AFC-100Drive Amplifier
3.4 Experimental Results
3.4.1 The Combustion Response Function
The collected data was post processed to produce Bode plots of the combustion response function. Here, the combustion response function is defined as:
(3-1) ( )
( ) ( ) p q s H s p s q
; si.
Nozzle Exit Velocity Ratio
Burner Equivalence Ratio
Ingestion Limit
Blow-out Limit
Cantera Computed LFS
0.60 0.70 0.80 0.90 1.00
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
In this relation, p and q are the average values of the pressure and heat release rate, respectively.
The fluctuating values of the pressure and heat release are represented by p s( ) and q s( ), which are complex valued. For the chemiluminescence experiments presented, the heat release rate is assumed to be approximately proportional to the intensity measured by the optically filtered PMT such that:
(3-2) ( ) ( )
( ) q s I s
q I s
,
where I is the optical intensity. Owing to effects such a collisional quenching and reabsorption it is known that, in general, q s( )I s( ) , const; however, it is conjectured that for moderate heat release rates (on average) and where fluctuations are small, q s( ) q s( ), that equation (3- 2) holds approximately. Given this, the combustion response function can be rewritten as:
(3-3) ( )
( ) ( ) ( ) p I s H s p s I s
; si.
The plots generated from the experimental data display the magnitude and phase of this function from 20 Hz to nearly 2000 Hz. Negative phase values represent an unsteady heat release that lags the unsteady pressure. Figure 3-4 shows a representative plot for the test case with an equivalence ratio of 0.85 and a nozzle exit velocity ratio on 4.0.
For the case shown, several features of interest can be noted. The combustion response magnitude peaks around 60 Hz and remains near the peak up to nearly 100 Hz. Ripples can be seen in the ascending side of the magnitude between 20 Hz and 40 Hz. In addition, ripples in the amplitude are also noticeable on the descending side of the amplitude curve between 100 Hz and 160 Hz. A notch in the amplitude response is present at 243 Hz with a corresponding jump in the phase. To the right of this, a broad peak is seen in the amplitude between 250 Hz and 500 Hz. This portion of the response corresponds with the Helmholtz resonance of the burner nozzle and cavity.
Figure 3-4: 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. Negative phase corresponds to the fluctuating heat release lagging the fluctuating pressure.
This can be seen in Figure 3-5 where the same peak is present in the signal from the internal nozzle microphone. This microphone is located in the side of the aerodynamic contraction near the bottom, just above the upper sintered disk. The plot of the microphone signal is on a logarithmic vertical axis; however, its scale is arbitrary and the curve is shown simply for comparison to the combustion response amplitude plot. Beyond this point, the combustion response appears to decay towards the noise floor.
The phase of the combustion response function rolls off smoothly from 20 Hz to 200 Hz.
As the notch in the response is approached, the phase jumps in the positive (leading) direction.
100.0 1000.0
1.0E-2 1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3
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20.0
Magnitude Phase (deg)
Frequency (Hz) Magnitude
Phase
Figure 3-5: Amplitude portion of the combustion response function as seen in Figure 3-4 plotted in parallel with the fluctuating pressure seen within the larger portion of the burner nozzle cavity. The peak in the combustion response function centered around 375 Hz corresponds to the Helmholtz resonance of the burner nozzle and cavity.
Beyond this (after the Helmholtz resonance range has been passed) the response settles into being in phase with the fluctuating pressure. It is quickly noticed that the phase behavior between 20 Hz and 200 Hz looks suspiciously like a transport delay. It is known that for “real” physical systems the phase must approach a multiple of 90° at zero frequency. Given this constraint, linear fits were produced by the least-squares method to the low frequency phase data for all datasets, and the intercept at the origin was determined in each case. These fits unequivocally indicated that the chemiluminescence response at zero frequency is in-phase with the varying pressure. With this
Frequency (Hz)
Magnitude
Combustion Response Magnitude
Burner Cavity Fluctuating Pressure
Helmholtz Response
100.0 1000.0
1.0E-3 1.0E-2 1.0E-1 1.0E+0 1.0E+1 1.0E+2 1.0E+3
20.0
Figure 3-6: Phase portion of the combustion response function as seen in Figure 3-4. Solid curve corresponds to the phase behavior of a 22.4 ms time or transport delay. Curve
generated by least-squares fit to data points from 20 Hz to 222 Hz.
information in hand, the pertinent phase data for the experiment under review was fit again, but with the constraint that the phase be equal to zero at the origin. The result was an estimated transport delay of 22.4 ms. This can be seen in Figure 3-6.
As was alluded to above, all of the collected chemiluminescence data exhibits transport delay phase behavior at low frequencies. At some distinct frequency the system’s phase behavior changes abruptly. In the case presented in Figure 3-6, this occurs at 222 Hz. This location is dubbed the “transition frequency.” The frequency range below the transition point is called the sub- or pre-transitional region. For frequencies above the transition point (called the post-transition
Phase (deg)
100.0 1000.0
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20.0
= Phase f = Frequency
= Transport Delay = 22.4 ms
Frequency (Hz)
region) the system phase changes are typically small and remain within one phase-wrap. Eventually the phase equilibriates to the in-phase condition at the highest frequencies.