Polarisation modelling

Sprik M and M L Klein 1988. A Polarisable Model for Water Using Distributed Charge Sites. Journal of Chemical Physics 89 7556-7560. 

Film and gel-polarisation models are developed for ultrafiltration. These models are also widely applied to cross-flow microfiltration. 

Intermolecular Energy decomposition analyses (EDA) are very useful approaches to calibrate force fields. Indeed, an evaluation of the different physical components of the interaction energy, especially of the many-body induction, is a key issue for the development of polarisable models. 

In the following section, film and gel-polarisation models are developed for ultrafiltration. These models are also widely applied to cross-flow microfiltration, although even these cannot be simply applied, and there is at present no generally accepted mathematical description of the process. 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

This is the same prediction for the limiting slope of a plot of J against In Cf as for the gel-polarisation model. The value of the slope of such plots at all other conditions is less in magnitude than hD. 

Transitions from a localized to an itinerant state of an unfilled shell are not a special property of actinides they can, for instance, be induced by pressure as they rue in Ce and in other lanthanides or heavy actinides under pressure (see Chap. C). The uniqueness for the actinide metals series lies in the fact that the transition occurs naturally almost as a pure consequence of the increase of the magnetic moment due to unpaired spins, which is maximum at the half-filled shell. The concept has resulted in re-writing the Periodic Chart in such a way as to make the onset of an atomic magnetic moment the ordering rule (see Fig. 1 of Chap. E). Whether the spin-polarisation model is the only way to explain the transition remains an open question. In a very recent article by Harrison an Ander-... 

It has been shown that shell models are capable of improving the description, but other shortcomings like the lack of exchange of ligands between the different potential regions restrain the simulation. Polarisable models on the other hand take into account only polarisation effects, while the non-Coulombic contributions remain unchanged. 

Fig. 9. Schematic diagrams of the three types of polarisable potentials. The left-hand diagram shows a point polarisability model (e.g. SK [35] and DC potentials [36]). The centre diagram shows die polarisation on the two 0-H bonds (e.g. NCC potential [37]). The right-hand diagram shows the all-atomic (or three-) polarisation models (e.g. Bernardo et al [44] and Burnham [26]). The lower diagram schematically illustrates the relative orientations of molecular dipole moments of the four nearest neighbour molecules would in possible to cancel out due to the ice rule and give rise a strong local field.

The ry is an interatomic distance, the parameters, z, a, b, c, are related to the atomic species and A B, r hoh, r and gr are related to 0-H pairs. The values of these parameters are given elsewhere [7]. The distances between the H atoms of adjacent water molecules are very different and could lead to considerable orientational variation of the force constants, much as one would expect to find in a polarisable model. However, the original parameters [7] failed to produce a stable structure for ice Ih [74]. A low value for was used in order to stabilise the structure and frequencies distribution fimction similar to earlier work were obtained [72]. 

In the seventies, most of the 37 papers (8-24) that we report are quantum chemical calculations, mainly on H502+ (8-14,20) or H30+(14-20) and a few on larger clusters with n=4-6 (8,9). However these last calculations are not accurate, obtained either from semi-empirical methods (8) or with small basis sets (DZ, 4-31G) and at the SCF level in ab initio calculations (9). The first accurate Cl calculations definitely establish the pyramidal geometry of the oxonium ion (15,16). The first ab initio determination of the barrier in H502+ appeared in 1970 (10). An attempt was made to study the effect of Cl on this barrier (11) and the abnormal polarizability of H502+ (12). At the end of this decade appeared the first Cl ab initio calculation on the excited states of H30+ (19) and the first CNDO calculations on excited states of larger clusters (20). In parallel to these quantum chemistry studies, a kinetic model (21) treats large systems with n=20 and 26, a polarisation model (22) is proposed, and a study on the liquid uses a continuum model (23). 

The Gel Polarisation Model is based on the fact that at steady state flux reaches a limiting value, where increases in pressure no longer increase the flux. According to the Gel Polarisation model, at this limiting value, the solubility limit of the solute in the boundary layer is reached and a gel formed. For 100% rejection, the expression for this hmidng flux (Jiim) is described by equation (3.10). cg is the gel concentration, beyond which the concentration in the boundary layer cannot increase. 

Tu et al. (1997) predicted NF flux by incorporating a gel-layer into the concentration polarisation model. Results corresponded well to filtration experiments with tannic acid. Gill %t al (1988) showed that viscosity effects in the boundary layer are more important than diffusivity. Concentration factors in UF of macromolecules were 40 to 400 times. A similar effect can be expected for large natural organic molecules. Kim et al (1992b) showed cake formation for low initial fluxes, and aggregation for high initial fluxes in protein UF. This demonstrated that solute-solute interactions in the boundary layer are important. 

Moaddeb and Koros (1997) described the deposition of silica on polymeric MF membranes as non-uniform. This means that cake characterisation is difficult as a cracks could vary the results. Meagher et al (1996) stated that attractive interaction between membranes and particles would cause a flux decline, even if the particles were aggregated. Aggregation reduced the flux decline if there was no attraction between the membranes and colloids. The authors outlined the restrictions of the gel polarisation model, as the porosity of the deposit is not accounted for in the model. It was also suggested that the resistance of the gel layer is more important than the particle-surface interaction (what is often referred to as adsorption). 

Li H., Fane A.G., Coster H.G.L., Vigneswaran S. (1999), An assessment of polarisation models of crossflow microfiltration by direct observation through the membrane, submitted to Journal of Membrane Science. 

Concentration profile for a gel-polarised UF membrane Figure 1.7 Concentration/gel polarisation model schematic. In the absence of a gel layer, Cg=Cw. 

According to Equations (1.3) and (1.6), flux is related to the boundary-layer mass transfer coefficient, k. Both k and are calculated from the data plot shown in Figure 1.9 based on the gel polarisation model (Region III in Figure 1.8). Wall concentration,... 

The basic flux relations and separation principles for UF process discussed above are applicable to cross-flow MF. The polarisation model discussed in Section 1.3 is also appH-cable to MF except that the diffusivities of larger particle and colloids are an order of magnitude smaller than those for macromolecules separation in UF. However, in the case of symmetric MF membranes, the separation mechanism is not always a simple sieve mechanism where, the particles, whose sizes are smaller than the pore size, flow freely through the pore while the larger particles are rejected. In many cases the particles to be separated are adsorbed onto the surface of the pore, resulting in a significant reduction in the size of the pore. Since particles also deposit on top of the membrane forming a cake-like... 

The processing time required for a given UF/MF operation is controlled by the membrane area, feed volume to be processed and solute concentration. Of these, solute concentration is the most important factor as it controls flux and the membrane gel concentration, which is determined by the well-known flux-concentration mass transfer limited gel-polarisation model introduced in Chapter 1 as ... 

The effect of slip coefficient on concentration polarisation (CP) was mathematically modeled for flat membrane and tubular membrane systems [12,13,15,16]. Lowering of CP due to slip coefficient as a function of product water recovery ( ) for different normalised diffusion coefficients (a) is shown in Figure 6.8. The data show that CP decreases both with and a. Since a is a measure of particle diffusion from the membrane surface to the bulk solution, slip-flow possibly augments diffusive back-transport of particles from the membrane surface to the bulk solution. Thus, the slip-flow velocity model possibly accounts for higher or actual UF/MF flux, which is under-predicted by the gel polarisation model discussed in Chapter 1. 

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