On the performance of multilayered membrane filters

Abstract

Multilayered membrane filters, which consist of a stack of thin porous membranes with different properties (such as pore size and void fraction), are widely used in industrial applications to remove contaminants and undesired impurities (particles) from a solvent. It has been experimentally observed that the performance of well-designed multilayer structured membranes is markedly better than that of equivalent homogeneous membranes. Mathematical characterization and modeling of multilayered membranes can help our understanding of how the properties of each layer affect the performance of the overall membrane stack. In this paper, we present a simplified mathematical model to describe how the pore-scale properties of a multilayered membrane affect the overall filter performance. Our results show that, for membrane stacks where the initial layer porosity decreases with depth, larger (negative) porosity gradients within a filter membrane are favorable for increasing throughput and filter lifetime, but at the expense of moderately poorer initial particle retention. We also found that the optimal layer thickness distribution that maximizes total throughput corresponds to a membrane stack with larger (negative) porosity gradients in which layer thickness increases slightly between successive layers in the depth of the membrane.

Appendix A: Constant flux case

Membrane fouling is often characterized in laboratory experiments by challenging the membrane with a constant pressure drop. However, many industrial micro-filtration applications operate at constant flux, and there are few papers that compare these modes of operation for multilayered membranes [42]. In this appendix, we briefly outline how the results change if boundary conditions of constant flux are imposed. We first outline the modifications to the model in Sect. 3, then present sample numerical simulations.

1.1 The model for specified flux

The original model (1)–(11) remains unchanged except that the boundary condition for the pressure at the upstream membrane surface is now time-dependent, $p_0(t)$ (the membrane resistance increases in course of time, therefore the imposed pressure at the membrane upstream should increase to sustain the flux). We nondimensionalize the model using the same scalings as in (12), except for

$$\begin{aligned} p=\frac{h \mu }{u_0\chi } p^*, \quad (u,u_\mathrm{f})=u_0 (1,u_\mathrm{f}^*). \end{aligned}$$

(29)

The dimensionless model is reduced to

$$\begin{aligned}&c=\text {exp}\Bigg [ - \int _0^x \tilde{\lambda } \phi ^{2/3} + \tilde{\delta } (1-\phi ^{1/3}) \;\mathrm{d}x'\Bigg ], \end{aligned}$$

(30)

$$\begin{aligned}&\frac{\partial \phi }{\partial t}= -\frac{\tilde{\lambda }}{\tilde{\delta }}\phi ^{2/3} c - (1-\phi ^{1/3}) c, \end{aligned}$$

(31)

where $\tilde{\lambda } = (h\bar{\lambda })/u_0$, $\tilde{\delta } = h\bar{\delta }$, $\tilde{\lambda }/\tilde{\delta } = \bar{\lambda }/u_0\bar{\delta }$, the initial condition is as in (24), and the modified Darcy pressure p within the membrane is given by

$$\begin{aligned} p=\int _x^1 \frac{(1-\phi )^2}{\phi ^3} \;\mathrm{d}x'. \end{aligned}$$

(32)

Note that the pressure at the upstream membrane surface p(0, t) can be calculated by setting $x = 0$ in (32).

1.2 Results

Figure 10a–e shows results for the same initial porosity profiles given in (27). Figure 10f shows the inverse pressure drop as a function of throughput for each of those porosity profiles. We observe that there is a significant difference in fouling behavior between the two operating modes. In contrast to the constant pressure simulations, the rate of fouling near the membrane inlet is observed to be less severe, which enables fouling to occur much more uniformly within the multilayered membrane. This is particularly true of membranes whose initial porosity profile is monotonically decreasing with depth, illustrated here by $\phi _2$. As the resistance increases with time due to fouling, the pressure drop must be increased to maintain constant flux. Thus we plot the inverse pressure drop instead of flux as a function of throughput in Fig. 10f to illustrate the fouling behavior, for the five initial porosity profiles given in Eq. (27). If the fluid is maintained at constant flux until total blockage is reached, the pressure must increase to infinity, which is of course not practical. In most industrial filtration systems, the fluid is pumped at constant flux (flow rate) till the maximum pressure (based on practical constraints of the system under consideration) is reached and then the fluid handling system is automatically switched to the constant pressure operating mode, with the pressure fixed at this maximal value (this can then be described by the model as discussed in Sect. 3). As with the constant pressure simulations, the case in which the initial porosity profile decreases monotonically with depth ($\phi _2$) gives significantly better performance (more total throughput and the longest time until total blockage is reached), while the membrane with initial porosity profile monotonically increasing along the filter depth ($\phi _3$) gives the least total throughput and the shortest life span.

Fig. 10

Simulations at constant flux: porosity profile and particle concentration as functions of dimensionless space at selected times up to final time ($t_\mathrm{f}$, indicated in the legends) for different initial porosity profiles given in Eq. (27): a $\phi _1(x,0)$, b $\phi _2(x,0)$, c $\phi _3(x,0)$, d $\phi _4(x,0)$, e $\phi _5(x,0)$, and f inverse pressure drop as a function of throughput for these initial porosity profile, with $\tilde{\lambda } = 1$, $\tilde{\delta } = 8$, and $r_0 = 1.5$