Membrane filtration with multiple fouling mechanisms

Manufacturers of membrane filters have an interest in optimizing the internal pore structure of the membrane to achieve the most efficient filtration. As filtration occurs, the membrane becomes fouled by impurities in the feed solution, and any effective model of filter performance must account for this. In this paper, we present a simplified mathematical model, which (i) characterizes membrane internal pore structure via permeability or resistance gradients in the depth of the membrane; (ii) accounts for multiple membrane fouling mechanisms (adsorption, blocking, and cake formation); (iii) defines a measure of filter performance; and (iv) for given operating conditions, is able to predict the optimum permeability or resistance profile for the chosen performance measure.

Figure 1

Schematic showing the pore, the small particles leading to adsorptive fouling, and the forming cake layer consisting of large particles. Here, $C(X,T)$ denotes the concentration of the small particles, while $A(X,T)$ denotes the radius of the pore, at depth $X$ and time $T$.

Figure 2

Simulation for initially cylindrical pore $a(x,0)=0.904$ with ${\lambda }_{\mathrm{m}}=2,{\lambda }_{\mathrm{c}}=0.5,{\beta }_{\mathrm{m}}=0.1,{\beta }_{\mathrm{c}}=0.1,{\rho }_{\mathrm{b}}=2,{\kappa }_{\mathrm{c}}=1$, and $\eta =0.15$. (a) Pore radius (black) and particle concentration (blue/red in membrane/cake) at several different times up to the final blocking time $(t_{\mathrm{f}}$, indicated in the legend). (b) Instantaneous flux through membrane (blue) and cake thickness (red) vs. throughput. Green and black dots correspond to $t=0.05t_{\mathrm{f}}$ and $t=0.1t_{\mathrm{f}}$, respectively.

Figure 3

Simulations for initially uniform pore $a(x,0)=0.904$, with ${\lambda }_{\mathrm{m}}=2,{\rho }_{\mathrm{b}}=2$, and ${\kappa }_{\mathrm{c}}=1$: (a) Flux (blue) and cake thickness (red) versus throughput with ${\beta }_{\mathrm{m}}=0.1$, for several values of ${\mathrm{\Delta }}_{\mathrm{p}}$ (effective large-particle volume), simulated by varying ${\beta }_{\mathrm{c}},{\lambda }_{\mathrm{c}}$, and $\eta$ [see Eqs. (30, 31, 32)]. (b) Throughput (blue) and $t_{\mathrm{f}}$ (red) versus $\eta$, corresponding to varying ${\mathrm{\Delta }}_{\mathrm{p}}$ as in (a). (c), (d) Flux (blue) and cake thickness (red) versus throughput with $\eta =0.15$ as key caking parameters are varied: (c) varying $G$ (large particle concentration) by varying ${\beta }_{\mathrm{m}}$ and ${\beta }_{\mathrm{c}}$ [see Eqs. (24) and (30)], with ${\lambda }_{\mathrm{c}}=0.5$; (d) varying ${\mathrm{\Lambda }}_{\mathrm{c}}$ (attraction coefficient between large and small particles within cake) by varying ${\lambda }_{\mathrm{c}}$ and ${\beta }_{\mathrm{c}}$ [see Eqs. (30) and (31)], with ${\beta }_{\mathrm{m}}=0.1$. Green and black markers correspond to $t=0.05t_{\mathrm{f}}$ and $t=0.1t_{\mathrm{f}}$, respectively. (e) Total throughput $v\left(t_{\mathrm{f}}\right)$, total deposited particle volume and final cake thickness $i\left(t_{\mathrm{f}}\right)$ versus ${\lambda }_c$ (varying ${\mathrm{\Lambda }}_c)$ with parameters as in (d).

Figure 4

Maximum throughput (at times $t_{\mathrm{f}}$ and $t_{20})$ versus initial net porosity ${\overline{\varphi }}_{\mathrm{m}0}$ [defined in Eqs. (46) or  (52)] for the optimum linear, quadratic, and cubic initial pore profiles, with ${\lambda }_{\mathrm{m}}=2,{\lambda }_{\mathrm{c}}=0.5,{\beta }_{\mathrm{m}}=0.1,{\beta }_{\mathrm{c}}=0.1,{\rho }_{\mathrm{b}}=2,{\kappa }_{\mathrm{c}}=1$, and $\eta =0.15$.

Figure 5

(a) Optimum pore profiles; and (b) flux versus throughput for optimum profiles, optimized over the classes of linear, quadratic and cubic pore profiles. Results are shown for initial net membrane porosities ${\overline{\varphi }}_{\mathrm{m}0}=0.2$ and 0.4 [defined in Eq. (46)], with ${\lambda }_{\mathrm{m}}=2,{\lambda }_{\mathrm{c}}=0.5,{\beta }_{\mathrm{m}}=0.1,{\beta }_{\mathrm{c}}=0.1,{\rho }_{\mathrm{b}}=2,{\kappa }_{\mathrm{c}}=1,$ and $\eta =0.15$.

Figure 6

(a) Optimum pore profiles based on maximizing $v\left(t_{\mathrm{f}}\right)$, (b) corresponding flux-throughput graphs, and (c) particle concentration at the membrane outlet $c_m(1,t)$, versus throughput over the classes of linear and quadratic initial pore profiles respectively, with initial net membrane porosities ${\overline{\varphi }}_{\mathrm{m}0}=0.4$, for caking dominated and negligible scenarios [corresponding to the dotted and solid curves in Fig. 3, respectively].