Cake filtration model, membrane

The Cake Filtration Model describes the filtration of particles which are much larger than the pores and will be retained, without entering the pores. The particles deposit on the membrane surface contributing to the boundary layer resistance. Included in this model is deposition due to concentration polarisation. 

As the membrane pores become smaller, the standard blocking law is no longer valid (Figure 6.55B). The cake filtration model remains valid, which indicates prevention of the ferric hydroxide precipitates from entering pores and subsequent internal deposition. Membrane pore radii were tabulated in Table 6.1 as 2,6, 4.8, and 9.1 nm for the 10, 30 and 100 kDa membranes, respectively. Lo and Waite (1998) reported iron hydroxide precipitates to be as small as 10 nm. Nevertheless, complete blocking (pore blocking by the particulates) is not a valid mechanism. Possibly cake formation prevents pore blockage. 

Cake Filtration model. This model treats both the membrane resistance and deposit resistances in a very similar way to that described in Chapter 2. Equation (10.10) is the starting point for fiirther development which is concerned initially with the in ease in cake resistance to some equilibnum value. This is analogous to accounting for the increasing cake d th in convoitional dead-end filtration, see Section 2.5. In membrane filtration a mass balance over the deposit layer is performed ... 

In this filtration model the solute is considered to form a cake or a deposition of panicles at the membrane wall of constant concentration see figure VII - 24). This cake-filtration model is frequently used to determine a fouling index. The total cake layer resistance (R ) is equal to the specific resistance of the cake (r ) multiplied by the cake thickness The specific cake resistance (r ) is assumed to be constant over the cake layer. 

To understand the flux decline in pressure-driven membrane operations, a number of models were developed. Two of the most widely smdied models are the resistance model and the concentration polarization model. The resistance model is the oldest and is based on the cake filtration theory, where it is assumed that a cake layer of rejected particles, which are too large to enter the membrane pores, is formed. The frictional drag due to permeation through these immobile particles leads to additional hydraulic resistance [21]. The cake layer and the membrane are considered as two resistances in series, and the permeate flux is described by Darcy s Law as... 

The mechanisms of membrane fouling can be predicted with empirical mathematical models, which are generally a function of TMP and flux [34, 35]. The fouling models, such as cake filtration, pore, standard, intermediate and complete blockage, are derived from Darcy s law [34, 36, 37]. Those fouling models can be applied for either constant flux [38, 39] or constant TMP [36, 40] operation. 

The MFI is based upon cake filtration theory that particles are retained on the membrane surface during filtration. According to the resistance in series model, the reduction in flux due to the presence of cake layer and the additional resistance from the membrane under constant operating filtration can be described as ... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

Membrane Formation. In earlier work. 2.) it was found that fumed silica particles could be dispersed in aqueous suspension with the aid of ultrasonic sound. Observations under the electron microscope showed that the dispersion contained disc-like particles, approximately 150-200 1 in diameter and 70-80 1 in height. Filtration experiments carried out in the "dead-end" mode (i.e., zero crossflow velocity) on 0.2 urn membrane support showed typical Class II cake formation kinetics, i.e., the permeation rate decreased according to equation (12). However, as may be seen from Figure 7, the decrease in the permeation rate observed during formation in the crossflow module is only t 1, considerably slower than the t 5 dependence predicted and observed earlier. This difference may be expected due to the presence of lift forces created by turbulence in the crossflow device, and models for the hydrodynamics in such cases have been proposed. 

Hermia (1982) introduced the Filtration Laws, which aim to describe fouling mechanisms. The models are valid for unstirred, dead-end filtration (deposition without cake dismrbance due to shear and no gravity settling) and complete rejection of solute by the membrane (but obviously allowing pore penetration). Under conditions where permeate drag dominates, the effect of stirring may be negligible. The constant pressure filtration law is shown in equation (3.11). 

Nowadays, ultrafiltration (UF) or microfiltration (MF) membrane processes are widely used because of their ability to remove particles, colloidal species and microorganisms from different liquids feeds. However a limitation inherent in the process is membrane fouling due to the deposition of suspended matter during filtration. Therefore the understanding of formation and transport properties of particle deposits responsible for membrane fouling is a necessary step to optimize membrane processes. Thus it is necessary to obtain local information in order to analyze and model the basic mechanisms involved in deposit formation and then to further predict the process operation. Besides, it is also useful to control the deposit formation and to plan preventive or curative actions with a controlled efficiency. Nonetheless, local parameters such as cake thickness and porosity are hardly reachable with conventional techniques. 

Here we have employed the filtration volume flux expression (6.3.136b) and the expression (7.2.106) for Vso- For an estimated value of the specific cake resistance Rcs, variation of Sc with z will yield the dependence of Vs on z one can then integrate Vs along z and obtain an expression for the length-averaged filtration flux for the membrane length. This is a major goal for any model. 

---