Impact of fuel supply impedance and fuel staging on gas turbine ...

Acoustics
can be rewritten in terms <strong>of</strong> the additional ports as follows
˙Q ′ (ω)
¯˙Q
= F u (ω) u′ m (ω)
ū m
+ F φ (ω)
( u
′
k (ω)
− u′ l (ω) )
K , (3.57)
ū k ū l
where K denotes the ratio between the <strong>fuel</strong> injected at this <strong>fuel</strong> injection stage
<strong>and</strong> the entire injected <strong>fuel</strong> in the combustion system (see Eqn. (2.10), (2.11)).
Using Eqn. (3.57) <strong>and</strong> the equation <strong>of</strong> state (Eqn. (3.7)), Eqn. (3.56) can be
transformed into relations for the Riemann Invariants upstream <strong>and</strong> downstream
<strong>of</strong> the flame
⎡
⎢
⎣
−r T M i
u 0i
u 0m
F u
u
r 0i T u 0m
F u
( ) T j
where r T = − 1
T i
⎡
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
⎤
1−r T M i 1+r T M i
−r T M i
u 0i
u 0k
F φ K
r T M i
u 0i
u 0m
F u
−r T
u 0i
u 0m
F u
r T
u 0i
u 0k
F φ K
r T M i
u 0i
u 0l
F φ K
u
−r 0i T F
u φ K
0l
⎤
⎥
⎦
[
fm
g m
]
=
r T M i
u 0i
u 0k
F φ K
−r T
u 0i
u 0k
F φ r m
−r T M i
u 0i
u 0l
F φ K
r T
u 0i
u 0l
F φ K
[ ρ0j c j
ρ 0i c i
ρ 0j c j
ρ 0i c i
1 −1
[
⎥ fi
⎦
g i
]
+
⎤
[ ]
⎥ fk
⎦ +
g k
⎤
[ ]
⎥ fl
⎦ +
g l
] [ ] f j
, (3.58)
g j
. Equation (3.58) can, <strong>of</strong> course, be extended easily to combustion
systems with more <strong>fuel</strong> injection stages.
3.3.1.3 Boundary conditions
To complete an acoustic network model appropriate boundary conditions are
required. In a combustion test rig the air or the <strong>fuel</strong> enter the plenum in many
cases through small holes, which generate a considerable pressure loss. If the
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