Role of caking in optimizing the performance of a concertinaed ceramic filtration membrane

Membrane filtration is a process of separating particles from fluids. Over time, particles are trapped within the membrane structure and on the membrane surface, forming a cake. In this paper, we develop a mathematical model for the transient blocking dynamics in a concertinaed filtration device composed of angled porous membranes and dead-ends. We examine how the inclusion of particles affects the flow dynamics, and we uncover potential inaccuracies in relying on flux–throughput curves to distinguish between caking and internal blocking dynamics. Moreover, we show that optimal filtration performance strongly depends on both the performance metric and the membrane configuration. Finally, to optimize the use of membrane area, we introduce a method for deriving a nonuniform permeance that ensures constant initial cake growth.

Figure 1

Schematic of the concertinaed filtration device. (a) The full three-dimensional device; reproduced from Ref. [4] with permission. (b) The two-dimensional domain of a single filtration module; the center of the membrane is $(\widehat{x},\widehat{z})=(\widehat{a},\widehat{Z}/2)$ and the center of the domain $(\widehat{x},\widehat{z})=(\widehat{X}/2,\widehat{Z}/2)$ is indicated by a cross.

Figure 2

Dimensionless filtration-module domain $(x,z)\in \left\{\right[0,1]\times [0,1\left]\right\}$. The membrane and cake layers are schematized in red and gray, respectively, with the evolved cake front denoted by $x=s(z,t>0)$. The membrane center is at $(x,z)=(a,1/2)$ and is specified by the parameter $a$. The membrane angle is given by $\beta$.

Figure 3

Analytical flux in $Q_1$ (3.3) in (a) and flux out $Q_2$ (3.4) in (b) plotted as functions of the fluid volume fraction for the case when $s(z,t)=m\left(z\right)$ and the particles are removed on contact with the membrane. Here $\beta =0$ and $a=0.5$ and ${\kappa }_m=0.5,1,1.5,2$.

Figure 4

Evolution of the cake profile $s(z,t)$ over time found by solving (2.36) for $\beta =0,0.4,0.8$ in panels (a), (b), and (c) respectively, with $a=0.5$, ${\varphi }_1^f=0.8$, ${\kappa }_m=k_c=1$, $Q_c={10}^{-2}$. The arrows indicated the direction of growth in time, and $T$ is the end time at which the flux $Q$ (2.37) drops below $Q_c$. Results are shown for eight time steps equally spaced between $[0,T]$ for each angle tested.

Figure 5

Flux–throughput relationship resulting from solving the system of Eqs. (2.36) and calculating the flux $Q$ [Eq. (2.37)] and throughput $V$ [Eq. (2.38)] for $a=0.5$, ${\varphi }_1^f=0.8$, ${\kappa }_m=1$, $Q_c={10}^{-2}$, and $\beta =0,0.2,0.4,0.6,0.8$ in (a). Convexity $C$ (4.1) of the flux–throughput curves indicated through $\text{sgn}\left(C\right)$ in the $({\kappa }_m,\beta )$ domain in (b).

Figure 6

The total throughput $V\left(T\right)$ (2.38) in (a), $\overline{Q}=V\left(T\right)/T$ in (b), and the configuration that maximizes $E\left(\alpha \right)$ (4.2) in (c) calculated from numerical solutions to (2.36) with $\beta \in [0,1]$, $a\in [\beta /2,1-\beta /2]$, ${\kappa }_m=1$, ${\varphi }_1^f=0.8$, and $Q_c={10}^{-2}$.

Figure 7

The upper bound of the constant velocity $U_m^{*}$ calculated in (5.5) using analytical pressure profiles (5.4) and $\beta \in [0,1]$ and $a\in [\beta /2,1-\beta /2]$ given in (a). The nonuniform membrane permeance profiles ${\kappa }_m\left(z\right)$ (5.2) calculated using (5.4) with $\beta =0,0.2,0.4,0.6,0.8$ at $t=0$ in (b) for $a=0.5$ and $U_m=0.02$. Both figures use ${\varphi }_1^f=0.8$.

Figure 8

Schematic of the dimensional unidirectional pipe flow through a membrane. Particles are filtered out by the membrane drawn in pink in the center of the pipe.

Figure 9

Analytical inflow flux (B8a) and outflow flux (B8b) through a pipe with dimensionless pressure drop $\mathrm{\Delta }p=1$ and porous membrane of permeance ${\kappa }_m=0.5,1,1.5,2$.