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

7.1 Setup <strong>of</strong> the acoustic network model
which have the same cross-section, are merged into one duct. Finally the network
model <strong>of</strong> the <strong>fuel</strong> injection stage contains a ”Closed end”, a simple duct
representing the ”Resonator tube” <strong>and</strong> ”Fuel plenum”, an area change (”AC
IV”) between <strong>fuel</strong> plenum <strong>and</strong> the ”Fuel injection tube” (simple duct), which
is connected by the ”Air-<strong>fuel</strong> T-junction” (”T-junction”) to the mixing section
<strong>of</strong> the combustor. To estimate the error, which is made with the simplified
model, a finite element-based acoustic model was set up using the commercial
s<strong>of</strong>tware package COMSOL. The model consists <strong>of</strong> the <strong>fuel</strong> injector including
the single injection tubes. An open end boundary condition (p ′ = 0)
is used as the termination on the injection tube side. The opposite side, the
resonator end, is realized as a rigid wall (u ′ = 0). The length <strong>of</strong> the resonator
tube was set exemplarily to L R = 0.3 m. As the finite element method solves
the homogeneous wave equation, mean flow effects <strong>and</strong> pressure losses are
not considered. Fig. 7.2 shows the first two eigenmodes <strong>of</strong> the <strong>fuel</strong> injector.
The scale <strong>of</strong> the legend is arbitrary <strong>and</strong> indicates the maximum <strong>and</strong> minimum
pressure fluctuations. The first eigenmode <strong>of</strong> the finite element model exhibits
nearly a λ/4 mode, whereas the second one approximates a 3λ/4 eigenmode.
The overall length corresponding to the λ/4 modes can be composed <strong>of</strong> the
geometrical length <strong>of</strong> the resonator <strong>and</strong> half <strong>of</strong> the circumference <strong>of</strong> the <strong>fuel</strong>
plenum. The acoustic field inside the <strong>fuel</strong> plenum ring is almost constant. The
acoustic characteristics <strong>of</strong> the <strong>fuel</strong> <strong>supply</strong> changes with increasing frequency
as the acoustics <strong>of</strong> the <strong>fuel</strong> plenum becomes more complex. This is shown
in table 7.1, in which the eigenfrequencies <strong>of</strong> the finite element-based model
are compared against the simplified network model with <strong>and</strong> without the effect
<strong>of</strong> mean flow <strong>and</strong> pressure losses. Here <strong>and</strong> in the following, the eigenfrequencies
are normalized with the Strouhal number, where the reference
speed is the mean axial velocity in the mixing section <strong>of</strong> case A. In the network
model the length <strong>of</strong> the <strong>fuel</strong> <strong>supply</strong> tube was set to L R = 0.355 m. This length
is slightly less than the one considered above, but results in a more accurate
match in terms <strong>of</strong> the first <strong>and</strong> second eigenfrequency. The mean flow <strong>and</strong> the
pressure loss have in general only a small effect on the eigenfrequencies. The
deviation <strong>of</strong> the first eigenfrequency between the network <strong>and</strong> finite element
model is rather low. As mentioned before, the error increases as the frequency
increases. Nevertheless, in the Strouhal range up to Sr = 2 the simplified net-
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