Empirical Fouling Models

The basis for the experiments as carried out was that the flux decline with time, which is an easily measured quantity, could be correlated with the foulant film growth by the model of Eq. (2), or some other appropriate semi-empirical model. In this way the fouling film thickness could be deduced indirectly. To show this we write from Eqs. (5)-(7)... 

Deo et al. [13] proposed an empirical model based on experimental results obtained with a small-scale spin-filter based bioreactor. They observed that the perfusion flux capacity, which was defined as the perfusion flux at which fouling begun to occur, was proportional to the inverse of the cell concentration and to the square of the tangential velocity at the screen surface (Eq. 21) ... 

Despite the numerous efforts in understanding the fouling mechanisms and the influence of operating parameters involved that lead to flux decline in filtration systems, there remain significant gaps in the present knowledge on this phenomenon. To date, none of the theoretical models, and their numerous modified forms, and empirical models sufficiently explain the influence of membrane-solute interactions and solute-solute interactions on real dairy systems. 

The catalyst activity factor (aj) is time-dependent. Several models have been proposed in the literature, depending on the origin of catalyst deactivation, i.e. sintering, fouling or poisoning (8). The following differential equation can represent semi-empirically different kinds of separable deactivation functions. 

Void [52] developed a variety of ballistic deposition models to simulate sedimentation processes. Void used ballistic models to determine deposition densities for spherical particles which traveled via vertical paths and were deposited on horizontal surfaces. Recently, Schmitz et al. [53] used a ballistic aggregation model to describe particle aggregation at the surface of a crossflow microfiltration membrane. Schmitz and co-workers were able to account for interfacial forces empirically, and demonstrated the influence of physical and chemical variables on the resulting morphology of the fouling deposits (such as aggregate density variation with depth, and influence of shear flow and re-entrainment properties on fouling deposit density and porosity). 

Figure 3. Comparison of multiplet model with empirical fouling correlation for values of A. Ref [6]. Reproduced by permission of Academic Press, hic.

Ideally any model of chemical reaction fouling should take account of these three aspects. Because in many industrial processes, the precise chemical mechanism may not be understood, particularly where the fouling is dependent on impurities, it will be necessary to make simplifying assumptions in order that resultant model has limited complexity. In some respects this approach may be regarded as empirical or semi-empirical. 

As with many model developments for fouling mechanisms the mathematical analysis is valuable since it draws attention to the salient effects of the variables. The present state-of-the-art models put forward, only go so far towards a complete mathematical solution that may be incorporated in the design of heat exchangers operating under chemical reaction fouling conditions. They are useful however, in suggesting a basis for an empirical formula to correlate experimental data that might so be used. 

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. 

Additionally, online monitoring methods have been developed to adapt off-line characterization methods into in situ (i.e., in-reactor) probes for determination of kinetics and monomer conversion with optical methods such as mass spectroscopy (MS), ESR, FTIR, near IR, and Raman spectroscopy. However, frequently, due to high turbidity and viscosity of the polymer reaction milieu, the optical surfaces are easily fouled, leading to frequent sensor failure. Furthermore, data acquired with these probes are model dependent the empirical and inferential calibration schemes used can be expensive and time consuming to develop and can drift and become unreliable as reactor conditions change and as sensors become fouled. Another limiting feature of these methods is that they usually measure only one characteristic of the reaction, such as monomer conversion and are not directly sensitive to polymer molecular mass and intrinsic viscosity. More detailed discussion of these techniques can be found in Chapters 6-10 of this book. 

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