Mass-transfer Models

The volatilization of low-molecular-weight by-products from molten PET can be described by using the classical two-film model or the penetration theory of interfacial transport [95], 

Rafler el al. [105] applied the two-film model to the mass transfer of different alkane diols in poly(alkylene terephthalate) melts and demonstrated a pressure dependency of the mass-transfer coefficient in experiments at 280 °C in a small 3.6L stirred reactor. They concluded that the mass-transfer coefficient kij is proportional to the reciprocal of the molecular weight of the evaporating molecule. 

Laubriet et al. [Ill] modelled the final stage of poly condensation by using the set of reactions and kinetic parameters published by Ravindranath and Mashelkar [112], They used a mass-transfer term in the material balances for EG, water and DEG adapted from film theory J = 0MMg — c ), with c being the interfacial equilibrium concentration of the volatile species i. 

This equilibrium concentration c, or the corresponding mole fraction x, of EG, water and DEG in the interface can be calculated from the vapour pressure and the activity coefficient y, derived from the Flory-Huggins model [13-17], Laubriet et al. [Ill] used the following correlations (with T in K and P in mm Hg) for their modelling  

Rieckmann et al. introduced a mass-transfer concept with a mass-transfer coefficient depending on the average molecular weight of the polymer, the melt 

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Bolles and Fair [129] present an analysis of considerable data in developing a mass-transfer model for packed tower design however, there is too much detail to present here. 

To simulate the empirical concentration profiles, an appropriate mass-transfer model has to be used. One of the simplest models is the model based on the equilibrium-dispersive model, frequently used in column chromatography [1]. It can be given by the following equation ... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

Experimental gas-solid mass-transfer data have been obtained for naphthalene in CO2 to develop correlations for mass-transfer coefficients [Lim, Holder, and Shah, Am. Chem. Soc. Symp. Ser, 406, 379 (1989)]. The mass-transfer coefficient increases dramatically near the critical point, goes through a maximum, and then decreases gradually. The strong natural convection at SCF conditions leads to higher mass-transfer rates than in liquid solvents. A comprehensive mass-transfer model has been developed for SCF extraction from an aqueous phase to CO2 in countercurrent columns [Seibert and Moosberg, Sep. Sci. Techrwl, 23, 2049 (1988) Brunner, op. cit.]. 

Bolles, W. L. and Fair, J. R. (1982) Chem. Eng., NY 89 (July 12) 109. Improved mass transfer model enhances packed-column design. 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

The remaining sections of this chapter provide examples of mass transfer models, presented using the systems approach described above. In many cases, the models are of such importance that they are regarded as theories in their own right. These basic models are also the foundation for the more specific applications in the subsequent chapters of this book. 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

Gouws, J.F., Majozi, T., Gadalla, M., 2008. Flexible mass transfer model for water minimization in batch plants. Chem. Eng. Process., 47 2323-2335... 

Gouws, J.F., Majozi, T., Gadalla, M., 2008. Flexible mass transfer model for water minimization in batch plants. Chem. Eng. Process., 47 2323-2335 Gouws, J.F., Majozi, T., 2009. Usage of inherent storage for minimisation of wastewater in multipurpose batch plants. Chem. Eng. Sci., 64 3545-3554 Majozi, T., 2005. Wastewater minimization using central reusable storage in batch plants. Comput. Chem. Eng., 29 1631-1646... 

When the two liquid phases are in relative motion, the mass transfer coefficients in either phase must be related to the dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive transfer to the Schmidt number. Another complication is that such a boundary cannot in many circumstances be regarded as a simple planar interface, but eddies of material are transported to the interface from the bulk of each liquid which change the concentration profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most industrial circumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass transfer model must therefore be replaced by an eddy mass transfer which takes account of this surface replenishment. 

Two-Film Mass-Transfer Model for Gas-Liquid Systems... 

Gekas, V. 2001. Mass transfer modelling. J. Food Engineer. 49, 97-102. 

Panagiotou, N.M., Karathanos, V.T., and Maroulis, Z.B. 1998. Mass transfer modeling of the osmotic dehydration of some fruits. Int. J. Food Sci. Technol. 33, 267-284. 

Dreybrodt W, Buhmann D. A mass transfer model for dissolution and precipitation of calcite from solutions in turbulent motion. Chem Geol 1991 107-122. 

The phase equilibria of the most important compounds will be described in the following section. In the sections thereafter, we will treat mass transport in melt-phase polycondensation, as well as in solid-state polycondensation, and discuss the diffusion and mass transfer models that have been used for process simulation. 

Roult s law is known to fail for vapour-liquid equilibrium calculations in polymeric systems. The Flory-Huggins relationship is generally used for this purpose (for details, see mass-transfer models in Section 3.2.1). The polymer-solvent interaction parameter, xo of the Flory-Huggins equation is not known accurately for PET. Cheong and Choi used a value of 1.3 for the system PET/EG for modelling a rotating-disc reactor [113], For other polymer solvent systems, yj was found to be in the range between 0.3 and 0.5 [96],... 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

Both the mass-transfer approach as well as the diffusion approach are required to describe the influence of mass transport on the overall polycondensation rate in industrial reactors. For the modelling of continuous stirred tank reactors, the mass-transfer concept can be applied successfully. For the modelling of finishers used for polycondensation at medium to high melt viscosities, the diffusion approach is necessary to describe the mass transport of EG and water in the polymer film on the surface area of the stirrer. Those tube-type reactors, which operate close to plug-flow conditions, allow the mass-transfer model to be applied successfully to describe the mass transport of volatile compounds from the polymer bulk at the bottom of the reactor to the high-vacuum gas phase. 

Before an in-depth discussion of mass transfer models and coefficients we need to be explicitly clear that all mass transfer models are approximations that allow us to solve the partial differential equations (pde) describing an adsorption problem. There are a great many sources that derive and present the partial differential equations that describe adsorption of gases appropriate for column separations. The Design Manual For Octane Improvement, Book I [7] was among the earlier works to show them. The forms as presented by Ruthven [2] are shown here owing to the consistent and compact nomenclature that he has employed. There are a wider array of forms to choose from in the literature including [6, 7]... 

A mass-transfer model was proposed to account for the function of the [BMIM]PF6/water biphasic system in the reaction (Scheme 19). According to this model, a small quantity of [BMIM]PF6 dissolved in water exchanged its anion with H2O2 to form Q OOH , which was transferred into the [BMIM]PF6 phase to initiate the epoxidation reaction. In this reaction system, the ring-opening... 

Figure 1 illustrates the physical mass transfer model for this situation (see the upper cubic particle where d

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