⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
d ( T)
⎞
T
d ( m)
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
m
⎠
d( W ) W d ( P) P d(ρ )
(
ρ
)
d Torel Torel
(
)
d Porel Porel
⎡ 2 W ⎢ −1 − 0 Cp ⋅ T ⎢ ⎢ 0 −1 ⎢ 1 ⎢ 0 1 0 ⎢ P 1 ⎢ 0 2 ⎢ ρ⋅W ⎢ U⋅ W ⋅ sin ( β ) ⎢ 0 0 ⎢ Cp ⋅ To ⎢ ⎢ ⎢ 0 0 0 ⎢ ⎢ T W ⋅ ( W − U⋅ sin ( β ) ) − 0 ⎢− To Cp ⋅ To ⎣
2
2⋅ h HV⋅ η b + U − W
0
0
0
1
0
0
0
1
0
0
−1
0
0
0
1
0
0
1
γ ⋅ M rel
0
0
0 −
Torel To
2
2⋅ Cp ⋅ T
2
γ ⋅ M rel
0
2
0
0
2
⎤ − 1⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
−1
U⋅ d ( U) ⎛ ⎞ − ⎜ Cp ⋅ T ⎜ ⎜ 0 ⎜ d (A) ⎜ − A ⎜ ⎜ 1 − ⋅ d ( CD) ⎜ 2 ⋅⎜ (56) d T ⎜ ( o) U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) − + ⎜ T Cp ⋅ To o ⎜ ⎜ 2 γ ⋅ M rel ⎜ − ⋅ d ( CD) ⎜ 2 ⎜ ⎜ − d( To) + U⋅ d ( U) − W ⋅ d ( U⋅ sin ( β ) ) ⎜ To Cp ⋅ To ⎝ ⎠
Initial flow values for the burner are the compressor exit properties. Solving Equation (56) numerically from r/r3 = 1 to r4/r3 in steps of d(δr/r3) gives the following flow properties: T
W
i
i
T
i− 1
W
i− 1
+
⎛ d ( T) ⎞ ⋅ T ⎜ T ⎝ ⎠ i−1
+
⎛ d(W ) ⎞ ⋅ W ⎜ ⎝ W ⎠ i−1
P
P
+
⎛ d ( P) ⎞ ⋅ P ⎜ ⎝ P ⎠ i−1
ρi
ρ i− 1 +
⎛ d( ρ ) ⎞ ⋅ ρ ⎜ i− 1 ⎝ ρ ⎠
i
i− 1
Porel + i− 1
Porel i
To
i
To
i− 1
+
⎛ d ( Porel) ⎞ ⋅ Porel ⎜ P i− 1 ⎝ orel ⎠ ⎛ d ( To ) ⎞ ⋅ To ⎜ T ⎝ o ⎠ i−1
⎛ m ⎞ ⎛ m ⎞ + ⎛ d ( m) ⎞ ⋅ ⎛ m ⎞ ⎜ ⎟⎜ ⎟ ⎜A ⎟ ⎜A ⎟ ⎝ 3 ⎠ i ⎝ 3 ⎠ i−1 ⎝ m ⎠ ⎝ A 3 ⎠ i−1 W M rel
i
i
γ i⋅ R⋅ T
i γi
Po i
s
i
⎛ To ⎞ i Porel ⋅ ⎜ i ⎜ Torel i⎠ ⎝
γ i−1
⎛ To i ⎞ ⎛ Poi ⎞ ⎜ ⎜ s 3 + Cp ⋅ ln − R⋅ ln ⎜ ⎜P i ⎝ To3 ⎠ ⎝ o3 ⎠
27
(57)