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

Numerical analysis <strong>of</strong> thermo-acoustic instabilities
2.2.2 Flame response to equivalence ratio fluctuations
In a practical premixed combustion system, where <strong>fuel</strong> is injected at a location
with non-negligible acoustic fluctuations <strong>of</strong> pressure or velocity, the equivalence
ratio φ i will also fluctuate, see Fig. 2.2. When the fluctuations, which are
transported convectively with the flow, arrive at the flame front, they affect
the heat release rate through a change <strong>of</strong> heat <strong>of</strong> reaction per unit mass <strong>of</strong> premixture,
burning velocity <strong>and</strong> flame area. All these processes can be captured
by another transfer function, which describes the impact <strong>of</strong> equivalence ratio
fluctuation φ i to heat release rate fluctuations
F φ,i (ω)≡ ˙Q ′ (ω)/ ¯˙Q
φ ′ i (ω)/ ¯φ , (2.5)
where ¯φ denotes the mean equivalence ratio <strong>of</strong> the entire system. φ i represents
the nominal equivalence ratio fluctuations, which is measured at the
point <strong>of</strong> injection. The transfer function represents again a SISO system with
input φ ′ i
<strong>and</strong> output ˙Q ′ . Similar to the discussion above the point <strong>of</strong> injection
is the physical correct reference location as equivalence ratio fluctuations
are the result <strong>of</strong> acoustic velocity fluctuations at this location. At the point
<strong>of</strong> injection the equivalence ratio fluctuations can be expressed in terms <strong>of</strong>
the acoustic velocity fluctuations <strong>of</strong> the main air mass flow <strong>and</strong> the <strong>fuel</strong> mass
flow, which is a very useful relationship. It provides in conjunction with the
transfer function F φ,i a closure relation for stability analysis <strong>of</strong> the combustion
system using e.g. a low-order network model as in the present work or
a finite-element model for a generalized Helmholtz-Equation [53, 84, 96, 101].
The derivation <strong>of</strong> the relationship is represented below. Care has to be taken
if equivalence ratio fluctuations are determined at a different position as the
convectively transported equivalence ratio fluctuations are dispersed on the
way to the flame. The dispersion <strong>of</strong> the <strong>fuel</strong> depends on the geometry <strong>of</strong> <strong>and</strong>
on the turbulent flow field in the mixing section or swirler. Additionally, in
jet-in-cross flow injector configurations for example, as shown in Fig. 2.2, the
depth <strong>of</strong> penetration can, depending on the fluctuations, also change. A different
depth <strong>of</strong> penetration results in a different radial <strong>fuel</strong>-air composition at
the burner exit. This may have an influence on the local heat release rate fluctuation
<strong>and</strong> thus on the transfer function as indicated by Armitage et al. [7]
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