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

Bibliographic revision
2
1
−
− ⎤ 3
γu
−1
⎡
dp 3 ⎛dx ⎞ ⎛ pi
⎞
⎛ p ⎞
= ⎢1 ( 1 x)
⎥
⎜ ⎟ −⎜ ⎟ − ⎜ ⎟
dt R ⎝dp ⎠ ⎢ ⎝ p ⎠ ⎥
⎝ pi
⎢
⎥
⎠
⎣
⎦
1
−
γu
S
u
(2.62)
A result first derived by O´Donovan and Rallis, (1959). Rearranging Eq. (2.62) gives:
S
u
2
1
−
⎤ 3
γu
1
⎡
−
R⎛dx ⎞ ⎛p
u
i
⎞ ⎛ p ⎞ γ
⎛dp⎞
= ⎢1 ( 1 x)
⎥
⎜ ⎟ − −
3 dp ⎢ ⎜ ⎟ ⎜ ⎟
p ⎥ p
⎜
i
dt
⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎢⎣
⎥⎦
(2.63)
tel-00623090, version 1 - 13 Sep 2011
In this equation, R is the effective radius of the vessel and γ u = c pu /c vu , the isentropic
exponent of the unburned mixture. Both x and its derivative dx/dp depend on pressure.
The advantage of this expression of the burning velocity is the possibility of exploring a
wide range of pressures and temperatures with one explosion. This is the main reason
of its utilization for burning velocity determination in engine conditions. Several x(p)
relations have been proposed in the literature. The most important of which are:
- O’Donovan and Rallis (1959)
Based on the same assumptions mentioned, O’Donovan and Rallis (1959) present an
x(p) relation that in the present symbols reads,
( γu
−1)/
( /
i )
( )
p−
pi( Tu / Ti)
T ⎛ p−
p
e
i
p p
x = =
pe( Tb / Te) − pi( Tu / Ti) T ⎜
b
⎝pe − ( Tb / Te) pi p/
pi
γu
( γu
−1)/
γu
⎞
⎟
⎠
(2.64)
Tb
is the mass averaged burnt temperature during combustion, with end value T
e
. T b is
determined for every burned shell from 3 consecutive increases. First its unburned
temperature is determined from adiabatic compression. Its burned temperature is then
obtained from energy conservation for the shell at constant pressure. Finally, the
burned shell is further compressed (and heated) adiabatically.
As a further simplification to their model, O’Donovan and Rallis assume that Tb
and T
e
are equal during the whole combustion period. In this case one finds
p−
pi( p/ pi)
x =
p − p ( p/ p )
( γu
−1)/
γu
( γu
−1)/
γu
e i i
(2.65)
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