Publisher Summary

The theory of chain self-ignition involves the following postulates: (1) the chain initiation rate, that is, the rate of production of active centers or chain carriers, is the lowest rate accessible to kinetic measurement; (2) the chains are material, that is, they are consisted of reacting atoms and radicals; (3) the active centers can react in three ways, that is, chain termination, chain propagation, and chain branching. The chain length is the number of elementary reactions that is produced by one active centre, generated in some fashion, which is not specified. The reaction rate in all chain processes equals the product of the initial chain generation rate and the total chain length. The theory of branched-chain reactions within the explosion limits is more complex than the theory of the self-ignition limits. The chain theory is developed in two stages. The first stage is related to photoreactions, and the second stage is related to thermal explosions.

1 GENERAL CONSIDERATIONS ON CHAIN SELF-IGNITION*

The theory of chain self-ignition was developed in the late 1920s and early 1930s, particularly by Soviet and English groups; it involves the following postulates, which have now been adequately verified by experiment.

The initiation rate is denoted by n0, while the probabilities of processes (a)−(c) at a given chain stage are β, α and δ respectively.

Chain length (v). This is the number of elementary reactions produced by one active centre, generated in some fashion not specified. The rate at which such centres form is n0; if there is no branching the active centres can only react in two ways, as in (a) and (b) above; in (b) a final product and one new centre is formed. Since (a) and (b) have been assumed to be the only reactions open to an active centre we must have α + β = 1. The probability of a chain having s links equals the product of the probability that propagation will occur (s − 1) times, i.e. αs−1, and the termination probability β:

s=αs−1β=αs−1(1−α) (1)

Since , v is just the mean value of s, namely

=s¯=∑I∞Pss=∑1∞sαs−1(1−α)=(1−α)−1=β−1 (2)

If branching can also occur the expression for v must incorporate δ, since this as it were reduces the effect of β. Thus if we have one act of branching per two terminations the net effect is that of one termination only, since two chains vanish but one new one appears. Hence the expression for v must contain β − δ instead of ß,* so

*=(β−δ)−1=v(1−δv)−1 (2′)

where v* is the length when δ ≠ 0 and v is as before.

The reaction rate in all chain processes equals the product of the initial chain generation rate and the total chain length, i.e.

=n0/(β−δ)=n0v/(1−δv) (3)

This simple formula already implies that ignition limits occur; suppose that β and δ vary in different ways with the external parameters (temperature, pressure, mixture composition, etc.) such that we can on occasion have δ = β, i.e. that v*, and hence w, tend to infinity. Then the condition

−δ=0 (4)

defines the boundaries to the chain ignition region. We might go further, and assume δ > β under some conditions, but we then get a nonsensical result, since v* and w become negative. This occurs because the above arguments only apply for v finite; in this connexion we must now examine what α, β and δ mean physically. We in fact find that they are merely the ratios of the numbers of reactions of the various species to the total number of reactions. The relations are in fact correct only at the limit δ → β, i.e. for v large, but finite.

Let us consider what occurs if the branching rate equals or exceeds the termination rate; for this purpose the time course of the process is required. For β > δ no argument is required, since if v is not too large the system is unchanged even over very long times, because a steady state with fixed active particle concentrations is soon set up, and these concentrations ensure a fixed v. When δ > β the number and concentration of active particles increase continuously, and so do w and v, as can be shown mathematically. The rate of change of active particle concentration is

n/dt=n0−(g−f)n (5)

where g and f are kinetic termination and branching coefficients proportional to β and δ respectively.* Integrating and putting n = 0 at t = 0, we have

=n0(1−exp{−(g−f)t})/(g−f) (6)

If (g – f) is sufficiently large the exponential term soon becomes negligible. Then

=n0/(g−f) (7)

Now w is the product of a kinetic factor a (for the chain propagation reaction) and the active centre concentration:

=an=an0/(g−f) (8)

Hence (8) taken with (3) gives δ = f/a and β = g/a. Thus when β > δ we soon get w constant, as would be expected. In the case of interest to us, f − g 0, we have that when f − g = 0 (5) and (7) go over to

n/dt=n0 (9)

and

=n0t (10)

respectively.

Then

=an=an0t (11)

the reaction rate slowly increasing with time (at small n0) so the process is no longer steady.

For f − g > 0 the integral to (5) is

=[n0/(f−g)][exp{(f−g)t}−1] (12)

and

=[an0/(f−g)][exp{(f−g)t}−1] (13)

hence w rises exponentially. Fig. 14 [2] shows how v = w/n0 varies with the dimensionless quantity at for various values of

f−g)/a=x

The amounts reacted will be

=∫0tw dt (14)

for

−g<0X=an0t/(f−g) (15)

−g=0X=an0t2/2 (16)

−g>0X=[an0/(f−g)2][exp{(f−g)t}−1]−an0t2/2 (17)

Thus even when f > g the rate never becomes negative, since the larger f − g the steeper the rate of rise. Formally speaking (13) would imply that w can become infinitely great, but in real (closed) systems the reactant concentrations begin to fall off as the rates become high.* Eventually the reactant concentrations reach the point where self-acceleration ceases and w begins to fall to zero from some maximum value. Ignition does not imply an infinite rate, but merely that all the reactant is transformed in a finite but very short time.

When n0 is very small the steady-state w for f − g < 0 is also very small, even if a is large and g −...