Multiple reaction-progress variables

The method just described for treating multiple reacting-progress variables has the distinct disadvantage that the upper bounds must be found a priori. For a complex reaction scheme, this may be unduly difficult, if not impossible. This fact, combined with the desire to include the correlations between the reacting scalars, has led to the development of even simpler methods based on a presumed joint PDF for the composition vector 

In Section 3.3, the general transport equations for the means, (3.88), and covariances, (3.136), of 0 are derived. These equations contain a number of unclosed terms that must be modeled. For high-Reynolds-number flows, we have seen that simple models are available for the turbulent transport terms (e.g., the gradient-diffusion model for the scalar fluxes). Invoking these models,134 the transport equations become 

the only unclosed terms are those involving the chemical source term. 

As discussed in Section 5.1, the chemical source term can be written in terms of reaction rate functions /, (0). These functions, in turn, can be expressed in terms of two nonnegative functions, (5.6), corresponding to the forward and reverse reactions  

Given the strong assumptions needed to model the joint PDF, the use of more refined models would not be justified. 

---

For reactor design purposes, the distinction between a single reaction and multiple reactions is made in terms of the number of extents of reaction necessary to describe the kinetic behavior of the system, the former requiring only one reaction progress variable. Because the presence of multiple reactions makes it impossible to characterize the product distribution in terms of a unique fraction conversion, we will find it most convenient to work in terms of species concentrations. Division of one rate expression by another will permit us to eliminate the time variable, thus obtaining expressions that are convenient for examining the effect of changes in process variables on the product distribution. 

For a single reaction this was called the fractional conversion X (or Xa), a number between zero and unity, because in a single reaction there is always a single variable that describes the progress of the reaction (we used Ca or X). For multiple reactants and multiple reactions there is not always a single species common to aU reactions to designate as A. However, there is fiequently a most valuable reactant on which to base conversion. We emphasize that by conversion Xj we mean the fractional conversion of reactant species j in all reactions. 

Given the existence of interphases and the multiplicity of components and reactions that interact to form it, a predictive model for a priori prediction of composition, size, structure or behavior is not possible at this time except for the simplest of systems. An in-situ probe that can interogate the interphase and provide spatial chemical and morphological information does not exist. Interfacial static mechanical properties, fracture properties and environmental resistance have been shown to be grealy affected by the interphase. Careful analytical interfacial investigations will be required to quantify the interphase structure. With the proper amount of information, progress may be made to advance the ability to design composite materials in which the interphase can be considered as a material variable so that the proper relationship between composite components will be modified to include the interphase as well as the fiber and matrix (Fig. 26). 

The progress of a given reaction depends on the temperature, pressure, flow rates, and residence times. Usually these variables are controlled directly, but since the major feature of a chemical reaction is composition change, the analysis of composition and the resetting of the other variables by its means is an often used means of control. The possible occurrence of multiple steady states and the onset of instabilities also are factors in deciding on the nature and precision of a control system. 

A series of experiments were performed in a diflFerent reactor to develop a kinetic model for the sulfur generation step see Reaction 2. The progress of the reaction was followed by analyzing the carbon for acid and sulfur content. Each run was made at a different combination of inlet hydrogen sulfide concentration and temperature. The ranges of variables tested were 250-325°F, 0-40% H,S, 0-30% H2O, and 0-24 lbs H2SO4/IOO lbs carbon. A rate equation was developed from these data by multiple-regression techniques ... 

We divide the chapter into two parts Part 1 Mote Balances in Terms of Conversion, and Part 2 Mole Balances in Terms of Concentration, C,. and Molar Flow Rates, F,." In Pan 1, we will concentrate on batch reactors, CSTRs, and PFRs where conversion is the preferred measure of a reaction s progress for single reactions. In Part 2. we will analyze membrane reactors, the startup of a CSTR. and semibatch reactors, which are most easily analyzed using concentration and molar How rates as the variables rather than conversion. We will again use mole balances in terms of these variables (Q. f,) for multiple reactors in Chapter 6. 

---