Asymptotical solution

In back-scattering, (n= - n ), and within the Bom approximation (mono-scattering), the asymptotic solution of (2) is ... 

We call this a partial M/ave expansion. To detennine tire coefficients one matches asymptotic solutions to the radial Scln-ddinger equation with the corresponding partial wave expansion of equation (A3.11.106). It is customary to write the asymptotic radial Scln-ddinger equation solution as... 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

We see here that the procedural solution is identical to the asymptotic solutions obtained earlierandthetimetodo n = 1000isonlyl.98secofCPUonaPentiumland0.22onthe... 

Note that the coefficients on temperatures sum to 1.0 in each equation. This is necessary because the asymptotic solution, > 1, must give 0 = for all i. Had there been a heat of reaction, the coefficients would be unchanged but a generation term would be added to each equation. 

In view of their effectiveness in reducing cold-start emissions, we are particularly interested in the behavior of small-volume electric heaters. For an uncatalyzed (inert) heater of sufficiently small size, the following explicit, analytical asymptotic solution can be obtained [10] ... 

This asymptotic solution suggests that the substitution t/r( ) = in Eq. (82) should be tried. If the resulting differential equation for C %) can be solved, die expression for might be valid for all values of the independent variable . 

Show that Vi 2 is an asymptotic solution to Eq. (83) that leads to Hermite s equation. 

In the more general problem in which V (r) 0, the previous boundary condition is not applicable. Thus, B((a) 0 and the asymptotic solution for lttge values of r is given by [Eq. (5-148)]... 

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

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... 

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

This is the constant-pattern simplification that enables many solutions to be obtained from what might otherwise be complex rate equations. It represents a condition that is approached as the wave becomes fully developed and leads to what are termed asymptotic solutions. 

In reality, Eqs. (13) and (14) should be solved simultaneously with Eqs. (8) and (9), but no analytical solution is available. However, we can examine the asymptotic solutions to Eqs. (13) and (14) to determine the bubble growth rate when heat transfer limits the growth, i.e., when P r) — Pq and r Tb so no inertial effects are present. For this extreme,... 

Zhang and Davis proposed an interpolation formula based on Churchill and Usagi s (1972) method of combining asymptotic solutions to obtain a correlation valid over the whole range of variables. Their correlation is... 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

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