54 Ransbased Models Of Reactive Flow Processes

Reynolds-averaged equations for momentum transport, are already discussed in Chapter 3. For modeling reactive flow processes, in addition to the solution of overall mass conservation equation described in Chapter 3, it is necessary to solve conservation equations for individual species. Following the practices of Reynolds averaging, an instantaneous concentration of species k, Ck, can be written as the sum of time-averaged species concentration Ck and a fluctuation around the time average, C'k . The time averaged conservation equation for species k can therefore be written:

dt (pCk) + V ■ (pUCk + pU7CTk) = V ■ (j) + Sk (5.11)

Note that unlike conventional molar units, concentration of species in the above equation is written in terms of mass fractions (Ck = mk ). Similar to the momentum equations discussed in Chapter 3, time averaging introduces unknown terms in the averaged equations. It is necessary to close the equation by modeling these unknown terms representing the turbulent flux of component k and the mean reaction rate. The scalar flux is dominated by the transport due to velocity fluctuations in the inertial sub-range of the energy spectrum. It is, therefore, primarily a term related to an integral scale of turbulence and is independent of molecular diffusivity. The gradient diffusion model is often employed to relate the scalar flux to the mean field:

where vT is turbulent momentum diffusivity and aTk is turbulent Schmidt number for component k. Although the gradient diffusion assumption may fail in some situations, it is typically employed to carry out engineering simulations of complex reactive flow processes.

The most difficult term to close in Eq. (5.11) is the reaction rate term. Reaction rates are seldom formulated by considering all the elementary reactions. More often than not, the reactive system is represented by a lumped mechanism, considering only a few species. The case of m components participating in n independent chemical reactions is usually represented by two two-dimensional matrices (m x n) of stoichiometric coefficients and order of reactions and two one-dimensional vectors (n) of frequency factors and activation energy. n chemical reactions are written:

Stoichiometric coefficients (zrk ) are generally considered positive for products and negative for reactants. Each chemical reaction is associated with its kinetics representing dependence of net rate of reaction on concentrations of participating species and temperature. Dependence on concentrations of participating species is represented by order of reaction, 'o'. The rate is represented by two parameters, frequency factor, k0, and activation energy, AE (see textbooks such as Levenspiel, 1972 for more discussion on these two parameters). The net rate of formation or consumption of component k due to reaction n is usually written:

where zkn is molar stoichiometric coefficient for species k in reaction n. k0n and AEn are frequency factor (pre-exponential factor) and activation energy for reaction n, respectively. R' is the universal gas constant. The product is taken over all participating reactants with oln is an order of reaction n with respect to reactant l. The net reaction source term for species k is calculated as a sum of the reaction sources over the n chemical reactions:

For any industrial reacting system, the relevant parameters appearing in the rate expression (Eq. (5.14)) need to be obtained by carrying out experiments under controlled conditions. It is necessary to ensure that physical processes do not influence the observed rates of chemical reactions. This is especially difficult when chemical reactions are fast. It may sometimes be necessary to employ sophisticated mathematical models to extract the relevant kinetic information from the experimental data. Some references covering the aspects of experimental determination of chemical kinetics are cited in Chapter 1. It must be noted here that in the above development, the intrinsic rate of all chemical reactions is assumed to follow a power law type model. However, in many cases, different types of kinetic model need to be used (for examples of different types of kinetic model, see Levenspiel, 1972; Froment and Bischoff, 1984). It is not possible to represent all the known kinetic forms in a single format. The methods discussed here can be extended to any type of kinetic model.

When chemical reactions are slow (with respect to mixing) it is not necessary to employ additional models to close the reaction source terms. For slow reactions (Da ^ 1), turbulent mixing will be complete before the reaction can take place. The contributions of fluctuating concentrations may be neglected. Therefore, the time-averaged reaction source term can be related to the time-averaged temperature and species concentrations:

For fast and intermediate reactions, the time-averaged reaction source term will contain some additional terms. These additional terms need to be modeled to close the set of equations. For example, consider the case of a single second-order reaction with instantaneous rate given by

The non-linearity in terms of concentrations and exponential factor containing temperature, make the task of closing the reaction source term quite difficult. Even for an isothermal system, the time-averaged reaction source term will contain a new term, the time average of the product of fluctuating concentrations ('c') of component 1 and component 2:

Closure models for terms like the second term in the bracket of the right-hand side are vital to the modeling of turbulent reactive flow processes. It must be noted that as the chemistry becomes more complicated, several such terms will appear, which will make the task of modeling more difficult. Various methods have been used to develop such closure models. These methods are classified into two groups, namely conventional closure models with or without using probability distribution functions

(PDF) and phenomenological models, which are not closure methods in the accepted sense.

5.4.1. Closure Models/PDF-based Models

If the chemical reactions are very fast compared to the mixing rate, it may be assumed that any mixed reactants are immediately reacted. No rate expression is therefore necessary. The simplest model to represent such cases is called the 'eddy break up (EBU) model' (Spalding, 1970; Magnussen and Hjertager, 1976). In the EBU model, the effective rate of chemical reactions is equated to the smaller of rate calculated based on kinetic model and that based on the eddy break-up rate. The eddy breakup rate is defined as the inverse of a characteristic time scale k/e. Therefore, for fast reactions, the rate of consumption or formation is proportional to the product of density, mass fraction and the eddy break-up rate (e/k). The model is useful for the prediction of premixed and partially premixed fast reactive flows. EBU, however, was originally developed for single-step chemical reactions. Its extension to multiple step reactive systems should be made with caution.

For complex chemistry, in many cases, a 'conserved scalar' or a 'mixture fraction' approach can be used, in which a single conserved scalar (mixture fraction) is solved instead of transport equations for individual species. The reacting system is treated using either chemical equilibrium calculations or by assuming infinitely fast reactions (mixed-is-reacted approach). The mixture fraction approach is applicable to non-premixed situations and is specifically developed to simulate turbulent diffusion flames containing one fuel and one oxidant. Such situations are illustrated in Fig. 5.6. The basis for the mixture fraction approach is that individual conservation equations for fuel and oxidant can be combined to eliminate reaction rate terms (see Toor, 1975 for more details). Such a combined equation can be simplified by defining a mixture nmF- mO + 1 Mixture fraction f = n+1

Individual species mass fraction is computed from mixture fraction and/ or assuming equilibrium chemistry

Individual species mass fraction is computed from mixture fraction and/ or assuming equilibrium chemistry

FIGURE 5.6 Mixture fraction approach.

where mF and mO are mass fractions of fuel and oxidant, respectively and n is the number of moles of oxidizer required to burn one mole of fuel. Since the conservation equation of mixture fraction, f does not contain reaction terms the time-averaged equation can be readily obtained as

The time-averaged mixture fraction can be related to time-averaged values of local mass fractions of fuel and oxidant by using the time-averaged form of Eq. (5.19). It can be seen that a knowledge of time-averaged mixture fraction is not sufficient to obtain values of time-averaged fuel and oxidant mass fractions (one equation and two unknowns). In addition to average mixture fraction, if the variance of the mixture fraction is known, it is possible to obtain values of individual mass fractions of fuel and oxidant (see Toor, 1975; Jones and Whitelaw, 1982). The variance of mixture fraction can be obtained by solving its transport equation:

This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aT, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971):

where Cg1 has a value of approximately 3. Spalding (1971) modeled the dissipation of variance as

where Cg2 is about 0.2. Corrsin (1964) modeled the dissipation as a function of a scalar length scale and the rate of turbulence energy dissipation in isotropic turbulence:

The scalar length scale, Ls, is assumed to be equal to k3/2/s in the above expression. For systems with low values of Schmidt number, Corrsin's model reduces to that of Spalding, albeit with higher coefficient (0.5). Corrsin's model is found to be useful even for shear flow turbulence (Patterson, 1985). Detailed models of the variance dissipation rate are difficult to formulate in a general manner due to the large range of time/length scales involved. Fox (1995) proposed an alternative multiscale model. Most of the published models, however, use Eq. (5.23) to model the dissipation rate of scalar variance. Knowledge of mean mixture fraction and its variance is sufficient to calculate local values of fuel and oxidant mass fractions (Toor, 1975).

An alternative way of relating concentrations (mass fractions) of individual species to f is the assumption of chemical equilibrium. An algorithm based on minimization of Gibbs free energy to compute mole fractions of individual species from f has been discussed by Kuo (1986). The equilibrium model is useful for predicting the formation of intermediate species. If such knowledge of intermediate species is not needed, the much simpler approximation of 'mixed-is-burnt' can be used to relate individual species concentrations with f . In order to calculate the time-averaged values of species concentrations the probability density function (PDF) approach is used.

The probability density function, written as p(f), describes the fraction of time that the fluctuating variable f takes on a value between f and f + Af . The concept is illustrated in Fig. 5.7. The fluctuating values off are shown on the right side whilep(f) is shown on the left side. The shape of the PDF depends on the nature of the turbulent fluctuations of f . Several different mathematical functions have been proposed to express the PDF. In presumed PDF methods, these different mathematical functions, such as clipped normal distribution, spiked distribution, double delta function and beta distribution, are assumed to represent the fluctuations in reactive mixing. The latter two are among the more popular distributions and are shown in Fig. 5.8. The double delta function is most readily computed, while the beta function is considered to be a better representation of experimentally observed PDF. The shape of these functions depends solely on the mean mixture fraction and its variance. The beta function is given as fa-1 (1 - a)ß-1

FIGURE 5.7 Graphical description of probability density function (PDF).
FIGURE 5.8 Shapes of commonly used probability density function. (a) Double delta function. (b) Beta function.

where a and p are given by:

The time-averaged values of scalar variables, temperature) is calculated as:

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