Colinearity

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

Now, using estimated thermal growth of Machines A and B, a cold target B line is drawn. This line is where shaft B should lie so that when hot it will be colinear with shaft A on the hot running line. 

In the simplest case the AB and BC bond axes will remain colinear throughout the reaction then the potential energy can be expressed as a function of the bond... 

Fig. A. Molecular-orbital representation of the 1-centrc F-Xe-F bond, (a) The possible combinations of colinear p, atomic orbitals, and (b) the energies of the resulting MOs (schematic).

Note that e is defined in terms of the components of the velocities, not the vector velocities, whereas the momentum balance is defined in terms of the vector velocities. To solve Equations 2-32 and 2-33 when all the velocities are not colinear, one writes the momentum balances along the principal axes and solves the resulting equations simultaneously. 

Shafts are considered to be in alignment when they are colinear at the coupling point. The term colinear refers to the condition when the rotational centerlines of two mating shafts are parallel and intersect (i.e., join to form one line). When this is the case, the coupled shafts operate just like a solid shaft. Any deviation from the aligned or colinear condition, however, results in abnormal wear of machine-train components such as bearings and shaft seals. 

Two perfectly aligned shafts are colinear and operate as a solid shaft when coupled. This condition is illustrated in Figure 54.1. However, it is extremely rare for two shafts to be perfectly aligned without an alignment procedure... 

Offset misalignment, also referred to as parallel misalignment, refers to the distance between two shaft centerlines, and is generally measured in thousandths of an inch. Offset can be present in either the vertical or horizontal plane. Figure 54.2 illustrates offset, showing two mating shafts that are parallel to each other, but not colinear. Theoretically, offset is measured at the coupling centerline. 

Couple imbalance is caused by two equal non-colinear imbalance forces that oppose each other angularly (i.e., 180° apart). Assume that a rotor with pure couple imbalance is placed on frictionless rollers. Because the imbalance weights or forces are 180° apart and equal, the rotor is statically balanced. However, a pure couple imbalance... 

The axial P-Os-P angle is 166°, so the three atoms are not quite colinear (Os—P 2.311 A) the unique Os-P bond is slightly longer at 2.347 A, showing a frans-effect of hydride. Os-H bonds average 1.663 A [164]. 

On reaction with 02, the P—Ir-P colinearity is maintained (174.5°), keeping these bulky ligands apart, but the Cl-Ir-C angle closes (to 97.4°) to accommodate the incoming 02 molecule. The geometry of the 02 adduct can be described as distorted octahedral, with Ir, Cl, C and the two O atoms essentially coplanar. As expected as the coordination number increases, there is a general increase in bond length (except for Ir—Cl). 

For a simple distillation column separating a ternary system, once the feed composition has been fixed, three-product component compositions can be specified, with at least one for each product. The remaining compositions will be determined by colinearity in the ternary diagram. For a binary distillation only two product compositions can be specified independently, one in each product. Once the mass balance has been specified, the column pressure, reflux (or reboil ratio) and feed condition must also be specified. 

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