Approach asymptotes

Then F( ) = S(/ -t d )- 2( ), and the density of states D E) = dS/d/ . A system containing a large number of particles N, or an indefinite number of particles but with a macroscopic size volume V, normally has the number of states S, which approaches asymptotically to... 

Interfacial Mass-Transfer Coefficients. Whereas equiHbrium relationships are important in determining the ultimate degree of extraction attainable, in practice the rate of extraction is of equal importance. EquiHbrium is approached asymptotically with increasing contact time in a batch extraction. In continuous extractors the approach to equiHbrium is determined primarily by the residence time, defined as the volume of the phase contact region divided by the volume flow rate of the phases. 

To achieve the very low initial fluorine concentration in the LaMar fluorination process initially a helium or nitrogen atmosphere is used in the reactor and fluorine is bled slowly into the system. If pure fluorine is used as the incoming gas, a concentration of fluorine may be approached asymptotically over any time period (Fig. 3). It is possible to approach asymptotically any fluorine partial pressure in this manner. The very low initial concentrations of fluorine in the system greatiy decreases the probabiUty of simultaneous fluorine coUisions on the same molecules or on adjacent reaction sites. 

Neutron reflectivity is ideally suited to this problem, since concentration profiles can be resolved on the nanometer level and since, for an infinitely sharp interface, Rkjf will approach asymptotically a constant value. In addition, neutron reflectivity is nondestructive and multiple experiments can be performed on the same specimen. Figure 4 shows a plot of Rk Q as a function of bilayer of protonated... 

Each curve therefore consists of three parts an initial and a final portion which are nearly horizontal for a finite part of their lengths, and an intermediate portion which slopes down comparatively rapidly from left to right. This means that the dissociation with rise of temperature is slow at first, then increases very rapidly, and then becomes increasingly slower as it approaches asymptotically to the limiting value for T = oo. The general form of curve so predicted corresponds exactly with the experimental curves, as will be seen from Fig. 66, which was drawn by Horstmann from the results of Wiirtz with amylene hydrobromide ... 

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

The first indication that NEMCA is due to electrochemically induced ion backspillover from solid electrolytes to catalyst surfaces came together with the very first reports of NEMCA Upon constant current application, i.e. during a galvanostatic transient, e.g. Fig. 5.2, the catalytic rate does not reach instantaneously its new electrochemically promoted value, but increases slowly and approaches asymptotically this new value over a time period which can vary from many seconds to a few hours, but is typically on the order of several minutes (Figure 5.2, galvanostatic transients of Chapters 4 and 8.)... 

Figure 2.10 shows a plot of 0a versus the partial pressure of A, p. . At low pressure, the coverage is small, and increases linearly with pressure the derivative of the plot equals the equilibrium constant, Ka- At high pressure, the surface becomes saturated, and the coverage approaches asymptotically its saturation value of 100 %. 

A comparison of the thus calculated with the measured specific rotations of the 0th- to 4th-generation dendrimers of this kind gave a close resemblance, with a curve, approaching asymptotically a limiting value (Fig. 26). It was also shown that the shape of this curve was independent of solvent, concentration and temperature. This was not the case when CD spectra of these dendrimers were compared (Fig. 27) in solvents such as CH2C12 and f-butyl methyl ether a constant rise of the Cotton effect was observed, which correlates with the increasing amount of benzene chromophores in the dendrimers. However, in the... 

Does the solution make sense from a steadystate point of view The new steadystate value of that is approached asymptotically by the exponential function can be found from either the solution [Eq. (6.48)], letting time t go to infinity, or from the original ODE [Eq. (6.46)], setting the lime derivative dCJdt equal to zero. Either method predicts that at the final steadystate... 

Obviously, the pairs (j4o,Bo) and (A, G) must be stabilizable and detectable, respectively. As we can see, controller (22) has the form of (5) and does not contain the mappings U (/x) and F (/x) thus, although the initial condition for 2 t) is not exactly known, the immersion observer (second expression in (22)) estimates the correct steady-state input and as a result, the controller is capable to drive the system towards the correct zero-error submanifold in spite of parametric variations. It can be seen from the first equation in (22) that as e t) approaches asymptotically zero, so does z. Notice also that the dynamics of Z2 is similar to immersion (21). It is important to point out that this design procedure does not require the exact calculation of mappings II (/x) and F (/x), but it suffices only to know the dimension of matrix S. 

For a bubble to grow, vapor must pass from the superheated liquid into the bubble. Thus latent heat of vaporization is removed from the surrounding liquid, and the liquid cools. The drop in liquid temperature near the bubble means a decrease in the driving force between liquid and bubble. This temperature drop strongly affects the bubble rate of growth. The rate can be shown to approach asymptotically a condition whereby the radius increases according to the square root of time. 

In direct contrast to the behavior of the singly doped europium samples, the decay curves for the doubly doped crystals cannot be characterized by a simple exponential behavior. It appears, however, that such behavior is approached asymptotically. 

Such an assembly of mixing and separating equipment is represented in Figure 14.3(a), and more schematically in Figure 14.3(b). In the laboratory, the performance of a continuous countercurrent extractor can be simulated with a series of batch operations in separatory funnels, as in Figure 14.3(c). As the number of operations increases horizontally, the terminal concentrations E1 and R3 approach asymptotically those obtained in continuous equipment. Various kinds of more sophisticated continuous equipment also are widely used in laboratories some are described by Lo et at. (1983, pp. 497-506). Laboratory work is of particular importance for complex mixtures whose equilibrium relations are not known and for which stage requirements cannot be calculated. 

Figure 14.3. Representation of countercurrent extraction batteries, (a) A battery of mixers and settlers (or separators), (b) Schematic of a three-stage countercurrent battery, (c) Simulation of the performance of a three-stage continuous countercurrent extraction battery with a series of batch extractions in separatory funnels which are designated by circles on the sketch. The numbers in the circles are those of the stages. Constant amounts of feed F and solvent S are mixed at the indicated points. As the number of operations is increased horizontally, the terminal compositions Et and R3 approach asymptotically the values obtained in continuous countercurrent extraction (Treybal, 1963, p. 360).

A steady-state (normalized) distribution function is approached asymptotically as t —> oo. This steady-state distribution, illustrated in Fig. 15.5, is approached by all initial distributions. The most frequent particle size in the steady-state distribution is 1.13(i ) and there will be no particles larger than 1.5(i ), the cut-off size. 

Figure 4.10 shows that two approximations, equations (4.3.22) and (4.3.23), become close in the middle of the intermediate region and both are approaching asymptotically equation (4.3.26). Note that the case of multipole... 

The formal theory of resonances due to Feshbach begins with the decomposition of the Hamiltonian in terms of a projection operator Q [8]. He defines Q as the projection onto the closed-channel space, just like the example of H discussed around Eqs. (4) and (5). Then, QBSs described well by the eigenfunctions Q4> of Eq. (5) with his Q may be called Feshbach resonances." A simplified picture would be that eigenstates Q are supported by some attractive effective potential approaching asymptotically the threshold energy of a closed channel. If this is the case, then the energies EQ of... 

This case, displayed in Fig. 1.17(d), occurs at a low inlet oxygen concentration or a high initial coke loading - that is, the combustion is oxygen-limited. The heat of combustion is carried away downstream by convection. For wr Wtherm the effective temperature rise approaches asymptotically the value... 

Because V is approached asymptotically in the saturation curve shown in Figure 5.5, the estimation of Km and is not very accurate. Equation (5.24) can be algebraically transformed into a linear form more suitable for plotting experimental data and obtaining Km and Vmax. 

Equation (5.5) is the classical equation quoted in textbooks indicating the exponential increase of concentration c (t) approaching asymptotically the saturation solubility cs. 

For low temperatures the curves must approach asymptotically the value 1. The physical meaning of these curves is as follows. It is possible to determine for each temperature in the transition range how many methyl groups in relation to the total number of methyl groups change from a frozen state to one of free rotation during a temperature increase... 

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