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

Bibliographic revision
where the subscript t denotes ‘total’, hence v t = V /m i and e t = U/m i where U is the total
internal energy. For ideal gases Eq. (2.69) can be written as
RT
u i
RT
b b
RT
u u
= x + ( 1− x)
(2.71)
p p p
i
The internal energy for a perfect gas can be written as e = e 0 + c v (T − T 0 ), where e 0 is
the chemical energy stored in the mixture at a reference temperature T 0 . Without loss
of generality we can write,
e ( T ) = C ( T − T ) +Δ e
(2.72)
u u vu u
0
e ( T ) = C ( T − T )
(2.73)
b b vb b
0
tel-00623090, version 1 - 13 Sep 2011
where Δe = e 0 , u −e 0 , b . Inserting Eqs. (2.72) and (2.73) into Eq. (2.70) and rearranging
gives
1
T = T + [ xΔ e+ C ( T −T ) −( 1−x) C ( T − T )]
(2.74)
b 0 vu i 0 vu u 0
xCvb
Now T b can be eliminated from Eq. (2.71) resulting in
p T R ⎛ T 1
= − x + x + xΔ e+ C T −T − −x C T −T
p T R ⎝ T C T
[ ]
u b 0
(1 ) ⎜
vu
(
i 0) (1 )
vu
(
u 0)
i i u i vb i
For the limiting case x =1 at p = p e , we have
p R ⎛T
1
p R ⎝T C T
[ e C ( T T )]
e b 0
= ⎜ + Δ +
vu i
−
i u i vb i
0
⎞
⎟
⎠
⎞
⎟
⎠
(2.75)
(2.76)
Inserting this expression for p e back into Eq. (2.75) gives
p Tu p ⎡
e
Rb Cvu Ti −T
⎤
u
= (1 − x) + x + (1 − x)
⎢ + ⎥
pi Ti pi ⎣Ru Cvb Ti
⎦
(2.77)
The term in square brackets can be rewritten as
Rb Cvu Ti −Tu γ
b
−1⎛
T ⎞ 1
u
+ = ⎜ − ⎟
(2.78)
R C T γ −1
⎝ T ⎠
u vb i u i
Inserting into Eq. (2.77) and rearranging for x we obtain
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