Progress variable using

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. 

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. 

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... 

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. 

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). 

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) ... 

Finally, note that all of the expressions developed above for the composition vector also apply to the reaction-progress vector ip or the reaction-progress variables Y. Thus, in the following, we will develop closures using the form that is most appropriate for the chemistry under consideration. 

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... 

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

This combustion model benefits from the low calculation time and memory usage of a flamelet model and is thought to improve the accuracy of gas-phase species and temperature predictions due to the use of a reaction progress variable. Moreover, turbulence-chemistry interactions are taken into account in a physical way. 

A 2-D CFD model has been set up using FLUENT4.5 in a joint EU JOULE project with FLUENT and ALSTOM. As turbulence models the k- model and Reynolds Stress model (RSM) have been applied. As chemistry models a chemical equilibrium model has been applied and on the other hand two models describing finite reaction chemistry, i.e. the laminar flamelet model and the reaction progress variable model. The comparison between experiments and the numerical results from the three chemistry models show that the chemical equilibrium model is sufficient to predict the combustion of LCV gas at elevated pressures, since deviation from chemical equilibrium is small due to the fast reactions. Hence no improvements are expected and have been observed from kinetically limited models. The RSM with constants Cl and C2 in the pressure-strain term proposed by Gibson and Younis [17] seems to yield the best predictions, however, the influence of the type of turbulence model (RSM or k- e) on the species concentrations and temperature predictions is not very large. 

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