Multiple reactions algorithms

Many of the models encountered in reaction modeling are not linear in the parameters, as was assumed previously through Eq. (20). Although the principles involved are very similar to those of the previous subsections, the parameter-estimation procedure must now be iteratively applied to a nonlinear surface. This brings up numerous complications, such as initial estimates of parameters, efficiency and effectiveness of convergence algorithms, multiple minima in the least-squares surface, and poor surface conditioning. 

Although we have indicated some applications of thermodynamics to biological systems, more extensive discussions are available [6]. The study of equilibrium involving multiple reactions in multiphase systems and the estimation of their thermodynamic properties are now easier as a result of the development of computers and appropriate algorithms [7]. 

There are a number of instances when it is much more convenient to work in terms of the number of moles (iV, N-g) or molar flow rates (Fj, Fg, etc.) rather than conversion. Membrane reactors and multiple reactions taking place in the gas phase are two such cases where molar flow rates rather than conversion are preferred. In Section 3.4 we de.scribed how we can express concentrations in terms of the molar flow rates of the reacting species rather than conversion, We will develop our algorithm using concentrations (liquids) and molar flow rates (gas) as our dependent variables. The main difference is that when conversion is used as our variable to relate one species concentration to that of another species concentration, we needed to write a mole balance on only one species, our basis of calculation. When molar flow rates and concentrations are used as our variables, we must write a mole balance on each species and then relate the mole balances to one another through the relative rates of reaction for... 

Table 8-3 gives the algorithm for the design of PFRs and PBRs with heat exchange for case A conversion as the reaction variable and case S molar flow rates as the reaction variable. The procedure in case B must be used when multiple reactions are present... 

The algorithm for multiple reactions with heat effects... 

In this chapter, we discuss reactor selection and general mole balances for multiple reactions. First, we describe the four baste types of multiple reactions series, parallel, independent, and complex. Next, we define the selectivity parameter and discuss how it can be used to minimize unwanted side reactions by proper choice of operating conditions and reactor selection. We then develop the algorithm that can be used to solve reaction engineering problems when multiple reactions are involved. Finally, a number of examples are given that show how the algorithm is applied to a number of real reactions. 

Closure. After completing this chapter the reader should be able to describe the different types of multiple reactions (series, parallel, complex. and independent) and to select a reaction system that maximizes the selectivity. The reader should be able to write down and use the algorithm for solving CRE problems with multiple reactions. The reader should also be able to point out the major differences in the CRE algorithm for the multiple reactions from that for the single reactions, and then discuss why care must be taken when writing the rate law and stoichiometric Steps to account for the rate laws for each reaction, the relative rates, and the net rates of reaction. 

Figure 11.8 Elementary algorithm for computing equilibrium compositions from multiple reactions occurring in isothermal-isobaric, vapor-Uquid situations. This algorithm combines the Rachford-Rice method for isothermal flash with the reaction-equilibrium method in Figure 11.7.

The algorithm for the design of the operation of batch reactors, with existent only one phase, multiple reactions taking place and operational constraints on temperature present, has been presented earlier (Papaeconomou et al., 2002). The key factor is the existence of desired reactions and of competing reactions, where the latter need to be suppressed. In addition to the operational constraints, there are also end constraints/objectives, namely the molefraction of the limiting reactant in the reaction of interest that should be as low as possible and the progress of the reaction of interest that should be as high as possible. At least one of the end constraints has to be satisfied. 

We then show how to modify our CRE algorithm to solve reaction engineering problems when multiple reactions are involved. Finally, a number of examples are given that show how the algorithm is applied to a number of real reactions. 

The multiple reaction algorithm can be applied to parallel reactions, series reactions, complex reactions, and independent reactions. The availability of software packages (ODE solvers) makes it much easier to solve problems using moles or molar flow rates Fj rather than conversion. For liquid systems, concentration is usually the preferred variable used in the mole balance equations. 

Anahsis In this example we applied our CRE algorithm for multiple reactions to the scries reaction A - B -> C. Here we obtained an analytical solution to find the time at which the concentration of the desired product B was a maximum and. consequently, the time to quench the reaction. We also calculated the concentrations of A. B. and C at this lime, along with the selectivity and yield. 

The algorithm for multiple reactions is own in Table 8S-1. As noted earlier in this chapter, equations for the Rates Step are the major change in our CRE algorithm. 

Salis and Kaznessis proposed a hybrid stochastic algorithm that is based on a dynamical partitioning of the set of reactions into fast and slow subsets. The fast subset is treated as a continuous Markov process governed by a multidimensional Fokker-Planck equation, while the slow subset is considered to be a jump or discrete Markov process governed by a CME. The approximation of fast/continuous reactions as a continuous Markov process significantly reduces the computational intensity and introduces a marginal error when compared to the exact jump Markov simulation. This idea becomes very useful in systems where reactions with multiple reaction scales are constantly present. 

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