Chemical time scale

In hindsight, the primary factor in determining which approach is most applicable to a particular reacting flow is the characteristic time scales of the chemical reactions relative to the turbulence time scales. In the early applications of the CRE approach, the chemical time scales were larger than the turbulence time scales. In this case, one can safely ignore the details of the flow. Likewise, in early applications of the FM approach to combustion, all chemical time scales were assumed to be much smaller than the turbulence time scales. In this case, the details of the chemical kinetics are of no importance, and one is free to concentrate on how the heat released by the reactions interacts with the turbulent flow. More recently, the shortcomings of each of these approaches have become apparent when applied to systems wherein some of the chemical time scales overlap with the turbulence time scales. In this case, an accurate description of both the turbulent flow and the chemistry is required to predict product yields and selectivities accurately. 

The chemical time scales are defined in Chapter 5. In general, they will be functions of the temperature, pressure, and local concentrations. 

In this section, we first introduce the standard form of the chemical source term for both elementary and non-elementary reactions. We then show how to transform the composition vector into reacting and conserved vectors based on the form of the reaction coefficient matrix. We conclude by looking at how the chemical source term is affected by Reynolds averaging, and define the chemical time scales based on the Jacobian of the chemical source term. 

The closure problem thus reduces to finding general methods for modeling higher-order moments of the composition PDF that are valid over a wide range of chemical time scales. 

The chemical time scales can be defined in terms of the eigenvalues of the Jacobian matrix of the chemical source term.27 For example, for an isothermal system the K x K Jacobian matrix of the chemical source term is given by d S... 

Another valid choice for the chemical time scales would be the non-zero singular values resulting from the S VD of the Jacobian matrix. 

The chemical time scales ra and the mixing time scale can be used to define the Damkohler number(s) Da, = /x . Note that fast reactions correspond to large Da, and... 

Due to the conservation of elements, the rank of J will lie less than or equal to K — E 1 In general, rank(J) = Ny < K - E, which implies that V = K — T eigenvalues of J are null. Moreover, since M is a similarity transformation, (5.51) implies that the eigenvalues of J and those of J are identical. We can thus limit the definition of the chemical time scales to include only the Nr finite ra found from (5.50). The other N components of the transformed composition vector correspond to conserved scalars for which no chemical-source-term closure is required. The same comments would apply if the Nr non-zero singular values of J were used to define the chemical time scales. 

As noted below, the set of the chemical time scales can be restricted to include only those corresponding to non-zero eigenvalues. For a non-reacting flow, the set would thus be empty. 

The limiting case where the chemical time scales are all large compared with the mixing time scale r, i.e., the slow-chemistry limit, can be treated by a simple first-order moment closure. In this limit, micromixing is fast enough that the composition variables can be approximated by their mean values (i.e., the first-order moments (0)). We can then write, for example,... 

Recall that the Jacobian of S will generate Ny chemical time scales. In the equilibrium-chemistry limit, all Ny chemical times are assumed to be much smaller than the flow time scales. 

The Nr eigenvalues of the Jacobian of S,p will be equal to the Nr non-zero eigenvalues of the Jacobian of Sc. Thus, in the equilibrium-chemistry limit, the chemical time scales will obey... 

An example of a smart tabulation method is the intrinsic, low-dimensional manifold (ILDM) approach (Maas and Pope 1992). This method attempts to reduce the number of dimensions that must be tabulated by projecting the composition vectors onto the nonlinear manifold defined by the slowest chemical time scales.162 In combusting systems far from extinction, the number of slow chemical time scales is typically very small (i.e, one to three). Thus the resulting non-linear slow manifold ILDM will be low-dimensional (see Fig. 6.7), and can be accurately tabulated. However, because the ILDM is non-linear, it is usually difficult to find and to parameterize for a detailed kinetic scheme (especially if the number of slow dimensions is greater than three ). In addition, the shape, location in composition space, and dimension of the ILDM will depend on the inlet flow conditions (i.e., temperature, pressure, species concentrations, etc.). Since the time and computational effort required to construct an ILDM is relatively large, the ILDM approach has yet to find widespread use in transported PDF simulations outside combustion. 

As discussed in Chapter 5, the chemical time scales can be found from the Jacobian of the chemical source term. 

Note that the dimensions of the fast and slow manifolds will depend upon the time step. In the limit where At is much larger than all chemical time scales, the slow manifold will be zero-dimensional. Note also that the fast and slow manifolds are defined locally in composition space. Hence, depending on the location of 0q], the dimensions of the slow manifold can vary greatly. In contrast to the ILDM method, wherein the dimension of the slow manifold must be globally constant (and less than two or three ), ISAT is applicable to slow manifolds of any dimension. Naturally this flexibility comes with a cost ISAT does not reduce the number (Ns) of scalars that are needed to describe a reacting flow.168... 

The de-excitation of the vibrationally excited state may take place within the time ofa few vibrations (< 10-12sec). It should be remembered that on the conventional chemical time scale one seldom considers intermediates with lifetimes < 10 12 sec thus most of these excited primary products may be overlooked from the chemical standpoint. One cannot, however, neglect these excited products when considering the detailed mechanism of the electron transfer. 

The slowest chemical time-scale which is decoupled must be faster than that of the physical processes. This is the only restriction which applies, however, there are no restrictions on the nature of the perturbation which may be caused by diffusion, convection, mixing processes etc. The technique has been successfully applied to laminar reacting flows and diffusion flames and could potentially be applied to autoignition systems, although it is unlikely that the degree of reduction will be as great as that found for diffusion flames [144]. 

Given ideal, well-stirred conditions, the heat release rate could be interpreted from (6.13) under non-stationary conditions, but accurate measurements of (dT/dt) would also be required. The rate of temperature change is always more important than the heat loss rate during the late stages of the development of ignition, because the chemical time-scale is much shorter than the Newtonian cooling time-scale. 

Spontaneous ignition and associated features of organic gases and vapours are a consequence of the exothermic oxidation chemistry discussed in Chapter 1, but the way in which events unfold is determined by the physical environment within which reaction takes place. The heat transfer characteristics are probably most important, as may be illustrated with respect to the different consequences of adiabatic and non-adiabatic operation in a CSTR (Section 5) [117]. The notion of adiabatic operation may seem remote from any practical application, but this idealized condition may be approached if the chemical time-scale is considerably shorter than the time-scale for heat losses. 

The front velocity v/ is the result of the interplay among the flow characteristics (i.e., intensity U and length scale L), the diffusivity D, and the production time scale x. In this chapter we shall study the problem of front propagation in the case of cellular flows. In particular, introducing the Damkohler number Da = L/(Ux) (the ratio of advective to reactive time scales) and the Pec let number Pe = UL/D (the ratio of diffusive to advective time scales), we shall discuss how the front speed can be expressed as a nondimensional function such as Vf/vo = < >(Da,Pe). A crucial role in determining i >(Da. Pe) is played by the renormalization of the diffusion coefficient and chemical time scale [13] induced by the advection. 

The amount of NO in the mixture when temperatures are returned to ambient values is not so much a function of the peak temperature reached, but depends on the temperature at which the chemical time-scale equals the cooling time-scale. If the cooling rate is very fast, the lowest temperature at which equilibrium is maintained will be higher -and the amount of NO present will be greater. The total amount of NO formed will also depend on the amount of air raised to high temperature and the extent to which other air is heated by entrainment, compression, radiation, and other factors. For example, in an automobile, the time-scale for cooling of the combustion mixture is about 0.05 s. The pressure of NO emitted from the tailpipe is about 0.0006 bar, far in excess of the ambient equilibrium value of about 3 x 10 bar. 

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