## Crossflow Microfiltration

Dead-end microfiltration, as stated, may suffer from dramatic flux loss because of deposition of particles on the membrane surface and fouling phenomena. Therefore microfiltration is often carried out in the cross-flow mode (Figure 2b). The tangential flow (cross-flow) 'scours' away particles from the membrane surface, and thus limits cake-layer build-up and fouling. Another advantage of cross-flow filtration is the possibility for continuous operation. Cross-flow filtration is used in most industrial large-scale microfiltration plants. For cross-flow microfiltration, screen filters are mainly used.

### Cake-layer Build-up and Fouling

During a cross-flow microfiltration process, a flux behaviour is often observed as shown in Figure 6. The flux declines at first rapidly with time; then the speed of flux decline decreases, and finally a steady state is reached where the flux does not decrease anymore. The decrease in flux is commonly ascribed to two phenomena: cake-layer build-up and fouling.

When filtering a suspension, the membrane retains suspended particles. The particle concentration near the membrane will therefore gradually increase. Cake-layer build-up will occur when the particle concentration near the membrane surface reaches the maximum packing density (0.6-0.7). Cake-layer build-up is thus caused by the particles that are retained by the membrane based on their size, independent of any specific interaction between these particles and the membrane. Cake-layer build-up in microfiltration is a phenomenon similar to concentration polarization in ultrafiltration.

Fouling, on the other hand, is based on a direct contact between solutes and the membrane surface. The term 'fouling' includes many processes, such as adsorption and deposition of macromolecules, bacteria, or small organic molecules on the membrane surface or within the pores. Fouling increases the hydraulic resistance against permeate flow, and thus reduces the capacity of the microfiltration process. Moreover, fouling in general increases the observed retention of the membrane as it reduces the effective pore size.

If one plots the steady-state flux for crossflow microfiltration versus transmembrane pressure (AP) often a curve as given in Figure 7 is obtained. Three regimes can be observed. For low values of AP, the flux increases linearly with AP and often equals the pure water flux. For higher values of AP, the flux curve bends, because of cake-layer build-up, and fluxes become less than the pure water flux. The point where the deviation from the straight line starts is often referred to as the critical flux. For even higher AP, the flux is independent of the pressure. This pressure independent flux value is referred to as the limiting flux.

Factors Influencing Membrane Fouling and Cake-layer Build-up

The extent of membrane fouling and cake-layer build-up depends on many parameters, which can be grouped in three main contributors:

• properties of the membrane,

• properties of the suspension, and

• properties of the process (hydrodynamics).

Membrane properties of importance are hydro-phobicity, surface charge (zeta potential), surface

Figure 7 Steady-state flux versus AP for the cross-flow microfiltration of a silica particle suspension.-, water; — □ —, silica particle suspension.

Figure 7 Steady-state flux versus AP for the cross-flow microfiltration of a silica particle suspension.-, water; — □ —, silica particle suspension.

roughness, and pore-size distribution. In general, macromolecular adsorption is more severe for hydrophobic than for hydrophilic membranes. Fouling by negatively charged colloids is less for negatively charged membranes than for uncharged or positively charged membranes. As most colloids in practical suspensions acquire a negative charge, negatively charged membranes are preferred in general. Membrane fouling is further reduced by choosing membranes with smooth surfaces, small pore sizes, and narrow pore size distributions.

Feed suspension properties of importance are particle concentration, particle charge (zeta potential), ionic strength, and overall composition. The amount of cake-layer build-up increases with particle concentration. Charge effects can reduce fouling by membrane-particle repulsion, and can reduce cake-layer build-up by particle-particle repulsion. Such charge effects are less pronounced at high ionic strength, as the ions present in solution 'shield' the charge of membrane and particles. Overall composition of the feed suspension is of great importance for the fouling behaviour. Fouling may be caused not only by the main particles retained, but also by macromolecules and small organic molecules, which 'geometrically' should pass through the pores easily.

Process properties of importance are the transmembrane pressure and the cross-flow velocity. Low fouling normally occurs at low transmembrane pressures and high cross-flow velocities. More detailed information is given in a later section.

### Calculating the Limiting Flux

To calculate the limiting steady-state flux, local mass balances near the membrane surface are used. It is then assumed that the limiting flux is reached when the amount of particles transported towards the membrane by the permeate flux (convection) equals the amount of particles transported away from the membrane by the cross-flow. The cross-flow can cause back-transport by at least four different mechanisms:

• Brownian diffusion,

• shear-induced diffusion,

• surface transport.

In the following, these mechanisms will be explained. It is assumed throughout this section that the particles are spherical and monodisperse, and that long-term fouling and physicochemical interactions are negligible.

Brownian diffusion If back-transport is caused by Brownian diffusion the standard concentration polarization theory can be used, employing the Brownian diffusion coefficient for spherical particles:

where k is the Boltzmann constant, T is temperature, and a is the particle radius. By numerical calculations, using a suspension viscosity that depends on the particle concentration, it can be shown that the flux is given by:

where tw is the wall shear stress and L is the membrane length.

Eqn [10] predicts fluxes of the right order of magnitude for suspensions of small particles (up to about 1° nm). It under-predicts fluxes by one or two orders of magnitude if applied to suspensions of larger particles. This discrepancy is called the 'flux paradox'. This paradox is explained by assuming that there are other mechanisms for back transport, apart from Brownian diffusion.

Shear-induced diffusion When a shear field is applied to a layer of particles, the particles will tumble over one another, leading to a more loosely packed layer. Obviously the particles must move perpendicular to the applied shear stress to achieve this. The resulting particle migration can be described by employing an effective diffusion coefficient, and is called shear-induced diffusion.

It can be calculated, using an empirical value of the shear-induced diffusion coefficient, that the limiting flux is given by:

VbL J

valid for < 0.2, i.e. for all practical applications. Eqn [11] has been shown to give good flux predictions for suspensions of hard spherical particles, and reasonable flux predictions for complex biofluids such as milk. Although eqn [11] is derived for local viscous flow, flux calculations have also been reported to be accurate for many turbulent flow processes.

Inertial lift If a diluted suspension of particles flows through a duct, particles present close to the wall will migrate towards the centre, perpendicular to the streamlines. This migration, caused by complex hy-drodynamic interactions, is called inertial lift. In cross-flow microfiltration, inertial lift may be able to prevent particles from depositing onto the membrane. To model this phenomenon, it is assumed that a cake layer builds up during microfiltration until the con-vective velocity towards the membrane (the flux J) equals the lift velocity, vL, away from the membrane:

Poaxj

obtained:

Ao tan 0(a2Rc)2/5

The inertial lift theory neglects the influence of a particle on the motion of another particle, resulting in a flux equation which does not depend on the particle concentration. The inertial lift model is therefore only valid for very low particle concentrations. As the flux predicted by eqn [12] increases with the cube of the particle size and the square of the wall shear stress, inertial effects are expected to be important only for large particles ( > 5 |im) and high cross-flow velocities (tw > 10 N m~2).

Surface transport A particle on top of a filter cake is subject to different forces, as shown in Figure 8. The horizontal drag force caused by the cross-flow Ft exerts a clockwise torque on the particle, and the vertical drag force caused by the permeate flux Fj exerts a counterclockwise torque. If the torque caused by the cross-flow is larger than the torque caused by the permeate flux, the particle can roll over the cake layer to the outlet of the membrane. This mechanism of transport is called surface transport.

Equating the clockwise torque with the anticlockwise torque, an equation for the limiting flux is where 0 is the angle of repose (see Figure 8).

Just as for the inertial lift model, the present model neglects the influence of a particle on the motion of another particle, resulting in a flux equation which does not depend on the particle concentration. Eqn [13] overpredicts fluxes for typical microfiltration conditions by an order of magnitude or more. Two of the models described above, the Brownian diffusion model and the shear-induced diffusion model, use a continuum approach. The other two, the inertial lift model and the surface transport model, are based on a single-particle approach. The single-particle approach is only valid for low particle concentrations and large particles.

In Figure 9, the fluxes predicted by the two continuum models are given as a function of particle size for typical cross-flow microfiltration conditions. The flux predicted by the inertial lift model is plotted in the same graph to indicate the order of magnitude of inertial effects. For small particle sizes, Brownian effects dominate and the flux decreases with particle size. For intermediate particle sizes, shear-induced diffusion dominates and the flux increases with particle size. For large particle sizes ( > 5 |im) inertial effects dominate causing the flux to increase even faster with particle size.

The combined effect of Brownian and shear-induced diffusion can be described by:

Figure 10 Flux and amount of matter deposited on the membrane as a function of time for the filtration of a suspension of 0.48-^m silica particles. Circles represent experimental values for a particle concentration of 1.7 kg m-3, a transmembrane pressure of 0.42 bar and a cross flow velocity of 1 m s_1; lines represent model calculations.

Figure 9 Flux calculated according to different models as a function of particle diameter. Calculations were performed for xw = 32 N m~2, <pb = 10 3, and L = 1.2 m, using eqns [10]-[12] for the Brownian, shear-induced, and inertial-lift models, and using eqn [14] for combining Brownian and shear-induced diffusion, ----, Brownian diffusion model; , shear-induced diffusion model; —, Brownian and shear-induced diffusion model; , inertial lift model.

0.1 1 10 Particle diameter (|tm)

Figure 9 Flux calculated according to different models as a function of particle diameter. Calculations were performed for xw = 32 N m~2, <pb = 10 3, and L = 1.2 m, using eqns [10]-[12] for the Brownian, shear-induced, and inertial-lift models, and using eqn [14] for combining Brownian and shear-induced diffusion, ----, Brownian diffusion model; , shear-induced diffusion model; —, Brownian and shear-induced diffusion model; , inertial lift model.

Figure 10 Flux and amount of matter deposited on the membrane as a function of time for the filtration of a suspension of 0.48-^m silica particles. Circles represent experimental values for a particle concentration of 1.7 kg m-3, a transmembrane pressure of 0.42 bar and a cross flow velocity of 1 m s_1; lines represent model calculations.

a steady-state cross-flow filtration model. This approach is illustrated in Figure 10, modelling the transient flux and cake-layer build-up in the cross-flow microfiltration of a suspension of silica particles.

where JBo is the flux according to Brownian theory, eqn [10], and JSI is the flux according to shear-induced theory, eqn [11]. Predictions according to eqn [14] are also given in Figure 9.

Calculating the Transient Behaviour of Cross-low Microfiltration

The time dependence of the flux can be predicted using an approach as outlined in the section on deadend filtration, yet allowing for back-transport according to the particle transport mechanisms described above. Such descriptions are rather complicated, and will not be treated here.

A simple but effective approach to model the transient behaviour of the permeate flux is the use of a combination of transient dead-end filtration theory and a cross-flow filtration model for the steady-state (limiting) flux. While the cake is initially developing, the effect of the cross-flow is small and can be neglected, so that cross-flow filtration theory can be approximated by dead-end filtration theory. Upon approaching the steady state, the cross-flow begins to arrest the cake growth and dead-end filtration theory is no longer accurate. However, near the steady state the flux shows only minor time dependence, and the flux can be approximated by its steady-state value.

The procedure to predict the total transient behaviour of the permeate flux is thus to use dead-end filtration theory (see the section on dead-end microfiltration) until the time the steady-state flux is reached and then use the steady-state flux predicted by

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