BURNER MODELING
5.2. Transfer Function Fitting
The simplest model for the presented burner is generated by fitting a transfer function to the combustion response data examined in chapter 3. The combustion response of the burner assembly was experimentally found to be linear in amplitude for sound pressure levels up to at least 116.8 dB. However, as will be seen, the response does not satisfy the necessary conditions of a linear, time-invariant (LTI) system in general.
The combustion response transfer function for the reference burner operating condition (φ
= 0.85, VR = 4.0) was originally shown in Figure 3-4 and is repeated here (with some additions) in Figure 5-1. Additional annotations on the plot highlight features of interest that were discussed earlier. The peak of the magnitude response is seen to begin around 60 Hz and continue (with a slight downward slope) up to 100 Hz. This appears to correspond to the peak receptivity of the reactant jet shear layer. As described by Davies, et. al. (1963), for a classical, free, laminar jet in an open half-space, the peak receptivity is seen to fall in a Strouhal number range from 0.42 to 0.48.
In the presented case, the jet is constrained by the stagnation plane located at an L/D = 1.5 above the nozzle exit plane. Nonetheless, the maximum receptivity is seen to occur at a frequency of about 60 Hz, corresponding to a Strouhal number of
(60 Hz)(14 mm)
St 0.50
(1670 mm/s) f D
= U = = .
At 100 Hz, where the peaked region ends, the associated Strouhal number is 0.84. The entirety of the pre-transitional region is marked by a roll-off in phase that is indicative of the transport delay of structures produced in the unsteady shear layer travelling from the nozzle exit to the reaction zone.
Figure 5-1: Combustion response function plotted for the flat-flame burner operating on premixed methane and air with an equivalence ratio of 0.85 and a nozzle exit velocity ratio of 4.0. Both magnitude and phase are presented along with notations identifying features of interest. Negative phase corresponds to the fluctuating heat release lagging the fluctuating pressure.
As was shown in chapter 3, an abrupt transition occurs where the character of the response changes dramatically. The transition is frequently marked with a notch in the magnitude response and an associated jump in the phase. For the reference case, the notch occurs at a drive frequency of approximately 240 Hz. Beyond the notch is the post-transitional region where the Helmholtz response of the burner cavity and nozzle is seen. This is centered at approximately 375 Hz.
Acknowledging the above features it is now possible to construct a simplified model transfer function for the described system. The process begins by predominantly (but not exclusively) focusing on the system magnitude response. The slope of the magnitude is upward at
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Frequency (Hz)
Magnitude Phase (deg)
Phase Peak
Magnitude
Helmholtz
Transition Transport Delay
low frequencies indicating improved coupling of the acoustic wave to the reactant flow and shear layer as the frequency is increased. Ripples are also seen in the response from 20 Hz to 40 Hz;
however, these will not be addressed in the model. To generate the initial upward slope, at least one zero has to be located at the origin in the s-plane. This is problematic, however, since (as it was shown earlier in chapter 3) the combustion response tends toward the in-phase condition as the frequency goes to zero. A single zero at the origin (unfortunately) produces a +90° contribution.
Therefore, to achieve the desired result a negated double zero must be used instead:
(5-1) G s1( )= −s2. (initial upward slope) As previously mentioned, the peak in receptivity of the shear layer appears between 60 Hz and 100 Hz. Four different approximations were tried for this portion of the response. These were a single complex pole pair, a double complex pole pair, a complex pole pair combined with a single real pole, and a complex pole pair combined with two real poles. The best agreement was found with the complex pole pair combined with a single real pole:
(5-2) 2
2 2
1 2 2 2
( ) 1
1 1
G s
s s s
ω ω ω Q
=
+ + +
. (receptivity peak)
After this peak, a steep drop in amplitude is present, accompanied by a notch in the response and subsequent recovery to a lower amplitude level. This is observed to be characteristic of the transition portion of all responses, although varying notch depths are seen depending on the burner operating conditions. The shape of the response is highly reminiscent of a low-pass notch filter where the characteristic frequency of the complex zero pair is higher than the characteristic frequency of the complex pole pair. The transfer function appears as
(5-3)
2 2
3 3 3
3 2
2
4 4 4
1 ( )
1
s s
G s Q
s s
Q
ω ω
ω ω
+ +
=
+ +
. (low-pass notch)
Finally, at drive frequencies from 300 to 450 Hz, the Helmholtz resonant response of the burner cavity and nozzle is observed. This is modeled simply with a complex pole pair. The given transfer function is
(5-4) 4 2
2
5 5 5
( ) 1
1 G s
s s
ω ω
Q=
+ +
. (Helmholtz resonance)
Combining the above relations along with a global gain parameter (α) into single response transfer function gives
(5-5)
2 2
2
3 3 3
5 2 2 2
2 2 2
1 2 2 2 4 4 4 5 5 5
1 ( )
1 1 1 1
s s
s Q
G s s s s s s s s
Q Q Q
α ω ω
ω ω ω ω ω ω ω
− + +
=
+ + + + + + +
.
The values for the eleven free parameters in this transfer function now need to be determined. They are selected using a modified, least-squares fitting algorithm against the collected data presented in Figure 5-1. The least-squares metric is used in conjunction with a progressive Monte Carlo technique to approach an optimal solution. The parameters obtained are shown in Table 5-1.
The magnitude response of the resulting transfer function is plotted in Figure 5-2 for comparison with the experimental data. The model transfer function is seen to be in good agreement with the experimental results for the magnitude behavior. However, to this point, no special treatment has been made for the phase; in particular, the previously detected time delay has
Param Value Group
α 0.0106 Global Gain
ω1 18.2 Hz (114 rad/s)
Receptivity ω2 65.2 Hz (410 rad/s)
Q2 2.70
ω3 244 Hz (1533 rad/s)
Low-Pass Notch
Q3 16.0
ω4 109 Hz (683 rad/s)
Q4 2.36
ω6 371 Hz (2330 rad/s) Helmholtz Response
Q6 0.724
Table 5-1: Optimized (fit) parameters for the trial transfer function, G5(s), presented as equation (5-5). Parameters were obtained using a modified least-squares metric in conjunction with a progressive Monte Carlo scheme.
Figure 5-2: Combustion response transfer function model versus experimental data for the reference burner test condition with an equivalence ratio of 0.85 and a nozzle exit velocity ratio of 4.0. Plot shows the magnitude response only. The solid line represents the model prediction while the circles indicate individual data points from the experimental data.
Frequency (Hz)
Magnitude
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Figure 5-3: Combustion response transfer function model versus experimental data for the reference burner test condition with an equivalence ratio of 0.85 and a nozzle exit velocity ratio of 4.0. Plot shows the phase response. The solid line represents the model prediction while the circles indicate individual data points from the experimental data.
not been addressed. Figure 5-3 shows the phase response of the model transfer function, G5(s), in comparison with the collected experimental data. It can be seen that, without additional corrections, the phase response of G5(s) falls woefully short of aligning with the experimental results. In an attempt to correct this, the aforementioned time delay is added. The revised transfer function is shown as G6(s) below:
(5-6)
2 2
2
3 3 3
6 2 2 2
2 2 2
1 2 2 2 4 4 4 5 5 5
1 ( )
1 1 1 1
s s s
s e
G s Q
s s s s s s s
Q Q Q
α
τω ω
ω ω ω ω ω ω ω
−
− + +
=
+ + + + + + +
. Frequency (Hz)
Phase (deg)
Phase – Model Transfer
Phase – Experimental Data
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