Rate equations, governing time dependence

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

We may now allow for reactant consumption explicitly by restoring the exponential decay, p = p0e- °. In dimensionless terms this means recognizing that p is a time-dependent parameter, p = /i0e . The governing rate equations are... 

There is another type of time dependence possible in this system. If the inflow concentration of the autocatalyst is adjusted so that b0 - a0, then the ignition and extinction points merge at trcs = (k1ao) 1, with ass = Iu0 Under these special conditions, the coefficient of the term in (Aa)2 in the rate equation, and hence in the denominator of eqn (8.21), becomes zero as well as those of the lower powers in A a. Thus the inverse time dependence disappears, and the only non-zero term governing the decay of perturbation is that in (Aa)3 ... 

The preceding remarks all apply to a steadily propagating one-dimensional detonation. It is a relatively simple task to solve the algebraic equation for the over-all steady-state motion. The differential equations for the structure may be solved in the steady state, but the task is tedious and, in addition, detailed knowledge of reaction rates needed in the equation is not available. It is not too difficult to solve the time-dependent over-all equation if a burning velocity for the flame as a function of temperature and pressure is assumed (J4). It is not practicable to solve the time-dependent equations which govern the structure of the wave with any certainty because of the lack of kinetic information, in addition to the mathematical difficulty. The acceleration of the slowly moving flame front as it sends forward pressure waves which coalesce into shock waves that eventually are coupled to a zone of reaction to form a detonation wave has been observed experimentally (LI, L2). 

The derivation of the time-dependent probability for a transition from nuclear state (Ei)) in the ground electronic state to state J f(Ef,n)) in the excited electronic state, Equation (2.21), and the corresponding transition rate (2.22) proceeds as in Section 2.1. Since a common phase factor governs the time dependence of all degenerate final states,... 

To determine fully the kinetic parameters governing upconversion using Eq. (10) it is imperative to know the laser-induced excitation densities, Nj and N2, in steady state, or Mi(0) and 2(0) in time-dependent studies. These parameters may be determined from careful measurement of the physical properties of the sample, the excitation configuration, and the experimentally absorbed power. These numbers are not easily reliably determined, however, and they are therefore more commonly estimated or taken as experimental unknowns in the use of these equations for simulations. The difficulty with which absolute excitation densities are determined is one of the practical limitations of this rate equation model. 

The time-dependent fiinction Hit ) is determined by the rate of increase or decrease in the bubble volume. The governing equations and boundary conditions that remain to be satisfied are (1) the radial component of the Navier Stokes equation (2) the kinematic condition, in the form of Eq. (2 129), at the bubble surface and (3) the normal-stress balance, (2 135), at the bubble surface with = 0. Generally, for a gas bubble, the zero-shear-stress condition also must be satisfied at the bubble surface, but xrti = 0, for a purely radial velocity field of the form (4-193), and this condition thus provides no usefirl information for the present problem. 

The importance of the time parameter in the study of reactions is clear already in the early kinetic studies, where it takes the form of the inverse rate constant. From the point of view of fundamental chemical theory, elementary chemical reactions are simply the detailed dynamics of electrons and atomic nuclei that constitute the total molecular system of reacting species, which is governed by the time-dependent Schrddinger equation... 

Fig. < . Population evolution of eight coupled states in a CHjF (0.17 torr)-argon (10 torr) mixture. The time dependence of the population is calculated from a numerical integration of the differential equations governing the rate processes of equations 19a, 19b, 20, 21, 28. The magnitudes of the rate constants are taken from Refs. 19, 35, and 55. All populations are shown normalized to 10,000 for the peak of the population of a given state. The laser pulse occurs at t> 0. Only V-V processes are included in the model.

Species formed from acetylene (Ay) adsorbed in zeolite Y, mordenite, beta and ZSM-5 have been studied by IR spectroscopy. The dynamics of Ay physisorption has been characterized by the frequency response method (FR). The rate of micropore diffusion governed the transport in Na-mordenite, while sorption was the rate limiting process step for all the H-zeolites. The equilibrium constants (Ka) of Ay sorption have been determined applying the Langmuir rate equation to describe the pressure dependence of the sorption time constants. The -octane hydroconversion activity of Pt/H-zeolites was found to increase linearly with the Ka of Ay sorption on the H-zeoIites. 

To describe the time-dependent behavior, a transient term should be added to the governing equations, as shown in Eq. (31.1), to account for the storage rates of mass, momentum, species, energy, and charges. In the catalyst layer, both the ionomer phase and void space can hold water, hence an effective factor, in Eq. (31.1) can be introduced to simplify the model expression [16] ... 

Transitions among the various spin states are governed by rate equations fliat represent two counteraeting probability funetions. The first of fliese fimctions pertains to the radiation-indueed transition and the probabihty that the time-dependent eleetric field of the ineident radiation eouples to the sample medium and induces transitions among the allowed spin states. The counterpart, spin relaxation, denotes the multifaceted dynamics of the spin population as it returns to thermal i.e., Boltzmann) equilibrium. The relative rates of these two processes determine the EMR signal intensity, and the associated spin dynamic processes serve as the basis for advanced EMR techniques and their specialized niches for determining super-hyperfine parameters. 

If a small pressure change LP from P to P2 is suddenly imposed there will be an immediate change in specific volume from an initial value i i to a valuer,- = v (1 - P/Kg) followed by a time-dependent change whose rate is governed by the differential equation corresponding to the model ... 

It will prove useful to write the governing equations in a non-dimensional format. However non-dimensionalisation of the equations is not unique. Here we chose to use the autocatalytic hydrolysis as a reference for all the other time-dependant processes (non-catalytic hydrolysis and TCP dissolution). This is because autocatalytic hydrolysis is expected always to occur in the polyesters considered in this chapter. The scission rate equation,... 

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