Time-lag theories

Many early analyses of time-dependent solid-propellant combustion employed the concept of a combustion time lag [4], [17], [57]-[60]. A given element of reactant material requires a certain amount of time to burn, and if a small pressure pulse is applied to a steadily burning propellant, then the regression rate will take a certain amount of time to reach its new steady level. As the simplest idealization of these observations, assume that a pressure change applied at time t has no effect on m until time t -f Tj, at which time m instantaneously assumes its steady-state value, at all x, appropriate to the new pressure. The interval T/ is the time lag, assumed here to be a known constant. If n is the pressure sensitivity [appearing, for example, in equation (7-41)], then in m(t) = m[l -f m (t)] and p(t) = p[l -f p (0] we have m (t) = np (t — t ), so that 

Thus the criterion in equation (53) becomes n cos(cot ) 1/y, which requires that n y and that (dx be in an appropriate range for amplification to occur typically (oxi is small, whence amplification is predicted at low frequencies, (o (l/r/)cos [l/(ny)]. Among the improvements to equation (54) is the introduction [17], [59] of a sensitive part of that varies with pressure and gives results more versatile than equation (54). 

A critique and more thorough review of the time-lag ideas has been published [7]. The principal value of time-lag concepts lies in the wide range of problems to which they can be applied with relative ease. Their principal deficiency lies in the difficulty of relating Xi to the fundamental processes occurring. If the one-dimensional, time-dependent conservation equations are linearized about any one of the steady-state solutions of Chapter 7 and p is calculated from a perturbation analysis, then it is found 

Although many different processes may be involved in solid-propellant combustion, estimates suggest that characteristic times for all of them may be short compared with the characteristic time for heat conduction in the solid. This latter characteristic time is 

The combustion mechanism addressed involves inert heat conduction in the solid, surface gasification by an Arrhenius process and a gas-phase deflagration having a high nondimensional activation energy. With the density, specific heat, and thermal conductivity of the solid assumed constant, the equation for energy conservation in the solid becomes 

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The control source he arising from combustion of the injected fuel can be treated as a distributed actuator, with its spatial distribution approximated by an array of M discrete sources [1]. If a generalized time-lag theory of Crocco and Cheng is used to model the process of the control fuel from injection to complete combustion, then h can be written as... 

Its main features are given by the use of a stream of inert carrier gas which percolates through a bed of an adsorbent covered with adsorbate and heated in a defined way. The desorbed gas is carried off to a detector under conditions of no appreciable back-diffusion. This means that the actual concentration of the desorbed species in the bed is reproduced in the detector after a time lag which depends on the flow velocity and the distance. The theory of this method has been developed for a linear heating schedule, first-order desorption kinetics, no adsorbable component in the entering carrier gas (Pa = 0), and the Langmuir concept, and has already been reviewed (48, 49) so that it will not be dealt with here. An analysis of how closely the actual experimental conditions meet the idealized model is not available. 

Although there has been a substantial body of pharmacological evidence in support of the monoamine theory of depression, clinical biochemical data have been less convincing (Luchins, 1976) this is where differences in the concentrations of NA and 5-HT and their metabolites or hormones, which are ultimately under the control of brain monoaminergic neurons (neuroendocrine markers), have been compared between depressed patients and normal controls. However, by the early 1970s a major difficulty with the theory was becoming apparent this was the time lag between the immediate... 

C(v,P). It is seen that the temp of the spike N is somewhat lower than half of the C-J temp. Since the temp is low, it is not expected that the chem reactions could occur to any appreciable extent in the short time required for the gas to pass thru the initial shock. Actually the temp at Njis so low that in many practical cases one would expect a time lag or a quenching zone before the reaction sets in. Hirschfelder inferred that accdg to NDZ theory some chem reaction can take place within the detonation front (in cases of unusually high reaction rates), and blunt the von Neumann spike, as can be seen in Fig 4 (Ref 7, pp 172-74)... 

Two theories have been described which would account for the independence of rate and pressure, one involving the assumption of a time-lag between activation and transformation, the other involving a chain mechanism in which the activated products give their energy only to, molecules of the reactant. Both theories have the... 

The gas-polymer-matrix model for sorption and transport of gases in polymers is consistent with the physical evidence that 1) there is only one population of sorbed gas molecules in polymers at any pressure, 2) the physical properties of polymers are perturbed by the presence of sorbed gas, and 3) the perturbation of the polymer matrix arises from gas-polymer interactions. Rather than treating the gas and polymer separately, as in previous theories, the present model treats sorption and transport as occurring through a gas-polymer matrix whose properties change with composition. Simple expressions for sorption, diffusion, permeation and time lag are developed and used to analyze carbon dioxide sorption and transport in polycarbonate. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

This problem was resolved in 1922 when Lindemann and Christiansen proposed their hypothesis of time lags, and this mechanistic framework has been used in all the more sophisticated unimolecular theories. It is also common to the theoretical framework of bimolecular and termolecular reactions. The crucial argument is that molecules which are activated and have acquired the necessary critical minimum energy do not have to react immediately they receive this energy by collision. There is sufficient time after the final activating collision for the molecule to lose its critical energy by being deactivated in another collision, or to react in a unimolecular step. 

It is the existence of this time lag between activation by collision and reaction which is basic and crucial to the theory of unimolecular reactions, and this assumption leads inevitably to first order kinetics at high pressures, and second order kinetics at low pressures. 

The crucial step in the development of unimolecular theory was the postulate of a time lag between the activation and reaction steps in the master mechanism for all elementary reactions given in Chapter 1. During this time an activated molecule can either be deactivated in a deactivating energy transfer collision, or it can alter configuration to reach the critical configuration and react. All elementary reactions involve three steps, two energy transfer steps and one reaction step, and for unimolecular reactions... 

In the Hinshelwood theory the time lag corresponds to the time taken for the activated molecule to rearrange configuration into the critical configuration of the activated complex. The Kassel theory deals explicitly with this process, and imposes a much more severe restriction than does Hinshelwood. Before an activated molecule can react there must be a flow of energy at least 0 into a... 

For heterogeneous propellants, the current situation is much less satisfactory. The complexity of the combustion process was discussed in Section 7.7. To employ a result like equation (66) directly is questionable, although attempts have been made to evaluate parameters like A and B of equations (67) and (68) from complicated combustion models for use in response-function calculations [81], [82]. Relatively few theories have been addressed specifically to the acoustic response of heterogeneous propellants [82]. Applications of time-lag concepts to account for various aspects of heterogeneity have been made [60], [83], a simplified model—including transient variations in stoichiometry—has been developed [84], and the sideways sandwich model, described in Section 7,7, has been explored for calculating the acoustic response [85], There are reviews of the early studies [7] and of more recent work [82],... 

In Lindemann s theory of active intermediates, decomposition of the intermediate does not occur instantaneously after internal activation of the molecule rather, there is a time lag, although infinitesimally small, during which the species remains activated. For the azomethane reaction, the active intermediate is formed by the reaction... 

The essential postulate is that an activated complex (or transition state) is formed from the reactant, and that this subsequently decomposes to the products. The activated complex is assumed to be in thermodynamic equilibrium with the reactants. Then the rate-controlling step is the decomposition of the activated complex. The concept of an equilibrium activation step followed by slow decomposition is equivalent to assuming a time lag between activation and decomposition into the reaction products. It is the answer proposed by the theory to the question of why all collisions are not effective in producing a reaction. 

Absence of a time lag. Current flows the moment light of this minimum frequency shines on the metal, regardless of the light s intensity. The wave theory, however, predicts that in dim light there would be a time lag before the current flowed, because the electrons had to absorb enough energy to break free. 

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