describe the numerical implementation of the multiscale model. First, we perform the microscale simulations as a preprocessing step and find the effective parameters, namely, the permeability $\boldsymbol{{\mathcal{K}}}$ , the effective diffusivity $\boldsymbol{{\mathcal{D}}}$ and the effective surface area  ${\mathcal{A}}$ , as functions of the porosity  $\unicode[STIX]{x1D719}$ . The description of the microscale simulations is presented as a schematic algorithm in figure 4. As input parameters we specify the microstructure type, the initial porosity and the fibre diameter distribution. Using this input, we generate one unit cell in the case of regular microstructures (square or hexagonal), and perform Monte Carlo simulations using multiple random instances of unit cells in the case of random microstructures. Once the unit cell for the given porosity is characterized (with the parameters of interest listed above), we decrease the porosity by a small porosity step $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ . Then, we increase the fibre radii until the new porosity is reached and, if necessary, apply the agglomeration algorithm described in § 2.2. Finally, we compute the effective parameters corresponding to the updated porosity value. We continue this process until we have reconstructed the whole dependence of the effective parameters on the porosity (see, for example, figures 6 and 7). We note that, if we were to take a microstructure configuration obtained at a later time from this algorithm and run the process in reverse (increasing rather than decreasing porosity), then we would not recover the earlier configurations since we lose information about the original microstructure upon merging fibres. We could, however, create a different algorithm to describe a scenario in which the obstacles decrease in size and divide.

As mentioned above and detailed in figure 4, in the case of random microstructures we need to perform Monte Carlo simulations using multiple samples of the unit cell for a given porosity value. Then, the resulting effective parameter $\boldsymbol{{\mathcal{P}}}\in \{\boldsymbol{{\mathcal{K}}},\boldsymbol{{\mathcal{D}}},{\mathcal{A}}\}$ is computed as an average over all samples. Our stopping criterion for the Monte Carlo algorithm reads

(2.21) $$\begin{eqnarray}e_{MC}=\frac{e_{\boldsymbol{{\mathcal{P}}}}}{e_{\boldsymbol{{\mathcal{P}}}}+S_{\boldsymbol{{\mathcal{P}}}}}\leqslant \unicode[STIX]{x1D700},\end{eqnarray}$$

where $e_{MC}$ is the dimensionless Monte Carlo error, $\unicode[STIX]{x1D700}$ is the accuracy, and $e_{\boldsymbol{{\mathcal{P}}}}$ and $S_{\boldsymbol{{\mathcal{P}}}}$ are computed as

(2.22a,b ) $$\begin{eqnarray}e_{\boldsymbol{{\mathcal{P}}}}=\frac{1.96\displaystyle \mathop{\sum }_{i,j}\unicode[STIX]{x1D70E}_{{\mathcal{P}}_{ij}}}{\sqrt{N}},\quad S_{\boldsymbol{{\mathcal{P}}}}=\mathop{\sum }_{i,j}|E_{{\mathcal{P}}_{ij}}|.\end{eqnarray}$$

Here, ${\mathcal{P}}_{ij}$ is the effective surface area with $i,j=1$ or the components of the tensor $\boldsymbol{{\mathcal{P}}}$ with $i,j=1,2$ in the case of the permeability or effective diffusivity and corresponds to the Monte Carlo samples $w$ ; $E_{{\mathcal{P}}_{ij}}$ and $\unicode[STIX]{x1D70E}_{{\mathcal{P}}_{ij}}$ are the mean and standard deviation of ${\mathcal{P}}_{ij}$ , respectively, and $N$ is the number of Monte Carlo samples.

We implement the microscopic algorithm in Python. We generate the unit cells and a triangular unstructured grid with refinement around the fibre surfaces using the open-source mesh generator GMSH (Geuzaine & Remacle Reference Geuzaine and Remacle2009). Then, we discretize the cell problems using the finite element library FEniCS (Alnæs et al. Reference Alnæs, Blechta, Hake, Johansson, Kehlet, Logg, Richardson, Ring, Rognes and Wells2015) using linear Lagrange elements to solve (2.17) and a mixed finite element method to solve (2.13) with linear and quadratic Lagrange elements for $\unicode[STIX]{x1D72B}$ and $\unicode[STIX]{x1D646}$ , respectively.

Figure 4.

Algorithm for microscale simulations.

Once we know the dependences of the effective parameters on the porosity, we use them as coefficients in the macroscale model (2.12), (2.14), (2.15) and (2.20). The system of macroscale equations is also implemented in Python, discretizing time with an implicit backward Euler scheme and space finite elements from the FEniCS library using quadratic Lagrange elements for the pressure and linear Lagrange elements for all other variables. Starting with an initial guess, we find the solution iteratively in time. For a given time step, we solve the mass-conservation equation for the contaminant, equation (2.15), and the equation for the contaminant adsorption, equation (2.20), as a fully coupled nonlinear system using the Newton method. Then, we find the corresponding pressure and velocity distributions from (2.12) and (2.14) using the obtained porosity and proceed to the next time step.

4.1 Filter performance criteria

The performance of the filter media can be evaluated by different criteria. In this study we consider the following metrics.

1. (i) Energy consumption: a small pressure drop $\unicode[STIX]{x0394}P(t)=P(0,t)-P(1,t)$ across the filter medium ensures economic use of energy. The pressure drop is determined by the permeability of the filter medium and the flow velocity.

2. (ii) Throughput: a high fluid velocity $\overline{U}$ achieves large throughput of contaminated fluid across the filter medium. We use the fluid velocity at the inlet $\overline{U}_{in}(t)=\overline{U}(0,t)$ .

3. (iii) Efficiency: to quantify how efficient the filter medium is at trapping the contaminants, we define the number efficiency $E$ as

   (4.1) $$\begin{eqnarray}E(t)=1-\frac{J_{C}(1,t)}{J_{C}(0,t)},\quad \text{with}~J_{C}(x,t)=-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}{\mathcal{D}}\frac{\text{d}C}{\text{d}x}+\unicode[STIX]{x1D701}\overline{U}C,\end{eqnarray}$$
   where $J_{C}$ is the concentration flux.

4. (iv) Dirt-holding capacity: this tells us how much contaminant the filter medium stores over time and, therefore, for how long it can be used. The dirt-holding capacity  $H$ is defined as follows:

   (4.2) $$\begin{eqnarray}H(t)=\unicode[STIX]{x1D70C}\int _{0}^{1}\unicode[STIX]{x1D719}(x,0)-\unicode[STIX]{x1D719}(x,t)\,\text{d}x.\end{eqnarray}$$

5. (v) Lifetime $T>0$ : different termination criteria of the filtration process can be used. For example, in filtration with a constant flow rate, the filter medium can be considered completely loaded when a critical pressure drop is reached. On the other hand, in filtration under a constant pressure drop, the lifetime can be determined when the flow rate becomes too small. We choose a criterion that suits both filtration regimes, namely, we set the lifetime $T$ as the time when a minimum porosity $\unicode[STIX]{x1D719}_{min}=0.5$ is reached at any location of the filter medium.

To quantify the influence of the microstructure on the filtration performance, we introduce some sensitivity characteristics based on the performance criteria discussed above. First, we consider how sensitive the dirt-holding capacity $H$ is to the microstructure. We compute the relative maximum difference in the dirt-holding capacity while all filter media are in operation:

(4.3) $$\begin{eqnarray}S_{H}=\frac{\max _{i,j=1,\ldots ,N}|H_{i}(T^{\ast })-H_{j}(T^{\ast })|}{\max _{i=1,\ldots ,N}H_{i}(T^{\ast })},\quad \text{where}~T^{\ast }=\min _{i=1,\ldots ,N}T_{i},\end{eqnarray}$$

where the subscripts denote characteristics corresponding to different microstructures and $N$ is the number of microstructures considered ( $N=5$ for us). Second, we quantify the deviations of the pressure drop and the fluid velocity depending on the microstructure:

(4.4) $$\begin{eqnarray}\{S_{\unicode[STIX]{x0394}P},S_{\overline{\boldsymbol{U}}}\}=1-\frac{\min _{i=1,\ldots ,N}|f_{i}(T_{i})-f_{i}(0)|}{\max _{i=1,\ldots ,N}|f_{i}(T_{i})-f_{i}(0)|},\quad \text{where}~f=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x0394}P\quad & \text{for}~S_{\unicode[STIX]{x0394}P},\\ \overline{U}_{in}\quad & \text{for}~S_{\overline{\boldsymbol{U}}}.\end{array}\right.\end{eqnarray}$$

Finally, we introduce a sensitivity characteristic for the lifetime of the filter medium in the same way that we introduced it for the pressure drop deviation:

(4.5) $$\begin{eqnarray}S_{T}=1-\frac{\min _{i=1,\ldots ,N}T_{i}}{\max _{i=1,\ldots ,N}T_{i}}.\end{eqnarray}$$

We note that the sensitivity characteristics $S_{H}$ , $S_{\unicode[STIX]{x0394}P}$ , $S_{\overline{\boldsymbol{U}}}$ and $S_{T}$ take values in $[0,1]$ . When they are close to zero, the respective performance criterion, that is, the dirt-holding capacity, the pressure drop, the fluid velocity or the lifetime, is not sensitive to the microstructures. In contrast, a sensitivity characteristic close to one implies there is an important difference in the performance criteria between different microstructures.

4.2 Advection–diffusion regime

Equation (2.15) describes the transport of contaminant by advection and diffusion. Typically for filtration regimes with advection, the time scale for the trapping of contaminant particles is much longer than the time scale for fluid to be advected through the filter medium. Therefore, $\unicode[STIX]{x1D6FD}\ll 1$ and the time derivative in the mass transport (2.15) can be neglected. Hence, we consider the steady state of (2.15) coupled with the contaminant adsorption (2.20). We supplement (2.15) with the following boundary conditions. At the inflow boundary $x=0$ , we specify the contaminant flux $J_{in}=1$ . At the outflow boundary $x=1$ , we use zero Neumann boundary condition for the concentration. Mathematically this reads (see Dalwadi, Griffiths & Bruna Reference Dalwadi, Griffiths and Bruna2015):

(4.6a ) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\left(-\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D719}{\mathcal{D}}\frac{\text{d}C}{\text{d}x}+\unicode[STIX]{x1D701}\overline{U}C\right)\right|_{x=0}=J_{in},\quad t\in [0,T], & \displaystyle\end{eqnarray}$$

(4.6b ) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\frac{\text{d}C}{\text{d}x}\right|_{x=1}=0,\quad t\in [0,T]. & \displaystyle\end{eqnarray}$$

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D719}(x,0)=\unicode[STIX]{x1D719}_{0},\quad x\in (0,1).\end{eqnarray}$$

When $\unicode[STIX]{x1D6FD}\ll 1$ , we can rewrite (2.15) using (2.20) as follows:

(4.8) $$\begin{eqnarray}-\frac{\text{d}J_{C}}{\text{d}x}+\frac{1}{\unicode[STIX]{x1D702}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}t}=0.\end{eqnarray}$$

Using (4.6) and setting $J_{in}=1$ , we can integrate (4.8) over the filter depth and time to obtain:

(4.9) $$\begin{eqnarray}H(t)=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D702}\int _{0}^{t}E(s)\,\text{d}s.\end{eqnarray}$$

Hence, the dirt-holding capacity $H$ becomes a cumulative measure of the number efficiency $E$ in filtration regimes when $\unicode[STIX]{x1D6FD}\ll 1$ and the boundary conditions (4.6) are used. For this reason, in what follows we mainly discuss the dirt-holding capacity  $H$ and not the number efficiency  $E$ in all filtration regimes except in the diffusion-only regime.

4.3 Advection regime

If diffusion is negligible in comparison to advection, we have that $\unicode[STIX]{x1D6FE}\ll 1$ in addition to $\unicode[STIX]{x1D6FD}\ll 1$ and the time derivative in (2.15) being negligible. We note that we must enforce $\unicode[STIX]{x1D6FF}\ll \unicode[STIX]{x1D6FE}\ll 1$ for the asymptotic analysis that we perform on system (2.9) to hold. (The case when $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$ are of comparable orders leads to a different distinguished limit and resulting multiscale model, and this is the subject of a future study.) Then, the Robin boundary condition (4.6a ) reduces to a Dirichlet boundary condition. We solve (2.15) and (4.6a ) and find the concentration distribution as

(4.10) $$\begin{eqnarray}C(x,t)=\frac{J_{in}}{\unicode[STIX]{x1D701}\overline{U}(x,t)}\exp \left(-\int _{0}^{x}\frac{{\mathcal{A}}(\unicode[STIX]{x1D719}(z,t))}{\unicode[STIX]{x1D701}\overline{U}(z,t)}\,\text{d}z\right).\end{eqnarray}$$

Thus, in the advection-only regime the model simplifies greatly. This is particularly useful for an extended parameter study, where the same problem must be solved many times for different sets of parameters.

4.4 Diffusion regime

When diffusion is the dominant transport mechanism of contaminants inside the filter, we neglect the advection ( $\unicode[STIX]{x1D701}\ll 1$ ) and keep all other terms in (2.15). We note that when the diffusion is important the time scale of the contaminant trapping can be comparable with the diffusive processes and so $\unicode[STIX]{x1D6FD}$ may not necessarily be small.

To facilitate the diffusion transport of contaminants across a filter medium, a large difference in concentrations at the opposite sides of the medium has to be maintained, in contrast to when advection is present. To model this set-up mathematically, we specify Dirichlet boundary conditions for the concentration at both sides of the medium:

(4.11a,b ) $$\begin{eqnarray}C(0,t)=C_{in}=1,\quad C(1,t)=0,\quad t\in [0,T].\end{eqnarray}$$

We note that, in this regime, the contaminant transport (2.15) decouples from the fluid flow (2.12a ) and (2.14). Therefore, we do not solve the fluid flow problem in this regime.

4.5 $\overline{U}_{in}=\text{const.}$ regime

To specify the constant flow rate, we use the following boundary conditions for the macroscopic equations (2.12a ) and (2.14):

(4.12a,b ) $$\begin{eqnarray}\overline{U}(0,t)=\overline{U}_{in},\quad P(1,t)=0,\quad t\in [0,T].\end{eqnarray}$$

The parameter  $\unicode[STIX]{x1D6FC}$ in (2.14) represents the ratio between the time scale of the fibre growth due to the contaminant deposition and the time scale of the fluid advection through the filter medium, which is usually very small. Using $\unicode[STIX]{x1D6FC}\ll 1$ and (4.12a,b ), the solution of the 1D flow model (2.12a ) and (2.14) is given by

(4.13a ) $$\begin{eqnarray}\displaystyle \overline{U}(x,t) & = & \displaystyle \overline{U}_{in},\hspace{45.96002pt}\quad x\in (0,1)~\text{and}~t\in [0,T],\end{eqnarray}$$

(4.13b ) $$\begin{eqnarray}\displaystyle \displaystyle P(x,t) & = & \displaystyle \int _{x}^{1}\frac{1}{{\mathcal{K}}(z,t)}\,\text{d}z,\quad x\in (0,1)~\text{and}~t\in [0,T].\end{eqnarray}$$

${\mathcal{K}}$

${\mathcal{K}}$

$E$

4.6 $\unicode[STIX]{x0394}P=\text{const.}$ regime

To model the filtration regime with a constant pressure drop, we use the following boundary conditions for the flow equations (2.12a ) and (2.14):

(4.14a,b ) $$\begin{eqnarray}P(0,t)=\unicode[STIX]{x0394}P,\quad P(1,t)=0,\quad t\in [0,T].\end{eqnarray}$$

In this case the solution of the 1D flow problem is

(4.15) $$\begin{eqnarray}\overline{U}(x,t)=\bar{{\mathcal{K}}}(t)\left(\unicode[STIX]{x0394}P+\int _{0}^{1}\frac{1}{{\mathcal{K}}(\unicode[STIX]{x1D719}(y,t))}\left(\unicode[STIX]{x1D6FC}\int _{0}^{y}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}(z,t)}{\unicode[STIX]{x2202}t}\,\text{d}z\right)\text{d}y\right)-\unicode[STIX]{x1D6FC}\int _{0}^{x}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}(y,t)}{\unicode[STIX]{x2202}t}\,\text{d}y,\end{eqnarray}$$

where

(4.16) $$\begin{eqnarray}\bar{{\mathcal{K}}}(t)=\left(\int _{0}^{1}\frac{1}{{\mathcal{K}}(\unicode[STIX]{x1D719}(y,t))}\,\text{d}y\right)^{-1}.\end{eqnarray}$$

Moreover, using again that $\unicode[STIX]{x1D6FC}\ll 1$ , we find

(4.17) $$\begin{eqnarray}\overline{U}(x,t)=\bar{{\mathcal{K}}}(t)\unicode[STIX]{x0394}P,\quad x\in (0,1),~t\in [0,1].\end{eqnarray}$$

Therefore, in this regime we see that the permeability  ${\mathcal{K}}$ has a direct impact on the fluid velocity and thus the efficiency.

5.1 Advection–diffusion and $\overline{U}_{in}=\text{const.}$ regime

The fluid flow is described by (4.13). The results of the numerical simulations for the five different microstructures are shown in figure 11(a–c). All microstructures exhibit similar filtration behaviour. The reduction of the pore space as contaminants are captured inside the filter medium results in the increase in contaminant capture and the number efficiency $E$ over time. Since we assume that the contaminants only adsorb and do not desorb, the dirt-holding capacity $H$ is also an increasing function, which shows how much contaminant is stored in the filter medium at a given time  $t$ . The dirt-holding capacity $H$ at the lifetime $T$ represents the maximum amount of contaminant that can be stored by this type of medium in this filtration regime.

Regarding the different microstructures, we observe some variations in the number efficiency  $E$ and the dirt-holding capacity  $H$ in figures 11(a) and 11(b), respectively. The random microstructure with no isolation distance and monodisperse fibres has the lowest number efficiency of all the microstructures considered. On the other hand, the random microstructure with the polydisperse fibre radii shows the best number efficiency until $t\approx 0.25$ , after which the regular microstructures exhibit the largest efficiency. Since the random microstructure with no isolation distance does not store as much contaminant as the other microstructures, it is not surprising that its lifetime is the longest (figure 10). However, overall the lifetime of a medium is not very sensitive to the microstructure: the sensitivity characteristic is $S_{T}=0.2$ (see table 3). The variations in the dirt-holding capacity  $H$ for the different microstructures (figure 11 b) are also small ( $S_{H}=0.09$ ). In contrast, the pressure drop $\unicode[STIX]{x0394}P$ (figure 11 c) shows significant variations depending on the microstructure ( $S_{\unicode[STIX]{x0394}P}=0.58$ ). For example, the square grid microstructure reaches a pressure of approximately 160 at the end of the lifetime of the filter medium, while the random microstructure with zero isolation distance reaches only around 60 at the same time.

Figure 11.

Multiscale simulation results for advection–diffusion (a–c) and for advection only (d–f), with $\overline{U}_{in}=\text{const.}$ We show the number efficiency  $E$  (a,d), the dirt-holding capacity  $H$  (b,e) and the pressure drop  $\unicode[STIX]{x0394}P$  (c,f) as functions of time  $t$ .

5.2 Advection and $\overline{U}_{in}=\text{const.}$ regime

Figure 11(d–f) shows the simulation results for the advection-only flow regime with constant flow rate for the different microstructures. The general filtration behaviour is the same as the previous filtration regime. We also observe that the dirt-holding capacity  $H$ is not significantly influenced by the microstructure type ( $S_{H}=0.1$ , figure 11 e), while the pressure drop  $\unicode[STIX]{x0394}P$ depends strongly on the microstructure model ( $S_{\unicode[STIX]{x0394}P}=0.5$ , figure 11 f). The sensitivity of the lifetime of the filter media is $S_{T}=0.26$ .

Comparing this regime with the advection–diffusion regime in § 5.1, we observe that the efficiency and dirt-holding capacity values are higher in the absence of the diffusive transport mechanism at any moment in time. This means that diffusion reduces the filtration efficacy for the same adsorption coefficient (see discussion in § 2.1.2 and further investigations in § 6). Nonetheless, the final dirt-holding capacity values are higher for the advection–diffusion regime than for the case of advection only when the filter media reach their lifetime. This means that the distribution of the dust within the filter depth improves in the presence of diffusion and the filter media can store more contaminants in total.

The pressure drop in the advection–diffusion regime is more sensitive to the different microstructure types than in the advection regime, while the dirt-holding capacity and the lifetime are slightly less sensitive to the microstructure. Overall, in these two filtration regimes with a constant flow rate, the permeability  ${\mathcal{K}}$ significantly impacts the pressure drop, but does not influence the efficiency performance for the reasons discussed in § 4.5. In contrast, variations across microstructures in the effective diffusivity  ${\mathcal{D}}$ and the effective surface area  ${\mathcal{A}}$ have a small effect on the efficiency and dirt-holding capacity. For both filtration regimes, the random microstructure with the polydisperse fibre radii shows the best efficiency and dirt-holding capacity while having an average lifetime. Although this microstructure initially has the highest pressure drop, during the lifetime of the filter medium the pressure drop does not increase as rapidly as for the regular microstructures, and so when the filter reaches its lifetime the pressure drop is as low as that for the random microstructure with isolation distance  $2r$ .

5.3 Advection–diffusion and $\unicode[STIX]{x0394}P=\text{const.}$ regime

In this filtration regime, the fluid flow is determined by (4.17). Figure 12(a–c) shows the simulation results for the different microstructure models. Unlike the previous two regimes, here we hold the pressure drop constant, which results in the decrease of the throughput $\overline{U}_{in}$ over time. This happens because of the reduction of the pore space available for the fluid flow as the contaminants are being stored inside the filter medium. If we continued the simulations, the throughput would keep decreasing until it became zero when the filter medium was completely blocked by the contaminants.

Figure 12.

Multiscale simulation results for advection–diffusion (a–c) and for advection only (d–f), with $\unicode[STIX]{x0394}P=\text{const}$ . We show the number efficiency  $E$  (a,d), the dirt-holding capacity  $H$  (b,e) and the inflow velocity  $\overline{U}_{in}$  (c,f) as functions of time  $t$ .

The number efficiency  $E$ and the dirt-holding capacity $H$ show strong dependences on the microstructure of the filter media in figures 12(a) and 12(b), respectively. The maximum difference in the dirt-holding capacity arises between the polydisperse random microstructure and the monodisperse random microstructure with no isolation distance, which have the best and worst dirt-holding capacity, respectively. The sensitivity of the dirt-holding capacity is characterized by $S_{H}=0.39$ . Again, the filter with the random microstructure with no isolation distance has a longer lifetime than all other microstructures because it does not store as much contaminant (figure 10). Overall, the lifetime of the filter media is quite sensitive to the microstructure ( $S_{T}=0.33$ ). Since the pressure drop is held constant during the filtration, the resulting pressure distribution for all microstructures is the same and is not shown here. Instead, we show the inflow velocity $\overline{U}_{in}$ as a function of time in figure 12(c), which demonstrates the throughput of the fluid. The random microstructure with polydisperse fibre radii has the smallest throughput until around $t=0.3$ at which point it switches with the regular square and hexagonal microstructures. The random microstructure with no isolation distance demonstrates the largest throughput for all time. In general, the throughput is sensitive to the microstructure ( $S_{\overline{\boldsymbol{U}}}=0.36$ ).

5.4 Advection and $\unicode[STIX]{x0394}P=\text{const.}$ regime

This regime is described by (4.10), (4.17) and (2.20). Figures 12(d) and 12(e) show that the number efficiency  $E$ and the dirt-holding capacity  $H$ are highly influenced by the microstructure. The sensitivity characteristic for the dirt-holding capacity is $S_{H}=0.48$ , the largest among the considered filtration regimes. Figure 10 shows the lifetime of the filter media, which varies between 0.28 for the random microstructure with polydisperse fibre radii and 0.5 for the random one with no isolation distance ( $S_{T}=0.43$ ). The throughput of the contaminated fluid has also a large sensitivity, $S_{\overline{\boldsymbol{U}}}=0.39$ .

The filtration regimes when the pressure drop is held constant (this and previous subsection) experience a large increase in the efficiency during the lifetime of the filter. Moreover, these regimes are influenced significantly by the microstructure of the filter medium (see table 3). This implies that the effective parameters in (2.15) have a significant impact on the overall filtration. Comparing the two filtration regimes with the constant pressure drop, we observe that the advection regime is more sensitive to the microstructure than the advection–diffusion regime, which is similar to what we observed for the regimes with constant flow rate.

In both regimes with constant pressure drop, the random microstructure with the polydisperse fibre radii has the highest efficiency and dirt-holding capacity values. However, its lifetime and throughput are one of the lowest. On the other hand, the random microstructure with no isolation distance has the highest throughput but it captures few contaminants. Therefore, the best filter medium in these regimes is one with the random microstructure with isolation distance $2r$ that balances all the performance criteria. This medium offers good efficiency while providing the second largest throughput.

5.5 Diffusion regime

In this regime, we solve (2.15) and (2.20) with boundary conditions (4.11) and parameters specified in table 1. Figures 13(a) and 13(b) show the number efficiency  $E$ and the dirt-holding capacity  $H$ , respectively. Generally, for all microstructures, these characteristics follow the same trends as discussed earlier for the advection–diffusion and $\overline{U}_{in}=\text{const.}$ filtration regime (see the first paragraph in § 5.1). Figure 10 shows the lifetime for all microstructure types, which varies between 0.24 and 0.32. The sensitivity characteristics for these two metrics are $S_{H}=0.17$ and $S_{T}=0.26$ (table 3).

Figure 13.

Multiscale simulation results for diffusion regime. We show the number efficiency  $E$  (a) and the dirt-holding capacity  $H$  (b) as functions of time  $t$ .

This regime is influenced more by the microstructure type than by the filtration regimes with the constant flow rate, but is less sensitive to the microstructure than regimes with the constant pressure drop. The best choice of the microstructure depends on the priorities of the filtration: the efficiency or the lifetime. The highest efficiency is exhibited by the random microstructure with polydisperse fibres; but if the lifetime is more important than the filtration efficiency, then the random microstructure with isolation distance $2r$ becomes appealing.

The lifetime of a filter medium $T$ is determined by the time at which a minimum prescribed value of the porosity is reached at any location of the filter medium. For our filtration set-ups, this location is always at the inflow boundary, $x=0$ , where the maximum concentration and respectively the maximum of the contaminant adsorption are. The diffusion filtration regime has constant concentration at the inflow (see (4.11)). The advection and $\overline{U}_{in}=\text{const.}$ regime also has constant inflow concentration due to the negligible diffusion term, which transforms the constant contaminant influx condition (4.6a ) into a Dirichlet boundary condition for the concentration. For both of these filtration regimes, the porosity at $x=0$ evolves in time according to (2.20) in exactly the same way due to the constant concentration. Therefore, we also observe that the dimensionless lifetime values for these regimes coincide exactly in figure 10. However, we note that the dimensional lifetime will be different for these filtration regimes because the time scaling ${\mathcal{T}}$ for diffusion- and advection-dominated processes are different.