Fins

In the foregoing development we derived relations for the heat transfer from a rod or fin of uniform cross-sectional area protruding from a flat wall. In practical applications, fins may have varying cross-sectional areas and may be attached to circular surfaces. In either case the area must be considered as a variable in the derivation, and solution of the basic differential equation and the mathematical techniques become more tedious. We present only the results for these more complex situations. The reader is referred to Refs. 1 and 8 for details on the mathematical methods used to obtain the solutions. 

To indicate the effectiveness of a fin in transferring a given quantity of heat, a new parameter called fin efficiency is defined by 

The fins discussed above were assumed to be sufficiently deep that the heat flow could be considered one-dimensional. The expression for mL may be written 

Harper and Brown [2] have shown that the solution in case 2 above may be expressed in the same form as Eq. (2-38) when the length of the fin is extended by one-half the thickness of the fin. A corrected length Lc is then used in all the equations which apply for the case of the fin with an insulated tip. Thus 

The error which results from this approximation will be less than 8 percent when 

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It is important to realize that while the uppennost diagonal elements of these matrices are numbers, the other diagonal element is a matrix of dimension N. Specifically, these are the matrix representations of Hq and Fin the basis q which consists of all the original set, apart from i.e. 

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

I iciuri- 4 J. Apparalus lo show clecfrofi tnwsier hetwt eu /fin.s ami ohilion... 

Total spin den sity reflects th e excess probability of fin din g a versus P electrons in an open-shell system. Tor a system m which the a electron density is equal to the P electron density (for example, a closed-shell system), the spin density is zero. 

IlypcrC hcm can calciilaic jiComcLi y opiinii/alion s (minimi/a-tioiis) with either molecular or qiiaiUiim mechanical methods. Geometry optinii/ation s fin d the coord In ates of a molecular stnic-mre that represent a potential energy minimum. 

To fin d a first order saddle poiri t (i.e., a trail sition structure), a m ax-imiim must be found in on e (and on/y on e) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) bein g varied. So, a tran sition structu re is ch aracterized by th e poin t wh ere all th e first derivatives of en ergy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. 

Symmetrical transition states are the lowest energy eon figuration w ithin th at syrn m etry. If a geometry optim i/ation starts off with in that sym m etry, th en th e calcu latio n can fin d th e trail sition state. 

It is also possible to simulate liquid droplets by surrouridiu g a solute by a fin ite ii urn ber of water moleeu les an d perform in g the sim -ulalion without a periodic box. The water, of course, eventually evaporates and moves away from the solute when periodic boundary con ditioii s arc n ot im posed. If the water is in itially added via periodic boundary con dition s, you rn ust edit the resu Itin g H IN file to remove th e periodic boti ruiary con ditioii s, if a droplet approach is desired. 

HyperChem uses single detenu in am rather than spin-adapted wave fn n ction s to form a basis set for th e wave Fin ciion sin a con -figuration interaction expansion. That is, HyperChem expands a Cl wave function, m a linear combination of single Slater deterniinants P,... 

HyperChcin s ah mitio calculations solve the Roothaan equations (.h9 i on page 225 without any further approximation apart from th e 11 se of a specific fin iie basis set. Th ere fore, ah initio calcii lation s are generally more accurate than semi-enipirical calculations. They certainly involve a more fundamental approach to solving the Sch riidiiiger ec nation than do semi-cmpineal methods. 

The synchniiuius Iran sit mclhod is com bined with quasi-Newton niethodslo find transition slates. Quasi-.Newlon m etliods are very rohu St an d effieien t in fin din g en ergy in in ini a. Based solely on local information, there is no unique way of moving uphill from eith er rcactari ts or products to reach a specific reaction state, sin ce all direcLion s away from a minimum go uphill. 

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