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176 CHAPTER 6. LAMINAR FLAMES
The compatibility of these three conditions determines, the “eigenvalue” of Λ, as established
in §9, and once it is determined,we obtain with it the value for the flame
velocity by virtue of Eq. (6.139).
The integration of the system is very simple using Kármán’s method, in the
following way. In the zone of interest, this is for θ close to 1, it is ε 3 ≫ X 3 and
ε 3 ≫ 1 − θ, therefore a first approximation of Eq. (6.157) is the following
where considering that for θ = 1 it is X 3 = 0, we obtain
dX 3
dθ ≃ − L q 3
, (6.161)
X 3 ≃ − L q 3
(1 − θ). (6.162)
If this value is taken into (6.155) and integrating between θ = 1 and θ = θ i ≃ 0 as
expressed in §9 of this chapter, we obtain the desired value of Λ.
Von Kármán and Penner have calculated the result for X 30 = 0.25, 0.40, 0.50,
0.75, and 1.00. Furthermore, these authors calculated as well the flame velocity for
the same values by means of the Lewis-von Elbe chemical scheme.
A comparison of results with those obtained through experimentation by Grosse
[31] proves that the approximation of the theoretical values reached by Kármán and
Penner is satisfactory since the deviations, in both cases, are under 25 per cent.
6.14 Hydrazine decomposition flame
Hydrazine combustion has been experimentally studied by Murray and Hall [32].
Tests were performed by burning mixtures of hydrazine and water vapours in a Bunsen
burner at ambient pressure. The flame propagation velocity was obtained by measuring
the area of the luminous cone of the flame.
The analysis of the combustion products shown that they correspond to the
overall reaction
N 2 H 4 ⇄ 1 2 H 2 + 1 2 N 2 + NH 3 . (6.163)
Murray and Hall also calculated the theoretical propagation velocity of the
flame for one of the mixtures experimentally analyzed (97.2% hydrazine). They assumed
that such velocity was determined by the unimolecular decomposition reaction
N 2 H 4 → 2NH 2 , (6.164)