Equilibrium-chemistry limit

For elementary chemical reactions, it is sometimes possible to assume that all chemical species reach their chemical-equilibrium values much faster than the characteristic time scales of the flow. Thus, in this section, we discuss how the description of a turbulent reacting flow can be greatly simplified in the equilibrium-chemistry limit by reformulating the problem in terms of the mixture-fraction vector. 

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The acid-base reaction is a simple example of using the mixture fraction to express the reactant concentrations in the limit where the chemistry is much faster than the mixing time scales. This idea can be easily generalized to the case of multiple fast reactions, which is known as the equilibrium-chemistry limit. If we denote the vector of reactant concentrations by and assume that it obeys a transport equation of the form... 

As noted earlier, the sum of the mass fractions is unity and thus Eq. (86) will be consistent with Eq. (85) only if the sum of the correction term b m over all chemical species a = 1,..., K is null. In general, this will not be the case if Eq. (89) is used. Another difficulty that can arise is that the mass fractions in two environments may be equal, e.g., i = a2, and thus the coefficient matrix in Eq. (89) will be singular. This can occur, for example, in the equilibrium-chemistry limit where the compositions depend only on the mixture fraction, i.e., (j) — co(0- F°r chemical species that are not present in the feed streams, the equilibrium values for = 0 and = 1 are zero, but for intermediate values of the mixture fraction, the equilibrium values are positive. This implies that the equilibrium values will be the same for at least two values of the mixture fraction in the range 0< < 1. Thus, in the equilibrium limit it is inevitable that two environments will have equal mass fractions for certain species at some point in the flow field. Since singularity implies an underlying correlation between... 

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. 

Having demonstrated the existence of a mixture-fraction vector for certain turbulent reacting flows, we can now turn to the question of how to treat the reacting scalars in the equilibrium-chemistry limit for such flows. Applying the linear transformation given in (5.107), the reaction-progress-vector transport equation becomes... 

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... 

Thus, for a turbulent reacting flow in the equilibrium-chemistry limit, the difficulty of closing the chemical source term is shifted to the problem of predicting / ( x, t). 

In a CFD calculation, one is usually interested in computing only the reacting-scalar means and (sometimes) the covariances. For binary mixing in the equilibrium-chemistry limit, these quantities are computed from (5.154) and (5.155), which contain the mixture-fraction PDF. However, since the presumed PDF is uniquely determined from the mixture-fraction mean and variance, (5.154) and (5.155) define mappings (or functions) from (I>- space ... 

In the equilibrium-chemistry limit, the turbulent-reacting-flow problem thus reduces to solving the Reynolds-averaged transport equations for the mixture-fraction mean and variance. Furthermore, if the mixture-fraction field is found from LES, the same chemical lookup tables can be employed to find the SGS reacting-scalar means and covariances simply by setting x equal to the resolved-scale mixture fraction and x2 equal to the SGS mixture-fraction variance.88... 

In summary, in the equilibrium-chemistry limit, the computational problem associated with turbulent reacting flows is greatly simplified by employing the presumed mixture-fraction PDF method. Indeed, because the chemical source term usually leads to a stiff system of ODEs (see (5.151)) that are solved off-line, the equilibrium-chemistry limit significantly reduces the computational load needed to solve a turbulent-reacting-flow problem. In a CFD code, a second-order transport model for inert scalars such as those discussed in Chapter 3 is utilized to find ( ) and and the equifibrium com-... 

An ad hoc extension of the method presented above can be formulated for complex chemistry written in terms of yip and . In the absence of chemical reactions, y>rp = 0. Thus, if a second limiting case can be identified, interpolation parameters can be defined to be consistent with the unconditional means. In combusting flows, the obvious second limiting case is the equilibrium-chemistry limit where yip = y>eq( ) (see Section 5.4). The components of the conditional reacting-progress vector can then be approximated by (no summation is implied on a)... 

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