Net branching factor

Because of difficulties associated with heat dissipation at the high velocity of the reaction, it was impossible to work at pressures much above the first limit. With the vessel used by Kowalsky [15] this limit for 2H2 + O2 at 480 and 535 °C occurred at 4 and 2.2 torr, respectively. The initial pressures used in the kinetic studies were 4 torr and a little above. The initial stages of the reaction were found to be represented fairly well by Ap = C e. The net branching factor 0 increased with temperature and... 

The experimental data of Kowalsky on 2H2 + O2 mixtures have been analyzed by Semenov [59] on the basis of reactions (i)—(v), together with surface destruction of H atoms. Because of the lower activation energies of reactions (i) and (hi) compared with reaction (ii), the concentrations of OH and O were assumed to be small compared with H. The variation of H atom concentration could thus be deduced by the method of partial stationary state concentrations [60], giving the net branching factor 0 at pressure p as... 

During the initial induction period the net branching factor 0 is n ative, giving a non-branched chain system (cf. Sect. 8.2). However, if induction period, because of the changes in concentration of NO2 and NO, then a sudden increase in chain centre concentration would occur as 0 passes through zero and becomes positive. This is taken to occur at Pe. 

Fig. 5.2. Variation of the quasi-steady-state radical concentration with net branching factor for the simple model of chain branching and chain termination showing [A ss as — 0 (note the steady-state only exists for <(> < 0).

The first term in this quadratic equation is the initiation reaction rate based on the inflow concentration of the reactant. The coefficient for the term in [X]ss has something of the character of the previous net branching factor. The above equation has a single positive solution for any set of rate constants, residence time and inflow concentration a typical variation of [.ATJss with [A]o is shown in Fig. 5.3(b) and shows a rapid increase in the vicinity of some critical concentration [A]o,cr-The behaviour can be quantified if we make the approximation of ignoring the (probably small) initiation terms, setting ki =0. The steady-state condition can then be written in the form... 

Provided step (5) is rate determining in this termination step then the termination rate r, = A 5[H][02][M] and the net branching factor will have... 

If on the other hand the net branching factor is positive, integration of (6.2.10) now yields ... 

Consequently, the rate of disappearance of methane is equal to what it would be in the case of a straight-chain reaction, namely 2ri(oi/at), multiplied by an exponential function of I times the net branching factor. 

Frequently it is indeed observed that the product of the induction period and the net branching factor is approximately a constant in a g ven system and this is a useful kinedc relation. 

Find the so-called first explosion limit in a cylindrical reactor I cm in diameter. This is the critical pressure at which the net branching factor will be equal to zero. Assume a value 7 = 10 for the probability of capture of hydrogen atoms at the wall. Assume that diffusion is not controlling and from the calculated explosion limit, verify the validity of this last assumption. In setting up the critical condition for explosion, reflect upon the fact that rt is now the rate of a surface reaction while the branching still occurs homogeneously throughout the volume of the reactor. 

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