Asymptotic solutions order

The asymptotic solution ( - large) for tj is [2/(n + l)]1/2/, of which the result given by 8.5-14c is a special case for a first-order reaction. The general result can thus be used to normalize the Thiele modulus for order so that the results for strong pore-diffusion resistance all fall on the same limiting straight line of slope - 1 in Figure 8.11. The normalized Thiele modulus for this purpose is... 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

In Fig. 7 the effectiveness factor is shown as a function of the generalized Thiele modulus pn for different reaction orders (flat plate). From this figure, it is obvious that, except for the case of a zero order reaction, the curves agree quite well over the entire range of interest. The asymptotic solution t = l/ pn is valid for any reaction order and for values of the modulus pn > 3. 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

This is a somewhat opaque and over-complicated result. Thus, to facilitate the extraction of useful kinetic information from experimental data, we return to equation (9.98) in order to identify simpler, asymptotic solutions. These are illustrated in the case diagram of Fig. 9.11. At the bottom of the diagram, where is small and r is therefore large, we obtain the asymptotic steady state solution ... 

An analysis of radial flow, fixed bed reactor (RFBR) is carried out to determine the effects of radial flow maldistribution and flow direction. Analytical criteria for optimum operation is established via a singular perturbation approach. It is shown that at high conversion an ideal flow profile always results in a higher yield irrespective of the reaction mechanism while dependence of conversion on flow direction is second order. The analysis then concentrates on the improvement of radial profile. Asymptotic solutions are obtained for the flow equations. They offer an optimum design method well suited for industrial application. Finally, all asymptotic results are verified by a numerical experience in a more sophisticated heterogeneous, two-dimensional cell model. 

In the following discussion, we assume that s <asymptotic solution in this thin-film limit, the governing equations for the leading-order approximation are... 

We seek an asymptotic solution of the leading-order thin-film equations, (5—134)— (5 136), in the form... 

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

For a first-order volume reaction and a translational Stokes flow past a spherical drop, the asymptotic solution of the inner problem (5.3.1), (5.3.2) as Pe -4 oo results in the following expression for the mean Sherwood number [104] ... 

The structure of MQDT involves a compact target state within a radius ro and an asymptotic solution, valid at large r, which is joined to the wavefunction of the inner region in order to determine amplitudes and phases. Thus, MQDT is simply a special form of scattering theory, specially adapted to handle bound Rydberg states of atoms and adjacent structure in the base of the ionisation continuum. 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

Next, we consider the two limiting cases in order to derive compact analytical expressions for Ux. The first is the case that the two pairs of transition points (/ /4) and (t2, t3) are well isolated from each other along the real axis and can be treated separately. Roughly speaking, this corresponds to b2 1 in both the LZ and NT cases. The connection matrix L which connects the coefficients of the asymptotic solutions are z - and is directly related to scattering matrix can be well approximated by... 

The aim of this chapter is to present the fundamentals of adsorption kinetics of surfactants at liquid interfaces. Theoretical models will be summarised to describe the process of adsorption of surfactants and surfactant mixtures. As analytical solutions are either scarcely available or very complex and difficult to apply, also approximate and asymptotic solutions are given and their ranges of application demonstrated. For particular experimental methods specific initial and boundary conditions have to be considered in these theories. In particular for relaxation theories the experimental conditions have to be met in order to quantitatively understand the obtained results. In respect to micellar solutions and the impact of micelles on the adsorption layer dynamics a detailed description on the theoretical basis as well as a selection of representative experiments will follow in Chapter 5. 

Figure 7.8 Nonisothermal effectiveness factor for first-order reaction in a sphere Ps — k/ [After P.B. Weisz and J.S. Hicks, Chem. Eng. Sci., 17, 265, with permission of Pergamon Press, Ltd., London, England, (1962).] Dotted lines are the asymptotic solution given by Petersen [E.E. Petersen, Chem. Eng. Sci., 17, 1987 (1962).]...

For second-order reactions, the asymptotic solution for infinitely rapid reaction [P.V. Danckwerts, Trans. Faraday Soc., 46, 701 (1950)] is given by the awkward parametric form... 

Figure 6.23. Thin solid line - numerical solution to Eq. (6.125), thick solid lines leading-order asymptotic solutions j = /Q (upper) and j=Q (lower). Dashed curve -matched as5mptotic solutions (6.134) (/ < 1) and (6.137) (/> 1).

The zero order asymptotic solution (Eq. 6.53) agrees reasonably well with the exact solution over most of the domain [0,1] except close to the origin. The first order asymptotic expansion (Eq. 6.55), even though it agrees with the exact solution better, also breaks down near the origin. This suggests that a boundary layer solution must be constructed near the origin to take care of this nonuniformity. We will come back to this point later. 

Ah Ax = A h/Abc = 0, noting that B can be set to zero by arbitrarily shifting the origin. It is then possible to match the curvature A hlAx of the asymptotic solution in Eq. 35 to the outer spherical cap solutions, which, to leading order, are static solutions of the Laplace-Young equation. Chang [17] showed that this leads to... 

As we have noted above, Edwards asymptotic solution for the SCF Eq. (3.101) is only approximate and improved solutions should be sought. However, the —4/3 power law in Eq. (3.101) does give the critical exponent 6/5 for the mean square end-to-end distance of a self-avoiding random walk in agreement with the extrapolations obtained from lattice calculations (Domb (1963)). Thus, the use of Eq. (3.102) is justified in order to obtain the qualitative results of this subsection. 

Nevertheless, the inclusion of axial dispersion may be interesting from the point of view of the numerical methods used to solve the conservation equations or in studies regarding the appearance of multiple steady-state solutions [141, 142], Petersen [81] presented an analysis for a ID reactor in terms of the dispersion factor E, which is the ratio between the length of a plug-flow reactor (no dispersion) and the one for a reactor with dispersion yielding the same conversion (Lm). Due to the coordinate transformations employed, F is a function of oP = kDAefu (isothermal first-order reaction). Asymptotic solutions for the dispersion ratio were obtained and are given by... 

If one takes into account the subsequent corrections of order C(e"), n = 1,2,..., then the asymptotic solutions can be matched up to order p(g( +i)/2) jjj common domain. Incidentally, similar relations to Eq. (2.6) apply for the previous trivial example. 

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