Series with Constant Terms

APPENDIX C. INFINITE SERIES Part 1. Series with Constant Terms... 

Arranging the solvents in separation-strength order, the so-called eluotropic series appeared. This term, introduced by Trappe, was related to the experience with bare silica, where a strong solvent is able to move polar solutes on a polar stationary phase. Later this was improved by the discovery of a direct proportion between the elution strength and the dielectric constant. Because silica is hydrophilic and highly polar, there was a correlation between the eluotropic series and the polarity of a solvent [16,18]. 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

This is the transient method for which most experience is available. It was introduced by Bowden and Rideal (1928). The name comes from that of Galvani4 and means, in fact, current. Thus, Galvanostatic transient means short-term constant current. The circuitry is simple. It consists of nothing more than a measurement cell in series with an adjustable resistance much larger in value than the resistance of the cell, a power source, a rapid action switch, and a cathode ray oscilloscope to record the variation in the potential of the working electrode with time. A typical potentialtime relation is shown in Fig. 8.6. 

A physical insight into eqs. (2)-(5) is gained by considering the equivalent circuit shown in Figure 4, which displays the same frequency response defined in eqs. (2)-(5). The membrane capacitance per unit area Cjj, appears in series with the access impedances p and Pa/2, while the term CTfl (1-1.5p) provides for the conductance of the shunting extracellular fluid. Hence the time constant T which determines the frequency where the impedances l/(jjCmR and (p + Pa/2) are equal is given by eq. (5). 

Consider steady-state heat transfer in a rectangular cross section, with constant thermal conductivity. Design the four nonhomogeneous boundary conditions such that the final expression of the temperature distribution consists only of four terms and not an infinite series of terms. Write down this expression. (Hint Look at the solutions of Ex. 4.3 and Prob. 4.9.)... 

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

Since we require %, which is given by (2.23), to be finite everywhere, we do not accept the minus signs in these expressions. Therefore the physically acceptable solutions /( ) and g g) are for small values of and g equal to (1+M)2 and (1+lml)2j respectively, times a power series in and g, respectively, with the constant term different from zero. 

For any allowed value of m, i.e., for m = 0, 1, 2,. .., one thus obtains the physically acceptable wave function y, which is finite everywhere, from the particular solution /( ) that is equal to (1+H)2 times a power series in with the constant term different from zero and from the particular solution g(g) that is equal to r/1+M)2 times a power series in g with the constant term different from zero. These particular solutions /( ) and g(rj), which are single-valued and uniquely determined except for arbitrary constant factors, obviously tend to zero as —> 0 and g —f 0, respectively. 

The partial derivative with respect to T is straightforward to calculate since it only appears in the first term where it is multiplied by a series of constants. Thus... 

Immediately after the introduction of a constant refractive index Bethe developed a dispersion theory of electron diffraction which is very closely related to the Darwin-Ewald theory. In this theory the propagation of de Broglie waves through a crystal is investigated, the potential being expanded in a triple Fourier series in terms of the contributions of the individual lattice planes hkl. Thus Vq in Schrodinger s equation is replaced by a triple Fourier series with the coefficients In accordance with this assumption, the solution... 

The permeability constant of the composite membrane is therefore represented by the harmonic average of the permeability constants of the individual layers, the respective weights being x /Ji., the ratio of layer thickness to the total. Although composite membranes Include layers of dense films or even liquid layers in series with films, in this discussion the term is being limited to those series in which at least one of the members is a phase inversion membrane of either the integrally-skinned or skinless variety. 

This is a common but confusing notation. In fact, it is concepmally more correct to classify these methods according to the terms excluded by the approximation. With the latter approach, a method that approximates the NDF with a constant value within each interval can be denoted a first-order method, since when approximating the NDF as a Taylor series the first term excluded is the first-order derivative term. 

Phase equilibria can be modeled in terms of equilibrium constants for the relevant reactions. Because of low mutual solubilities of the phases, extraction reactions appear to be heterogeneoue. Some reactions eshibil slow chemical kinetics, with die reaction step constituting a resistance to extraction in series with intraphase mass transfer. 

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