Rigorous models

A rigorous model in batch distillation involves consideration of column dynamics along with the reboiler and condenser dynamics. Distefano presented detailed analysis of the characteristics of differential mass and energy balances associated with the complete dynamics of a multicomponent batch distillation column. Distefano s work forms the basis for almost all of the later work on rigorous modeling of batch distillation columns, and this model is presented below. 

The system of equations governing the batch distillation process is difficult to solve as the plate holdup is generally much smaller than reboiler holdup resulting in severe transients. Stiff equation solver is necessary to solve these type of equations. The stiffness of the system is reduced considerably when one considers zero plate holdup. This results in semirigorous model for batch distillation. This model is similar to what was used earlier with McCabe-Theile method (except with additional energy balance equations whenever necessary). 

For columns where the plate dynamics are significantly faster than the reboiler dynamics (due to very small plate holdups and/or wide boiling components), the stiff integrator often fails to find a solution. The solution to this problem is to split the system into two levels (a) the reboiler, where the dynamics are slower, can be represented by differential equations (Equations 4.12-4.13), and (b) the rest of the column can be assumed to be in the quasi-steady state. Thus, the composition changes in the condenser and accumulator (dx /dt), the composition changes on plates (dx j /dt), and the enthalpy changes in the condenser and on plates (Stlo 

The holdup effects can be neglected in a number of cases where this model approximates the column behavior accmately. This model provides a close approximation to the Rayleigh equation, and for complex systems (e.g., azeotropic systems) the synthesis procedures can be easily derived based on the simple distillation residue curve maps (trajectories of composition). However, note that this model involves an iterative solution of nonlinear plate-to-plate algebraic equations, which can be computationally less efficient than the rigorous model. 

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The EVB approach described in this chapter provides a convenient way for estimating the energetics of chemical reactions in various solvents. However, the approximation involved in eq. (2.21) cannot be justified without detailed studies by more rigorous models. Such models will be described in Chapter 3. 

The reader already familiar with some aspects of electrochemical promotion may want to jump directly to Chapters 4 and 5 which are the heart of this book. Chapter 4 epitomizes the phenomenology of NEMCA, Chapter 5 discusses its origin on the basis of a plethora of surface science and electrochemical techniques including ab initio quantum mechanical calculations. In Chapter 6 rigorous rules and a rigorous model are introduced for the first time both for electrochemical and for classical promotion. The kinetic model, which provides an excellent qualitative fit to the promotional rules and to the electrochemical and classical promotion data, is based on a simple concept Electrochemical and classical promotion is catalysis in presence of a controllable double layer. 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

In the mathematical optimization based approaches first a superstructure is created which has embedded a large number of alternative designs. Then mathematical techniques like MINLP are used to find the optimum process within the specified superstructure. For the products considered here there are two big hurdles preventing the large scale use of these techniques (Hill, 2004). Firstly a lot of the physico-chemical phenomena occurring are not completely understood. This makes rigorous modeling difficult. Secondly there is a lack of relevant property models for structured products. 

Such relationship can be obtained using the approaches of rigorous modeling, order-of-magnitude analysis, or black box analysis as suggested by Wibowo and Ng [5], Examples of how SA relates to OV are given in Table 12. Note that due to the complex phenomena involved in these unit operations, shortcut empirical models that have some physical basis are often the most practical to describe the relationship. Unfortunately, such models are rarely available, making it difficult to quantify the relationship. For this reason, this part of the procedure is not emphasized in this article. 

In Figure 2.4, data for the equilibrium constants of esterification/hydrolysis and transesterification/glycolysis from different publications [21-24] are compared. In addition, the equilibrium constant data for the reaction TPA + 2EG BHET + 2W, as calculated by a Gibbs reactor model included in the commercial process simulator Chemcad, are also shown. The equilibrium constants for the respective reactions show the same tendency, although the correspondence is not as good as required for a reliable rigorous modelling of the esterification process. The thermodynamic data, as well as the dependency of the equilibrium constants on temperature, indicate that the esterification reactions of the model compounds are moderately endothermic. The transesterification process is a moderately exothermic reaction. 

Table 2.10 Reactions which should be considered in rigorous modeling of PET synthesis, PET processing and PET recycling...

Kinetic experiments and rigorous modelling of the mass-transfer controlled polycondensation reaction have shown that even at low melt viscosities the diffusion of EG in the polymer melt and the mass transfer of EG into the gas phase are the rate-determining steps. Therefore, the generation of a large surface area is essential even in the prepolycondensation step. 

The simpler models diseussed above (sueh as eases A and B) are usually good enough for continuoits-flow systems where the charrges in liquid arrd vapor holdups and temperatures are not very large. Bateh systems may require the more rigorous models (cases C and D) because of the big variations of most... 

B. RIGOROUS MODEL. Dynamics can be included in a number of ways, with varying degrees of rigor, by using models similar to those in Sec. 3.7. Let us merely indicate how a rigorous model, like case C of Sec. 3.7, could be developed. Figure 3.8 shows the system schematically. 

Thirty years ago these computed variables were calculated using pneumatic devices. Today they are much more easily done in the digital control computer. Much more complex types of computed variables can now be calculated. Several variables of a process can be measured and all the other variables can be calculated from a rigorous model of the process. For example, the nearness to flooding in distillation columns can be calculated from heat input, feed flow rate, and... 

I described a simple method suitable for rigorous modelling of nonlinear one-dimensional structures in the frequency domain. The method was applied to model COST Pll task on slow-wave optieal structures. It was demonstrated that the use of slow-wave structures significantly decreases bistablity threshold. 

Hence, supersaturated conditions exist, which are only physically realistic if water is assumed to be a mist or suspended in the gas. More physical and rigorous models have also been generated. 

P. Durand and J.P. Malrieu Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling., in K.P. Lawley (ed.) Ah Initio Methods in Quantum Chemistry., J. Wiley sons, New York, pp 321 12 (1987). 

Catalyst deactivation is primarily caused by the blockage of active sites due to the coke formed from these olefinic intermediates. Higher hydrogen pressures suppress the diolefin formation, making the selectivity between olefinic intermediates and liquid products (in contrast to coke products) more favorable. However, higher pressures reduce selectivity to aromatics in the desired liquid product. Thus, a rigorous model must accurately predict not only the rates of product formation, but also the formation of coke precursors... 

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