One-dimensional models

We studied previously a one-parameter ladder model (46-48) with non-degenerate singlet ground state. The exact ground state wave function of the cyclic ladder was written in the MP form (39). Now we write the wave function I o in a form more suitable for subsequent generalization to other types of lattices [31]. 

We consider a ladder of N = 2M spins 1/2. The wave function of this system is described by the Nth-rank spinor 

We partition the system into pairs of spins located on rungs of the ladder. The wave function can then be written as the product of M second-rank spinors 

We now form a scalar from Eq. (52), simplifying the latter with respect to index pairs  

Here subscripts correspond to the covariant components of the spinor, which are related to the contravariant components (superscripts) through the metric spinor 

Typical for atoms in metals is that they bond in many directions, even if only a few orbitals are available per atom. For example, in sodium metal there is only a single valence electron, which binds to the other atoms, primarily with the help of the 3s and 3p orbitals. We first limit ourselves to the one-dimensional case. 

The one-dimensional system has a lot of similarities with the planar n-systems treated using the Hiickel model in Chapter 3. In the case of solids, the Hiickel model becomes the tight-binding model. Let us first assume that we have a onedimensional metal containing a great number of atoms. Each atom provides one loosely bound electron, as in a linear tt-system. Our experience tells us that instead of a linear system, we could use a cyclic system. The wave functions are different just at the end points, and our system is assumed to be so large that the end points do not matter. 

For benzene (N = 6) there is a single orbital for f = 0. The next two, for f = , are degenerate. In Chapter 3, we added and subtracted them to get two real orbitals. In the present case, it is more convenient to keep the exponential form. The result agrees precisely with Equation 16.7 above and we may define momentum. Using 

The energy difference between the HOMO and the LUMO (after some calculations) is 

Since only the lower orbitals of the band are occupied, there is a gain of enthalpy equivalent to a metal bonding, creating the cohesive forces of the crystal. 

The uniaxial nonlinear viscoelastic constitutive equation of Schapery( can be written for an isotropic material as 

e(t) represents uniaxial kinematic strain at current time t, cr(t) is the Cauchy stress at time t is the elastic compliance and D (vl ) is a transient creep compliance function. The factor defines stress and temperature effects on elastic compliance and is a measure of state-dependent reduction (or increase) in stiffness, gQ = g ia, ). The transient (or creep) compliance factor g has similar meaning, operating on the creep compliance component. The factor g2 accounts for the influence of load rate on creep, and depends on stress and temperature. The function represents a reduced time-scale parameter defined by 

The transient creep compliance, D (i i), can be expressed in the following exponential form  

If the product g2cr is expressed as G and the integrand on the right-hand side of Eq. (25) is simplified, then we obtain 

The third integration term on the right-hand side of Eq. (26) is now separated into two parts, the first part having limits from zero to (t - AO and the second integral spanning only the current load step, i.e., from (t - AO to t. Hence 

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Kao M, Uhlenbeck G E and Hemmer P 1963 Gn van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one-dimensional model J. Math. Phys. 4 216... 

Figure A3.13.12. Evolution of the probability for a right-handed ehiral stmetnre (fiill eiirve, see ( equation (A3,13.69))) of the CH eliromophore in CHD2T (a) and CHDT2 ( ) after preparation of ehiral stnietures with multiphoton laser exeitation, as diseussed in the text (see also [154]). For eomparison, the time evolution of aeeording to a one-dimensional model ineluding only the bending mode (dashed enrve) is also shown. The left-hand side insert shows the time evolution of within the one-dimensional ealeulations for a longer time interval the right-hand insert shows the time evolution within the tln-ee-dimensional ealeulation for the same time interval (see text).

Table 3. One-Dimensional Model of Propellant Burning Process ...

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

While the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES, have been performed so far. Nonetheless these pioneering studies have established a number of novel features of tunneling reactions, which do not show up in the effectively one-dimensional models. 

Although the inadequacy of the one-dimensional model [Goldanskii 1959, 1979] is well by now understood, we start with discussing the simplest one-dimensional version of the theory, since it... 

The Arrhenius plot of k(T) for H and D transfer is presented in fig. 15. Qualitatively, the conclusions about the isotope effect drawn here on the basis of the one-dimensional model remain correct for more dimensions, but turns out to depend more weakly on m than In k This... 

The discussion so far has dealt with one-dimensional models which as a rule do not directly apply to real chemical systems for the reasons discussed in the introduction. In this section we discuss how the above methods can be extended to many dimensions. In order not to encumber the text and in order to make physics more transparent, we conflne ourselves to two dimensions, although the generalization to more dimensions is straightforward. 

A straightforward calculation of A has been performed by Fraser [1989] within the framework of the one-dimensional model of concerted interconversion. The diabatic terms were taken in the form... 

Krumbhaar. Solidification in the one-dimensional model for a disordered binary alloy under diffusion. Eur Phys J. B 5 663, 1998. 

The Chapman-Jongnet (CJ) theory is a one-dimensional model that treats the detonation shock wave as a discontinnity with infinite reaction rate. The conservation equations for mass, momentum, and energy across the one-dimensional wave gives a unique solution for the detonation velocity (CJ velocity) and the state of combustion products immediately behind the detonation wave. Based on the CJ theory it is possible to calculate detonation velocity, detonation pressure, etc. if the gas mixtnre composition is known. The CJ theory does not require any information about the chemical reaction rate (i.e., chemical kinetics). 

One-dimensional model Onsanger cavity field Onsanger equation Orbital polarization Ordered phase Ordered state... 

The quasi-one-dimensional model of two-phase flow in a heated capillary slot, driven by liquid vaporization from the interface, is described in Chap. 8. It takes... 

The quasi-one-dimensional model of laminar flow in a heated capillary is presented. In the frame of this model the effect of channel size, initial temperature of the working fluid, wall heat flux and gravity on two-phase capillary flow is studied. It is shown that hydrodynamical and thermal characteristics of laminar flow in a heated capillary are determined by the physical properties of the liquid and its vapor, as well as the heat flux on the wall. 

Peles el al. (2000) elaborated on a quasi-one-dimensional model of two-phase laminar flow in a heated capillary slot due to liquid evaporation from the meniscus. Subsequently this model was used for analysis of steady and unsteady flow in heated micro-channels (Peles et al. 2001 Yarin et al. 2002), as well as the study of the onset of flow instability in heated capillary flow (Hetsroni et al. 2004). 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

Chapter 8 consists of the following in Sect. 8.2 the physical model of the process is described. The governing equations and conditions of the interface surface are considered in Sects. 8.3 and 8.4. In Sect. 8.5 we present the equations transformations. In Sect. 8.6 we display equations for the average parameters. The quasi-one-dimensional model is described in Sect. 8.7. Parameter distribution in characteristic zones of the heated capillary is considered in Sect. 8.8. The results of a parametrical study on flow in a heated capillary are presented in Sect. 8.9. 

Significant simplification of the governing equations may be achieved by using a quasi-one-dimensional model for the flow. Assume that (1) the ratio of meniscus depth to its radius is sufficiently small, (2) the velocity, temperature and pressure distributions in the cross-section are close to uniform, and (3) all parameters depend on the longitudinal coordinate. Differentiating Eqs. (8.32-8.35) and (8.37) we reduce the problem to the following dimensionless equations ... 

From the frame of the quasi-one-dimensional model it is possible to determine the hydrodynamic and thermal characteristics of the flow in a heated capillary, accounting for the influence of the capillary force. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The quasi-one-dimensional model described in the previous chapter is applied to the study of steady and unsteady flow regimes in heated micro-channels, as well as the boundary of steady flow domains. The effect of a number of dimensionless parameters on the velocity, temperature and pressure distributions within the domains of liquid vapor has been studied. The experimental investigation of the flow in a heated micro-channel is carried out. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

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