Interfacial area mathematical models

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

Surface and Interfacial Tension. Some properties of liquid surfaces are suggestive of a skin that exercises a contracting force or tension parallel to the surface. Mathematical models based on this effect have been used in explanation of surface phenomena, such as capillary rise. The terms surface tension (gas—liquid or gas—solid interface) and interfacial tension (liquid—liquid or liquid—solid) relate to these models which do not reflect the actual behavior of molecules and ions at interfaces. Surface tension is the force per unit length required to create a new unit area of gas—liquid surface (mN/m (= dyn/cm)). It is numerically equal to the free-surface energy. Similady, interfacial tension is the force per unit length required to create a new unit area of liquid—liquid interface and is numerically equal to the interfacial free energy. 

To analyze the behavior of the system with the mathematical model proposed in the previous section, three parameters are needed the chemical equilibrium parameters and of the extraction and of the stripping chemical reactions, respectively, and the product, Afci, of the interfacial area of the emulsion and the kinetic constant of the forward stripping reaction. Estimation of the parameter values needs deep experimental analysis in the literature the following set of parameter values... 

This equation was used to estimate the interfacial adhesion in comparison with the acid-base properties of glass fibers in LDPE. The effect of surface treatment of glass beads on their interfacial adhesion to PET was also estimated from a mechanical property measurement. A mathematical model describing the adsorption of polymers on filler surfaces related coupling density to the average area available for coupling between rubber and filler surface. ... 

In multiphase reactive flows, the interfacial transfer fluxes of momentum, heat and species mass are of great importance. These interfacial transfer fluxes are generally modeled as a product of the interfacial area concentration and a mean interfacial flux. It is normally assumed that the mean interfacial fluxes are, in turn, given as the product of the difference in the phase values of the primitive variables (driving force) multiplied by the transfer (proportionality) coefficients. Mathematically, a generic flux Ifc can be expressed on the form ... 

Reeves, P.C., and M.A. Celia. 1996. A functional relationships between capillary pressure, saturation, and interfacial area as revealed by a pore-scale network model. Water Resour. Res. 32 2345-2358. Rice, J.A. 1995. Mathematical statistics and data analysis. 2nd ed. Duxbury Press, Belmont, CA. Skopp, J. 1985. Oxygen uptake and transport in soils Analysis of the air-water interfacial area. Soil Sci. Soc. Am. J. 49 1327-1331. 

The mathematical models for different kinds of gas-liquid reactors are based on the mass balances of the components in the gas and liquid phases. The bulk gas and liquid phases are divided by thin films where chemical reactions and molecular diffusion occur. The flux of component i from the gas bulk to the gas film is, and the flux from the liquid film to the liquid bulk is, Vf(. The fluxes are given with respect to the interfacial contact area (A)... 

Erickson, Humphrey, and Prokop (1969) have developed mathematical models which can be used to describe batch growth in hydrocarbon fermentation with two types of dispersed system. In the first type of fermentation, the growth-supporting hydrocarbon substrate is dissolved in inert hydrocarbons, and it is assumed that substrate utilization from the dispersed phase causes little or no change in the interfacial area. 

The interfacial contact area is known, constant, and independent of the operation conditions, thus facilitating the mathematical modeling of mass transfer phenomena. 

The mathematical description considered in Section 10.3.3 was used as a modeling basis for the specially developed completely rate-based simulator [80]. This tool consists of several blocks including model libraries for physical properties, mass and heat transfer, reaction kinetics and equilibrium as well as specific hybrid solver and thermodynamic package. It also contains different hydrodynamic models (e.g., completely mixed liquid - completely mixed vapor, completely mixed liquid - vapor plug flow, mixed pool model, eddy diffusion model [80]) and a model library of hydrodynamic correlations for the mass-transfer coefficients, interfacial area, pressure drop, holdup, weeping and entrainment that cover a number of different column internals and flow conditions. 

The geometrical stmctures of packed beds are too complex for simple mathematical description. Therefore, the real bed stmctures are replaced by model stmctures, for instance by a system of parallel channels or a system of dispersed particles. The model stmctures should have the same interfacial area a and the same porosity s as the packed bed. From both conditions follows the hydraulic diameter df, of the channel model stmcture ... 

The practical response to these difficulties has been to lump the interfacial area with the intrinsic rate constant for the mass transfer process. The lumped parameter is known as the effective mass transfer coefficient. This approach permits useful analysis of data such as effluent concentration histories from laboratory columns or wells in the field, and it permits the construction of tractable mathematical models of macroscopic transport (1-5). However, an effective mass transfer coefficient determined in this way for one set of conditions may not apply in a different soil, for a different volume fraction of NAPL, at a different flow rate, etc. 

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