Untitled - Aerobib - Universidad Politécnica de Madrid

6.11. IGNITION TEMPERATURE 161
6.11 Ignition temperature
We have established that, except when activation temperature θ a is zero or very small,
the ignition temperature θ i bears no influence on the flame velocity unless it has a
value close to the temperature of the burnt gases θ = 1, or very close to initial temperature
θ 0 .
1.6
1.2
θ a
=0
Λ −1/2
0.8
θ a
=2
0.4
θ a
=4
θ a
=8
0.0
0.0 0.2 0.4 0.6 0.8 1.0
θ =0.125 0
θ i
Figure 6.14: Flame velocity as a function of the ignition temperature θ i for different values
of the activation temperature θ a.
This property is illustrated by Fig. 6.14, where the law of variation of Λ −1/2
(to which u 0 is proportional as before said), as a function of θ i is shown for a typical
case, corresponding to a first-order reaction, with a Lewis-Semenov number equal to
unity and a temperature of the unburnt gases θ 0 = 0.125. The figure represents the
solutions corresponding to values of θ a comprised between 0 and 8 and it is seen that
except for the first case (θ a = 0), there exists a wide range on values of θ i in which
Λ −1/2 is practically constant, as it was previously announced. 7
This property is of a general nature and it justifies the assumption of a ignition
temperature which eliminate the difficulty of the cold boundary.
7 For θ a ≫ 1 and θ i very close to θ 0 , it can be shown by means of asymptotic techniques that Λ −1/2 =
„
(1 − θ 0 ) ln` 1 ´ 1 − θ 0 ´«1/2
exp`−θa , so that when θ i → θ 0 there exists an exponentially
θ a(θ i − θ 0 )
θ 0
thin layer in which the transition for Λ −1/2 = constant to Λ −1/2 = ∞ occurs, Ed.