Page 338 - Advanced thermodynamics for engineers

15.3 EXPLOSION LIMITS 327

The kinetic processes involved in the H 2 –O 2 reaction are
H þ O 2 /OH þ O
which leads to the further branching steps
(15.1)
O þ H 2 /OH þ H
OH þ H 2 /H 2 O þ H
The first two of the three steps are called branching steps and produce two radicals (highly reactive
ions) for each one consumed. The third step does not increase the number of radicals. Since all steps
are necessary for the reaction to occur the multiplication factor (i.e. the number of radicals produced
by the chain) is between 1 and 2. The first step is highly endothermic (it requires energy to be supplied
to achieve the reaction), and will be slow at low temperatures. This means that an H atom can survive a
lot of collisions without reacting, and can be destroyed at the wall of the container. Hence, H 2 –O 2
mixtures can exist in the metastable state at room temperatures, and explosions will only occur at high
temperatures, where the first step proceeds more rapidly.

15.3.1 THE EFFECT OF MULTIPLICATION FACTOR ON THE TENDENCY TO EXPLODE
The effect of the multiplication factor can be examined in the following way. Assume a straight chain
8
3
3
reaction has 10 collisions/s, and there is 1 chain particle/cm , with 10 19 molecules/cm . Then all the
molecules will be consumed in 10 11 s, which is approximately 30 years. However, if the multiplication
N
19
factor is now 2 then 2 ¼ 10 , giving N ¼ 62. This means that all the reactants molecules will be
consumed in 62 generations of collisions, giving a total reaction time of 62 10 8 s, or 0.62 ms: a
extremely fast reaction! If the multiplication factor is only 1.01 then the total reaction time is still only
10 ms. Hence, the speed of the reaction is very dependent on the multiplication factor of the reactions,
but the overall multiplication factors do not have to be very high to achieve rapid combustion.
A general branched chain reaction may be written

k 1
M/R initiation
k 2
R þ M/aR þ M chain branching; a_1
k 3
R þ M/P product formation; removes radical (15.2)
9
k 4
>
>
R/destruction >
wall =
chain termination
k 5 >
>
R/destruction >
gas
;
where M is a molecule, R is a radical and P is a product. a is the multiplication factor. The value of a
necessary to achieve an explosion can be evaluated. The rate of formation of the product, P, is given by
d½PŠ
¼ k 3 ½RŠ½MŠ: (15.3)
dt