Propagation steady-state

The above description is of a thermally propagating steady-state wave. It must be emphasized, however, that the basic feature of a thermal mechanism is not altered by the superposition of molecular diffusion onto the diffusional transport of heat. This applies not only to interdiffusion of reactants and products but also to the diffusion of chain carriers participating in the chemical reaction, provided that the chains are unbranched. The reason for this is that in a wave driven by a diffusion process, the source strength of an entering mass element must continue to grow despite the drain by the adjacent sink region. This growth can occur only if the reaction rate is increased by a product of the reaction, which may be temperature as well as a material product. 

The ratio describes the relative reactivity of polymer chain M toward monomer M and monomer M2. Likewise, describes the relative reactivity of polymer chain M2 toward M2 and M. With a steady-state assumption, the copolymerisation equation can be derived from the propagation steps in equations 3—6. 

Eigenvalue problems. These are extensions of equilibrium problems in which critical values of certain parameters are to be determined in addition to the corresponding steady-state configurations. The determination of eigenvalues may also arise in propagation problems. Typical chemical engineering problems include those in heat transfer and resonance in which certain boundaiy conditions are prescribed. 

The overall rate of a chain process is determined by the rates of initiation, propagation, and termination reactions. Analysis of the kinetics of chain reactions normally depends on application of the steady-state approximation (see Section 4.2) to the radical intermediates. Such intermediates are highly reactive, and their concentrations are low and nearly constant throughout the course of the reaction ... 

The result of the steady-state condition is that the overall rate of initiation must equal the total rate of termination. The application of the steady-state approximation and the resulting equality of the initiation and termination rates permits formulation of a rate law for the reaction mechanism above. The overall stoichiometry of a free-radical chain reaction is independent of the initiating and termination steps because the reactants are consumed and products formed almost entirely in the propagation steps. 

Following the steady-state approximation, both propagation steps must proceed at the same rate otherwise, the concentration of A- or C- would build up. By substituting for the concentration of the intermediate C-, we obtain ... 

Photoinitiation is an excellent method for studying the pre- and posteffects of free radical polymerization, and from the ratio of the specific rate constant (kx) in non-steady-state conditions, together with steady-state kinetics, the absolute values of propagation (kp) and termination (k,) rate constants for radical polymerization can be obtained. 

The kinetics and mechanism of living radical polymerization have been reviewed by Fischer,21 Fukuda et at.,22 and Goto and Fuktida.23 In conventional radical polymerization, new chains are continually formed through initiation w hile existing chains are destroyed by radical-radical termination. The steady state concentration of propagating radicals is 10"7 M and an individual chain will have a lifetime of only 1-10 s before termination within a total reaction lime that is... 

Monomers not amenable to direct homopolymerization using a particular reagent can sometimes be copolymcrizcd. For example, NMP often fails with methacrylates (e.g. MMA, BMA), yet copolymerizalions of these monomers with S are possible even when the monomer mix is predominantly composed of the methacrylate monomer,15j This is attributed to the facility of cross propagation and the relatively low steady state concentration of propagating radicals with a terminal MMA (Section 7.4.3.1). MMA can also be copolymerized with S or acrylates at low temperature (60 C).111 Under these conditions, only deactivation of propagating radicals with a terminal MMA unit is reversible, deactivation of chains with a terminal S or acrylate unit is irreversible. Molecular weights should then be controlled by the reactivity ratios and the comonomer concentration rather than by the nitroxide/alkoxyamine concentration. 

In the case of cooperative processes, the formation of a nucleus, already discussed from the kinetical point of view, plays a crucial role. The steady state described by Eq. (1) depicts the formation of a triple helix as the simplest model by the formation of a nucleus Hx through fast pre-equilibria and subsequent propagation steps, Hx in this case is a triple-helical intermediate with x tripeptide units (that means x hydrogen bonds) in the helical state. The final product H3n 2 possesses two hydrogen bonds less than tripeptide units because the three single chains are staggered at one amino add residue each. 

It has been possible to study the asymptotic stability of the system, i.e., the manner in which the fluctuations of motion of the lead car are propagated down the line of cars. The steady state behavior is easily derived. Because of velocity control between the following cars, a steady state is eventually reaohed in which each car moves with speed u, and, hence, A = 0 i.e., the study of the system involves no time lag. 

The work done in the deformation zone can be estimated assuming the deformation zone propagates together with the crack and maintains its steady state displacement profile [86]... 

Also, the rates of the propagation steps are equal to one another (see Problem 8-4). This observation is no surprise The rates of all the steps are the same in any ordinary reaction sequence to which the steady-state approximation applies, since each is governed by the same rate-controlling step. The form of the rate law for chain reactions is greatly influenced by the initiation and termination reactions. But the chemistry that converts reactant to product, and is presumably the matter of greatest importance, resides in the propagation reactions. Sensitivity to trace impurities, deliberate or adventitious, is one signal that a chain mechanism is operative. 

Biernath et al. concluded that phenolic novolac and epoxidized cresol novolac cure reactions using triphenylphosphine as the catalyst had a short initiation period wherein the concentration of phenolate ion increased, followed by a (steady-state) propagation regime where the number of reactive phenolate species was constant.85 The epoxy ring opening was reportedly first order in the steady-state regime. 

Steady State Population Density Distributions. Representative experimental population density distri-butions are presented by Figure 1 for two different levels of media viscosity. An excellent degree of theoretical (Equation 8) / experimental correlation is observed. Inasmuch as the slope of population density distribution at a specific degree of polymerization is proportional to the rate of propagation for that size macroanion, propagation rates are also observed to be independent of molecular weight. 

In this example, an initial steady-state solution with a = 0 is propagated downstream. At the fourth axial position, the concentration in one cell is increased to 16. This can represent round-off error, a numerical blunder, or the injection of a tracer. Whatever the cause, the magnitude of the upset decreases at downstream points and gradually spreads out due to diffusion in the y-direction. The total quantity of injected material (16 in this case) remains constant. This is how a real system is expected to behave. The solution technique conserves mass and is stable. 

The line 9 is given by the steady-state, back-pressure drive flame propagation theory [29], which assumes the momentum flux balance between the upstream and downstream positions on the center streamline and the angular momentum conservation on each streamline. 

The steady states of such systems result from nonlinear hydrodynamic interactions with the gas flow field. For the convex flame, the flame surface area F can be determined from the relation fSl = b zv, where Sl is the laminar burning velocity, the cross-section area of the channel, and w is the propagation velocity at the leading point. 

Nevertheless, despite all these remarkable achievements, some open questions still remain. Among them is the influence of the molecular transport properties, in particular Lewis number effects, on the structure of turbulent premixed flames. Additional work is also needed to quantify the flame-generated turbulence phenomena and its relationship with the Darrieus-Landau instability. Another question is what are exactly the conditions for turbulent scalar transport to occur in a coimter-gradient mode Finally, is it realistic to expect that a turbulent premixed flame reaches an asymptotic steady-state of propagation, and if so, is it possible, in the future, to devise an experiment demonstrating it ... 

From the practical point of view, the most important aspects of the accelerated flame phenomenon are with respect to the steady-state propagation of very highspeed flames, transition to detonation, and propagation of sub-CJ detonations (quasi-detonations). 

The high-speed flames propagate in a tube with repeat-able obstacles with the steady-state velocity, which is maintained for the rest of their passage over the obstacles. In some cases, the steady-state flame propagation velocity of the combustion products may approach the... 

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