One-dimensional theory

The above expressions are defined in terms of adiabatic potentials E R) and E2(R). The treatment in the adiabatic state representation is generally more accurate than the diabatic state representation. Since the diabatic state representation is, however, sometimes more convenient, the corresponding expressions in terms of the diabatic states are also given here for convenience  

FIGURE 5.4 Nonadiabatic tunneling type potential curve crossing. Ej(R)U = 1. 2) is the j th adiabatic potential curve. Various notations are used in the text. (Taken from Reference [48] with permission.) 

The Zhu-Nakamura theory provides analytical expressions separately for the following three energy regions E[ (top of the lower adiabatic potential), iif E Eb (bottom of the upper adiabatic potential), and Et E. The formulas given here contain some empirical corrections so that the formulas can cover even those small regions of parameters in which the original formulas are not necessarily very accurate. Thus the formulas given below can be directly applied to practical problems. 

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Figure 1. Schematic diagram of a photoacoustic cell used to develop the one-dimensional theory of microphonic PAS by Rosencwaig and Gersho.

Delayed-, After-, or Post-Reactions in Detonation. There are two general types those which occur within a confined space such as in a closed bomb, and those which involve reaction with external air and are known as "afterburning". Accdg to classical one-dimensional detonation theory, chemical equilibrium is achieved and reaction ceases at the CJ (Chapman-Jouguer) plane, which terminates the reaction zone. In some cases, however, as noted by Craig (Ref 3, p 863), the sharp shock wave and the reaction zone of falling pressure are followed by a further rapid pressure drop which is not predicted by an extrapolation of the one-dimensional theory... 

Nonlinear Theory of Unstable One-Dimensional Theory of Detonation. See Detonation, Nonlinear Theory of Unstable... 

The decay zone, characterized by a rate of fall in pressure intermediate between those of two zones, is not predicted by any extrapolation of one- dimensional theory. In the liquid explosive NMe, the pressure at the head of the decay zone is essentially independent of the charge size but the extent of the zone is very strongly dependent on the charge diameter and length. The decay zone increase in length as the detonation runs. 

Recent work on spatial stabilization has been directed towards the production of one-dimensional flames (Fll, P10). These may be either flat, cylindrical, or spherical. The primary purpose of such flames has been to measure velocities accurately and to provide a flame that can be described by a one-dimensional theory. The measurement of temperature and composition profiles is meaningful, of course, only in a flame in which the geometry is known. One-dimensional geometry greatly reduces the labor required to analyze such profiles in order to study kinetics. 

Fig. 10.16 Scaled microwave ionization field, nAE, for H plotted against the scaled microwave frequency n3co experimental ( ) one dimensional theory (x) two dimensional theory (O). n3co = 0.05 corresponds to n = 32 and n3aj = 0.6 corresponds to n = 73. Note the decline from n4E = 1/9 at n = 30 to progressively lower values as n approaches 60 (from...

In the one-dimensional theory of NM we can imagine only a flat flame front the temperature varies only as a function of the coordinate along which the flame propagates, and the direction of the temperature gradient coincides with the direction of propagation. The gradient is small, as is the surface through which heat is transferred (it is equal to the tube cross-section). 

In the one-dimensional theory we may calculate the velocity at the limit as a function of a quantity characterizing the heat transfer. 

Practical application of the one-dimensional theory developed to the calculation of the effects of losses on the detonation velocity is limited by the fact that even at the limit the reaction time is small and heat transfer and braking do not cover the entire cross-section of the tube. At the same time, in the vast majority of cases, long before the limit is reached one observes the so-called spin—a spiral-like or periodic propagation of detonation which is not described by our theory. Some thoughts are given concerning the dimensionless criteria on which the spin depends. 

Let us consider the propagation of a detonation wave in a tube, taking account of heat transfer and braking against the side walls of the tube. We will restrict ourselves to the one-dimensional theory in which heat transfer and drag are uniformly distributed over the entire cross-section of the tube. We denote by x the coordinate measured from the detonation wave front toward the unreacted gas, in the direction of wave propagation. It is in fact on this coordinate alone in the steady and one-dimensional theory that all the following quantities depend ... 

Continuing the refinement, we could take into account the increased heat transfer and braking at the beginning of the motion compared to stabilized motion with steady temperature and velocity profiles. But here we already overstep the boundary of applicability of the theory from the moment that we begin to speak of a profile (distribution over the radius), a one-dimensional theory is no longer possible. We cannot distribute the heat transfer and braking in the short segments of interest to us over the entire cross-section. 

In view of this the boundaries of applicability of the one-dimensional theory developed above, particularly for establishing the influence of losses on the detonation velocity, are in need of additional study. 

The one-dimensional theory developed is compared with available experimental data and peculiarities of the theory and limits of its applicability are indicated. 

Tarassov (1955) and also Desorbo (1953) have considered these ideas in relation to a onedimensional crystal in which case the one-dimensional frequency distribution function predicts a T dependence of the specific heat at low temperatures. In the case of crystalline selenium, however, it has been found necessary to combine the one-dimensional theory with the three-dimensional Debye continuum model in order to obtain quantitative agreement with the data below about 40° K. Tem-perley (1956) has also concluded that the one-dimensional specific heat theory for high polymers would have to be combined with a three-dimensional Debye spectrum proportional to T3 at low temperatures. For a further discussion of one-dimensional models see Sochava and TRAPEZNrKOVA (1957). 

When the mass of the tunneling particle is extremely small, it tunnels through a one-dimensional static barrier. With increasing mass, the contribution from the intermolecular vibrations also increases, and this leads to a weaker mass dependence of kc than that predicted by the one-dimensional theory. That is why the strong isotope H/D effect is observed along with weak dependence kc(m) for heavy transferred particles, as illustrated in Figure 2.16. It is this circumstance that makes the transfer of heavy reactants (with masses 20-30) possible. 

D. B. Spalding, A One-Dimensional Theory of Liquid Fuel Rocket Combustion, A.R.C. Tech. Rept. No. 20-175, Current Paper No. 445 (1959). 

One of the drawbacks of the above one-dimensional theory is that the boundary layer height H = j is, in fact, unknown and was taken as H = 2 - 3. This difficulty does not occur for the pressure-driven duct flow. Therefore, the similar calculations were performed for the latter, [223] although no measurements are known for such a... 

Laboratory modeling and theoretical investigations of an abstract easily penetrable roughness, as well as the in-situ measurements overviewed in Section 1.4, were carried out to perform a real object, an extended spraying cooling system for a project of the Zaporizhzhya nuclear power plant shown in Fig. 1.12. The interpretation of its droplet layer as an EPR was done in Section 1.4. It is the aim of this section to briefly present the main results of a one-dimensional theory. 

The classical theory of activated rate processes has been developed rather extensively during the past decade. This chapter has outlined progress made using the Hamiltonian equivalent approach for the dynamics of the GLE. The one-dimensional theory has matured in a sense. Some of the important open questions such as the derivation of a... 

The authors analyzed the problem by means of numerical analysis using the Galerkin finite element method, as well as a one-dimensional theory for viscoelastic filaments. Their findings were successfully used to interpret existing experimental data on Newtonian and viscoelastic jet... 

The multidimensional view on enei etical states, outlined earlier, is better able to accommodate a number of experimental observations that are difficult to explain in a one-dimensional concept. A one-dimensional theory cannot explain ... 

The quasi-one-dimensional theory of capillary breakup of pseudoplastic jets provides an explanation of the phenomenon of sausage-like breakup [29, 107]. In the case of the power law liquids, the continuity and momentum balance equations of straight jets have the form (1.49) and (1.50), whereas, based on (1.62), (1.51) is replaced by a more general one ... 

The one-dimensional theory for steady incompressible fluid flow in collapsible tubes (when P — P < 0) was outlined by Shapiro (1977) in a format analogous to that for gas dynamics. The governing equations for the fluid are that for conservation of mass,... 

Shapiro (1977) defined the speed index, S = ulc, similar to die Mach number in gas dynamics, and demonstrated different cases of subcritical (5 < 1) and supercritical (S > 1) flows. It has been shown experimentally in simple experiments with compliant tubes that gradual reduction of the downstream pressure progressively increases the flow rate until a maximal value is reached (Holt, 1969 Conrad, 1969). The one-dimensional theory demonstrates that for a given tube (specific geometry and wall properties) and boundary conditions, the maximal steady flow that can be conveyed in a collapsible tube is attained for 5 = I (e.g., when u = c) at some position along the tube (Dawson Elliott, 1977 Shapiro, 1977 Elad et id., 1989). In this case, the flow is said to be choked and further reduction in downstream pressure does not affect the flow upstream of the flow-limiting site. Much of its complexity, however, is still unresolved either experimentally or theoretically (Kamm et al., 1982 Kanun and Pedley, 1989 Elad et al., 1992). 

To start, we consider one-dimensional diffusion model. This is not only because water diffusion into polymer matrix is the first step but the mathematics of two- and three-dimensional diffusion is more complicated. In fact, the results obtained from one-dimensional diffusion model has been practically used whenever diffusion kinetics under investigation. The kinetic models obtained from one-dimensional theory are enough to cover the kinetic models often used by most researchers in the world. Two- or three-dimensional diffusion follows the same principle. 

Vortmeyer D, Schaefer RJ. Equivalence of one- and two-phase models for heat transfer processes in packed beds one dimensional theory. Chemical Engineering Science 1974 29 485-491. 

The analogy between cold drawing and a phase transition emerges clearly from Ericksen s often cited discussion [6] of instabilities that can occur in a one-dimensional theory of the equilibrium of bars under tension in that discussion it is assumed that the total tensile force T at a cross-section is given by a function T of the local stretch ratio 1. and that the material function t has the non-monotone, single-loop, form shown here in Figure 1. 

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