Modeling mass transfer

Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

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. 

Main axis experimental conversions ( ) intrinsic kinetic model (solid line) kinetic model + mass transfer kinetics (dashed line). Secondary axis variation of computed k,o with flow rate. 

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.]. 

Part I treats the fundamentals of modelling (mass balance ei ualions, involving reaction kinetics and mass-transfer rates), making them readily understandable to those new in the field. 

Here, I review a one-dimensional model for melt moving relative to solid following the work of Spiegelman and Elliott (1993). The physical model described provides the parameters used in the equations tracking residence times differences for decay chain nuclides and thus generating disequilibria. Assuming steady state, the transfer of mass between the solid and melt is described by ... 

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... 

Interfacial area measurement. Knowledge of the interfacial area is indispensable in modeling two-phase flow (Dejesus and Kawaji, 1990), which determines the interphase transfer of mass, momentum, and energy in steady and transient flow. Ultrasonic techniques are used for such measurements. Since there is no direct relationship between the measurement of ultrasonic transmission and the volumetric interfacial area in bubbly flow, some estimate of the average bubble size is necessary to permit access to the volumetric interfacial area (Delhaye, 1986). In bubbly flows with bubbles several millimeters in diameter and with high void fractions, Stravs and von Stocker (1985) were apparently the first, in 1981, to propose the use of pulsed, 1- to 10-MHz ultrasound for measuring interfacial area. Independently, Amblard et al. (1983) used the same technique but at frequencies lower than 1 MHz. The volumetric interfacial area, T, is defined by (Delhaye, 1986)... 

The intrinsic constitutive laws (equations of state) are those of each phase. The external constitutive laws are four transfer laws at the walls (friction and mass transfer for each phase) and three interfacial transfer laws (mass, momentum, energy). The set of six conservation equations in the complete model can be written in equivalent form ... 

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. 

As with the stirred tank model, mass balance equations can be developed to describe mass transfer in plug flow. In this case, it is convenient to define the system as a differential cylindrical section of the tube, with length Az and volume nD2 Az/4, where D is the tube diameter. This system is fixed in space and may... 

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... 

Fig. 2.1. Schematic diagram of a reaction model. The heart of the model is the equilibrium system, which contains an aqueous fluid and, optionally, one or more minerals. The system s constituents remain in chemical equilibrium throughout the calculation. Transfer of mass into or out of the system and variation in temperature drive the system to a series of new equilibria over the course of the reaction path. The system s composition may be buffered by equilibrium with an external gas reservoir, such as the atmosphere.

In this chapter we consider how to construct reaction models that are somewhat more sophisticated than those discussed in the previous chapter, including reaction paths over which temperature varies and those in which species activities and gas fugacities are buffered. The latter cases involve the transfer of mass between the equilibrium system and an external buffer. Mass transfer in these cases occurs at rates implicit in solving the governing equations, rather than at rates set explicitly by the modeler. In Chapter 16 we consider the use of kinetic rate laws, a final method for defining mass transfer in reaction models. 

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. 

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