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

Numerical analysis <strong>of</strong> thermo-acoustic instabilities
input u ′ b <strong>and</strong> output ˙Q ′ as shown in Fig. 2.3. The flame response <strong>of</strong> a perfect
premixed combustion system, in which no equivalence ratio fluctuations exist,
is completely described by such a SISO system.
u' b
.
Q'
Figure 2.3: SISO system representing a perfect premixed flame
A proper choice <strong>of</strong> the reference location "b" for the velocity fluctuation u ′ b is
crucial to obtain a flame transfer function which is independent <strong>of</strong> the acoustic
<strong>impedance</strong> at the burner. The acoustic <strong>impedance</strong> at the burner is characterized
by the acoustic properties <strong>and</strong> the acoustic boundary conditions <strong>of</strong> the
combustion system. As the flame dynamics is mainly controlled by the flame
front kinematics, the proper reference location is the burner exit. At this location
acoustic fluctuations generate vortical perturbations in the flow field
(shear layer), which are transported with the speed <strong>of</strong> the flow to <strong>and</strong> through
the flame, modulating the flame area <strong>and</strong> turbulent flame speed. The strength
<strong>of</strong> the disturbance is directly related to the amplitude <strong>of</strong> the velocity fluctuation
at this position. A velocity signal, which is taken at a different location,
may be influenced in a spurious manner by the acoustics <strong>of</strong> the combustion
system. If the distance between burner exit (x b ) <strong>and</strong> the reference location (x ˜b)
is not much smaller than a acoustic wave length (speed <strong>of</strong> sound divided by
frequency), the acoustic velocities at both locations differ in terms <strong>of</strong> amplitude
<strong>and</strong> phase. If now the acoustics <strong>of</strong> the system is changed, e.g. by changing
the upstream boundary condition, the amplitude <strong>and</strong> phase difference between
the acoustic velocities at both locations is modified. This is clarified
in Fig. 2.4, which shows the spatial amplitude <strong>of</strong> the acoustic velocity <strong>of</strong> two
cases a) <strong>and</strong> b) with a different upstream boundary condition Z 1 <strong>and</strong> Z 2 . As the
velocity fluctuation at x b is the driving parameter, a high velocity value at x ˜b
would result in almost no heat release rate fluctuation in case a). However, the
contrary is demonstrated in case b), in which a low value at x ˜b corresponds to
a strong response <strong>of</strong> the flame. The modification <strong>of</strong> the system changes there-
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