Explosions branched-chain

In straight chain propagation one radical is destroyed while another is produced, so that the net change in the number of radicals is zero. 

For branching to occur a 1, and can often be 2 or 3, though rarely greater. There are occasions (see Section 6.15 on degenerate branching) where a is just slightly greater than unity, and under these circumstances a different situation will occur. 

Question. In a branched chain reaction with a = 2, there is a consequent build-up of radicals with each cycle of branched chain propagation  

At time t, when 30 cycles have occurred, from the table 

Although the nonlinearity Cj in the propagation prevents an exact solution to this system from being obtained analytically, the qualitative behavior of the solution may be deduced from the differential equations. 

From equation (44) we see that the reactant concentration decreases monotonically with time from its initial value C (0), while that of the product increases monotonically. The concentration of the chain carrier exhibits a more complicated behavior, according to equation (45). If — 0 initially, then there is a linear increase of with time at early times by initiation. If /Cj (a — l)kpCj (0) (which applies, for example, to straight-chain reactions), then dc /dt decreases continually with time, and after a sufficient amount of reactant depletion and carrier buildup, begins to decrease and eventually decays exponentially through termination with a time constant /c that is, Cq However, there are other ranges of parameters in 

Since this equals /Cj/[(a — l)/c ] at the propagation-termination balance, the approximate formula for the induction time is 

It may be noted from this formula that an increase in (a — l)kpCj (0X in /c-(less strongly), or in k (much less strongly) produces a decrease of the induction time. There are elements of arbitrariness in the definition of induction time as well as in judgments of whether explosion occurs. 

Since it has been seen that dc /dt decreases with t if k (a - 1)/c/r(0) but may exhibit an explosionlike growth if k (a — l)k P/ (0), it may be reasonable to identify the equality k = (a — l)kpCj (0) with the boundary of explosion. Alternative reasoning that leads to this same result makes use of the steady-state approximation for the intermediary C. Vanishing of the right-hand side of equation (45) yields 

It may be noted from equation (48) that for a = 1, the overall rate is proportional to P the number of propagation steps per termination step, which may be termed the chain length. Since chain reactions typically have 

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Their reactions are explosive without appreciable self-heating (branched chain explosion without steady temperature rise). Explosion usually occurs when passing from region 1 to region 2 in Fig. 3.9. Explosions may occur in other regions as well, but the reactions are so fast that we cannot tell whether they are self-heating or not. 

Belles prediction of the limits of detonability takes the following course. He deals with the hydrogen-oxygen case. Initially, the chemical kinetic conditions for branched-chain explosion in this system are defined in terms of the temperature, pressure, and mixture composition. The standard shock wave equations are used to express, for a given mixture, the temperature and pressure of the shocked gas before reaction is established (condition 1 ). The shock Mach number (M) is determined from the detonation velocity. These results are then combined with the explosion condition in terms of M and the mixture composition in order to specify the critical shock strengths for explosion. The mixtures are then examined to determine whether they can support the shock strength necessary for explosion. Some cannot, and these define the limit. 

H +C>2 - OH+ O, so called because the disappearance of one chain carrier leads to the appearance of two. If chain carriers are produced at a rate faster than they are removed (by chain-breaking or chain-terminating reactions), a branching-chain explosion can occur without any preliminary temperature rise at all (hence "isothermal )... 

Formaldehyde, in sufficient quantities, can suppress cool-flame formation. Jost (27) presents evidence indicating that cool flames are a form of branched-chain explosions. It has been suggested that the cool-flame reaction is quenched by its own reaction product, formaldehyde, and arrested short of complete release of chemical enthalpy. This seems unlikely, however, because in systems exhibiting multiple cool flames the concentration of formaldehyde after the first cool flame does not drop in some cases it increases, and yet does not suppress subsequent cool flames. Bardwell (5), and Bard well and Hinshelwood (4) explain cool flame phenomena by a modified theory of Salnikov. This thermal theory is further supported by the results of Knox and Norrish (30) in the ethane-oxygen system. The key intermediate is presumed to be a peroxide by Bardwell and Hinshelwood (4). Formaldehyde is considered an inert, stable product with little effect on the reaction. 

I he occurrence of a spontaneous explosion in a chemically reacting system is a complicated process. However, the events that lead to explosion can be characterized as being either of a branching chain or of a thermal nature. Branching-chain explosions occur in systems that react by a chain mechanism, the details of which allow the chain carrier concentration, and hence, the over-all reaction rate to increase without limit, even under isothermal conditions. Such a condition is possible only if one or more of the steps in the reaction chain results in a multiplication of chain carriers—i.c., X + A — Y + Z + , where X, F, and Z arc chain carriers. 

Induction Periods in Branching Chain Explosions. Induction periods in the case of chain-branching explosions are often observed (22, 23, Ifl, 41) and may be interpreted in any one of the following ways ... 

One of the remarkable features of these branched chain explosions is the astounding speed at which the system moves from a zero rate, through the steady state, through the build-up and thence to explosion. This induction period is short and can have a duration ranging down from seconds to milliseconds, or even less. 

The Branching Chain Explosion Upper and Lower Limits... 

Systems showing the characteristic of a branching chain explosion (e.g., oxidation of P4, PH3, NH3, H2, etc.) exhibit critical explosion limits of the... 

Few reactions have been studied as extensively as the classical reaction of hydrogen and oxygen. Because of its relative chemical simplicity, it has served as a prototype and proving ground for theories of branching chain explosions. 

Above 400°C the rate of production of water becomes measurably fast (above 100 mm Hg total pressure), and between about 400 and 600°C the reaction shows all the characteristics of a branching chain explosion replete with three explosion limits. Figure XIV.4, taken from work of Lewis and Von Elbe, illustrates this behavior for a stoichiometric mixture (2112 02) in a 7.4-cm-diameter pyrex vessel coated with KCl (volume = 220 cc). 

The model scheme described by equations (44) and (45) represents a simplification of mechanisms that apply to real branched-chain explosions. Initiation steps are seldom ever unimolecular R X 2C X or 2R carriers would be more realistic, and the latter would introduce further nonlinearity. The stable species produced in the termination step are not all products but instead include substantial amounts of other molecules, such as reactants. Finally, there are generally more than one reactant,... 

Examples of catalysts in combustion reaction include the effect of H2O on the carbon monoxide oxidation reaction CO H- 2 C02- Nitric oxide also catalyzes CO oxidation through the mechanism 2 NO 4-O2 2NO2 (overall) and NO2 -f CO NO H- CO2. In both of these examples, an intermediate compound (for example, NO2) is formed and then destroyed. The addition of a small amount of NO2 to an H2 — O2 mixture leads to a branched-chain explosion by introducing the relatively rapid initiation step NO2 H- X NO H- O H- X, with the O atoms so produced generating the usual H2 — O2 chain. The NO2 also participates in the efficient termination step NO2 H- O NO H- O2, which is sufficiently important at large concentrations of NO2 to cause a slow reaction to be... 

Critical explosion limits for a typical branching chain explosion showing explosion pe-ABCD represents explosion limits. The region ABC is called the explosion peninsula. 

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