Triangular correlations

Figure 8 shows the results for the individual analysis of the sand-sized fractions from Units B, N, and C as a triangular correlation diagram for Cr, Hf, and Th. Ignoring the samples from Unit C, which is a wadi sediment that may be representative of very old Nile deposits, the material from Unit... 

Counter-current flow. Structured packings. Gauze-type with triangular flow channels, Bravo, Rocha, and Fair correlation... 

For the first time through a liqmd-liquid extrac tion problem, the right-triangular graphical method may be preferred because it is completely rigorous for a ternary system and reasonably easy to understand. However, the shortcut methods with the Bancroft coordinates and the Kremser equations become valuable time-savers for repetitive calculations and for data reduction from experimental runs. The calculation of pseudo inlet compositions and the use of the McCabe-Thiele type of stage calculations lend themselves readily to programmable calculator or computer routines with a simple correlation of equilibrium data. 

Zhao and Bi (2001b) measured pressure drop in triangular conventional size channels d = 0.866—2.866 mm). The variations of the measured two-phase frictional multiplier with the Martinelli parameter X for the three miniature triangular channels used in experiments are displayed, respectively, in Fig. 5.29a-c. In Fig. 5.29 also shown are the curves predicted by Eq. (5.25) for C = 5 and C = 20. It is evident from Fig. 5.29 that the experimental data are reasonably predicted by the Lockhart-Martinelli correlation, reflected by the fact that all the data largely fall between the curves for C = 5 and C = 20, except for the case at very low superficial liquid velocities. 

Data Analysis Because of the danger of false conclusions if only one or two parameters were evaluated, it was deemed better to correlate every parameter with all the others, and to assemble the results in a triangular matrix, so that trends would become more apparent. The program CORREL described in Section 5.2 retains the sign of the correlation coefficient (positive or negative slope) and combines this with a confidence level (probability p of obtaining such a correlation by chance alone). 

Calculate the correlation coefficient r for every combination of columns, and display the results in a triangular matrix (an absolute value just under 1.00 indicates a strong correlation between the measurements in columns i and j a minus sign indicates that the slope is negative). 

If the site correlations are extended to triangular clusters, then... 

The most effective reply to this criticism is a reference to the two-dimensional Ising model for which an analytic solution is available, and coefficients can be computed for all values of n. Taking the triangular lattice, for example, predictions based on values of n up to 15 are quite adequate to represent the asymptotic behavior of the coefficients relating to thermodynamic and correlation properties to a high degree of accuracy. 

Figure 14.1. Equilibria in a ternary system, type 1, with one pair of partially miscible liquids A = 1-hexene, B = tetramethylene sulfbne, C = benzene, at 5(TC (JR.M. De Fre, thesis, Gent, 1976). (a) Equilateral triangular plot point P is at 20% A, 10% B, and 70% C. (b) Right triangular plot with delines and tieline locus, the amount of A can be read off along the perpendicular to the hypotenuse or by difference, (c) Rectangular coordinate plot with tieline correlation below, also called Janecke and solvent-free coordinates.

We have also performed calculations for higher diffusion rates (D = 100) and for the triangular lattice (coordination number z — 6). The qualitative behaviour is in complete agreement with the calculation presented here. For the case S = 1 the increase of the diffusion rate or change of the lattice structure leads to a very small shift of the phase transition point y to higher values of j/n- This trend is clear because the reactivity of the H atoms is increased by the larger mobility. For S < 1 nearly no effect is observed which means that the system s behaviour is mainly dominated by the number of activated sites. The correlation of the adsorbed particles are rather small as expected for S < 1. 

The multireference results of Table 5.12 were all based on full valence CASSCF calculations with eleven electrons in twelve active orbitals. This produces a large configuration expansion (about 85 000 CSFs) so it is not possible to perform MRCI(CAS) calculations. Reference configuration lists were selected at the cyclic and linear geometries (taken from MP2 optimized structures) and then merged. The core electrons were not correlated in any of the calculations. One complication in the CASSCF calculations should be pointed out. Since the cyclic state arises as the 2B component, in C3v symmetry, of a 2E state in the D3h symmetry (equilateral triangular) structure, it would be desirable to obtain MOs with D3h symmetry and equivalence restrictions... 

A n can be any number of window shapes, but the two most commonly used are a triangular window, and a cos2 window. In both cases, the window should sum to one when overlapped with itself by W-72 samples, and should be unity at the center when the waveform segments that are overlapped correlate highly. If the overlapped waveform segments have a very low correlation, then it is preferable to use an equal power crossfade, where the sum of the windows is 1.414. This strategy tends to keep the RMS energy close to constant across the cross fade. 

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