Radial coordinate

Figure 7 IN718 sample with segregation, SQUID-Signal versus rotation angle, scan radius is constant and equal to radial coordinate of segregation. Measurements are performed after demagnetization.

Another example of the difficulty is offered in figure B3.1.5. Flere we display on the ordinate, for helium s (Is ) state, the probability of finding an electron whose distance from the Fie nucleus is 0.13 A (tlie peak of the Is orbital s density) and whose angular coordinate relative to that of the other electron is plotted on the abscissa. The Fie nucleus is at the origin and the second electron also has a radial coordinate of 0.13 A. As the relative angular coordinate varies away from 0°, the electrons move apart near 0°, the electrons approach one another. Since both electrons have opposite spin in this state, their mutual Coulomb repulsion alone acts to keep them apart. 

The numerical calculations have been done on a two-coordinate system with q being a radial coordinate and <() the polar coordinate. We consider a 3 x 3 non-adiabatic (vector) mabix t in which and T4, aie two components. If we assume = 0, takes the following form,... 

The adiabatic functions are characterized by two interesting features (1) they depend only on the angular coordinate (but not on the radial coordinate) and (2) they are not single valued in configuration space because when

adiabatic wave functions back to their... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Note that 0" < A< 60". The invariants A , and form a cylindrical coordinate system relative to the principal coordinates, with axial coordinate / A equally inclined to the principal coordinate axes, with radial coordinate /3t, and with angular coordinate The plane A" = 0 is called the II plane. Because the principal values can be ordered arbitrarily, the representation of A through its invariants in n plane coordinates has six-fold symmetry. 

The radial coordinate r is also replaced by the nondimensional param... 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

A mixed DVR (discrete variable representation)43 for all the radial coordinates and basis set representations for the angular coordinates are used in the wavepacket propagation.44... 

Optical fiber measurement of local solids concentrations of FCC catalyst fluidized in a 9-cm-i.d. column gave the results shown typically in Fig. 26. Analysis of these data showed that the radial voidage profile could be described solely by the cross-section-average voidage e, calculated as shown in Sec. 5.1, and the reduced radial coordinate r/R ... 

---