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

Chapter 2
This is compatible with the assumption that the (mass averaged) burned temperature
does not vary over time in this model. O’Donovan and Rallis agree that this is a severe
simplification, and suggest to measure the burned temperature at the centre of the
bomb T b,c immediately after the start of combustion to evaluated T e /T b .
- Lewis and von Elbe (1987)
The most cited x(p) relation in literature equates the burnt mass fraction to the
fractional pressure rise originally proposed by Lewis and von Elbe, (1987):
p − pi
x =
p − p
e
i
(2.66)
tel-00623090, version 1 - 13 Sep 2011
Because its appealing simplicity and reasonable accuracy, the linear relation has been
used a lot for obtaining laminar burning velocities. Dahoe et al., (1996) give a derivation
similar to the above, but immediately start from the linear relation (2.66). Eq. (2.62)
then turns into
2
1 3
⎡
⎤
dp 3( p ) u
e
−pi ⎛ pi
⎞γ
( pe
−p)
⎛ p ⎞
= ⎢1− ⎥
dt R ⎢ ⎜ ⎟
⎜ ⎟
⎝ p ⎠ ( pe − pi)
⎥
⎝pi
⎢
⎥
⎠
⎣
⎦
1
γu
S
u
(2.67)
- Nagy et al., (1969)
Nagy et al., (1969) assume isentropic compression of the burned mixture and have
used the following x(p) expression:
p − p
xp ( ) =
p p p
1/ γu
1/ γu
i
1/ γ b (1/ γu−1/ γb) 1/ γu
e
−
i
(2.68)
- Luijten et al., (2009)
Luijten et al. proposed an extended linear analytical two-zone model, where both the
burned and unburned zones have uniform temperatures and compositions.
Conservation of specific volume v and internal energy e is expressed by
v = xv + ( 1− x)
v
(2.69)
t b u
e = xe + ( 1− x)
e
(2.70)
t b u
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