Equivalent section

Electromagnetic flow meters ate available in essentially all pipe si2es, ie, 1 mm to 3 m, and provide measurement accuracy of 1% of rate or better over wide ranges. The meters are obstmctionless, have no moving parts, and are extremely mgged. Pressure loss is that of an equivalent section of pipe. 

On a horizontal belt filter, the equivalent sections extend across the filter in narrow strips. Therefore, changes in vacuum do occur rapidly and may be considered as happening at a particular point along the length of the filter. 

It is becoming common practice to have the cross-section of a plastic moulding made up of several different materials. This may be done to provide a permeation barrier whilst retaining attractive economics by having a less expensive material making up the bulk of the cross-section. To perform stress analysis in such cases, it is often convenient to convert the cross-section into an equivalent section consisting of only one material. This new section will behave in exactly the same way as the multi-layer material when the loads are applied. A very common example of this type of situation is where a solid skin and a foamed core are moulded to provide a very efficient stiffness/weight ratio. This type of situation may be analysed as follows ... 

Solution The first step in analysing the foamed sandwich type structure is to calculate the second moment of area of the cross-section. This is done by converting the cross section to an equivalent section of solid plastic. This is shown in Fig. 2.18. 

From the equivalent section the second moment of area can then be calculated as... 

Once the foamed plastic moulding has been converted to an equivalent section of solid plasdc then the long term design procedures illustrated in the... 

The rules for resonance stress that the greatest stabilization is expected when the contributing structures are equivalent (Section 6-5B). Therefore we can conclude that the resonance energy of a carboxylate anion should be... 

The addition of unstabilized a-nitrile carbanions to a,3-unsaturated carbonyl acceptors affords predominantly 1,2-addition products, 56 while lithiated acetonitrile derivatives having a-alkoxy, a-aromatic, a-dialkylamino, a-phenylselenyl, a-phenylthio or a-trimethylsilyl substituents afford 1,4-adducts. However, some of these are acyl anion equivalents (Section 1.2.2.3.2) so this discussion is limited to a-stabi-lized nitriles in which the nitrile function is retained after removal of the activating group. Notable examples are trimethylsilylacetonitrile (208),157 phenylthioacetonitriles (209),158a b phenylselenylacetoni-... 

The synthesis of 3-benzylcyclobutanone (3) is an illustration of an overall intramolecular alkylation of an acyl anion equivalent (Section 5.9). The a,a>-dihalide is 2-benzyl-l,3-dibromopropane, and the acyl anion equivalent is methyl methylthiomethyl sulphoxide2 the product is 1-methylsulphinyl-l-methylthio-3-benzylcyclobutane which is obtained as a mixture of cis/trans isomers [(9) and (10)] (Expt 7.3). Aqueous acid hydrolysis in ethereal solution unmasks the carbonyl group. The possible mechanism of the reaction is via a Stevens-type rearrangement of the intermediate sulphur ylide, which may proceed in a pericylic, radical or ion pair fashion. 

The ring system, which has an axis of symmetry, consists of two interchangeable ortho, two interchangeable meta protons, and one para proton, all in the characteristic region for ring protons. The alert student, having absorbed the concept of chemical-shift equivalence, (Section 3.8.3) and the description of the... 

The following similar achiral molecules contain diastereotopic methylene protons 3-hydroxyglutaric acid, glycerol, citric acid, diethyl acetal, and a cyclic acetal (respectively, Figure 3.43, structures f, g, h, i, and j) structure j involves the additional concept of magnetic equivalence (Section 3.9). 

Synthetic equivalent (Section 10.2) A species that is used in place of another reagent in a synthesis. A synthetic equivalent gives the same final product, usually in better yield, as the other reagent, although more steps are required. 

For example, the 60-MHz H spectrum of cyclopropane derivative 10-10 dissolved in carbon tetrachloride exhibits only one sharp signal at 8 1.45 ppm,9 indicating that the two sets of hydrogens (six methyl and two methylene) are accidentally equivalent (Section 4.4) ... 

Another application of solid-state NMR is in the study of solid-state conformation. In solution molecules tumble rapidly, and rotations about single bonds are fast recall that in solution spectra the three methyl protons are normally equivalent (Section 4.2). In the solid state many of these motions are restricted or suppressed entirely, and many of the chemical shift degeneracies to which we have become accustomed are now split. For example, pnra-dimethoxybenzene, structure 15-1, exhibits a solution l3C spectrum in which the two methoxy carbons are equivalent to each other, as are the two substituted aromatic carbons, and each set gives rise to a single resonance ... 

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

This is exactly what one would expect for binding of deprotonated enzyme to protonated inhibitor. The pH of maximal inhibition ( 7) is well removed from the pH optimum of catalysis (<5). However, as with all pH variation studies (see the discussion of kinetic equivalence, Section 3.7.4), information is provided on the composition of the complex, not its structure. If we set B = AK j Ki, then eqn. (5.28) holds, giving the appearance of the binding of protonated enzyme to deprotonated inhibitor. pH variation studies cannot by themselves make the distinction between the two modes of binding. 

Protons in a set must be both chemical-shift equivalent and magnetic equivalent (Section 4.13). 

The spin system for p-fluoroacetophenone is AA MM X because of magnetic non-equivalence (Section 4.13) it is not first order. 

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