Simple Sigma Theory

This theory concerns clarification in cylindrical bowl centrifiiges, such as the tubular bowl machine. It was first published by Hebb and Smith [1948], and is generally known as Ambler s sigma theory [Ambler, 1952]. 

The critical particle trajectory is the particle diameter where the trajectory goes from the top surfrce (inner radius) of the centrifiige bowl to the bottom surface (outer radius) in the residence time within the machine. Particles of a similar diameter entering the 

The residence time of the particle in the axial direction (from left to right) will be  

Equation (8.5) has the dimensions of area, and is equivalent to the required plan area of a settling tank operating under ideal conditions needed to perform the same clarification duty as the centrifuge (see Equation 7.4). 

The right-hand side of Equation (8.5) is called the machine parameters, and is the theoretical area value at 100% efficiency. The left-hand side is called the process parameters. Both are given the symbol S. Tiie machine agma term is conq rised solely of physical characteristics of the centiifiige, being a measure of the clarification ability of the machine. The settling velocity and volume flow rate to give the desired clarification duty are ftmctions of the process. 

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Because the 2p orbitals have a node at the atomic nucleus, the electron distribution of the VB wavefunction formed by their side-on overlap does not have cylindrical symmetry, but instead changes sign when rotated by 180° about the bond axis. Bonds that have this property are called pi (tt) bonds. Thus, the double bond in the oxygen molecule consists of a sigma bond and a pi bond, which are not equivalent. The existence of two different types of bonds in double bond formation is something that is not predicted by simple Lewis theory. A triple bond such as that in N2 is described within the VB approach as a sigma bond formed from the head-on overlap of 2p orbitals and two pi bonds formed by the overlap of the 2py and 2p orbitals on one N atom with their counterparts on the other atom (Figure 3.7). 

It is informative to compare methylene and nitrene in terms of simple molecular orbital theory and then to extend this comparison to their aromatic derivatives phenyl carbene and phenyl nitrene. Hoffman et al. [28] have described the molecular orbitals of CH2 in the following way. Methylene has two nonbonding molecular orbitals, one is an in-plane, a type, hybrid orbital, the other is an out-of-plane pure p type n orbital. Singlet methylene has a bond angle of 105° and the in-plane, sigma, orbital is doubly occupied. Triplet methylene has both a singly occupied n and a orbital and a bond angle of 135°. 

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