Etude de la combustion de gaz de synthèse issus d'un processus de ...

Chapter 2
Where C 1 is a constant to be determined. The Lambert function W is defined as the
W
inverse function of fW ( ) = We . By definition, dW W ( Z )
=
dZ Z(1 + W ( Z)) for −1 Z ≠ and W( Z)
≠− 1 .
e
If L b >0, Z>0, and consequently W 0 (Z) is a positive real number. Thus L b , S and C 1 can
be found by minimizing the following equation:
0
n
N
N
2 2
( r ( t original
) − r ( t )) = ( r original
( t ) −2 L W b 0( Z ))
1 1
∑ ∑ (2.43)
The flame speed and the stretch could also be linked based on a nonlinear
methodology proposed by Buckmaster, (1977):
tel-00623090, version 1 - 13 Sep 2011
2 2
⎛S
⎞ ⎛⎛ n
S ⎞ ⎞
n
2Lbκ
⎜ ln
0 ⎟ ⎜⎜ 0 ⎟ ⎟ =
0
(2.44)
⎝Sn ⎠ ⎜⎝S ⎟
n ⎠ S
⎝ ⎠
n
The derivation of this equation is based on the asymptotic hypothesis. The structure of
the flames considered is characterized by a small flame thickness to which all the
reaction is confined. Moreover, deformations occur on a long scale of the order of the
dimensionless activation energy.
Based on Eq. (2.44), unstretched flame speed and burned gas Markstein length can be
deduced by a minimization of the following expression:
2 2
n ⎡⎛S
⎞ ⎛⎛ n
S ⎞ ⎞
n
2Lbκ
⎤
∑ ⎢⎜ ln
0 ⎟ ⎜⎜ 0 ⎟ ⎟+ ⎥
0
(2.45)
1 ⎢⎝S ⎜
n ⎠ ⎝S ⎟
n ⎠ Sn
⎥
⎣ ⎝ ⎠ ⎦
Where n is the number of flame images.
2.5.2 Burning velocity measurement methods
Burning velocity is a physicochemical constant for a given mixture. It is the velocity,
relative of unburned gas, which a plane, one-dimensional flame front travels along the
normal to its surface. Clearly, it is the volume of combustible mixture, at its own
temperature and pressure, consumed in unit time by unit area of flame front. It is
independent of flame geometry, burner size and flow rate. As indicated above, the
burning velocity is essentially a measure of the overall reaction rate in the flame and is
important, both in the stabilization of flames and in determining rates of heat release.
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