Composition vector

CONVERT COMPOSITION VECTORS TO DIMENSION 20 TO MATCH LOWER LEVEL... 

CONVERT COMPOSITION VECTOR TO DIMENSIONS 20 TO MATCH LOVER LEVEL... 

When dealing with high molecular weight polymers it is convenient instead of 1 to introduce the composition vector Cwith components a=lall(a=l,...,m) which, in combination with the chemical size Z, is completely equivalent to 1. For such copolymers the variables Z and ta may be thought of as continuous and recourse can be made to a expression which relates SCD/W(1) to the size distribution (SD) /W(Z) and the composition distribution (CD) W( C l) of macromolecules of given size Z. 

The results reported above have been extended to the general case of irreversible polycondensation of an arbitrary mixture of monomers (characterized by arbitrary matrix of functionalities f and the composition vector v) under the conditions of the applicability of the FSSE model [26]. 

The reaction rates Rt will be functions of the state variables defining the chemical system. While several choices are available, the most common choice of state variables is the set of species mass fractions Yp and the temperature T. In the literature on reacting flows, the set of state variables is referred to as the composition vector [Pg.267]

Note that if the mass fractions are used to define the composition vector, then by definition... 

The reader will recognize these terms as having of the same form as the correction terms in the two-environment model discussed earlier. With N — 1, 6 j = 0 and the model reduces to the laminar-chemistry approximation. With N —2, additional information is obtained concerning the second-order moments of the composition vector. Likewise, by using a larger N, the Mh-order moments are controlled by the DQMOM correction terms found from Eq. (89). 

The NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting flows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the finite-dimensional composition vector laminar flow where the PDFs are not needed, the NDF still introduces an extra dimension (1) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is discretized. Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed ... 

By defining the composition vector to include the species concentrations and the moments as follows ... 

Note that the correction terms are proportional to fT and result from turbulent velocity fluctuations (represented by a gradient-diffusion model). For the multi-environment model the composition vector is defined by... 

The composite vector is seen to spiral around the z-axis and in projection moves anti-clockwise in a circle around the z-axis. The other component which is the mirror image of the first, performs a clockwise circular motion in projection along z. The decomposition into circularly polarized components can also be formulated in complex notation, Er = E0e t6. 

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. 

Figure 5.2. The composition vector c can be partitioned by a linear transformation into two parts c,., a reacting-scalar vector of length /VT and cc, a conserved-scalar vector of length N. The linear transformation is independent of x and t, and is found from the singular value decomposition of the reaction coefficient matrix Y.

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. 

We will assume throughout this section that the composition vector, as well as the initial and boundary conditions, have been previously transformed into the reacting/conserved sub-spaces. 

Thus, the non-stationary turbulent reacting flow will be completely described by the first (Nr +N m) components of turbulent reacting flow, c(0) can be replaced by any of the inlet composition vectors c(l) for any /el,..., A m. For this case, if N m = Nin, then = Nin - 1 and thus one less conserved-variable scalar will be required to describe completely the stationary turbulent reacting flow. 

If AW AW the process of finding a linear-mixture basis can be tedious. Fortunately, however, in practical applications Nm is usually not greater than 2 or 3, and thus it is rarely necessary to search for more than one or two combinations of linearly independent columns for each reference vector. In the rare cases where A m > 3, the linear mixtures are often easy to identify. For example, in a tubular reactor with multiple side-injection streams, the side streams might all have the same inlet concentrations so that c(2) = = c(iVin). The stationary flow calculation would then require only AW = 1 mixture-fraction components to describe mixing between inlet 1 and the Nm — I side streams. In summary, as illustrated in Fig. 5.7, a turbulent reacting flow for which a linear-mixture basis exists can be completely described in terms of a transformed composition vector ipm( defined by... 

Note that, given yrp and , the inverse transformation, (5.109), can be employed to find the original composition vector c. In order to simplify the notation, we will develop the theory in terms of y>rp. However, it could just as easily be rewritten in terms of c using the inverse transformation. 

Finally, note that all of the expressions developed above for the composition vector also apply to the reaction-progress vector ip or the reaction-progress variables Y. Thus, in the following, we will develop closures using the form that is most appropriate for the chemistry under consideration. 

The method just described for treating multiple reacting-progress variables has the distinct disadvantage that the upper bounds must be found a priori. For a complex reaction scheme, this may be unduly difficult, if not impossible. This fact, combined with the desire to include the correlations between the reacting scalars, has led to the development of even simpler methods based on a presumed joint PDF for the composition vector... 

However, care must be taken to avoid the singularity that occurs when C is not full rank. In general, the rank of C will be equal to the number of random variables needed to define the joint PDF. Likewise, its rank deficiency will be equal to the number of random variables that can be expressed as linear functions of other random variables. Thus, the covariance matrix can be used to decompose the composition vector into its linearly independent and linearly dependent components. The joint PDF of the linearly independent components can then be approximated by (5.332). 

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