Progress variable, defined

In order to facilitate the calculation of the composition of the reaction mixture at equilibrium, it is convenient to introduce an intensive progress variable defined by the relation... 

Kinetic interpretation of the recombination portion of the [OH] profile proceeds by means of the partial equilibrium approximation, by which the course of change of the entire system composition with recombination is computed, using the measured shock wave speed and initial gas composition and known thermochemical properties of the expected species. To place the progress of recombination in perspective, the change in N from its original value of unity to the value at full equilibrium, A, is reckoned in terms of the normalized progress variable defined as... 

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). 

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. 

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... 

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. 

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 ... 

Despite these difficulties, the multi-environment conditional PDF model is still useful for describing simple non-isothermal reacting systems (such as the one-step reaction discussed in Section 5.5) that cannot be easily treated with the unconditional model. For the non-isothermal, one-step reaction, the reaction-progress variable Y in the (unreacted) feed stream is null, and the system is essentially non-reactive unless an ignition source is provided. Letting Foo(f) (see (5.179), p. 183) denote the fully reacted conditional progress variable, we can define a two-environment model based on the E-model 159... 

Following the description given in the previous section, the detailed chemical system is mapped on two controlling variables the mixture fraction (Z) and a reaction progress variable (c). The mixture fraction describes the mixing of the species and enthalpy, while the progress variable follows the advance of the chemical reaction. Hence, the species mass fractions, temperature, density and species chemical source terms become functions of Z and c. The mixture fraction is defined according to [19] as ... 

The problem in equation (83) is well-posed and solvable by matrix methods, with all eigenvalues being negative, and the derivatives dXi/dz at t = 1 are readily computed from equation (83). These derivatives provide linear relationships between and t in the reaction zone that enable the rates w,-there to be expressed explicitly in terms of t. A progress variable e for the effective one-step process may be defined, which changes from 0 to 1 across the reaction zone, and an equation of the form de/dz = Aoy(T) may be derived from equation (80), where A is inversely proportional to and cd(z) is known. Then... 

In our most straightforward implementation of VTST for gas-phase reactions, rather than allow arbitrary orientations of the dividing surface, we consider a one-parameter sequence of dividing surfaces that are defined in terms of a reaction path [12,13]. This procedure is applicable to complex problems, and it immediately provides a practical improvement over the conventional choice of placing the dividing surface at the saddle point. A robust choice for the reaction path is the minimum energy path (MEP), that is, the path of steepest descent in the mass-scaled coordinates [14]. The coordinates on this path are denoted q (j ) as a function of a progress variable s, and the path is defined by... 

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