Radial term

Delete the radial convection term but otherwise run the full simulation. This gives avgC = 0.5197. Now add the radial term to get 0.5347. The change is in the correct direction since velocity profile elongation hurts conversion. 

Then the radial terms of this expansion can be connected to those of the momentum density expansion of Eq. (5.36) by... 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

Not-Roundness. The size normalized radial standard deviation is illustrated in the NR template shown in Figure 4. Thus, no matter how the profile deviates from a circle the NR value will only indicate the statistical property of the radial distribution. There is no information in this term concerning the sequence of radial terms. 

The orthogonality of the spherical harmonics, V) m, insures that there will be no mixing of radial terms, in the resulting integral. Since the spherical harmonics are also normalized to unity, simplification of Eqn.(23) will produce the following functional form. 

Then, in order to solve the CETO integral problem the general radial term ... 

As can be seen, these orbitals are the product of a radial part R dependent on r, the distance from the electron to the nucleus, and Y, called a spherical harmonic, a function detailing the angular dependence of the atomic wavefimction. For all many-electron atoms, the radial term must be approximated because of the aforementioned problem of electron-electron repulsions. 

Here S.Q represents the imbalance of the electrons in the valence p orbitals of the atom, r )np is an average radial term, where r is the distance of the np electrons from the nucleus, and AE is the lowest energy transition of the allowed type or some average of the lowest lying transitions.)... 

The quantity AE is the average energy of excitation required to reach certain excited states. The radial term average distance r of the 2p electrons from the nucleus and serves as a measure of electron density. Finally, Qij is a measure of tt bonding to carbon. The negative sign in the equation indicates that shielding is in the opposite direction from cr. ... 

The radial term in (3-211) adjusts itself to maintain the correct overall heat balance, plus the local balance between conduction of heat in the radial direction and convection in the axial direction. We shall see that the analysis in this section is very similar to that used for the solution of the Taylor dispersion problem, which is discussed in the next section. 

Let us ignore the R2 rotational part (R=J-L-S) of this operator, which leads to off-diagonal matrix elements that are proportional to J(J + 1) but still very small compared to the matrix elements of the remaining radial term (Leoni, 1972). The effect of the derivatives with respect to R on the electronic and vibrational wavefunctions, both of which depend on R, is given by... 

In fact, the two cases are surprisingly similar and the differences can be concentrated into a new, more general, form of each of the two main integrals the angular and radial terms. 

Manipulation of the angular (i.e., second) term in the spherical coordinate Lapla-cian operator reveals cos 9 dependence, analogous to the radial term. Hence, in operator notation. 

Now, it is possible to predict the curvature correction factor F r]) for the radial term in the equation of continuity at any position in the fluid (see Table 11-1) ... 

TABLE 11-1 Exact Calculations of the Curvature Correction Factor and the Error Embedded in the Radial Term of the Locally Flat Equation of Continuity"... 

Hence, neglecting the curvature correction factors of in the radial term of the equation of continuity corresponds to a 14% underestimate of (l/r )[d(r Vr)/dr] at the outer edge of the mass transfer boundary layer on the equator of the pellet. 

One should realize that these calculations are based on an expression for Vr which corresponds to potential flow past a stationary nonde-formable bubble, as seen by an observer in a stationary reference frame. However, this analysis rigorously requires the radial velocity profile for potential flow in the Uquid phase as a nondeformable bubble rises through an incompressible liquid that is stationary far from the bubble. When submerged objects are in motion, it is important to use liquid-phase velocity components that are referenced to the motion of the interface for boundary layer mass transfer analysis. This is accomplished best by solving the flow problem in a body-fixed reference frame which translates and, if necessary, rotates with the bubble such that the center of the bubble and the origin of the coordinate system are coincident. Now the problem is equivalent to one where an ideal fluid impinges on a stationary nondeformable gas bubble of radius R. As illustrated above, results for the latter problem have been employed to estimate the maximum error associated with the neglect of curvature in the radial term of the equation of continuity. 

The homogeneous diffusion model is slightly more complex in cyUndrical coordinates relative to the model described above in rectangular coordinates. Additional complexity arises because the radial term of the Laplacian operator (V V = V ) accounts for the fact that the surface area across which radial diffusion occurs increases linearly with dimensionless coordinate r/ as one moves radially outward. Basic information for = f(t]) is obtained by integrating the dimensionless mass balance with radial diffusion and chemical reaction ... 

The coefficients are combined with the radial term to construct a simplified Hamiltonian describing the d-orbital energies. Note that represents a radial correction due to LFSE only, and that the primary contribution to the radial dependence of the ligand-metal interaction energy comes from the standard... 

---