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

Chapter 2
γb−1
Tu γu −γb
( γ 1) ( 1 )
u − Ti γu
−
γb−1
Tu γu −γ
⎡
b
( γ −1) ( γ −1
)
p− p ⎡
i
+
x =
⎣
p − p +
⎣
e i u Ti u
⎤
⎦
⎤
⎦
(2.79)
As a final step, the above result is written fully in terms of pressure using Eq. (2.62).
For later use, the result is given in a more concise form:
p − pf
i
( p)
x =
p − pf( p)
e
i
(2.80)
Where
⎛γb −1⎞ ⎛γu −γ
⎞⎛
b p ⎞
f( p)
= ⎜ ⎟+
⎜ ⎟⎜ ⎟
⎝γu −1⎠ ⎝ γu −1
⎠⎝ pi
⎠
( γ −1/
) γ
u
u
(2.81)
tel-00623090, version 1 - 13 Sep 2011
Differentiation is straightforward, yielding
Where
'( )[ −
i
( )]
[ − ] 2
dx 1 − pf
i
'( p)
pif p p pf p
= +
dp p − p f ( p) p pf( p)
e i e i
⎛γu
− γ ⎞⎛
b
p ⎞
pf
i
'( p)
= ⎜ ⎟⎜ ⎟
⎝ γ
u ⎠⎝pi
⎠
−1/
γu
(2.82)
(2.83)
The above methodology was extended to multiple zones by Luijten et al., (2009)
without considerable improvement in the results.
The linear approximation, introduced by Lewis and Von Elbe, is still the most
widespread analytical relation to interpret burning velocity data. Differences in laminar
burning velocities between the varieties of x(p) were quantified by Luijten et al., (2009)
for the example case of stoichiometric methane–air combustion, demonstrating that
deviations between burning velocities from bomb data and other methods can at least
partly be ascribed to the limited accuracy of the linear approximation. For the example
case, differences up to 8% were found.
2.5.2.4 Constant pressure method
Laminar burning velocity and Markstein length can be deduced from schlieren
photographs as described by Bradley et al., (1998). For a spherically expending flame,
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