Column-pattern

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

We can now define the rank of the column-pattern as the number of linearly independent columns or rank of X. If all 50 points are coplanar, then we can reconstruct each of the 50 columns, by means of linear combinations of two independent ones. For example, if x, and Xj 2 linearly independent then we must have 48 linear dependences among the 50 columns of X ... 

The elements of m represent the coordinates of the row-centroid, which corresponds with the center of mass of the column-pattern P of points formed in... 

Similarly, Fig. 31.2b shows the column-pattern F of the p columns of the data table X by means of an elliptical envelope in the dual n-dimensional row-space 5". The ellipses should be interpreted as (hyper)ellipsoidal equiprobability envelopes of multinormal data. In practice the data are rarely multinormal and the centroid (or center of mass) of the pattern does not generally appear at the origin of space. An essential feature is that the equiprobability envelopes are similarly shaped in Figs. 31.2a and b. The reason for this will become apparent below. Note that in the previous section we have assumed by convention that n exceeds p, but this is not reflected in Figs. 31.2a and b. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

The vector of column-means nip defines the coordinates of the centroid (or center of mass) of the row-pattern P" that represents the rows in column-space Sf . Similarly, the vector of row-means m defines the coordinates of the center of mass of the column-pattern that represents the columns in row-space S". If the column-means are zero, then the centroid will coincide with the origin of SP and the data are said to be column-centered. If both row- and column-means are zero then the centroids are coincident with the origin of both 5" and S . In this case, the data are double-centered (i.e. centered with respect to both rows and columns). In this chapter we assume that all points possess unit mass (or weight), although one can extend the definitions to variable masses as is explained in Chapter 32. 

But for some liquids exists the third stage of liquid s penetration inside conical capillary, which was established in [5]. During this stage a channel is filling both from its entrance and from its closed top. Two liquid columns arise and are growing towards each other till the complete channel s filling (fig. 2). The most intriguing pattern can be observed when we exclude direct liquid s access to channel s entrance. It corresponds to the cases... 

Figure Cl.5.14. Fluorescence images of tliree different single molecules observed under the imaging conditions of figure Cl.5.13. The observed dipole emission patterns (left column) are indicative of the 3D orientation of each molecule. The right-hand column shows the calculated fit to each observed intensity pattern. Molecules 1, 2 and 3 are found to have polar angles of (0,( ))=(4.5°,-24.6°), (-5.3°,51.6°) and (85.4°,-3.9°), respectively. Reprinted with pennission from Bartko and Dickson [148]. Copyright 1999 American Chemical Society.

Under constant pattern conditions the LUB is independent of column length although, of course, it depends on other process variables. The procedure is therefore to determine the LUB in a small laboratory or pilot-scale column packed with the same adsorbent and operated under the same flow conditions. The length of column needed can then be found simply by adding the LUB to the length calculated from equiUbrium considerations, assuming a shock concentration front. 

Favorable and unfavorable equihbrium isotherms are normally defined, as in Figure 11, with respect to an increase in sorbate concentration. This is, of course, appropriate for an adsorption process, but if one is considering regeneration of a saturated column (desorption), the situation is reversed. An isotherm which is favorable for adsorption is unfavorable for desorption and vice versa. In most adsorption processes the adsorbent is selected to provide a favorable adsorption isotherm, so the adsorption step shows constant pattern behavior and proportionate pattern behavior is encountered in the desorption step. 

Fig. 13. Bubble column flow characteristics (a) data processing system for split-film probe used to determine flow characteristics, where ADC = automated data center (b) schematic representation of primary flow patterns.

Fig. 15. Flow pattern in a crossflow plate distillation column.

This overall flow pattern in a distillation column provides countercurrent contacting of vapor and hquid streams on all the trays through the column. Vapor and liquid phases on a given tray approach thermal, pressure, and composition equilibriums to an extent dependent upon the efficiency of the contac ting tray. 

Other control methods. A cychng procedure can be used to set the pattern for column operation. The unit operates at total reflux until equilibrium is established. Distillate is then taken as total draw-... 

When straight or serrated segmental weirs are used in a column of circiilar cross secdion, a correction may be needed for the distorted pattern of flow at the ends of the weirs, depending on liquid flow rate. The correction factor F from Fig. 14-33 is used direcdly in Eq. (14-112) or Eq. (14-119). Even when circular downcomers are utilized, they are often fed by the overflow from a segmental weir. When the weir crest over a straight segmental weir is less than 6 mm V in), it is desirable to use a serrated (notched) weir to provide good liquid distribution. Inasmuch as fabrication standards permit the tray to be 3 mm Vh in) out of level, weir crests less than 6 mm V in) can result in maldistribution of hquid flow. 

Another type of distributor, not shown in Fig. 14-64, is the spray nozzle. It is usually not recommended for hquid distribution for two reasons. First, except for small columns, it is difficult to obtain a uniform spray pattern for the packing. The fuU-cone nozzle type is usually used, with the need for a bank of nozzles in larger columns. When there is more than one nozzle, the problem of overlap or underlap arises. A second reason for not using spray nozzles is their tendency toward entrainment by the gas, especially the smaller droplets in the spray size distribution. However, some mass transfer in the spray can be expected. 

During backwash the larger, denser panicles will accumulate at the base and the particle size will decrease moving up the column. This distribution yields a good hydraulic flow pattern and resistance to fouling by suspended solids. 

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