Chemical reactions conserved scalars

In Section 5.1, we have seen (Fig. 5.2) that the molar concentration vector c can be transformed using the SVD of the reaction coefficient matrix T into a vector c that has Nr reacting components cr and N conserved components cc.35 In the limit of equilibrium chemistry, the behavior of the Nr reacting scalars will be dominated by the transformed chemical source term S. 36 On the other hand, the behavior of the N conserved scalars will depend on the turbulent flow field and the inlet and initial conditions for the flow domain. However, they will be independent of the chemical reactions, which greatly simplifies the mathematical description. 

The chemical-source term does not appear in equation (71) because (like P or Zj) Z is a conserved scalar, a scalar quantity that is neither created nor destroyed by chemical reactions. 

The cascade concepts may be applied to intensities of scalar fields as well as to the turbulent kinetic energy. Passive scalars (those that do not influence the velocity field) in the absence of chemical reactions can experience spectral transfers as a consequence of the convective terms in their conservation equations, and an inertial-convective subrange can exist in which the integrand of equation (26) exhibits a power-law dependence on k analogous to that of e(k) [66]. The average rate of scalar dissipation,... 

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... 

All these dipoles are conservative ones with respect to the entity number, the basic quantity in this case of capacitive dipoles. The interesting feature is that the separability is linked to the symmetry between energies-per-entity (here efforts) When the dipole is inseparable, both efforts are equal in magnitude but opposed in direction (for vectors) or value (for scalars). The converse is not true such symmetry may be found in peculiar cases for separable dipoles. For instance, the two potential values Vi and V2 of a capacitor may be equal in magnitude and opposite however, this happens only in case of equal pole capacitances. This is a frequent case in electrodynamics when the two capacitor plates are strictly identical, as in the case of planar capacitor, but this is not general as nonidentical shapes or geometries can also be found. Note that, in physical chemistry, this never happens, because it would correspond to identical partners in a chemical reaction of to identical phases in an interface ... 

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). 

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