Scalar mean

Starting with the scalar transport equation ((1.28), p. 16), Reynolds averaging leads to the transport equation for the scalar means  

The second term on the left-hand side of this expression can be further decomposed into two terms  

The final form for the scalar mean transport equation in a turbulent reacting flow is 

The first term on the right-hand side of this expression is the molecular transport term that scales as Sc Re 1. Thus, at high Reynolds numbers,26 it can be neglected. The two new unclosed terms in (3.88) are the scalar flux Uj f a), and the mean chemical source term (Sa( f )). For chemical reacting flows, the modeling of (5 a,(0)) is of greatest concern, and we discuss this aspect in detail in Chapter 5. 

The reader will recognize ScRe as the Peclet number. Here we will assume throughout that the Schmidt number is order unity or greater. Thus, high Reynolds number will imply high Peclet number. 

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Here, since the measurements were done in an integral reactor, calculation must start with the Conversion vs. Temperature function. For an example see Appendix G. Calculation of kinetic constants starts with listed conversion values as vX and corresponding temperatures as vT in array forms. The Vectorize operator of Mathcad 6 tells the program to use the operators and functions with their scalar meanings, element by element. This way, operations that are usually illegal with vectors can be executed and a new vector formed. The v in these expressions indicates a vector. 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

Because die outlet concentrations will not depend on it, micromixing between duid particles can be neglected. The reader can verify this statement by showing that die micromixing term in the poorly micromixed CSTR and the poorly micromixed PFR falls out when die mean outlet concentration is computed for a first-order chemical reaction. More generally, one can show that die chemical source term appears in closed form in die transport equation for die scalar means. 

For homogeneous binary mixing of an inert scalar, the scalar mean will remain constant so that (e/>(x, t)) = (c/Tx, ())) = p. Thus, the rate of scalar mixing can be quantified in terms of the scalar variance ([Pg.84]

It is imperative that the same closure for the scalar flux be used in (4.70) to find the scalar mean and the scalar-variance-production term Vj,. 

The Reynolds-averaged chemical source terms can be written in terms of the scalar means and covariances. For example,... 

Chapter 3 will be employed. Thus, in lieu of (x, t), only the mixture-fraction means ( ) and covariances ( , F) (/, j e 1,..., Nm() will be available. Given this information, we would then like to compute the reacting-scalar means and covariances (require additional information about the mixture-fraction PDF. A similar problem arises when a large-eddy simulation (LES) of the mixture-fraction vector is employed. In this case, the resolved-scale mixture-fraction vector (x, t) is known, but the sub-grid-scale (SGS) fluctuations are not resolved. Instead, a transport equation for the SGS mixture-fraction covariance can be solved, but information about the SGS mixture-fraction PDF is still required to compute the resolved-scale reacting-scalar fields. 

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... 

Figure 5.21. Scatter plot of concentration in a turbulent reacting flow conditioned on the value of the mixture fraction. Although large fluctuations in the unconditional concentration are present, the conditional fluctuations are considerably smaller. In the limit where the conditional fluctuations are negligible, the chemical source term can be closed using the conditional scalar means. 

The scalar mean conditioned on the mixture-fraction vector can be denoted by... 

By definition, the unconditional scalar means can be found from Q(C x, t) and the mixture-fraction-vector PDF ... 

If the conditional fluctuations p p are neglected, the homogeneous conditional scalar mean Q(C t) = ( jpIO is governed by (Klimenko 1990 Bilger 1993) (summation is implied with respect to j and k)... 

If (5.303) is disregarded and the functional form for the conditional scalar dissipation rate is chosen based on other considerations, an error in the unconditional scalar means will result. Defining the product of the conditional scalar means and the mixture-fraction PDF by... 

Integrating (5.307) results in the governing equation for the scalar means in a homogeneous flow 124... 

Note that if the conditional scalar dissipation rate is chosen correctly (i.e., Z = Z), then the first term on the right-hand side of this expression is null. However, if Z is inconsistent with /f, then the scalar means will be erroneous due to the term... 

I) The molecular mixing model must leave the scalar mean unchanged. 

In this context, where the scalar mean is constant, universal implies that the shape of the scalar PDF at the same value of the scalar variance is identical regardless of the initial scalar spectrum. 

Note that (6.190) contains a number of conditional expected values that must be evaluated from the particle fields. The Lagrangian VCIEM model follows from (6.86), and has the same form as the LIEM model, but with the velocity, location-conditioned scalar mean (0 U, X )(U. X. t) in place of location-conditioned scalar mean ( X )(X. t) in the final term on the right-hand side of (6.190). 

The estimation of statistical quantities for each cell is straightforward. For example, the estimated scalar mean in the /th cell is just... 

Note that A depends only on the conditional scalar means ([Pg.396]

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

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