Macroscopic mass balance models

Johnson and Swindell [77] developed a method for evaluating the complete particle distribution and its effect on dissolution. This method divided the distribution into discrete, noncontinuous partitions, from which Johnson and Swindell determined the dissolution of each partition under sink conditions. The dissolution results from each partition value were then summed to give the total dissolution. Oh et al. [82] and Crison and Amidon [83] performed similar calculations using an expression for non-sink conditions based on a macroscopic mass balance model for predicting oral absorption. The dissolution results from this approach could then be tied to the mass balance of the solution phase to predict oral absorption. 

The macroscopic mass balance model by Tsotsis et al. [1992], when applied to the reaction of ethane dehydrogenation, compares well with experimental data and both show higher conversions than the corresponding equilibrium values based on either tube-or shell-side conditions (pressure and temperature). This is clearly a result of the equilibrium displacement due to the permselective membrane. The conversion, as expected, increases with increasing temperature. 

III. STEADY-STATE MODEL A. Macroscopic Mass Balance Approach... 

Although the pH-partition hypothesis and the absorption potential concept are useful indicators of oral drug absorption, physiologically based quantitative approaches need to be developed to estimate the fraction of dose absorbed in humans. We can reasonably assume that a direct measure of tissue permeability, either in situ or in vitro, will be more likely to yield successful predictions of drug absorption. Amidon et al. [30] developed a simplified film model to correlate the extent of absorption with membrane permeability. Sinko et al. [31] extended this approach by including the effect of solubility and proposed a macroscopic mass balance approach. That approach was then further extended to include facili-... 

The macroscopic approach, in which it is not taken into account what happens inside the cell in detail, but only an overall view of the system is described. In fact, the system is considered as a black box from the fluid dynamic point of view and then, it is assumed that the cell behaves a mixed tank reactor (the values of the variables only depend on time and not on the position since only one value of every variable describes all positions). This assumption allows simplifying directly all the set of partial differential equations to an easier set of differential equations, one for each model species. For the case of a continuous-operation electrochemical cell, the mass balances take the form shown in (4.5), where [.S, ]... 

Chaotic behavior and synchronization in heterogeneous catalysis are closely related. Partial synchronization can lead to a complex time series, generated by superposition of several periodic oscillators, and can in some cases result in deterministically chaotic behavior. In addition to the fact that macroscopically observable oscillations exist (which demonstrates that synchronization occurs in these systems), a number of experiments show the influence of a synchronizing force on all the hierarchical levels mentioned earlier. Sheintuch (294) analyzed on a general level the problem of communication between two cells. He concluded that if the gas-phase concentration is the autocatalytic variable, then synchronization is attained in all cases. However, if the gas-phase concentration were the nonautocatalytic variable, then this would lead to symmetry breaking and the formation of spatial structures. When surface variables are the model variables, the existence of synchrony is dependent upon the size scale. Only two-variable models were analyzed, and no such strict analysis has been provided for models with two or more surface concentrations, mass balances, or heat balances. There are, however, several studies that focused on a certain system and a certain synchronization mechanism. 

Macroscopic mass, energy, and momentum balances provide the simplest starting point for reactor modeling. These equations give little spatial detail, but provide a first approximation to the performance of chemical reactors. This section builds on Chapter 22 of Bird, Stewart, and Llghtfoot (2002). A table of notation is given at the end of the current chapter. 

We shall be interested in a broad, macroscopic, view of reactors—one where elementary mass balances are applied over entire process units. That is, we are not interested in modeling the system in detail, and attainable region (AR) theory does not demand the use of microscopic transport equations for instance. 

The process of diffusion is central to much of what we describe as mass transfer. It manifests in a variety of ways and needs to be looked at repeatedly, even at this introductory level, in order to grasp its full ramifications. This was done by first introducing the reader, in Chapter 1, to the notion of the rate of diffusion, enshrined in Pick s law, and to the linear driving force mass transfer rates derived from it. In Chapter 2 we demonstrated how these rate expressions are incorporated into mass balances leading to models of various mass transfer processes. Chapter 3 took up the topic of diffusivities, the all-important proportionality constants in Pick s law, and examined them both at the molecular and macroscopic level. 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

The rate-based models usually use the two-film theory and comprise the material and energy balances of a differential element of the two-phase volume in the packing (148). The classical two-film model shown in Figure 13 is extended here to consider the catalyst phase (Figure 33). A pseudo-homogeneous approach is chosen for the catalyzed reaction (see also Section 2.1), and the corresponding overall reaction kinetics is determined by fixed-bed experiments (34). This macroscopic kinetics includes the influence of the liquid distribution and mass transfer resistances at the liquid-solid interface as well as dififusional transport phenomena inside the porous catalyst. 

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