Untitled - Aerobib - Universidad Politécnica de Madrid

174 CHAPTER 6. LAMINAR FLAMES
Steady state assumption
In spite of the simplification introduced into the system, it is still very difficult to
perform its integration by semi-analytical methods.
In an effort to find further simplifications, von Kármán and Penner checked the
applicability of the steady state assumption for the distribution of atomic oxygen. This
assumption is expressed through nullifying the reaction velocity of the oxygen atoms,
thus obtaining
e −θ a1/θ X 3 − e −θ a3/θ X 1 X 3 = 0, (6.153)
from which we derive the following distribution of atomic oxygen throughout the
flame
X 1 = e − θ a1 − θ a3
θ . (6.154)
10
9
8
7
Hirschfelder
Karman−Penner
6
10 4 X 1
5
4
3
2
1
0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
θ
Figure 6.16: Mole fraction of atomic oxygen as a function of the temperature for the ozone
decomposition flame.
Figure 6.16 taken from [6] compares the distribution of X 1 corresponding to
this assumption, with that obtained by Hirschfelder through numerical integration of
the flame equations, without making use of the assumption. It is seen that the result is
completely satisfactory and it plainly justifies the assumption of Kármán and Penner,
which is not applicable just within a narrow zone very close to the maximum temperature,
at which, since the concentration of ozone is practically reduced to zero there
are other reactions, different from 1 and 3 in (6.142) controlling the formation of O.