Mixture fraction, conservation

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). 

Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

An alternative approach might be to address solely the mixture fraction / (mass of original fuel atoms per mass of mixture) since it has been established that there is a firm relationship between y, and/for a given fuel. Note that/moves from 1 to 0 for the start and end of the fire space and / is governed by Equation (12.45) for y, = 0. This then conserves the fuel atoms. Under this approach it is recognized that... 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

The interest in reformulating the conserved-variable scalars in terms of the mixture-fraction vector lies in the fact that relatively simple forms for the mixture-fraction PDF can be employed to describe the reacting scalars. However, if < /Vmf, then the incentive is greatly diminished since more mixture-fraction-component transport equations (Nmf) would have to be solved than conserved-variable-scalar transport equations (/V, << ). We will thus assume that N m = Nmf and seek to define the mixture-fraction vector only for this case. Nonetheless, in order for the mixture-fraction PDF method to be applicable to the reacting scalars, they must form a linear mixture defined in terms of the components of the mixture-fraction vector. In some cases, the existence of linear mixtures is evident from the initial/inlet conditions however, this need not always be the case. Thus, in this section, a general method for defining the mixture-fraction vector in terms of a linear-mixture basis for arbitrary initial/inlet conditions is developed. 

The determination of a mixture-fraction basis is a necessary but not a sufficient condition for using the mixture-fraction PDF method to treat a turbulent reacting flow in the fast-chemistry limit. In order to understand why this is so, note that the mixture-fraction basis is defined in terms of the conserved-variable scalars pcv without regard to the reacting scalars pT. Thus, it is possible that a mixture-fraction basis can be found for the conserved-variable scalars that does not apply to the At reacting scalars. In order to ensure that this is not the case, the linear transformation Mr defined by (5.30) on p. 149 must be applied to the (K x VIM ) matrix... 

The rank of (bj,11 is N m = 2 = Ain, and hence both conserved scalars are variable. In addition, for this example, the mixture-fraction basis will automatically be a linear-mixture... 

Note that the reaction-progress vector in the first column is non-zero. Thus, as we suspected, the mixture-fraction basis is not a linear-mixture basis. The same conclusion will be drawn for all other mixture-fraction bases found starting from (5.118). For these initial and inlet conditions, a two-component mixture-fraction vector can be found however, it is of no practical interest since the number of conserved-variable scalars is equal to Nq,m = 1 (k e 0, 1, 2). In conclusion, although the mixture fraction can be defined for the... 

The fact that no two-component mixture-fraction vector exists does not, however, change the fact that the flow can be described by two conserved scalars. 

This boundary condition does not ensure that the unconditional means will be conserved if the chemical source term is set to zero (or if the flow is non-reacting with non-zero initial conditions Q( 0) 0). Indeed, as shown in the next section, the mean values will only be conserved if the conditional scalar dissipation rate is chosen to be exactly consistent with the mixture-fraction PDF. An alternative boundary condition can be formulated by requiring that the first term on the right-hand side of (5.299) (i.e., the diffusive term) has zero expected value with respect to the mixture-fraction PDF. However, it is not clear how this global condition can be easily implemented in the solution procedure for (5.299). 

Note, however, that in order to conserve probability and to keep the mixture-fraction mean constant, the product Z f and its derivative with respect to J must be null at both f = 0 and f = 1. 

This discussion can be easily extended to the mixture-fraction vector or any other conserved linear combination of the composition variables. 

This equation shows that the mixture fraction is a conserved scalar—its transport equation has no... 

Equation (71) is a conservation equation for the mixture fraction, and equations (76), (77), and (79) relate the other variables to this quantity and Yp. To study the structure of the reaction sheet, we therefore need an additional conservation equation—for example, that for fuel. From equations (1-4), (1-8), (1-9) and (1-12) we find that, under the present assumptions, this additional equation can be written as... 

Coupling functions play a central role in reducing problems of turbulent diffusion flames to problems of nonreacting turbulent flows. In Section 1.3 and in Chapter 3, we emphasized that coupling functions are helpful for analyses of nonpremixed combustion. From the analysis of Section 3.4.2, it may be deduced that their utility extends to turbulent flows. Mixture fractions, which are conserved scalars (Section 10.1.5), were defined and identified as normalized coupling functions in Section 3.4.2. The presentation here will be phrased mainly in terms of the mixture fraction Z of equation (3-70). 

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... 

For the description of mixtures of substances, FLUENT provides the species model. This model calculates the convection, diffusion, and reaction equations for each component in a mixture. This allows the volumetric reactions, surface reactions, and reactions at phase boundaries to be modeled. For the analysis of one-phase mixtures, the conservation of mass equation for a component i can be formulated by accounting for the local mass fraction Yj ... 

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