Reaction progress variable

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. 

Except for the chemical source term, these equations have the same form as those used for the mixture fraction. Note that the chemical source term (S oo) is evaluated using the mixture fraction and reaction-progress variable in the particular environment. The average chemical source term (S2oo(Y2) will thus not be equal to S2cc, (( ), (Y2)) unless micromixing occurs much faster than the second reaction. 

By definition of the reaction-progress variables, Y2 and T22 are zero for the inlet streams, and nonnegative inside the reactor due to the chemical source term. Once the CFD model has been solved, the reactant concentrations in each environment n are found from... 

Fig. 8. Reaction-progress variables F21 and F22 in the cross-section of the confined impinging-jets reactor.

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

The procedure for tracing a kinetic reaction path differs from the procedure for paths with simple reactants (Chapter 13) in two principal ways. First, progress in the simulation is measured in units of time t rather than by the reaction progress variable . Second, the rates of mass transfer, instead of being set explicitly by the modeler (Eqns. 13.5-13.7), are computed over the course of the reaction path by a kinetic rate law (Eqn. 16.2). 

In Section 5.5, we also introduce reaction-progress variables for simple chemistry that are defined differently than 5rp. Although their properties are otherwise quite similar, the reaction-progress variables are always nonnegative, which need not be the case for the reaction-progress vector. 

Note that, due to the choice of c(1) as the reference vector, the mixture-fraction vector l (third and fourth components of y> ) is null. The first component of the mixture-fraction vector thus describes mixing between the initial contents of the reactor and the two inlet streams, and the second component describes mixing with the second inlet stream. For a stationary flow (0) -> 0, and only one mixture-fraction component ( 2) will be required to describe the flow. Note, however, that if c(0) had been chosen as the reference vector, a similar reduction would not have occurred. As expected, the inlet and initial values of the two reaction-progress variables are null. 

Chemical reactions for which the rank of the reaction coefficient matrix T is equal to the number of reaction rate functions R, (i. e 1,..., I) (i.e., Nr = I), can be expressed in terms of / reaction-progress variables Y, (i. e 1,...,/), in addition to the mixture-fraction vector . For these reactions, the chemical source terms for the reaction-progress variables can be found without resorting to SVD of T. Thus, in this sense, such chemical reactions are simple compared with the general case presented in Section 5.1. 

The dimensionless vector of reaction-progress variables Y is then defined to be null in the initial and inlet conditions, and obeys... 

Note that the reaction-progress variable is defined such that 0 < Y < 1. However, unlike the mixture fraction, its value will depend on the reaction rate k = /> Bo. The solution to the reacting-flow problem then reduces to solving two transport equations ... 

Starting from (5.176), the limiting cases of k = 0 and k = oo are easily derived (Burke and Schumann 1928). For the first case, the reaction-progress variable is always null. For the second case, the reaction-progress variable can be written in terms of the mixture fraction as... 

Thus, just as was the case for equilibrium chemistry, the statistics of the reaction-progress variable depend only on the mixture fraction in this limit. In the infinite-rate chemistry... 

For the non-isothermal case, the reaction-progress variable is proportional to the temperature. 

One-step chemistry is often employed as an idealized model for combustion chemistry. The primary difference with the results presented above is the strong temperature dependence of the reaction rate constant k T). For constant-property flows, the temperature can be related to the mixture fraction and reaction-progress variable by a linear expression of the form... 

Using the same inlet/initial conditions as were employed for the one-step reaction, this reaction system can be written in terms of two reaction-progress variables (Fi, Y2) and the mixture fraction f. A linear relationship between c and (co, Y, f) can be derived starting from (5.162) with y = Y2 = A0B0/(A0 + B0) ... 

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