London Equations

Mitchell, A. R. (1969). Computational Methods in Partial Differential Equations. London John Wiley. 

Cercignani, C. 1975 Theory and Application of the Boltzmann Equation. London Scottish Academic Press. 

Corduneanu, C., Principles of Differential and Integral Equations, London Chelsea Pnblishing, 1977. 

Noble, B., Methods Based on The Wiener-Hopf Technique for the Solution of Partial Differential Equations, London Chelsea Publishing, 1988. 

G. B. Airy s little book,. An Elementary Treatise on Partial Differential Equations, London, 1873, will repay carefnl study in connection with the geometrical interpretation of the solutions of partial differential equations. 

For the ingenious general methods of Charpit and G. Monge, the reader will have to consult the special text-books, say, A. B. Forsyth s A Treatise on Differential Equations, London, 1903. There are some special variations from the general equation which can be solved by short cuts... 

I have sometimes found it convenient to evade a tedious demonstration by reference to the regular text-books . In such cases, if the student wants to dig deeper, one of the following works, according to subject, will be found sufficient B. Williamson s Differential Calculus, also the same author s Integral Calculus, London, 1899 A. R. Forsyth s Differential Equations, London, 1902 W. W. Johnson s Differential Equations, New York, 1899. 

In the classical model of superconductivity, the London equations (London and London 1935) are equivalent to Ohm s law j = o-E for a normal electric conductor. The first of the London equations [Eq. (E.l)] represents a conductor with R = 0, while the second [Eq. (E.2)] is equivalent to the Meissner-Ochsenfeld effect (Figure E.l), and describes the decay of a magnetic field within a thin surface layer characterized by the penetration depth,... 

Watts R C 1972 Integral equation approximations in the theory of fluids Specialist Periodical Report vol 1 (London Chemical... 

The Schrodinger equation contains the essence of all chemistry. To quote Dirac The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known. [P.A.M. Dirac, Proc. Roy. Soc. (London) 123, 714 (1929)]. The Schrodinger equation is... 

Noncircular Ducts Equations (5-50 ) and (5-50/ ) may be employed for noncircular ducts by using the equivalent diameter D = 4 X free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-HiU, New York, 1984) give charts for various noncircular duels encountered in compact heat exchangers. 

For turbulent flow, equations by Ito (J. Basic Eng, 81,123 [1959]) and Srinivasan, Nandapnrkar, and Holland Chem. Eng. [London] no. 218, CE113-CE119 [May 1968]) may be used, with probable aecnracy of 15 percent. Their equations are similar to... 

Witze Am. In.st. Aeronaut. Astronaut. J., 12, 417-418 [1974]) gives equations for the centerline velocity decay of different types of subsonic and supersonic circular free jets. Entrainment of surrounding fluid in the region of flow establishment is lower than in the region of estabhshed flow (see Hill, J. Fluid Mech., 51, 773-779 [1972]). Data of Donald and Singer (T/V7/1.S. In.st. Chem. Eng. [London], 37, 255-267... 

Equations (12-31), (12-32), and (12-33) hold only for a slab-sheet solid whose thickness is small relative to the other two dimensions. For other shapes, reference should be made to Crank The Mathematics of Diffusion, Oxford, London, 1956). 

The specific resistance coefficient for the dust layer Ko was originally denned by Williams et al. [Heat. Piping Air Cond., 12, 259 (1940)], who proposed estimating values of the coefficient by use of the Kozeny-Carman equation [Carman, Trans. Inst. Chem. Fng. (London), 15, 150 (1937)]. In practice, K and Ko are measured directly in filtration experiments. The K and Ko values can be corrected for temperature by multiplying by the ratio of the gas viscosity at the desired condition to the gas viscosity at the original experimental conditions. Values of Ko determined for certain dfists by Williams et al. (op. cit.) are presented in Table 17-5. 

If the overlap integral is neglected, the Heitler-London equation becomes... 

Potential energy surfaces calculated by means of the London equation (5-15) cannot be highly accurate, but the results have been very useful in disclosing the general shape of the surface and the reaction coordinate. The London equation also forms the basis of some semiempirical methods. 

The semiempirical methods combine experimental data with theory as a way to circumvent the calculational difficulties of pure theory. The first of these methods leads to what are called London-Eyring-Polanyi (LEP) potential energy surfaces. Consider the triatomic ABC system. For any pair of atoms the energy as a function of intermolecular distance r is represented by the Morse equation, Eq. (5-16),... 

In this equation, AG°CS is taken to be negligible for p- and y-cyclodextrin systems and to be constant, if there is any, for the a-cyclodextrin system. The AG W term is virtually independent of the kind of guest molecules, though it is dependent on the size of the cyclodextrin cavity. The AG dw term is divided into two terms, AG°,ec and AGs°ter, which correspond to polar (dipole-dipole or dipole-induced dipole) interactions and London dispersion forces, respectively. The former is mainly governed by the electronic factor, the latter by the steric factor, of a guest molecule. Thus, Eq. 2 is converted to Eq. 3 for the complexation of a particular cyclodextrin with a homogeneous series of guest molecules ... 

Another generally useful equation for the London energy is that... 

Table II shows clearly the large differences between various theories for many-electron systems. The Kirkwood-Muller equation always yields somewhat too large coefficients for the atoms which are the only spherical systems but the London equation deviates by a greater amount on the low side. The Slater-Kirkwood equation gives a high value for He but yields coefficients smaller than the empirical ones for all other cases.

In view of the complications of the intermolecular potential (as compared to the interatomic potential of the rare gas atoms) the comparisons for molecules in Tables II, III, and IV should be judged with caution. The apparent discrepancies from the theories for single atoms can be misleading. An example is the calculation for CH4 on the Slater-Kirkwood theory where Table IV shows the absurd value of 24 for the effective number of electrons. Pitzer and Catalano32 have applied the Slater-Kirkwood equation to the intermolecular potential of CH4 by addition of all the individual atom interactions and, with N = 4 for carbon and 1 for hydrogen, obtained agreement within 5 per cent for the London energy at the potential minimum. 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

In addition to the calculations in Table VI which are based upon Eq. 29, one may use the Slater-Kirkwood equation (Eq. 39). If one takes effective N values less by one than those given in Table IV for the rare gas atom adjacent to each halogen, the resulting London energies are very nearly the same as those in Table VI. 

Hydrodynamic Theory of Detonation, I. Thermochemistry And Equation of State of The Explosion Products of Condensed Explosives , Res (London) 1, 132-44 (1947) CA 44, 10321 (1950) 66) J. Svadeba, Impact Sens -... 

Effectiveness factors q are plotted against number of transfer units N with (Gj CPI jG2CPl) as parameter for a number of different configurations by Kays and London 251. Examples for countercurrent flow (based on equation 9.235) and an exchanger with one shell pass and two tube passes are plotted in Figures 9.85a and b respectively. 

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