Membrane filtration with complex branching pore morphology

Membrane filters are in widespread industrial use, and mathematical models to predict their efficacy are potentially very useful, as such models can suggest design modifications to improve filter performance and lifetime. Many models have been proposed to describe particle capture by membrane filters and the associated fluid dynamics, but most such models are based on a very simple structure in which the pores of the membrane are assumed to be simple circularly cylindrical tubes spanning the depth of the membrane. Real membranes used in applications usually have much more complex geometry, with interconnected pores that may branch and bifurcate. Pores are also typically larger on the upstream side of the membrane than on the downstream side. We present an idealized mathematical model, in which a membrane consists of a series of bifurcating pores, which decrease in size as the membrane is traversed. Feed solution is forced through the membrane by applied pressure and particles are removed from the feed by adsorption within pores (which shrinks them). Thus, the membrane's permeability decreases as the filtration progresses. We discuss how filtration efficiency depends on the characteristics of the idealized branching structure.

Figure 1

Magnified membranes with various pore distributions and sizes from (a) Ref. [13] and (b) and (c) Ref. [4]. Photographs in (b) and (c) show samples of width $10\mu \mathrm{m}$.

Figure 2

Schematic of (a) symmetric and (b) asymmetric branching structures with three layers ($m=3$), thicknesses $D_1$, $D_2$, and $D_3$, and specified pressure drop $P=P_0$. In (b) the radius of the $j\mathrm{th}$ pore in layer $i$ and the pressure at the downstream end of this pore are $A_{ij}$ and $P_{ij}$, respectively.

Figure 3

Symmetric branching model. (a) and (b) Pore radius evolution at the top of each pore $a_i(x_i,t)$ for equal (initial) resistance membrane structures, where the initial pore radii in subsequent layers are geometrically decreasing, with geometric coefficient $\kappa$: (a) $\kappa =0.6$ and (b) $\kappa =0.8$. Also shown is the throughput $v\left(t\right)$ vs (c) total flux $q\left(t\right)$ and (d) particle concentration at outlet $c_m(1,t)$, for several different $\kappa$ values shown in the legend. In all cases $r_0=1$, $\lambda =30$, and the number of layers $m=5$.

Figure 4

Symmetric branching model: total throughput $v\left(t\right)$ versus geometric coefficient $\kappa$ with $\lambda =30$ for (a) several different values of dimensionless initial resistance $r_0$ with number of layers $m=5$ and (b) several different numbers of layers $m$, with $r_0=6.66$.

Figure 5

Symmetric branching model: (a) total throughput $v\left(t\right)$ and (b) initial particle concentration at the pore outlet $c_m(1,0)$, versus the geometric coefficient $\kappa$ for several different values of $\lambda$, with $m=5$ and $r_0=1$ (orange curves) and $m=10$ and $r_0=6.66$ (black curves).

Figure 6

Symmetric branching model: total throughput $v\left(t\right)$ versus membrane layer thickness geometric coefficient ${\kappa }_{\mathrm{d}}$ with $\lambda =30$, $r_0=1$, $m=5$ layers, and geometric coefficients $\kappa =0.65,0.707,0.75,0.8$.

Figure 7

Asymmetric branching model: results for membranes with initial dimensionless resistance $r_0=1$ and ratio of right and left branch geometric coefficients ${\kappa }_R/{\kappa }_L=0.8$. (a) and (b) Inlet pore radius evolution in each layer [left $L$ (black) and right $R$ (red)]$a_{R/L,i}(x_i,t)$ for different values of the top layer initial pore radius $a_1\left(0\right)$: (a) $a_1\left(0\right)=0.2512$ and (b) $a_1\left(0\right)=0.1008$. (c) Total flux $q\left(t\right)$ vs throughput $v\left(t\right)$ for $a_1\left(0\right)=0.2512,0.1887,0.1472,0.1194,0.1008$ (red curves) and also for the corresponding symmetric cases of Fig. 3 (black curves). (d) Particle concentration at outlet $c_{mj}(1,t)$ for $j=1,...,2^{m-1}$ versus throughput for the left (black curves) and right (red curves) subbranches, with $\lambda =30$ and $m=5$.

Figure 8

Asymmetric branching model: total throughput $v\left(t\right)$ versus geometric coefficient ratio ${\kappa }_R/{\kappa }_L$, for several branching structures with different initial top pore radii $a_1\left(0\right)$, but the same initial resistance $r_0=1$. In all cases $\lambda =30$ and $m=5$.