Asymptotic shape

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be self-sharpening. (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded oed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.)... 

Figure 3 Periodic boundary conditions realized as the limit of finite clusters of replicated simulation cells. The limit depends in general on the asymptotic shape of the clusters here it is spherical. Cations are presented as shaded circles anions as open circles.

When a recovery well is located within a contaminant plume and the pump is started, the initial concentration of contaminant removed is close to the maximum level during preliminary testing. As the pump continues to operate, cleaner water is drawn from the plume perimeter through the aquifer pores toward the recovery well. Some of the contaminant is released from the soil into the water in proportion to the equilibrium coefficient. For example, if the Kd is 1000, at equilibrium, 1 part is in the water and 1000 parts are retained in the soil. If the water-soil contact time is sufficient, complete equilibrium will be established. After the first pore volume flush (theoretically), the concentration in the water will be 0.9 and that on the soil will be 999. With each succeeding flush, the 1000 1 ratio will remain the same. If the time of water-soil contact is not sufficient to establish equilibrium, the recovered water will contain a lesser concentration. A typical decline curve is shown on Figure 9.2. Note the asymptotic shape of the curve where the decline rate is significantly reduced. 

There is a theoretical study on the asymptotic shape of probability distribution for nonautocatalytic and linearly autocatalytic systems with a specific initial condition of no chiral enantiomers [35,36]. Even though no ee amplification is expected in these cases, the probability distribution with a linear autocatalysis has symmetric double peaks at 0 = 1 when ko is far smaller than k -,kototal number of all reactive chemical species, A, R, and S. This can be explained by the single-mother scenario for the realization of homo chirality, as follows From a completely achiral state, one of the chiral molecules, say R, is produced spontaneously and randomly after an average time l/2koN. Then, the second R is produced by the autocatalytic process, whereas for the production of the first S molecule the... 

The value of this parameter was determined on the basis of soap adsorption on AAH precipitates at pH = 6.0-6.4. For soap solutions between 0.1-0.3 g/m , an asymptotic shape of the isotherm of adsorption was found and the value of adsorption capacity was determined as 4.5 eq/kg AI2O3. Assuming that an oriented adsorption takes place and that the area covered by a carboxyl group is 30.5 A, the specific surface area of AAH was calculated ... 

The fluorescence and isotherm measurements are consistent with the generalized phase diagram for fatty acids shown schematically in Fig. 12. Presumably the G-LE coexistence region ends in a critical point but none of the measurements that have been carried out can fix and define the asymptotic shape of the coexistence curve with any precision. The measurements show a clear narrowing of the range of LE-LC coexistence, which suggests that this may end at some sort of critical point as well, but here again experiments do not yet provide much direct information. 

Equation (120) shows that at large values of x and t, the main part of the signal travels with the equilibrium sound speed x/t = a Q, and the width of the signal broadens (as x and t increase) in proportion to l/y/t. Since f is the dimensionless time variable appearing in equation (119), the asymptotic shape given by equation (120) is attained earlier (at smaller values oft and x) for smaller values of the reaction time t. 

As an alternative to direct heat transfer measurements it is possible to use changes in pressure drop brought about by the presence of the deposit. The pressure drop is increased for a given flow rate by virtue of the reduced flow area in the fouled condition and the rough character of the deposit. The shape of the curve relating pressure drop with time will in general, follow an asymptotic shape so that the time to reach the asymptotic fouling resistance may be determined. The method is often combined with the direct measurement of thickness of the deposit layer. 

In the effort to determine transient flux shapes certain interesting characteristics of the flux behavior become evident. If the power rise is not checked the flux eventually attains a stable or asymptotic shape. The method of harmonics also provides a means of determining the asymptotic flux shape and also the elapsed time in the attainment of this stable shape. After the asymptotic shape has been achieved there is, at least from a theoretical point of view, no further change in the flux shape and the kinetics problem becomes separable. 

Graphs are provided in Figures 7, 8, and 9 corresponding to three values of ASp. In each case there is exhibited the steady state flux, some of the transient fluxes, and the asymptotic flux, all normalized to equal areas. Also provided is the stabilization time for each case, that is, the total time elapsed in attainment of the asymptotic shape. The criterion used to specify a stabilization time is discussed at the end of this section. The step changes in reactivity exhibited in the three figures were computed by a conventional steady state two-group criticality calculation with the aid of the Wanda Code (IBM-704) [19]. For the calculations under discussion = A eff — 1-... 

It is also of interest that the coefficients in the asymptotic flux are almost identical in the two cases. Moreover, the stable periods are almost the same. Thus, not only are the asymptotic shapes identical, for all practical purposes,... 

The potential (7ex originates from the exchange of light fermions and thus can be treated as an exchange interaction. It is purely repulsive and, according to Equation 10.44, has the asymptotic shape of a Yukawa potential at large R. Direct calculations show that f/ex is a very good approximation of Ueff for / > 1,5a. 

For practical manipulations, the asymptotic shapes of the Bessel function J ( ) of Eq. (5. 236) will be considered (Mathews Walker, 1969) ... 

The asymptotic shape of the dependence Uq = (pCf) (Fig. 8.7a) demonstrated the existence of an isotropy frequency /q at which Ae returns to zero and above which it can assume negative values. The piesence of this effect, well known... 

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