Mathematical model for filtration and drying in filter membranes

A filter membrane may be frequently used during its lifetime, with filtration and drying processes occurring in the porous medium for several cycles. During these cycles, the concentration and distribution of molecules or contaminants as well as the medium morphology evolve. As a consequence, the filter performance ultimately deteriorates after several cycles. In this work, we formulate a coupled mathematical model for the filtration and drying dynamics in a porous medium occurring consecutively. Our model accounts for the porous medium internal morphology (internal structure, porosity, etc.), the contaminant deposition, and the evolution of dry-fluid interfaces due to evaporation. An asymptotic model is derived based on the small aspect ratio of the thin filter membrane. The reduced model provides insights to the overall porous medium evolution over cycles of filtration and drying processes and predicts the timeline to discard the filter based on its optimum performance. Given the complexity of fluid boundary movements due to the filtration and drying processes, the reduced model still acts as an efficient prediction tool offering a tremendous reduction in computational costs.

Figure 1

Schematics of a membrane region $\widehat{\mathrm{\Omega }}=[0,\widehat{L}]\times [0,\widehat{H}]$. Panels (a) and (b) represent the filtration and the drying (or evaporation) processes, respectively. The filtration process takes place under constant pressure drop across the membrane, i.e., the pressure at $\widehat{y}=0$ and $\widehat{y}=\widehat{H}$ are prescribed as ${\widehat{p}}_{\mathrm{inlet}}$ and 0, respectively. Particle deposition occurs within the membrane with the inlet particle concentration being ${\widehat{c}}_{\mathrm{inlet}}$ at $\widehat{y}=0$. During the drying process, the domain consists of dry ${\widehat{\mathrm{\Omega }}}_d$ and fluid ${\widehat{\mathrm{\Omega }}}_f$ regions in the $\widehat{x}-\widehat{y}$ plane. $\widehat{y}={\widehat{h}}_i(\widehat{x},\widehat{t}), i\in \{T,B\}$ are the dry-fluid interfaces at the top and bottom of the membrane respectively, which evolve in time $\widehat{t}$ due to evaporation.

Figure 2

Numerical simulations for filtration and drying processes. (a) Dynamic solution of the leading-order asymptotic model (52, 53, 54) for the filtration process starting from initial porosity defined in (62). The system parameters are ${\lambda }_f=1, {\delta }_f=8$, and ${\alpha }_f={\lambda }_f/{\delta }_f=0.125$. (b) Dynamic solution of the leading-order model (56, 57, 58) for the drying process starting from the initial porosity and particle concentration defined in (63). The system parameters are $c_{\mathrm{sat}}=0.1, T_d=1, {\lambda }_d=0.1, {\gamma }_d=1, {\eta }_d=0.5, {\alpha }_d=1$, and $\mathrm{\Psi }=0.1$.

Figure 3

(a) Plots of the effective evaporation rates $({\overline{E}}_T,{\overline{E}}_B)$ at the dry-fluid interfaces, as well as the average porosity and particle concentration $(\overline{\varphi },\overline{c})$, respectively, in the fluid region versus time $t$, corresponding to the drying process shown in Fig. 2. (b) The average porosity and particle concentration $(\overline{\varphi },\overline{c})$, respectively, for several values of the evaporation coefficient ${\eta }_d$. The other parameters are identical to those use in Fig. 2. In (a) we have ${\eta }_d=0.5$, and for both (a) and (b) we set ${\gamma }_d=1, T_d=1, {\alpha }_d=1, c_{\mathrm{sat}}=0.1, \mathrm{\Psi }=0.1$, and ${\lambda }_d=0.1$.

Figure 4

The average porosity in the $x$ direction, ${\langle \varphi \rangle }_x\left(y\right)={\int }_0^1\varphi (x,y)dx$, at times $t=0$ (black dotted curve) and $t=1$ for several values of the pore shrinkage coefficient ${\alpha }_d=0.5,0.8,1$. Panel (a) shows the results obtained by solving the drying system in Case I (56, 57, 58); panel (b) corresponds to the results from the reduced system in Case II (57)–(59) where ${\tilde{\lambda }}_d=0.1$. The other parameters are identical to those used in Fig. 2.

Figure 5

(a) Plots of flux $\mathcal{Q}\left(t\right)$ and throughput $\mathcal{V}\left(t\right)$ in time over three filtration-drying cycles with the initial porosity profile given in (62). The evaporation parameter ${\eta }_d=0.5$ and the length of individual filtration/drying period $\mathrm{\Delta }\mathcal{T}=1$. (b) The flux-throughput curves over three filtration-drying cycles for several values of the evaporation coefficient ${\eta }_d$. Other system parameters are identical to those used in Fig. 2.

Figure 6

A plot of the final throughput ${\mathcal{V}}_{\mathrm{overall}}$ as a function of $\mathrm{\Delta }\mathcal{T}$, after a varying number coupled filtration and drying cycles, showing the dependence of the filter performance on the length of individual filtration/drying period $\mathrm{\Delta }\mathcal{T}$ and the total number of cycles $N_{\mathrm{cycle}}$. The initial porosity is given by (62). The system parameters are identical to those used in Fig. 2.