Profile infinite linear

Homogeneous function profiles 16-6 Separable profiles 16-7 Example Infinite linear profile 16-8 Example Double parabolic profile... 

Fig. 16-2 Contours of constant refractive index for (a) the infinite linear profile of Eq. (16-24) and (b) the double parabolic fiber profile of Eq. (16-30), where d is the separation of the fiber centers.

The power-law profiles satisfy Eq. (16-19b), and are therefore homogeneous function profiles. Consequently, the above expression is consistent with the more general result of Eq. (16-20). In the case of the infinite linear profile ( = 1), Eq. (17-6) reduces to the exact eigenvalue equation of Eq. (16-29), apart from a change in the multiplicative constant on the ri t from 1.018 to 3/(2 t ) S 1.024, a relative error of 0.6%. This excellent agreement is independent of and and demonstrates the accuracy of the Gaussian approximation for arbitrary eccentricity. 

The simulation of other electrochemical experiments will require different electrode boundary conditions. The simulation of potential-step Nernstian behavior will require that the ratio of reactant and product concentrations at the electrode surface be a fixed function of electrode potential. In the simulation of voltammetry, this ratio is no longer fixed it is a function of time. Chrono-potentiometry may be simulated by fixing the slope of the concentration profile in the vicinity of the electrode surface according to the magnitude of the constant current passed. These other techniques are discussed later a model for diffusion-limited semi-infinite linear diffusion is developed immediately. 

The exact solution for the time-dependence of the current at a planar electrode embedded in an infinitely large planar insulator, the so-called semi-infinite linear diffusion condition, is obtained. Solving the diffusion equation under the proper set of boundary and initial conditions yields the time-dependent concentration profile. 

Electrode geometry — Figure 3. Concentration profiles at an array of inlaid disks at different time in response to an electrochemical perturbation. a Semi-infinite planar diffusion at short times b hemispherical diffusion at intermediate times c semi-infinite linear diffusion due to overlap of concentration profiles at long times... 

These principles are valid regardless of the electrode employed, as long as semi-infinite linear diffusion can be assumed and renewal of the concentration profile can be accomplished in each cycle. For a stationary planar electrode, the relationships worked out above apply directly. For an SMDE, they apply to the extent that /d,oc is the Cottrell current for an electrolysis of duration r and is not disturbed by the convection associated with the establishment of the drop. For a DME, the picture is complicated by the steady expansion of area, but it turns out (47, 48) that (7.3.8) is still a good approximation if /d,DC is understood as the Ilkovic current for time r [(7.3.1) or (7.3.4)] and the pulse width is short compared to the preelectrolysis time [i.e., (r — r )/ t < 0.05]. 

Let us now consider the prototypical case in which the electrode reaction O ne R exhibits reversible kinetics and the solution contains O, but not R, in the bulk. The solution has been homogenized and the initial potential E is chosen well positive of, so that the concentration profiles are uniform as the SWV scan begins. The experiment is fast enough to confine behavior to semi-infinite linear diffusion at most electrodes, and we assume its applicability here. These circumstances imply that we can invoke Pick s second law for both O and R, the usual initial and semi-infinite conditions, and the flux balance at the electrode surface, exactly as in (5.4.2)-(5.4.5). The final boundary condition needed to solve the problem comes from the potential waveform, which is linked to the concentration profile through the nemstian balance at the electrode. It is convenient to consider the waveform as consisting of a series of half cycles with index m beginning from the first forward pulse, which has m = 1. Then,... 

Figure 4.1 shows the variation of the reactant concentration with the distance to the electrode, x, (that is, the concentration profile) in a Cottrell experiment under semi-infinite linear diffusion. A large overpotential is applied to the working electrode such that the reactant species is consumed... 

Therefore, for large optimal control problems, the efficient exploitation of the structure (to obtain 0(NE) algorithms) still remains an unsolved problem. As seen above, the structure of the problem can be complicated greatly by general inequality constraints. Moreover, the number of these constraints will also grow linearly with the number of elements. One can, in fact, formulate an infinite number of constraints for these problems to keep the profiles bounded. Of course, only a small number will be active at the optimal solution thus, adaptive constraint addition algorithms can be constructed for selecting active constraints. 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Figure 6.13 illustrates the streamline patterns and velocity profiles for two rotation rates. The outer flow for the rotating disk is seen to be quite different from the semi-infinite stagnation-flow situation. In the rotating-disk case, the inviscid flow outside the viscous boundary layer has only uniform axial velocity. In the stagnation flow, the axial velocity varies linearly with the distance from the stagnation surface z and the scaled radial velocity v/r is a constant (cf. Fig. 6.6). The rotating-disk solutions reveal that as the rotation rate increases, the axial velocity increases in the outer flow and the boundary-layer thickness decreases as fi1/2 and f2-1/2, respectively.

Consider the two axial-velocity profiles in Fig. 17.10 that correspond to the low-strain solution in Fig. 17.9. While at the symmetry plane both solutions must have zero velocity, the inlet-velocity boundary conditions are quite different. In the finite-gap case (here the gap is 3.5 mm), the inlet velocity is specified directly (here as 250 cm/s). In the semiinfinite case, the inlet cannot be specified. Instead, the velocity gradient a = du/dz is specified, with the velocity itself growing linearly away from the surface. In the finite-gap case the strain rate is determined by evaluating the velocity gradient just ahead of the flame, where there is a region in which the velocity gradient is reasonably linear. In the semi-infinite case, the velocity gradient is specified directly, whereas in the finite-gap case it must be evaluated from the solution. 

FEM versus Analytical Solution of Flow in a Tapered Gap Consider isothermal pressure flow of a constant viscosity Newtonian fluid, between infinite plates, 10 cm long with a linearly decreasing gap size of 1.5 cm at the entrance and 1 cm at the exit. The distance between the entrance and the exit is 10 cm. The pressure at the inlet and outlet are 2 atmospheres and zero, respectively, (a) Calculate the pressure distribution invoking the lubrication approximation, (b) Calculate the pressure profile using the FEM formulation with six equal-sized elements, and compare the results to (a). 

As a first example of low-density heat transfer let us consider the two parallel infinite plates shown in Fig. 12-14. The plates are maintained at different temperatures and separated by a gaseous medium. Let us neglect natural-convection effects. If the gas density is sufficiently high so that A — 0, a linear temperature profile through the gas will be experienced as shown for the case of A. As the gas density is lowered, the larger mean free paths require a greater distance from the heat-transfer surfaces in order for the gas to accommodate to the surface temperatures. The anticipated temperature profiles are shown in... 

Within the present model some semiquantltatlve measure of the Interfacial thickness can be given. Strictly speaking it is infinitely thick, but most of the density variation takes place over a distance t, obtained by extrapolation of the linear part. A simplification of the real profile would be to replace it by the linear step function. This linear part is mathematically represented by the linear term p (z) = z/ (recall tanhx = x - x /3 + 2x /15 -. ..). We find... 

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

The ideal model of chromatography, which has great importance in nonlinear chromatography, has little interest in linear chromatography. Along an infinitely efficient column, with a linear isotherm, the injection profile travels unaltered and the elution profile is the same as the injection profile. We also note here that, because of the profound difference in the formulation of the two models, the solutions of the mass balance equation of chromatography for the ideal, nonlinear model and the nonideal, linear model rely on entirely different mathematical techniques. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

Finally, Kvaalen et al have shown that the system of equations of the ideal model for a multicomponent system (see later, Eqs. 8.1a and 8.1b) is strictly h5q3er-bolic [13]. As a consequence, the solution includes two individual band profiles which are both eluted in a finite time, beyond the column dead time, to = L/u. The finite time that is required for complete elution of the sample in the ideal model is a consequence of the assumption that there is no axial dispersion. It contrasts with the infinitely long time required for complete elution in the linear model. This difference illustrates the disparity between the hyperbolic properties of the system of equations of the ideal model of chromatography and the parabolic properties of the diffusion equation. 

The apparent plate munber can be calculated from the experimental profiles [27]. However, this number depends on the fractional height at which the bandwidth is measured. The value of Nth is calculated from the profiles predicted, under the same experimental conditions, by the ideal model. Finally, Nion is derived from the band profiles recorded in linear chromatography, e.g., with a very small sample size, using the relationships valid for Gaussian profiles. From Eqs. 7.24 and 7.26, we can derive the band width at half height, Wi/2, and the retention time of the band profile, ty, obtained with an infinitely efficient column. In the case of a Langmuir isotherm, we obtain [31]... 

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