Reference vector

We have chosen the initial conditions as the reference vector. For steady-state problems, o = 0 and any non-zero inlet vector can be chosen. 

The superscript on is used as a reminder that c(0> was chosen as the reference vector. 

Since the numbering of the initial/inlet conditions is arbitrary, the reference vector can always be renumbered such that / = 0. 

Note that 7 depends on the choice of both the reference vector and the linearly independent columns. 

Note that this condition admits the case where one or more columns of B is null. This would occur if the reference vector corresponded to more than one initial/inlet condition. 

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... 

Any of the columns of d> can be chosen as the reference vector. However, we shall see that there is an advantage in choosing one of the two inlet streams. Thus, letting c(1) be the reference vector, the matrix is easily found to be65... 

Note that, due to the choice of c(1) as the reference vector, the mixture-fraction vector l (third and fourth components of y> ) is null. The first component of the mixture-fraction vector thus describes mixing between the initial contents of the reactor and the two inlet streams, and the second component describes mixing with the second inlet stream. For a stationary flow (0) -> 0, and only one mixture-fraction component ( 2) will be required to describe the flow. Note, however, that if c(0) had been chosen as the reference vector, a similar reduction would not have occurred. As expected, the inlet and initial values of the two reaction-progress variables are null. 

Again, note that

[Pg.189]

Note that, due to the choice of c(1) as the reference vector, is again null. We also see that... 

In order to show that no mixture-fraction basis exists, it is necessary to check all possible reference vectors. For each choice of the reference vector, there are three possible sets of linearly independent vectors that can be used to compute B. Thus, we must check a total of 12 possible mixture-fraction bases. Starting with c(0) as the reference vector, the three possible values of B(0) are... 

None of these matrices satisfies (5.96). Thus, we next take c(1) as the reference vector and find the three possible values of B(1) ... 

Reference vector. After calibration a reference vector is created. This vector is used to match all the same fractions together in a data matrix and is created by ordering of all existing calibrated retention time values. The number of the values is reduced by replacing each value that falls into a small retention time "window" with its mean value. The "window" was set wide enough to reduce the number of values to 250. 

Matching to form a data matrix. Each value of the reference vector is compared with the retention time values for the fractions in every chromatographic run. Only a small variation in the values is tolerated thus no doublex matching should occur. 

The reference vectors R span the/-dimensional space, and thus form a basis in Ms-A general element of this space, denoted RE), may be expanded as... 

Eq. (133) is simply the operator version of the matrix expression in Eq. (30). In Eq. (133) the state S> is orthogonal to the reference state 0> and the norm of S> determines the angle of rotation between these two states. Eq. (134) assumes that a basis for the orthogonal complement to the reference vector exists and that the coefficients p are the expansion coefficients of the state S> in this basis. Three bases have been used for this expansion. The Hamiltonian eigenvector basis " , which satisfies = is linearly... 

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