Multistage operation modeling

Unit operation model (black box models such as mixers, separators, component splitters, etc. models of phase separation and relaxation, heat-transfer model, multistage models, pumps and compressors, reactor models such as equilibrium reactor, stoichiometric reactor, tubular reactor, etc. see Chapter 2). 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

If RMCLs were to be set at a nonzero level, use of the linearized multistage model would often appear to be more appropriate than others to meet the congressional intent. The conservative nature of the model could actually mean that the real risk of exposure was probably lower (e.g., 10-7 or 10-8) if any risk actually exists (assuming a nonthreshold mechanism was operative) because the model was structured to be conservative and because of the nature of many of the assumptions in the model. 

Needless to say, kinetic parameters established in this way are empirical factors dependent on the definition of other variables,e.g., averaging of particle size distribution to determine R. Nevertheless, the model provides a rational approach to the complex physical and chemical phenomena of multistage, multireaction leaching. It is intended to expand the model to include countercurrent operations. 

The apparent transfer coefficient of the cathodic reaction, ac, is a measure of the sensitivity of the transition state to the drop in electrostatic potential between electrolyte and metal [109,112]. According to Ref. 113, it is ac = 0.75 for the O2 reduction on Pt in aqueous acid electrolytes. In Ref. Ill the value ac = 1.0 was reported instead. Since the cathodic reaction is a complex multistep process, it might follow several reaction pathways, and the competition between them is affected by the operation conditions (rj, p, T). Therefore, different values of ac have been reported in different regimes of operation. Although in the simple reactions the transfer coefficient is a microscopic characteristic of the elementary act [112], for complex multistage reactions in fuel cell electrodes it is rather an empirical parameter of the model. The dependence of effective a for methanol oxidation on the catalyst layer preparation was recently studied [114]. 

Wardius [51] extended the advancing front model to be employed to multistage mixer-settler systems for liquid membrane operations. They presented a zero order solution to the perturbation equations based on the model developed by Ho et al. [29]. The emulsion globule residence time distribution in each mixer was assumed to be exponential and the fractional utilization of internal reagent was given by... 

Multistage separation columns will operate at unsteady-state conditions during startup or shutdown, or when any of the operating variables change. While the condition of steady-state operation is a basic model assumption for most of the solution methods, it is an assumption that represents an operation that in reality may apply only to limited periods of time, in which steady-state conditions actually prevail. As column conditions change with time, a new steady-state solution will be required. Whereas steady-state models can simulate the column performance at a point in time, dynamic models can simulate the column performance on a continuous time basis. 

Earlier chapters use simplified and binary models to analyze in a very informative manner some fundamentals such as the effect of reflux ratio and feed tray location, and to delineate the differences between absorption/stripping and distillation. Following chapters concentrate on specific areas such as complex distillation, with detailed analyses of various features such as pumparounds and side-strippers, and when they should be used. Also discussed are azeotropic, extractive, and three-phase distillation operations, multi-component liquid-liquid and supercritical extraction, and reactive multistage separation. The applications are clearly explained with many practical examples. 

Then they simulated the extraction and scmbbing behavior from the leaching solution of phosphor waste shown in Table 8 by the equilibrium model for the multicomponent and multistage system. Using simulations, yttrium, europium, and terbium were found to be obtained with the efficiencies and purities shown in Table 9. As such, the simulation based on the equilibrium constant of each element is thought to be useful as a predictive tool and would considerably save the experimental effort to conduct multistage extraction operations. 

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