Vector mixing

The first steps in DTLD are unfolding the K I x J matrices into a K x (I J) matrix and employing the SVD algorithm to calculate the first two vectors, Uj and u2, of the row space in this unfolded matrix. These two vectors serve as mixing vectors to construct the two pseudosamples, Gj and G2, from linear combinations of the K matrices where... 

We can then have the situation illustrated in fig 7.3, from which it is clear that, instead of just one transition to the sp excited configuration, the A selection rule will, in effect, allow transitions to a state one would label pd, since one cannot distinguish quantum mechanically between the mixed vectors of the ground state. Thus, if the ground state label is... 

CSTR. The mixing vector v is collinear with the scaled reaction vector because c-c = rr(c). 

FIGURE 3.13 The separation vector, S, is tangent to the residue curve at x. The mixing vector, M, points in the direction of Xa. The vector sum is tangent to the column profile at x. 

Deviation is observed in the region of high curvature due to the local behavior of the separation and the mixing vectors. 

Notice that the DPEs describing the scenarios in Cases 1 and 2 (Equations 9.18 and 9.20) are mathematically similar to each other. Each comprise of the addition of two vectors, namely the separation vector, S = x —y, and the mixing vector, M = Xa — X. The only difference between the two cases is that the difference point (Xa) is equal to the composition of the product being removed in Case 1. To avoid repetition, these two cases will be discussed simultaneously, due to the similar nature of their equations. 

Mixture concentrations are commonly viewed as a linear combination of vectors. In AR theory, the vector difference C2 - Cl is frequently referred to as the mixing vector... 

Experiment 1 Mixing Amounts from Beakers 2 and 3 From Section 2.3.1.3, mixtures obtained from beakers 2 and 3 will result in a mixing vector lying in a direction given by the vector (C2 - C3). The corresponding vector expression for this mixture is then... 

Visualizing Rate Vectors Rate vectors are plotted at the point of evaluation. Since r(C) depends on C, each coordinate in concentration space can be assigned a unique vector given by r(C). This assignment is useful as rate vectors may then be incorporated with concentration and mixing vectors, where all vectors can be visualized on a single plot in concentration space. 

Note that parentheses are placed around the left-hand side terms to emphasize the fact that the quantity (C - Cf) is itself a vector. Sometimes, it is convenient to refer to this as the mixing vector v. 

Solution via the geometric interpretation follows a somewhat different methodology. We shall use the fact the mixing vector between the CSTR effluent concentration and the feed vector Cf is collinear with the rate vector evaluated at the exit concentration r(C). That is we have... 

Each point on the CSTR locus is collinear (and points in the same direction) with the mixing vector (C-Cf). 

The tangent to each point on the DSR trajectory lies in a direction somewhere in between the rate vector and the mixing vector (C — C). 

The Jacobian of the rate vector, dr(C), and mixing vector v(C) may be computed giving... 

This expression may be substituted directly into the DSR expression along with the mixing vector = [1, 0, 0]. In this instance, observe that a(C) is undefined when Cb = Omol/L or Ca = 1 mol/L, and thus a(C) is undefined at C for the particular kinetics and C used. 

Mj has columns composed of powers of dr(C) and the mixing vector (C - Cf). From the CSTR expression... 

CSTR effluent concentrations are considered to lie on the AR boundary (they are considered to be critical points) when A(C) = 0. Note that we have expressed the controllability expression in terms of r(C) instead of the mixing vector v(C). This is the preferred method as it does not involve the specification of an appropriate mixing point C on the AR boundary, which is unknown at present. 

Note that this is analogous to the DSR expression in concentration space. The variable p used in Equation 9.18 is the mass fraction equivalent of a. Note the geometric interpretation of a DSR operating in mass fraction space is also equivalent—a DSR solution trajectory in mass fraction space is resultant from the linear combination of a reactive component Wr(C) and a mixing vector (z° — z). 

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