Progress variable

A great many reactions in solution can profitably be treated by an extension of the earlier ideas to include more than one progress variable. We will introduce the idea with the general reaction... 

The essential goal is to locate the transition state on the RIP diagram. This involves speeifying two eoordinates, so two quantitative progress variables are required. One approaeh, fairly widely applied, ean be illustrated with the study by Hill et al. of the general aeid-eatalyzed addition of substituted anilines to diey-anamide. The overall reaetion is... 

For a system describable with a single progress variable, we derived the Marcus equation, Eq. (5-76). 

Let X be the normalized progress variable in a system subject to the Marcus equation (Eq. 5-69), so jc = -t- AG°/8AGo, where has the significance of a in Eq. (5-67). Then deduce this equation, which describes the energy change, relative to the reactant, over the reaction coordinate ... 

By analogy with Problem 8, we write for a system having two progress variables... 

It is further assumed that the mesophase layer consists of a material having progressively variable mechanical properties. In order to match the respective properties of the two main phases bounding the mesophase, a variable elastic modulus for the mesophase may be defined, which, for reasons of symmetry, depends only on the radial distance from the fiber-mesophase surface. In other words, it is assumed that the mesophase layer consists of a series of elementary peels, whose constant mechanical properties differ to each other by a quantity (small enough) defined by the law of variation of Ej(r). 

The Eulerian approach requires a measurement of the temperature or the progress variable at many sample points at a given normal distance from the ignition plane, at a given time elapsed since ignition. The progress variable introduced here can be for instance a normalized temperature or concentration that varies from... 

In this simplified situation, can we really consider that the mean flame structure and thickness are steady, after certain delay and distance from initiation, and then the "turbulent flame speed" is a well-defined intrinsic quantity Indeed, with the present state of knowledge, there is no certainty in any answer to this question. Of course, it is hardly possible to build an experiment with nondecaying turbulence without external stirring. In deca)dng turbulence, the independence of the turbulent flame speed on the choice of reference values of progress variable has been verified in neither experiment nor theory. 

Turbulent flame-front regime. Eddy-like contortions of fhe flame preheaf and burned gases zones give rise to "ouf of fronf" islands and peninsula sfructures of intermediate progress variable values. Scalar transport becomes gradient-like. 

Radial and axial components of the Favre mean flux of progress variable obtained in a Bunsen burner geometry for different operating conditions. (Reproduced from Chen, Y.C. and Bilger, R., Combust. Sci. Tech., 167,18 2001. With permission. Figure 19, p. 218, copyright Gordon Breach Science Publishers (Taylor and Francis editions).)... 

Y.C. Chen and R. Bilger 2001, Simultaneous 2-D imaging measurements of reaction progress variable and OF radical concentration in turbulent premixed flames Instantaneous flame front structure. Combust. Sci. Tech. 167 187-222 (more informations through www.infor-maworld.com). 

J2.2.2 Methods of Following the Course of a Reaction. A general direct method of measuring the rate of a reaction does not exist. One can only determine the amount of one or more product or reactant species present at a certain time in the system under observation. If the composition of the system is known at any one time, then it is sufficient to know the amount of any one species involved in the reaction as a function of time in order to be able to establish the complete system composition at any other time. This statement is true of any system whose reaction can be characterized by a single reaction progress variable ( or fA). In practice it is always wise where possible to analyze occasionally for one or more other species in order to provide a check for unexpected errors, losses of material, or the presence of side reactions. 

The chemical composition of many systems can be expressed in terms of a single reaction progress variable. However, a chemical engineer must often consider systems that cannot be adequately described in terms of a single extent of reaction. This chapter is concerned with the development of the mathematical relationships that govern the behavior of such systems. It treats reversible reactions, parallel reactions, and series reactions, first in terms of the mathematical relations that govern the behavior of such systems and then in terms of the techniques that may be used to relate the kinetic parameters of the system to the phenomena observed in the laboratory. 

Swain (7) has discussed the general problem of determining rate constants from experimental data of this type and some of the limitations of numerical curve-fitting procedures. He suggests that a reaction progress variable for two consecutive reactions like 5.3.2 be defined as... 

If we let and t2 represent the times corresponding to reaction progress variables and <5J, respectively, the time ratio t2/tl for fixed values of <5 and <5 will depend only on the ratio of rate constants k. One may readily prepare a table or graph of <5 versus k t for fixed k and then cross-plot or cross-tabulate the data to obtain the relation between k and ktt at a fixed value of <5. Table 5.1 is of this type. At specified values of <5 and S one may compute the difference log(fe1t)2 — log f) which is identical with log t2 — log tj. One then enters the table using experimental values of t2 and tx and reads off the value of k = k2/kv One application of this time-ratio method is given in Illustration 5.5. 

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. 

The partial pressures of the various species are numerically equal to their mole fractions since the total pressure is one atmosphere. These mole fractions can be expressed in terms of a single reaction progress variable-the degree of conversion-as indicated in the following mole table. 

The description of Eqs. (58) and (59) in terms of the mixture fraction and reaction-progress variables is described in detail by Fox (2003). Here we will consider a variation of Eq. (59) wherein the acid acts as a catalyst in the second reaction (Baldyga et al., 1998) ... 

The microscopic transport equations for the reaction-progress variables can be found from the chemical species transport equations by generalizing the procedure used above for the acid-base reactions (Fox, 2003). If we assume that Fa Fb Fc, then the transport equations are given by... 

Note that since the reaction rates must always be nonnegative, the chemically accessible values of the reaction-progress variables will depend on the value of the mixture fraction. We will discuss this point further by looking next at the limiting case where the rate constant Ay is very large and k2 is finite. 

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