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Measurement of the transfer function of a combustor exit nozzle
Dominic Carolan
Submitted in total fulfillment of the degree of Master of Engineering Science
Supervisor: Dr. M. Brear
The University of Melbourne Department of Mechanical Engineering
Abstract The interaction of convected entropy disturbances with choked nozzles has been shown to be an important contributor to jet engine noise and thermoacoustic instability in gas turbine combustors. In order to be able to reduce noise and control or eliminate instability in premixed combustors, a better understanding of the physical mechanisms involved in thermoacoustic instability is required. An important part of the thermoacoustic stability problem is the frequency response of the combustor exit nozzle. The provision of experimental evidence demonstrating the eect of a choked downstream boundary condition has been limited by the diculties associated with making dynamic measurements in a hot turbulent gas ow. This thesis presents experimental results to allow comparison with the established linear theory of choked nozzle dynamics. An experimental premixed combustor tted with fast response thermocouples and pressure transducers was used for the experiments. Tests were carried out with both an open and choked exit nozzle in order to determine the nozzles response to pressure and entropy disturbances using a modied two microphone method. Results for both the open and choked exit response to pressure disturbances showed good agreement with the theory. The analysis of the results for the open and choked exit response to entropy disturbances revealed the limitations of the method. Spectral analysis techniques were successfully applied to the open exit data to demonstrate the existence of traveling entropy waves moving downstream at the convective velocity of the ow. However, the frequency range in which these temperature disturbances were found to be spatially correlated was limited to frequencies below the working range of the modied two microphone method.
This is to certify that the thesis is less than 30,000 words in length, exclusive of words in tables, gures, bibliographies and appendices.
I would like to thank my supervisor, Dr Michael Brear for his help and encouragement. I will also note here my gratitude to all those who helped in a variety of ways: Tony Kitchener, for his support of the research; Don, for making the bits and pieces as required; Dr Peter Hield, Ashley Wiese and Dennis for their help with the experiments; Nader Karimi, for his kind sharing of the lab test cell; Will, for his help with the theory and the analysis; Dad and Joe, for proof reading the draft; and Sujata, for her help and encouragement.
1 Introduction 1.1 1.2 1.3 15
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Aims of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . 19 20
2 Literature Review 2.1 2.2 2.3 2.4
Thermoacoustic instability . . . . . . . . . . . . . . . . . . . . 20 Lean premixed combustors . . . . . . . . . . . . . . . . . . . . 22 Previous experimental work on nozzles . . . . . . . . . . . . . 24 Acoustic reection coecients . . . . . . . . . . . . . . . . . . 26 2.4.1 2.4.2 2.4.3 Reection from an open end . . . . . . . . . . . . . . . 28 Including the eects of radiation from an open end . . 28 Reection from a closed end . . . . . . . . . . . . . . . 30
Reection from a choked nozzle . . . . . . . . . . . . . . . . . 30 Experimental determination of reection coecients . . . . . . 33 2.6.1 2.6.2 2.6.3 The two microphone method of Seybert and Ross [1] . 33 Response to an entropy disturbance . . . . . . . . . . . 35 Fast response temperature measurement . . . . . . . . 37
Summary of the literature . . . . . . . . . . . . . . . . . . . . 40 42
3 Experimental method 3.1
Description of the experimental rig . . . . . . . . . . . . . . . 42 3
3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9
Air and fuel preparation . . . . . . . . . . . . . . . . . 42 Flame holder . . . . . . . . . . . . . . . . . . . . . . . 43 Working section . . . . . . . . . . . . . . . . . . . . . . 44 Assembly of the thermocouples . . . . . . . . . . . . . 45 Thermocouple signal conditioning . . . . . . . . . . . . 47 Thermocouple cold compensation . . . . . . . . . . . . 49 Thermocouple time constant estimation . . . . . . . . 49 Estimating the time constant using basic theory . . . . 50 Estimating the thermocouple time constant using Strahles [2] method . . . . . . . . . . . . . . . . . . . . . . . . . 52 Estimating the thermocouple time constant using Tagawas [3] method . . . . . . . . . . . . . . . . . . . . . . . . . 54 Comparing Strahles [2] and Tagawas [3] methods on simulated data . . . . . . . . . . . . . . . . . . . . . . 55 Comparing Strahles [2] and Tagawas [3] methods on Bunsen burner test data . . . . . . . . . . . . . . . . . 59
Thermocouples . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.10 Comparing Strahles [2] and Tagawas [3] methods on the combustor rig data . . . . . . . . . . . . . . . . . . 64 3.2.11 Compensation of thermocouple measurements . . . . . 65 3.3 Pressure transducers . . . . . . . . . . . . . . . . . . . . . . . 69 3.3.1 3.3.2 3.4 3.4.1 3.4.2 3.5 Pressure transducers and signal conditioning . . . . . . 69 Pressure transducer testing . . . . . . . . . . . . . . . 70
Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Data acquisition phase shift . . . . . . . . . . . . . . . 73 Noise considerations . . . . . . . . . . . . . . . . . . . 74
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 79
Evidence of traveling entropy disturbances . . . . . . . . . . . 79 Uniformity of the temperature disturbances at the exit . . . . 89 Results for the two microphone one thermocouple method . . 98 4.3.1 4.3.2 4.3.3 4.3.4 Response of the open exit to pressure and entropy disturbances . . . . . . . . . . . . . . . . . . . . . . . . . 98 Response of a choked exit to a pressure disturbance . . 99 Response of the choked exit to an entropy disturbance 100 Coherence analysis of the data . . . . . . . . . . . . . . 105
Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . 107 109
5 Conclusions 5.1
Recommendations for further Work . . . . . . . . . . . . . . . 111 120
A Additional Data and Results
A.1 Commissioning data . . . . . . . . . . . . . . . . . . . . . . . 120 A.2 Measured mean data . . . . . . . . . . . . . . . . . . . . . . . 123 B MATLAB and SIMULINK les 127
B.1 Import pressure and thermocouple data . . . . . . . . . . . . . 127 B.2 Import Mean data . . . . . . . . . . . . . . . . . . . . . . . . 128 B.3 Import R-type thermocouple lookup table . . . . . . . . . . . 130 B.4 Calculate variables . . . . . . . . . . . . . . . . . . . . . . . . 132 B.5 Compute Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 134 B.6 Apply two microphone method . . . . . . . . . . . . . . . . . 138 B.7 Normalized cross-correlation . . . . . . . . . . . . . . . . . . . 140 B.8 Plotting the spectrogram . . . . . . . . . . . . . . . . . . . . . 141 B.9 Strahle method . . . . . . . . . . . . . . . . . . . . . . . . . . 142 B.10 Thermocouple compensation . . . . . . . . . . . . . . . . . . . 142 B.11 Thermocouple simulation . . . . . . . . . . . . . . . . . . . . . 143 B.12 Two microphone method simulation . . . . . . . . . . . . . . . 143
2.1 2.2 2.3 2.4 2.5 2.6 Flammability limits of the experimental combustor designed by Hield [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Theoretical power reection coecient for an open end [5] . . 29 Two microphone method for determining an unknown impedance from Seybert and Ross [1] . . . . . . . . . . . . . . . . . . . . 34 Two microphone and one thermocouple method for determining an unknown impedance from Hield [4] . . . . . . . . . . . 36 Two input single output system from Newland [6] . . . . . . . 37 Experimental apparatus used by Forney and Fralick to test thermocouple response [7] . . . . . . . . . . . . . . . . . . . . 40 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 The combustor rig designed by Hield [4] . . . . . . . . . . . . 43 Section through the ame holder . . . . . . . . . . . . . . . . 44 Modied combustor working section . . . . . . . . . . . . . . . 45 Spot welding of thermocouple (A) using graphite electrode (B) 46 Thermocouple circuit from Analog Devices [8] . . . . . . . . . 47 Raw thermocouple signal . . . . . . . . . . . . . . . . . . . . . 48 Amplied thermocouple signal . . . . . . . . . . . . . . . . . . 48 Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 2 ms without noise . . . . . . . . . 56
Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 2 ms with incoherent noise (S/N ratio=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 1.25 ms with incoherent noise (S/N ratio=2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.11 Time constant for simulation TC1s (+) and TC2s () versus maximum frequency used (S/N ratio=10000) 3.12 Experimental setup . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . . . . 60
3.13 TC1 and TC2 signal at start of data record . . . . . . . . . . 60 3.14 Spectral density for TC1 () and TC2 (.) and noise oor (Bunsen test 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.15 Spectral density for TC1 () and TC2 (.) and noise oor (Bunsen test 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.16 Spectral density for TC1A () and TC2A (.) and noise oor (Bunsen test 23) . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.17 Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.18 Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.19 Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 23) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.20 Spectral density of the thermocouple signal premixed combustor tests compared to the Bunsen burner tests . . . . . . . . . 64 3.21 Ri (+), Rr () and 2 (.) versus frequency for TC4 (open exit) 65 3.22 Ri (+), Rr () and 2 (.) versus frequency for TC4 (choked exit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.23 Calculated time constant TC1 (+) and TC2 () versus maximum frequency used (Bunsen test 6) . . . . . . . . . . . . . . 66
3.24 Calculated time constant TC1 (+) and TC2 () versus maximum frequency used (Bunsen test 12) . . . . . . . . . . . . . 67 3.25 Calculated time constant TC4 (+) and TC5 () versus maximum frequency used (Open exit) . . . . . . . . . . . . . . . . 67 3.26 Calculated time constant TC4 (+) and TC5 () versus maximum frequency used (Open exit) . . . . . . . . . . . . . . . . 68 3.27 Pressure transducer signal conditioning circuit adapted from Analog Devices [8] . . . . . . . . . . . . . . . . . . . . . . . . 70 3.28 High pass lter frequency response showing phase shift due to the data acquisition system . . . . . . . . . . . . . . . . . . . 71 3.29 Pressure transducer high pass lter frequency response . . . . 71 3.30 Amplied pressure transducer output versus time during the speaking test . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.31 Spectrogram of pressure transducer output during the speaking test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.32 Data acquisition system frequency response for the same sinusoidal input signal with all channels referenced to channel 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.33 Corrected data acquisition system frequency response for the same sinusoidal input signal with all channels referenced to channel 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.34 Spectra of M1 for the open and choked exit tests compared to the noise oor . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.35 Spectra of M2 for the open and choked exit tests compared to the noise oor . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.36 Spectra of T2 for the open and choked exit tests compared to the noise oor . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.37 Spectra of T4 for the open and choked exit tests compared to the noise oor . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Locations of thermocouples and pressure transducers . . . . . 80 Pressure spectra (open exit) for M = 0.034, = 1.0. The location of M1 and M2 is shown in Figure 4.1 . . . . . . . . . 81 Pressure spectra (choked exit) for M = 0.1, = 1.0. The location of M1 and M2 is shown in Figure 4.1 . . . . . . . . . 81 Temperature spectra compared to the estimated isentropic temperature spectra estimated from pressure signal (open exit). The location of the thermocouples and pressure transducers is shown in Figure 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 82
Temperature spectra compared to the estimated isentropic temperature spectra estimated from pressure signal (choked exit). The location of the thermocouples and pressure transducers is shown in Figure 4.1 . . . . . . . . . . . . . . . . . . . 83
Amplitude and phase relationship of T to Amplitude and phase relationship of T to
(open exit) of (choked exit)
the downstream measurement station shown in Figure 4.1 . . . 84
of the downstream measurement station shown in Figure 4.1 . 85 Cross-correlation of downstream pressure transducer (M1) with downstream thermocouple (TC4) (open exit) . . . . . . . . . . 86 Cross-correlation of downstream pressure transducer (M1) with downstream thermocouple (TC4) (choked exit) . . . . . . . . . 86 4.10 Cross-correlation of the upstream (TC2) and downstream (TC3) thermocouple signals for the open and choked exit tests . . . . 88 4.11 Amplitude and phase of TC2 relative to TC4 (open exit) . . . 89 4.12 Amplitude and phase of TC2 relative to TC4 (choked exit) . . 90 4.13 Cross-correlation of thermocouples TC4 and TC5 (open exit) . 91 4.14 Cross-correlation of thermocouples TC4 and TC5 (choked exit) 91 4.15 Cross-correlation of thermocouples TC4 and TC1 (open exit) . 92 4.16 Cross-correlation of thermocouples TC4 and TC1 (choked exit) 92
4.17 Cross-correlation of thermocouples TC4 and TC3 (open exit) . 93 4.18 Cross-correlation of thermocouples TC4 and TC3 (choked exit) 93 4.19 Amplitude and phase of TC5 relative to TC4 (open exit) . . . 95 4.20 Amplitude and phase of TC5 relative to TC4 (choked exit) . . 95 4.21 Amplitude and phase of TC1 relative to TC4 (open exit) . . . 96 4.22 Amplitude and phase of TC1 relative to TC4 (choked exit) . . 96 4.23 Amplitude and phase of TC3 relative to TC4 (open exit) . . . 97 4.24 Amplitude and phase of TC3 relative to TC4 (choked exit) . . 97 4.25 Experimental (.) open exit response to a pressure disturbance compared to theory (-) . . . . . . . . . . . . . . . . . . . . . . 99 4.26 Experimental (.) open exit response to an entropy disturbance (average TC1, TC3, TC4) . . . . . . . . . . . . . . . . . . . . 100 4.27 Experimental (.) choked exit response to a pressure disturbance compared to theory (-) . . . . . . . . . . . . . . . . . . 101 4.28 Experimental choked exit response to an entropy disturbance (TC3 only) compared to theory (-) . . . . . . . . . . . . . . . 102 4.29 Experimental choked exit response to an entropy disturbance (TC4 only) compared to theory (-) . . . . . . . . . . . . . . . 102 4.30 Experimental choked exit response to an entropy disturbance (TC1, TC3, TC4) compared to theory (-) . . . . . . . . . . . . 103 4.31 Experimental choked exit response to an entropy disturbance (TC0, TC1, TC3, TC4, TC5) compared to theory (-) . . . . . 103 4.32 Ordinary coherence functions for the open exit . . . . . . . . . 106 4.33 Ordinary coherence functions for the choked exit . . . . . . . . 106 4.34 Raw downstream pressure signal versus time for the choked exit107 A.1 R-type Thermocouple voltage versus temperature . . . . . . . 123 A.2 Amplied upstream (TC2) and downstream (TC3) thermocouple signals (open exit) . . . . . . . . . . . . . . . . . . . . . 125
A.3 Amplied upstream (TC2) and downstream (TC3) thermocouple signals (choked exit) . . . . . . . . . . . . . . . . . . . 126 B.1 Thermocouple simulation . . . . . . . . . . . . . . . . . . . . . 143 B.2 Two microphone method simulation . . . . . . . . . . . . . . . 143
A.1 Scale factors for mean measurements . . . . . . . . . . . . . . 121 A.2 Zero check for mean measurements . . . . . . . . . . . . . . . 121 A.3 Span check for mean measurements . . . . . . . . . . . . . . . 122 A.4 Open exit test mean values . . . . . . . . . . . . . . . . . . . . 124 A.5 Open exit mean temperatures (20 s of data) . . . . . . . . . . 124 A.6 Choked exit mean temperatures (20 s of data) . . . . . . . . . 124
a A B Be Br c cp C D f h H j k k M Nu p Pr Q R R Rxy Rxy Re Pipe radius Downstream traveling pressure wave amplitude Upstream traveling pressure wave amplitude Frequency bandwidth Half power point resonant peak bandwidth Speed of sound Specic heat at constant pressure Capacitance Diameter Frequency Convective heat transfer coecient Frequency response function Imaginary number Wave number Thermal conductivity Mass Nusselt number Pressure Prandtl number Quality factor Gas constant Resistance Cross-correlation Normalized cross-correlation Reynolds number
List of symbols continued...
s S S Sxy M t T u U x Z Entropy Entropy wave amplitude Pipe cross-sectional area Cross-spectral density Mach number Time Temperature Velocity Volume velocity Distance Acoustic impedance
2 xy
Ratio of specic heats Coherence function Density Entropy wave amplitude Time constant Radian frequency
Subscripts and Superscripts 1 2 g i r s Upstream of nozzle Downstream of nozzle Gas Incident Reected Entropy Complex conjugate
In many heat engines and propulsion devices, heat is added to a continuous ow of compressed gas directly by a ame in a combustor. This process of heat addition may occur at approximately constant pressure. The heated gases can then be expanded through a turbine to obtain useful mechanical work or through a nozzle to obtain thrust. Examples of such continuous ow thermodynamic devices tting this denition include jet engines, turbofan engines, gas turbines, afterburners, ram jets and rocket engines. All of these continuous ow devices can be susceptible to so-called thermoacoustic instability and unwanted noise resulting from the interaction of unsteady heat addition and pressure oscillations in the combustor. Furthermore, often these devices operate such that the ow is choked downstream of the combustor. That is, the ow is accelerated to sonic velocity at a point of minimum ow area. This thesis aims to determine the dynamic response of such a combustor exit to incident pressure disturbances and convected hot spots in the ow. Particular attention is paid to the importance of these convected hot spots,
which may also be considered as entropy waves generating sound inside the combustor as they are accelerated through the pressure gradient at the choked exit. Marble and Candels [9] linear theory for the response of the exit nozzles is widely cited in the literature and used as a downstream boundary condition for numerical models. However, experimental data supporting the theory is limited [4]. Furthermore, the actual importance of convected entropy disturbances to thermoacoustic instability and the generation of noise is currently an area of contention [10]. The main diculties in taking dynamic measurements in a combustor with a choked exit are a result of a number of complicating factors such as the limited viewing access and the elevated pressure. The relative diculty of these experiments may explain the absence of experimental data in the literature. The application of a frequency response method to determine a choked nozzles response to entropy disturbances is a novel approach having rst been adopted by Hield [4]. Hield used two microphones and a single thermocouple. The method relies heavily on the use of ne wire thermocouples to measure uctuations in gas temperature. The ability of thermocouples to directly measure the actual gas temperature is limited to frequencies below the cuto frequency of the thermocouple. However, if the thermocouple cuto frequency is known then the frequency response may be compensated for to give the true gas temperature. The actual cuto frequency depends not only on the size of the thermocouple but also on the convective conditions of the ow. Therefore an in-situ technique for time constant measurement is thought to be the most appropriate. The thermocouple cut-o frequency will be determined using two dierent in-situ techniques. The results will be compared to data obtained in tests done outside the combustor using a Bunsen burner and checked using
basic empirical relations for the heat transfer coecient.
Aims of this thesis
1. To determine the response of an open and choked combustor exit to pressure and entropy disturbances at dierent amplitudes of forcing Usually in real thermodynamic devices such as gas turbines the ow is choked downstream of the combustor. Hield and Brear [11] have studied experimentally the dynamics of acoustically open and choked exits. They have shown that the dynamics of a choked combustor exit dier signicantly from an open exit. However, their results for the response to an entropy disturbance do not agree with the established linear theory for choked downstream boundary conditions. The experiments will be repeated using a dierent approach in an attempt to improve on the accuracy of the technique. The measurement of smaller amplitude forcing would allow testing of the hypothesis that smaller amplitude forcing will result in a more linear response. Achieving dierent amplitudes of pressure and entropic forcing may require modication of the existing experimental combustor and improvement in the signal to noise ratio of the instrumentation. Once obtained, the experiment results may be compared to the established linear theory. 2. To investigate the existence of traveling entropy waves at the combustor exit The importance of traveling entropy waves on combustor noise and thermoacoustic instability is an area of contention in the literature. The existence of entropy waves traveling downstream from the ame in the experimental combustor during thermoacoustic resonance is assumed. However, the frequency and amplitude of these waves are not known. Furthermore, the 17
entropy waves are expected to dissipate through the eects of convection and diusion in the turbulent ow as they travel downstream [10]. The rate of diusion is unknown and therefore the signicance of the remaining entropy disturbances at the exit nozzle is uncertain. The measurement of the convected entropy waves is complicated by the simultaneous existence of isentropic temperature uctuations resulting from large amplitude pressure disturbances in the combustor. Therefore the convected and isentropic (acoustic) temperature uctuations should be separated in the analysis so that their relative signicance may be determined. 3. To investigate the uniformity of the temperature prole at the combustor exit Previous work by Hield [4] assumed axisymmetric, one dimensional ow inside the combustor working section. A single thermocouple was used to measure temperature disturbances at the combustor exit. Using a number of thermocouples arranged at dierent radial locations around the duct will give an indication of the axisymmetry of the temperature prole. Furthermore, using the instantaneous average value of these thermocouples in the analysis of the forced response of the combustor exit may improve the accuracy of the technique. 4. Apply existing techniques for in-situ thermocouple time constant measurement The accurate measurement of the amplitude of entropy disturbances at higher frequency requires compensation for the eects of the thermocouples time lag. Measuring the actual uctuating gas temperature can be dicult at higher frequencies because of the limited speed of response of the thermocouple and the low sensitivity of thermocouples suited to the temperature range expected in a combustor. The frequency response of a thermocouple is a di18
cult parameter to estimate based on empirical data because the mechanism of heat transfer is primarily convective and therefore highly dependent on the local properties of the ow which determine the boundary layer thickness around the thermocouple [12]. It is therefore preferable that the frequency response be measured in-situ. However estimating the frequency response or time constant of the system is dicult if the input function is unknown. In the present case the actual uctuating temperature is unknown. A dierent approach is therefore required. If a rst order system response is assumed for a pair of adjacent thermocouples measuring the same uctuating temperature, but with dierent sizes, the time constants of both thermocouples can be determined [2].
Many real continuous ow devices can be susceptible to thermoacoustic instability and unwanted noise resulting from the interaction of unsteady heat addition and pressure oscillations in the combustor. Many of these devices feature a choked nozzle downstream of the combustor. This experimental study aims to determine the dynamic response of a choked combustor exit to incident pressure disturbances and convected hot spots in the ow after rst investigating the existence and uniformity of these convected hot spots at the combustor exit. A frequency response technique will be used to obtain the response from simultaneous ne-wire thermocouple and microphone data taken inside the combustor. The time constant of these ne-wire thermocouples will be estimated using an in-situ measurement technique.
Chapter 2 Literature Review
2.1 Thermoacoustic instability
Thermoacoustic oscillations have been studied for over two hundred years. However, there is no general formulation to describe the coupling of the wave equation with the energy equation [13]. That depends on the details of the heat release mechanism, which dier for each situation or thermoacoustic device. One such thermoacoustic device is the so-called Rijke tube, named after Rijke who rst reported observing that a sound may be produced by placing a source of heat inside an open vertical tube [14]. The location of the heat source is known to be very important in determining the occurrence of the sound as well as its frequency and amplitude. For the relatively simple case of the Rijke tube with a heated gauze as a heat source the unsteady heat release may be coupled to the acoustic velocity uctuation. The unsteady volumetric expansion occurring as a result of the heat release may drive the acoustic oscillations. Saito [15] used 25 m diameter platinum wires to measure both temperature and velocity in an experiment to investigate the heat induced vibrations
in a Rijke tube. By using the same wire diameter for the velocity and temperature measurements the frequency response was assumed to be matched. The author studied a 40 m diameter platinum wire, heated with a 50 Hz alternating current to provide temperature uctuations of 100 Hz frequency. Saito also repeated the experiment with a premixed ame and showed experimentally that an air column may be excited if heat is added to a point in the pipe where the pressure lagged behind the velocity by one quarter of a cycle. The point of strongest excitation was found to be midway between a pressure node and antinode. Dowling [13] has described various models for the calculation of the frequency of thermoacoustic oscillations for dierent coupling mechanisms between the unsteady heat input and the velocity perturbation. The thermoacoustic feedback mechanism may be complicated in the case of a premixed ame by the existence of a time lag between the velocity uctuation and the heat release due to the nite rate of the chemistry as well as the ames dynamics, and in particular its interaction with vortices shed by the ame holder. Matveev and Culick [16] have developed a detailed model to describe how vortices of burning fuel shed by a premixed ame holder may drive pressure oscillations in the combustor. Some early examples of devices that have utilized the principle of thermoacoustic resonance to produce sound are the thermophone [17], which is a thin conductor that produces a sound when an alternating current is applied to it; the redamp detector [18], a singing ame device used to detect explosive mine gases and the pyrophone [19] an instrument purported to produce musical tones by the vibration of a hydrogen ame. More recently the principles of thermoacoustics have been applied to the design of heat pumps [20, 21]. Although there may be a few examples of useful themoacoustic devices, the most frequently cited motivation for the study of thermoacoustics has
been the negative consequences of thermoacoustic instability in devices such as rocket engines, ramjets, aircraft jet engines, gas turbines, burners and furnaces [22]. Of particular interest to researchers at present is the occurrence of thermoacoustic instability in lean premixed combustors. These will be discussed in the following section.
Lean premixed combustors
In the past two decades nearly all major manufacturers of industrial gas turbines have adapted their combustors to operate in a lean premixed conguration to minimise emissions of oxides of nitrogen (N OX ). In a lean premixed combustor the ratio of fuel to air is less than the stoichiometric ratio and the air and fuel are well mixed before combustion takes place. There has been a general trend away from the more stable but higher N OX emitting diusion ame combustors, although these are still used in aircraft. The downside of lean premixed combustors is that they are more susceptible to thermoacoustic instability. The main cause of nitrogen oxide emissions in a conventional diusion type ame is the so-called Thermal N O mechanism. Thermal N O refers to nitric oxide formed by reactions of nitrogen with oxygen at elevated temperature. The mechanism of thermal N O formation is also referred to as the extended Zeldovich mechanism after Yakov B. Zeldovich who rst studied it in 1947 [23]. The reactions are as follows O + N2 N O + N N + O2 N O + O N + OH N O + H (2.1) (2.2) (2.3)
The rate of the thermal N O mechanism is limited by the rst reaction which is highly endothermic. The overall reaction is therefore strongly temperature 22
dependent so that lower ame temperatures and shorter combustor residence times result in signicantly reduced N O emissions by this mechanism. In 1993, a US Environmental Protection Agency report [24] compared the most common methods used to control N OX emissions from operational stationary gas turbines. The report showed that on a cost per unit weight of N OX reduction basis, lean premixed combustor (LPM) design are more eective than steam and water injection - the other common method at that time. At the time lean premixed combustor designs were sometimes referred to as dry low emission (DLE) combustors to distinguish them from the steam or water injection types. In this work the term lean premixed (LPM) combustor will be used. Nitric oxide N OX emissions may be reduced in a lean premixed burner by burning a lean premixed fuel air mixture to achieve a lower ame temperature and by the early addition of dilution air to minimise the time that the combustion products are at an elevated temperature. However the stable combustion of hydrocarbons is limited to a relatively narrow range of equivalence ratios known as the ammability limits. Figure 2.1 shows a plot of equivalence ratio versus combustor Mach number for the experimental premixed burner used by Hield [4]. The ammability limits are plotted as a series of points at which the ame blew out as the equivalence ratio was varied. In order to eectively minimise N OX emissions, a lean premixed combustor must operate close to the lower ammability limit (also known as the lean blow out limit). LPM burners are more susceptible to instability caused by coupling between the unsteady heat release and pressure oscillations in the combustor. This leads to the problems of noise and instability. See Lieuwen [25] and Candel [26] for reviews of various approaches to modeling and controlling acoustic wave interactions in premixed combustors.
Figure 2.1: Flammability limits of the experimental combustor designed by Hield [4]
Previous experimental work on nozzles
The exit nozzle denes the downstream boundary condition for the combustor and thus the response of the nozzle to the inlet conditions aects the thermoacoustic stability of the system. The earliest experiments investigating the response of supersonic nozzles to uctuating inlet conditions were described by Zukoski and Auerbach in 1974 [27]. The experimental apparatus used was an electrically heated nitrogen gas stream. A total of 375 nichrome wires of 100 m diameter were positioned inside a duct and were periodically heated to provide a uctuating source of heat. The frequency was varied from 200 to 1200 Hz. A temperature uctuation of only 0.5 K was produced in a gas stream of mass ow rate 0.5 kg/s. The uctuating gas temperature was measured using a 10 m hot wire anemometer. A rotating bleed valve was used to bleed o precisely the required amount of gas at the correct phase, frequency and amplitude to cancel out the pressure increase 24
caused directly by the heat addition. The processing speed of the data acquisition system at that time was limited such that the data could only be sampled continuously for 0.1 second periods. Phase averaged data was collected by triggering the start of each sample with respect to the phase of the bleed valve or heater input signal. A 10 kHz sampling frequency was used and the 0.1 s length of data used gave a frequency resolution of 10 Hz. Results were presented only for the forcing frequency of 400 Hz. The authors stated that the results obtained for these very small amplitude disturbances in a supersonic nozzle were in agreement with the theory of Marble and Candel [9]. Muthukrishnan and Strahle investigated hydrodynamic, entropy, and combustion noise from a gas turbine combustor [28]. A full size gas turbine combustor was used for the experiments. The authors categorized core noise in a jet engine as direct combustion noise and indirect entropy noise. Combustion noise was dened as the noise produced by pressure uctuations caused by unsteady heat release. Entropy noise was dened as noise caused by hot or cold spots passing through pressure gradients in the turbine. The authors found that in a choked combustor at low frequency the entropy and direct combustion noise were in phase and inseparable. However as the frequency approaches the inverse of the particle residence time in the combustor the entropy noise became dominant. It was suggested that another possible mechanism of noise generation in a combustor was vorticity-nozzle interaction noise. The measurement of non-propagating hydrodynamic noise or pseudosound was minimized by locating the pressure transducers in an S-tube outside of the ow eld. An alternative approach to minimise the eect of apparent pressure uctuations caused by turbulence on the analysis may be using a pair of microphones separated far enough apart that the turbulence uctuations are uncorrelated. A paper by Corcos [29] provides the relevant formula for the degree of attenuation of background turbulent noise in such
situations. Recently experiments were done to measure pressure and entropy disturbances downstream of both a combustor and an electrically heated grid [30]. By using a periodic current pulse to heat the grid it was demonstrated that downstream propagating pressure disturbances due to the direct heating were followed by pressure disturbances generated by the convected hot spot passing through the nozzle. Importantly the authors concluded that the entropy noise was more signicant than direct combustion noise in the exhaust downstream of the nozzle. The experimental setup was very similar to the experiment of Saito [15] described above. The experimental aims where to measure the external noise generation rather than the internal reected noise. This present work follows on from directly from Hields [4, 11] experiments. Hield extended the two microphone method to also allow the determination of the response of a nozzle to an entropy disturbance. Hield applied the new method to an open and choked exit. For the open exit both the acoustic and entropic results showed good agreement with classical theory. However in the case of the choked exit neither the pressure nor entropy response agreed with Marble and Candels [9] theory. Hield concluded that large amplitude forcing in the experiment may have led to a non-linear nozzle response and recommended the reevaluation of the nozzle transfer function under more controlled conditions. This recommendation was taken as an aim of this thesis. The analysis will be explained below.
Acoustic reection coecients
Sound is usually considered to be isentropic because the spatial pressure gradients are so gradual that heat transfer between the compressed and rareed gas is small [31]. However, for large pressure disturbances such as shock
waves the pressure gradients are very steep and the isentropic assumption is no longer valid. In this study, the pressure disturbances are assumed to be isentropic. The inviscid momentum equation, or Eulers equation, can be written as u u 1 p +u = , t x x (2.4)
For the limit of low Mach number, the second term in equation2.4 becomes zero. Integration yields u= 1 p t , x (2.5) (2.6) (2.7)
p+ = Aejkx , p = Bejkx .
Dening the downstream and upstream characteristics as pressure and substiting 2.6 and 2.7 into 2.5 yields u+ = 1 1 1 Aejkx t = Aejkx = p+ . (2.8)
Similarly the velocity of the upstream traveling wave is given by 1 u = p . Often the volume velocity is used for the quantity U =Su, (2.10) (2.9)
where S is the pipe cross-sectional area. The acoustic impedance may be dened as pressure divided by volume velocity Z= p . U (2.11)
Using equations 2.6-2.10, for a pipe with an incident and reected pressure wave the impedance therefore is given by Z= c Aejkx + Bejkx ( ). S Aejkx Bejkx 27 (2.12)
Where c is the speed of sound. Dening Z0 as the impedance in the pipe at a particular point x = 0, then equation 2.12 becomes Z0 = c A + B ( ). S AB (2.13)
Solving equation 2.13 for the reection coecient B/A yields B Z0 c/S = . A Z0 + c/S (2.14)
In this way the reection coecient is related to the complex impedance. The reection from open and closed ends of a pipe will be considered in the following sections.
Reection from an open end
For a plane wave in a pipe with a change in cross-sectional area from S1 to S2 the reection coecient may be obtained by setting Z0 = c/S2 in equation 2.12 to yield S1 S2 B = . A S1 + S2 (2.15)
In the case of an open end the plane wave analysis may be used with S2 = . The resulting impedance is zero and the reection coecient is -1. This classical approximation is clearly not perfect as it implies that at low frequency no sound will leave the pipe. The following section describes a more accurate result which describes the impedance due to radiation from an open end.
Including the eects of radiation from an open end
The theory of Levine and Schwinger [5] for the radiation of sound from an open unanged circular pipe was used to obtain an approximation for the impedance of an open pipe at low frequency. Figure 2.2 shows the power reection coecient plotted against non-dimensional frequency or Helmholtz 28
Figure 2.2: Theoretical power reection coecient for an open end [5] number ka for an open end, where k is the wave number and a is the pipe diameter and the power reection coecient is dened as the square of the reection coecient. At low frequency ka << 1 (i.e. frequencies less than roughly 2000 Hz for a 50 mm pipe as in this study) an approximation for the unanged open end impedance is given by [32] as Z= c S k 2 a2 + 0.6jka 4 . (2.16)
The corresponding reection coecient may be determined by substituting the result for Z into equation 2.14. The resulting system transfer function is given by B b1 (j)2 + b2 j 1 = , A b1 (j)2 + b2 j + 1 (2.17)
where the constants are given by b1 = a2 /4c2 and b2 = 0.6a/c.
Reection from a closed end
By letting the area S2 = 0 in equation 2.15 for a closed end, the impedance becomes innite. The corresponding reection coecient for a closed end is 1. Unlike an open end, reection from a closed end occurs without change of phase.
Reection from a choked nozzle
Marble and Candel [9] formulated a linear theory for the response of nozzles to convected gas non-uniformities and acoustic disturbances. Cumpsty and Marble [33] also described the interaction of these entropy spots with turbine blade rows in which the steady change of pressure in the stream-wise direction is large. These are referred to as entropy uctuations rather than temperature uctuations because the temperature uctuations are made up of both an entropy component and the isentropic temperature change associated with a pressure uctuation, s p T = cp + ( 1) p . T (2.18)
The entropy disturbance is assumed to convect with the ow such that p s = eit , = cp p (2.19)
where s(t) is the entropy wave amplitude and p (x, t) is the uctuating pressure given by p (x, t) = eit (A(t)ejki x + B(t)ejkr x ) , (2.20)
where A(t) is the incident pressure wave amplitude and B(t) is the reected pressure wave amplitude. By denition the Mach number at the throat of a choked nozzle is equal to one. The nozzle can be assumed to be compact. Therefore the Mach 30
number at the nozzle entrance must also be constant to satisfy the equation of continuity of mass through the pipe, M =0 Squaring equation 2.21 and substituting c2 = yields u2 p =0. (2.23) p (2.22) (2.21)
Equation 2.23 may expanded into its steady and uctuating parts, u+u + u2 = p ( + p ) p
Retaining only the rst order terms this equation may be rearranged to obtain the linear boundary condition of Marble and Candel [9] for a compact, supercritical nozzle u p = +2 . p u Response of a choked nozzle to a pressure disturbance The above boundary condition will rst be used to determine the response of a choked nozzle to a downstream traveling pressure wave of amplitude A in the absence of entropy disturbances. The non-dimensional pressure and velocity at a point is given by the sum of the upstream and downstream traveling acoustic wave as follows p = (Aejki x + Bejkr x )ejt p and u = (Aejki x Bejkr x )ejt . c 31 (2.27) (2.26) (2.25)
The linearized equation 2.19 is s p = . cp p When there are no entropy disturbances, p = . p Substitution for / in equation 2.25 yields u M p ( 1) = . c 2 p (2.30) (2.29) (2.28)
Substitution of 2.30 into equation 2.27 and equation 2.26 at x=0 and solving for the response of a choked nozzle to a pressure disturbance Rp yields 1 1 ( 1)M1 B 2 . Rp = = 1 A 1 + 2 ( 1)M1 (2.31)
For small Mach numbers this reection coecient becomes the same as that for reection from a closed end. Response of a choked nozzle to an entropy disturbance The choked nozzle response to an entropy disturbance of amplitude will be derived here. The convected entropy disturbance is s = ej(tx/u) . cp (2.32)
In this case the amplitude of the incident pressure disturbance A = 0. Substitution of 2.28 into 2.25 allows equation 2.26 and 2.27 to be written as p = Bejt p and M1 p p ( + ejt ) = Bejt . 2 p p 32 (2.34) (2.33)
Solving equations 2.33 and 2.34 yields the reection coecient Rs , the response of a choked nozzle to an entropy disturbance as follows Rs = 1 M1 B = ( ). 1 2 1 + 2 ( 1)M1 (2.35)
Superposition of the results for the pressure disturbance caused by incident pressure wave and the pressure disturbance caused by the entropy wave yields the total upstream traveling pressure wave B. As the nozzle entry Mach number approaches zero the eect due to the convected entropy disturbance becomes negligible compared to reected incident pressure disturbance.
Experimental determination of reection coecients
There are several forms of so-called two microphone methods for the experimental determination of acoustic impedance or reection coecients [1, 34, 35]. The method assumes a single input single output (SISO) system with only the eect of linear acoustic disturbances considered. The discussion of this method will be extended to include the response to an entropy disturbance traveling with the ow in the following section.
The two microphone method of Seybert and Ross [1]
Two microphones are located at positions x1 and x2 as shown in Figure 2.3. The wave numbers for the incident and reected waves are given by ki = and kr = . cu 33 (2.37) c+u (2.36)
Figure 2.3: Two microphone method for determining an unknown impedance from Seybert and Ross [1] The pressure at any point is given by the sum of the contributions due to the incident and reected pressure disturbances at that point as follows p (x, t) = ejt (A(t)ejki x + B(t)ejkr x ) , (2.38)
where A(t) is the incident pressure wave amplitude and B(t) is the reected pressure wave amplitude. The quantity of interest is the ratio of the Fourier transforms of these amplitudes B() and A() at each frequency, which may be found using the frequency response method SAB A B B Rp = = = . SAA AA A
The Fourier transform of the pressure signal from each microphone P1 and P2 are used to obtain the auto spectral density functions S11 and S22 and the cross spectral density functions S12 and S21 . These known auto-spectral and cross-spectral densities are then used to solve for the unknowns, SAA , SBB , SAB and SBA at x = 0. Using S11 = P1 P1 = (A ejki x1 + B ejkr x1 )(Aejki x1 + Bejkr x1 ) , which may be simplied to obtain S11 = A A + A Bejki x1 +jkr x1 + B Aejki x1 jkr x1 + B B , 34 (2.41) (2.40)
which may also be written in terms of spectral density functions as S11 = SAA + SAB ejki x1 +jkr x1 + SBA ejki x1 jkr x1 + SBB . Similarly S12 , S21 and S22 are given by S12 = SAA ejki x1 jk1 x2 + SAB ejki x1 +jkr x2 + SBA ejki x1 jkr x2 + SBB ejki x1 +jk1 x2 , (2.43)
S21 = S12 ,
(2.44) (2.45)
S22 = SAA + SAB ejki x2 +jkr x2 + SBA ejki x2 jkr x2 + SBB .
Equations 2.42 to 2.45 are solved to obtain SAB and SAA . The pressure reection coecient Rp , is equivalent to the frequency response function for the system with the incident pressure wave amplitude A as the input and the reected pressure wave amplitude B as the output. Therefore for a single input single output (SISO) system the pressure reection coecient is given by Rp = SAB B = . SAA A (2.46)
Response to an entropy disturbance
The results obtained above for the amplitude of the upstream and downstream traveling pressure waves at the exit A and B are also valid in the presence of a linear entropy disturbance in a homogeneous ow. The two microphone signals may be used along with a single temperature signal to solve for the cross-spectral densities associated with the entropy and pressure disturbances at the nozzle. This method has been developed by Hield [4] and is discussed here. In a pipe with an unsteady heat source there may be an upstream and downstream traveling pressure wave and a downstream traveling entropy wave as shown in Figure 2.4. The entropy wave is assumed to be convected at the mean velocity of the ow. A second input for the entropy waves or hot 35
Figure 2.4: Two microphone and one thermocouple method for determining an unknown impedance from Hield [4] spots is included in the system model. Linear superposition of the reected pressure wave and the response due to the entropy disturbance gives the total upstream traveling pressure wave. The wave number, describing the convection of the entropy with the ow is . (2.47) u For the simple case in which the thermocouple and one of the pressure transks = ducers are located at the same distance x from the nozzle, the Fourier trans form of the non-dimensional entropy Sx may be calculated directly by taking Fourier transforms and rearranging equation 2.18 to yield ( 1) Sx = Tx Px , (2.48)
where Tx and Px are the Fourier transforms of the non-dimensional temperature and pressure at point x. The auto-spectral density function SSS for the entropy disturbance at the exit (where x = 0) may be estimated to obtain SSS Sx Sx . (2.49)
The cross-spectral density functions of the incident and upstream traveling pressure waves with the entropy disturbance at the exit, SAS and SBS , may be obtained by solving S13 = SAS ejki x1 jks x3 + SBS ejkr x1 jks x3 , 36 (2.50)
Figure 2.5: Two input single output system from Newland [6] and S23 = SAS ejki x2 jks x3 + SBS ejkr x2 jks x3 . (2.51)
The overall system is assumed to be a linear system with two inputs and a single output as shown in Figure 2.5. The result for the frequency response function for a two input single output system [6] is used. The frequency response function Rp between the rst input A, the incident pressure wave, and the output B, the reected pressure wave, is given by Rp = SSS SAB SSB SAS B . = A SAA SSS SAS SSA (2.52)
The frequency response function, RS between the second input = s/cp , the convected entropy wave, and the output B is given by Rs = B SAA SSB SAB SSA = . SAA SSS SAS SSA (2.53)
Fast response temperature measurement
The experimental measurement of entropy disturbances in a combustor usually requires the use of ne wire thermocouples and some knowledge of their frequency response. Early experimental work by Scadron and Warshawsky [12] resulted in comprehensive heat transfer models for ne-wire thermocouples over a range of Reynolds and Mach numbers. Time constants and 37
correction factors for steady state errors due to radiation and conduction losses for various geometries were obtained. The Reynolds number range based on the gas properties at the total temperature and the thermocouple wire diameter was an order of magnitude higher than the Reynolds number range encountered in these experiments. The Mach number range of the current experiments coincides with the lower limit of the range studied by Scadron and Warshawsky. Most thermocouple systems are typically modeled as a rst order lag systems however at low frequency the response can be attenuated due to conduction along the wire. The eects of conduction are treated in detail in a recent paper by Kobus [36]. Strahle and Muthukrishnan [2] describe a frequency domain technique to estimate thermocouple time constants using a pair of closely spaced thermocouples. The technique was applied to 25 m type-K thermocouples while measuring the exhaust gas temperature from a jet engine combustor. Reconstructed temperature results were presented up to a frequency of 1200 Hz [28]. The Strahle technique will be demonstrated in chapter 3. The length scale was estimated based on cross-correlation analysis of two thermocouple signals. One thermocouple was xed at a point and the other was positioned at varying distances from the xed thermocouple. The results revealed a steep drop in the cross-correlation coecient for the signals beyond the spacing of the length scales of temperature uctuations in the rig. Cambray [37] also claimed to have developed a new method of measuring thermocouple time constants. However the method requiring curve tting of the actual time series is inferior to the Strahle method which saves computation time due to the use of the fast Fourier transform. Talby calibrated a ne thermocouple using a cold wire [38]. A 50 micron thermocouple was positioned on the edge of a hot turbulent jet using a 0.63 micron platinum cold wire. The thermal response of the cold wire was esti-
mated to be of order 10 kHz for the particular turbulent ow. It was found that the compensation was eective to about 1 kHz. At higher frequencies comparison of the compensated thermocouple temperatures with the cold wire results showed that the amplication of noise became signicant. Tagawa et al. [3] compared fast response thermocouple measurements to a ne wire resistance thermometer (cold wire) of 0.6 to 3.1 microns diameter in the turbulent wake of a heated cylinder. Two thermocouple combinations were used. In the rst experiment, a pair of thermocouples of 40 and 100 m diameter were used. In the second experiment a pair of thermocouples 25 and 60 m were used. The compensated temperatures agreed closely with temperatures measured by the cold wire. Initially the method was applied using curve tting of the time series. The authors also applied a curve tting approach in the frequency domain to the same experimental data to obtain very similar results [3]. Forney and Fralick [7] showed by experiment that the two thermocouple method results in a good estimation of the true temperature for a controlled temperature uctuation in a constant ow. The temperature of the gas surrounding the thermocouple was varied by using a rotating wheel to switch between a hot and a cold gas stream as shown in Figure 2.6. The method was eective up to a frequency of 30 Hz. Wire diameters of 50 and 75 m were used. The cold air temperature was 303 K and the hot air temperature was 328 K. The air velocity was 18 m/s. The cuto frequency of the 50 m thermocouple under these conditions was found to be 5 Hz. Eckstein [10] applied the two thermocouple method to a pair of thermocouples of 50 and 75 m to make compensated dynamic temperature measurements inside an experimental combustor at temperatures up to 1700 K. Tashiro, Biwa and Yazaki [39] have described an original method of thermocouple calibration using the isentropic temperature oscillations caused by
Figure 2.6: Experimental apparatus used by Forney and Fralick to test thermocouple response [7] periodically pressurising and depressurising gas in a tube. The maximum forcing frequency in the experiment was 18 Hz. There was no mean ow of gas in the tube during the test.
Summary of the literature
Thermoacoustic oscillations are a potential problem in a range of continuous ow thermodynamic devices such as premixed gas turbine combustors, rocket engines and after-burners. To understand and eliminate the problem requires a better understanding of the physical mechanisms involved in the interaction between the unsteady heat release, the pressure oscillations and the boundary conditions of the combustor. The interaction of convected entropy disturbances with a choked downstream boundary condition is an important factor in the assessment of the
stability of real combustors. The linear theory of Marble and Candel [9] for the response of exit nozzles has become well established in the literature. However, experimental validation of the theory is limited. The two microphone method of Seybert and Ross [9] has been extended by Hield [4] to allow experimental determination of the response of a choked combustor exit nozzle to both pressure and entropy disturbances. In this way the previously limited experimental evidence may be obtained from simultaneous microphone and ne-wire thermocouple data measurements taken inside a combustor.
Chapter 3 Experimental method
3.1 Description of the experimental rig
The experimental premixed combustor, designed and built by Hield [4] is shown in Figure 3.1. The parts of the experimental combustor rig will be explained here.
Air and fuel preparation
A steady ow of compressed air was taken from a large air receiver. Fuel was taken from three liqueed petroleum gas cylinders manifolded together. It was assumed that the fuel was primarily composed of propane although an analysis was not done. Both fuel and air were injected upstream of a choke plate, which was used as a calibrated sonic nozzle to measure the air and fuel mass ow rate and to prevent equivalence ratio uctuations. The combustors estimated maximum air and fuel mass ow rates were 0.16 kg/s and 0.02 kg/s respectively [4].
Figure 3.1: The combustor rig designed by Hield [4]
Flame holder
Flame stabilisation A premixed ame may be stabilised in a number of ways, including positioning a blu body into the ow to anchor the ame in the recirculating turbulent wake and by introducing swirl into the ow. Flame ashback into the premixing section is one potential problem that has to be considered when designing lean premixed burners. Unlike a diusion ame combustor, in which the ame is burned at essentially the same point where the fuel is injected, in a lean premixed combustor a ammable mixture exists in the pre-chamber upstream of the ame. A section through the blu body ame holder designed by Hield [4] is shown in Figure 3.2. The ame holder diameter was 25 mm. The ammability limits of the combustor were determined empirically by Hield for open and choked combustor exits. The results for ammability limits for the case 43
of an open exit are plotted in Figure 2.1. This ame holder arrangement was found to allow combustion to occur over a range of equivalence ratios centered about the stoichiometric ratio. In most cases operation resulted in continuous and very loud thermoacoustic instability. The results show that operating below an equivalence ratio of 0.7 is impossible for this particular centre body stabilised ame. With the addition of an exit nozzle of area ratio 0.1, the range was eectively reduced to a single operating point. This point occurred at roughly stoichiometric equivalence ratio at a Mach number sucient to cause the downstream nozzle to become choked.
Figure 3.2: Section through the ame holder
Working section
The original working section designed by Hield [4] was intended to allow easy replacement with a glass viewing tube as required. However dierential heating and expansion of the stainless steel tube relative to the tie rods lead to signicant plastic deformation of the anges and steel tube. Sealing of the tube was also a problem. For these reasons the removable working section was replaced with a simpler anged pipe section for the experiments. Sockets were welded to the tube to allow tting of the thermocouples and pressure transducers. The ends of the thermocouples were positioned in the ow at approximately 10 mm from the inside wall of the pipe, which had an internal 44
diameter of 50 mm. Figure 3.3 shows the modied working section.
Figure 3.3: Modied combustor working section
R-type thermocouples (platinum/platinum 87 %, rhodium 13%) were used. The melting temperature of pure platimum is 2045 K. Tables of R-type thermocouple voltages for a large temperature range are available for download from the NIST online database [40]. The temperatures encountered downstream of the premixed ame in the experiment was well below the thermocouple melting point. The maximum temperature observed during the experiments was approximately 1700 K.
Assembly of the thermocouples
The simple spot welding method described by Hart and Elkin [41] was used to connect the ne thermocouple wires. A DC circuit with a graphite negative electrode was used. The thermocouple junction connected to the positive electrode was momentarily touched with the graphite electrode as shown in 45
Figure 3.4: Spot welding of thermocouple (A) using graphite electrode (B) Figure 3.4. Hart and Elkin mentioned that the polarity of the circuit was important for preventing carbon deposits in the weld. Multi-strand copper wires were pulled through ceramic tubing and tied to the 50 micron diameter thermocouple wires. This was found to be the most reliable method of joining the wires. The copper wires were then pulled back inside the ceramic tubes. The small thermocouple junctions were made by twisting together the pair of 0.002 inch (50 micron) diameter wires and then spot welding them together. A DC power supply was used and a 0.5 mm graphite pencil led (attached to the negative terminal of the power supply) was momentarily touched on the twisted wires (attached to the positive electrode). It was found that by using this method and keeping the voltage to a about 12V no large beads were formed. At higher voltages (about 24V) occasionally beads did form at the end of the wires, although this was not repeatable and may have had little eect on the time constant of the wires. Bigger thermocouple beads were made by using an oxyacetylene ame using the procedure described by Hield [4]. However this method resulted in a very slow response thermocouple unless the beads were attened.
Figure 3.5: Thermocouple circuit from Analog Devices [8]
Thermocouple signal conditioning
Thermocouple cold junction compensation may be done using hardware or software. Historically an ice bath was used to provide the reference temperature. For practical reasons these have been replaced by electronic ice points and software compensation. The obvious advantage is that there is no need to replace the ice. A useful note on the application of thermocouples including cold junction compensation is provided by Potter [42]. A thermocouple signal conditioning board was constructed for the six thermocouples. The system consisted of six AD621 instrument ampliers with preset gain of 100. The circuit is shown in Figure 3.5. Testing the thermocouple signal conditioning The signal from an amplied thermocouple was compared to the raw signal of an adjacent and nearly identical thermocouple to observe the eect of the signal conditioning. A short test signal was provided by lighting the ame in the combustor and then turning the fuel o. Figure 3.6 and Figure 3.7 show the raw and amplied signals, respectively. The eect of the instrumentation ampliers in improving the signal to noise ratio is clear.
Figure 3.6: Raw thermocouple signal
Figure 3.7: Amplied thermocouple signal
Thermocouple cold compensation
The temperature of the cold junctions inside the signal conditioning box was measured using a LM335 solid state temperature sensor. The resulting cold junction temperature measurement was converted to an equivalent thermocouple voltage using the NIST R-type thermocouple lookup table as was used for converting from thermocouple voltage to temperature. The thermocouple cold compensation is done as part of post processing by adding this cold junction voltage to the thermocouple signal before conversion to temperature. The MATLAB code for importing the lookup table and conversion between temperature and voltage is included in Appendix B. The thermocouple voltage is plotted against temperature in Figure A.1.
Thermocouple time constant estimation
Estimation of the actual uctuating gas temperature at higher frequencies is complicated by the nite speed of response of the thermocouple. The frequency response of a thermocouple is a dicult parameter to estimate based on empirical data because the heat transfer mechanism is primarily convective, and therefore the rate of heat transfer is dependent on the local properties of the ow which determine the boundary layer thickness around the thermocouple. It is preferable then that the frequency response be measured directly. Traditional methods used to measure frequency response usually require the application of a known impulse, step function, periodic or even random signal to the input of the system. The output from the system is measured and along with the input used to estimate the systems response. However in the case of the thermocouple located in a ow downstream of a ame, the input function is unknown as this is the actual uctuating gas temperature. Therefore a dierent approach is required. If a rst order system response is assumed for a pair of thermocouples of diering sizes used
to measure the temperature at a point, then the data obtained from the two thermocouples subject to approximately the same unknown, randomly varying gas temperature may be used to solve for the time constants of the thermocouples [2]. In practice the problem is complicated further since the sensitivity of thermocouple types suited to the temperature range expected in a combustor is such that electrical noise may be signicant at higher frequencies.
Estimating the time constant using basic theory
The thermcocouples used in the experiment may be assumed to behave as a rst order system. Thus, the change in temperature of the thermocouple T is assumed to follow the rst order dierential equation obtained using the rst law of thermodynamics dT + T = Tg , dt (3.1)
where Tg is the change in the gas temperature and is the time constant of the system determined by the thermal resistance R and capacitance C of the thermocouple/gas system = RC . (3.2)
The time constant was estimated using a lumped thermal capacitance model. The thermal capacitance is due to the density and specic heat cp of the thermocouple metal and the volume of the thermocouple. The geometry of a thermocouple junction formed without a substantial weld bead may be approximated as a cylinder of same diameter as the wire [43]. The thermal capacitance for a section of cylindrical wire of equal length and diameter is D3 cp . (3.3) 4 The thermal resistance between the uid and the wire is due to the convective C= and conductive resistances in series. The resistance to thermal conduction 50 given by
may be neglected for a very small thermocouple in a highly convective ow. Assuming the thermocouple wire is an innite cylinder, the thermal resistance of a section of wire of equal length and diameter is R= 1 , hD2 (3.4)
where h is the heat transfer coecient which will depend on the properties of the uid and the thickness of the thermal boundary layer as will be discussed in the next section. The time constant is given by = RC = Dcp . 4h (3.5)
The heat transfer coecient for the case of the choked exit was estimated using a theoretical heat transfer model. The denition of the Nusselt number N u may be rearranged to give the convective heat transfer coecient h h= kg Nu , D (3.6)
where kg is the thermal conductivity of the gas evaluated at the free stream stagnation temperature. A Nusselt number relation for an innite cylinder in a ow was determined by Collis and Williams [44]. Heitor et al. [43] found that this relation was reasonably accurate when used for predicting ne wire thermocouple time constants in a turbulent premixed ame. The relation is N u = 0.24 + 0.56Re0.45 , (3.7)
and is applicable to Reynolds number Re based on diameter in the range 0.02 to 44. The properties of air at 1300 K, a reasonable estimate of velocity of 80 m/s and a wire diameter of 50 m were used to obtain the Reynolds number, Re = 20. Substitution of these values into equations 3.7 and 3.6 resulted in N u = 2.47 and h = 4050 W/m2 . This heat transfer coecient was used with the thermophysical properties of pure platinum to obtain = 9 ms using equation 3.5 corresponding to a cut o frequency of 18 Hz. 51
The Nusselt number relation used here yields a similar result to the relation used by Hield [4] taken from Reference [45] which is N u = 2 + 0.6Re 2 P r 3 ,
where P r is the Prandtl number. For air at 1300 K, P r = 0.719. Using this relation yields a time constant of 5 ms corresponding to a cuto frequency of 33 Hz. It will be shown in the next section that the time constant estimated here is consistent with the range of values obtained experimentally using the Strahle method.
Estimating the thermocouple time constant using Strahles [2] method
The technique developed by Strahle and Muthukrishnan [2] was implemented to estimate the thermocouple time constant. The results were compared with the technique described by Tagawa, Kato and Ohta [3]. In both methods it is assumed that two thermocouples of dierent time constant are spaced closely enough that they are exposed to the same randomly varying temperature. The advantage of both the Strahle and Tagawa methods over traditional methods of thermocouple time constant measurement is that the time constants are estimated using data from the actual experiment. Most other methods of thermocouple time constant estimation require a separate test setup with contrived conditions to replicate the conditions of the experiment. The basic steps of the Strahle method will be explained here. The thermocouples are assumed to behave as a rst order system. The thermocouple signals from two closely spaced thermocouples T C1 and T C2, of diering time constant were used. The Fourier transform of the signals Y1 and Y2 are
related to the Fourier transform of the gas temperature Tg by Y1 = Y2 = Tg , 1 Y1 + 1 Tg . 2 Y2 + 1 (3.9) (3.10)
The Fourier transform of the thermocouple signals Y1 and Y2 was used to estimate the power spectrum S11 and S21 of thermocouples. The ratio R of S11 to S21 is given by R= Y Y1 S11 1 . S21 Y2 Y 1 (3.11)
Substitution of equations 3.9 and 3.10 into 3.11 yields R= (1 2 Y2 )(1 + 2 Y2 ) . (1 1 Y1 )(1 + 1 Y1 ) (3.12)
Taking the imaginary part of equation3.12 yields Ri = (1 2 ) . 2 1 + 2 1 (3.13)
Dierentiating equation 3.13 and setting the result to zero allows the location of the frequency of the extrema e (the cuto frequency of T C1 ) by
2 2 2 2 (1 2 )(1 + e 1 ) 2e 1 (1 2 ) =0. 2 (1 + 2 1 )2
The extrema is found at e = 1 . 1 (3.15)
The procedure used to determine the thermocouple time constant began with plotting Ri against frequency. Then the frequency at the extremum e of Ri was used to determine the time constant 1 of the rst thermocouple. It may also be shown that the real part of the ratio Rr corresponds to the ratio of 1 to 2 as . The real part was also plotted so that the time constant of the second thermocouple could be obtained using the result for the rst thermocouple and this ratio. 53
The coherence function 2 was also plotted for each test. The coherence function (also known as the coherence squared function) is dened as 2 = S12 S21 . S11 S22 (3.16)
The coherence function is always less than or equal to 1. It may be proved that [46] S12 S21 S11 S22 . (3.17)
A value for the coherence function close to one therefore indicates that the two signals are related by a single input single output (SISO) linear transfer function. A value of coherence less than one may indicate that random noise is present in the signals or that a SISO model is insucient to describe the overall system. The plots of coherence function versus frequency consistently showed a value of the coherence function near one at the frequency at which the time constant was determined.
Estimating the thermocouple time constant using Tagawas [3] method
Like the Strahle method the Tagawa [3] method also assumes a rst order response for the thermocouple output. The Fourier transform of each of the two thermocouple voltages are obtained. The values of 1 and 2 are then calculated so as to minimize the frequency averaged error between the Fourier transforms of the estimates for the true gas temperature T g1 and T g2 . The frequency range over which the averaging is done must be chosen carefully so as to exclude high frequency parts of the spectrum that are corrupted by noise. Therefore the choice of the maximum frequency should be related to the signal to noise ratio at various frequencies. It has also been shown that the frequency range used in the averaging need not exceed the cuto frequency. The noise oor was used as the reference to determine an 54
appropriate maximum frequency. The maximum frequency may be chosen arbitrarily at any point below the frequency at which the signal crosses the noise oor. It was found that result for the time constant was highly dependent on the maximum frequency chosen in cases with even small amounts of noise. In the noise free simulation the Nyquist frequency (10 kHz) was used as the maximum frequency and the correct result for the time constants was obtained. However using the Nyquist frequency to implement the Tagawa method on real data resulted in erroneous results. A dierent approach was adopted as used by Tagawa. This approach involved plotting the calculated time constant versus the maximum frequency used for the method. Tagawa obtained the correct results using a maximum frequency of about ten times the calculated time constant.
Comparing Strahles [2] and Tagawas [3] methods on simulated data
The Strahle method was rst tested using simulated data. An array of random numbers was generated to represent the raw temperature signal for both thermocouples. This signal was ltered through two separate rst order low pass lters with diering time constants representing the fast and slow thermocouple. The radian frequency e of the trough in the imaginary part of the ratio R versus frequency in Figures 3.8 corresponds to the time constant 1 of 1 millisecond of the faster thermocouple according to equation 3.15. For signals free of incoherent noise both methods showed good agreement for the estimated and actual time constant used in the simulation. The procedure was repeated using a dierent array of random numbers added to one of the signals to represent incoherent random noise. This has the eect of obscuring the extrema as shown in Figures 3.9 and 3.10. Various
Figure 3.8: Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 2 ms without noise ratios of thermocouple time constants were tested. In the case of thermocouple pairs with very similar time constants the extrema is almost completely obscured for a signal to noise ratio of 25 as can be seen in Figure 3.10. It was determined from the simulation results that a large time constant ratio would be preferred for the actual measurements. In practice the time constant ratio was dicult to control using thermocouples of the same wire diameter. Dierent techniques were used to make the large and small thermocouple beads. The techniques were described in Section 3.2.1. Without the addition of noise the Tagawa method gives the correct time constant for both thermocouples. However the method is very sensitive to incoherent noise. This is shown in Figure 3.11. The same pair of time constants 1 and 1.25 ms were used but the result is very dependent on the maximum frequency used for the computation. In this case a reasonably good estimate of the time constant would be possible by choosing the frequency
Figure 3.9: Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 2 ms with incoherent noise (S/N ratio=2)
Figure 3.10: Ri (+), Rr () and 2 (.) versus frequency for a simulation with time constants 1 and 1.25 ms with incoherent noise (S/N ratio=2) 57
Figure 3.11: Time constant for simulation TC1s (+) and TC2s () versus maximum frequency used (S/N ratio=10000) range for the computation to be within a decade of the cuto frequency. Compared to the Strahle method, the Tagawa method computes the result directly without the need to locate an extrema in a plot. This has an advantage of removing subjectivity from the analysis but is less illustrative of the uncertainty in the result. The Strahle method is far more useful in highlighting cases with incoherent noise in the signals which lead to erroneous results. The Tagawa method is also very sensitive to the frequency range used to apply the method and therefore it is considered inferior to the Strahle method.
Comparing Strahles [2] and Tagawas [3] methods on Bunsen burner test data
A bunsen burner ame was used to heat a pair of thermocouples outside of the combustor as a further test of the methods. The thermocouple pair were positioned above a Bunsen burner ame and a jet of air was directed through the top of the ame onto the thermocouples as shown in Figure 3.12. With the introduction of the air jet the Bunsen ame became shorter, noisier and visibly more turbulent. The bandwidth of the measured temperature uctuations was also increased. Figure 3.13 shows the thermocouple signals from the start of the data record. The rst six seconds of data were taken before the ame was lit. This data was used as the reference for the noise oor during the experiments. The Bunsen burner tests were repeated 24 times while varying the fuel and compressed air ow in an attempt observe the eect on the bandwidth of the temperature disturbances. The spectral density for the thermocouple signals are shown in Figures 3.14, 3.15 and 3.16 for three of the tests. The most turbulent test case is shown in Figure 3.16. In this test the thermocouples were spaced very closely together and the spectra are almost identical. This test is typical of the majority of test results which did not show a well dened extrema in Ri . The corresponding extrema in Ri for each of the tests are shown in Figures 3.18, 3.17 and 3.19. The two tests giving the clearest indication of an extrema are shown in Figures 3.14 and 3.15. In the most turbulent case, the plot of Ri shown in Figure 3.19 does not have an extremum.
Figure 3.12: Experimental setup
Figure 3.13: TC1 and TC2 signal at start of data record
Figure 3.14: Spectral density for TC1 () and TC2 (.) and noise oor (Bunsen test 6)
Figure 3.15: Spectral density for TC1 () and TC2 (.) and noise oor (Bunsen test 12) 61
Figure 3.16: Spectral density for TC1A () and TC2A (.) and noise oor (Bunsen test 23)
Figure 3.17: Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 6) 62
Figure 3.18: Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 12)
Figure 3.19: Ri (+), Rr () and 2 (.) versus frequency for TC1 (Bunsen test 23) 63
Figure 3.20: Spectral density of the thermocouple signal premixed combustor tests compared to the Bunsen burner tests
Comparing Strahles [2] and Tagawas [3] methods on the combustor rig data
In this section the data from the choked and open exit combustor tests is used to estimate the thermocouple time constant of one of the thermocouples (TC4). A dierent pair of thermocouples were used in the combustor tests compared to the Bunsen burner tests. The pair used in the combustor tests were of signicantly dierent diameter bead sizes. The large and small beads were made by the gas ame and welding methods respectively. The application of the Strahle method to the data for the open and choked exit cases gave the clearest indication of the thermocouple time constant. Figure 3.20 shows the thermocouple spectral density for the open choked exit tests compared to the Bunsen burner test and the noise oor measured with the thermocouples installed in the combustor. 64
Figure 3.21: Ri (+), Rr () and 2 (.) versus frequency for TC4 (open exit) Both the open and choked exit cases resulted in a distinct trough in Ri at around 40 Hz to 100 Hz as shown in Figures 3.21 and 3.22. The value that Rr approached at high frequency showed the ratio of the thermocouple time constants was approximately 1.5 to 2. Therefore a time constant of 40 Hz to 100 Hz was assumed for the faster thermocouple and a time constant of 20 Hz was assumed for the slower thermocouple.
Compensation of thermocouple measurements
The thermocouple response was compensated for by assuming a rst order system response with a time constant based on the results of the previous section. Ballantyne and Moss [47] discuss various dierent methods of thermocouple time constant compensation and the errors due to the uncertainty of the time constant in the ame region. Here the thermocouples were compensated as part of post processing with the objective of allowing the thermocouples 65
Figure 3.22: Ri (+), Rr () and 2 (.) versus frequency for TC4 (choked exit)
Figure 3.23: Calculated time constant TC1 (+) and TC2 () versus maximum frequency used (Bunsen test 6) 66
Figure 3.24: Calculated time constant TC1 (+) and TC2 () versus maximum frequency used (Bunsen test 12)
Figure 3.25: Calculated time constant TC4 (+) and TC5 () versus maximum frequency used (Open exit) 67
Figure 3.26: Calculated time constant TC4 (+) and TC5 () versus maximum frequency used (Open exit) signals to be related to the pressure transducer signals with the correct amplitude and phase. The relevance of the thermocouple response compensation is limited mainly to the analysis of the choked exit response to an entropy disturbance. For this reason a single thermocouple cuto frequency was assumed as 40 Hz for the average of the three small bead thermocouples measurements used. This value was reasonable, given the results presented in Figures 3.21 and 3.22. The thermocouple signal was compensated by dividing the Fourier transform of the raw signal by the analytical frequency response function as follows T () Tc () = , H() transfer function given by (3.18)
where Tc () is the corrected Fourier transform and H() is the rst order
H() =
1 . i + 1
Generally this will have the eect of greatly amplifying the noise oor at high frequency, but this should not have a signicant eect on the signal at frequencies below about 100 Hz.
Pressure transducers and signal conditioning
The pressure disturbances were measured using Kulite WCT-312M pressure transducers. The Kulite sensors pressure linear measurement range is 170 kPa gauge with a maximum over pressure of 340 kPa. The sensors natural frequency is 240 kHz, well above the range of the measurements. The sensor principle of operation is a four arm Wheatstone bridge. In order to minimize noise from power supplies the bridge power was supplied by a nine volt battery. The pair of pressure transducers were dynamically (AC) coupled to a pair of AD621 instrument ampliers as shown in Figure 3.27. The ampliers gain was preset to 100. Dynamic coupling allowed for better measurement resolution since the mean pressure was removed from the signal and this also had the additional eect of reducing low frequency noise. The high pass lter resistor and capacitor combination was estimated to give a (-3dB) bandwidth of 7 Hz. The actual frequency response of the lter was measured by the frequency response function technique. A signal generator was used to generate a sine wave output which was fed to the two high pass ltered pressure transducer ampliers and one of the thermocouple ampliers (without lter) simultaneously. The sine wave signal frequency was varied continuously from zero to close to the Nyquist frequency. Noise in the measurements was negli-
Figure 3.27: Pressure transducer signal conditioning circuit adapted from Analog Devices [8] gible relative to the signal power. The signals from the three ampliers were sampled at 20 kHz by the NI data acquisition card. The resulting amplitude and phase plots are shown in Figure 3.28.
Pressure transducer testing
The pressure transducer sensitivity was tested by counting loudly from one to ve into the exit end of the working section pipe as shown in Figure 3.30. The amplitude of the background electrical noise in the amplied signal during the test was about 0.5 mV as can be seen in the gure which in terms of sound equates to a noise level of 110dB SPL. As comparison the maximum thermoacoustic pressure waves measured during the experiments were of order 170 dB SPL, showing that very high sensitivity was achieved. The spoken words can also be identied in the resulting spectrogram (Figure 3.31). The data sheet for the pressure transducers states that the sensors range when dynamically coupled is limited by the few microvolts of noise generated by the silicon gauge and other internal components. A root mean square value of noise of 1 to 10 microvolts equates to a sound of 99 to 118 dB SPL. Therefore the total noise observed in these measurements is close to the noise expected from the pressure transducers alone. 70
Figure 3.28: High pass lter frequency response showing phase shift due to the data acquisition system
Figure 3.29: Pressure transducer high pass lter frequency response
Figure 3.30: Amplied pressure transducer output versus time during the speaking test
Figure 3.31: Spectrogram of pressure transducer output during the speaking test 72
The national instruments 6013 data acquisition card was used. Eight dierential analogue inputs were used for the six thermocouples and two pressure transducer signals. A sampling frequency of 20 kHz per channel was chosen. It was found that by using a sampling frequency of 20 kHz and a total data length of around 20 to 30 seconds provided an average spectra that was approximately independent of sample length and resulted in a frequency resolution of 4.88 Hz. The data were stored in a binary le in 16 bit integer format. The resolution of the measurements was determined by choosing the input voltage limits. For most of the experiments these limits were set at -0.5 V to +2 V giving a resolution of 38 V. In terms of temperature scale this equates to a resolution of 0.03 K and a maximum temperature of 1900 K. In terms of pressure this equates to a resolution of 0.6Pa and a maximum amplitude of 8.6 kPa or 170 dB SPL. The pressure transducer signal was AC coupled to the instrumentation amplier. This allowed greater resolution for tests in which the mean pressure was above atmospheric.
Data acquisition phase shift
Because the ADC sequentially samples each input channel there was an inevitable phase shift associated with this which was most obvious at higher frequencies. A frequency domain technique was found to be a simple method for correcting for the phase shift. The phase dierence was corrected by applying a frequency response calibration to the Fourier transforms of the signals. The frequency response correction is Xnc () = Xn ()ein , (3.20)
where n refers to the channel number (0 to 7), c refers to the corrected Fourier transform and is the time dierence between two consecutive channels 73
Figure 3.32: Data acquisition system frequency response for the same sinusoidal input signal with all channels referenced to channel 0 determined experimentally. In this way channel zero was arbitrarily set as the reference channel. For the case of sampling all eight channels at 20 kHz is equal to exactly 6.25 microseconds. This time delay corresponds to a phase shift of one eighth of 180 degrees at the Nyquist frequency. Figure 3.32 shows the frequency response for channels one through to eight before correction. Figure 3.33 shows the same data after the phase correction has been applied.
An accurate estimate of signal to noise ratio was important to the implementation of the spectral methods in order to determine the maximum frequency at which meaningful conclusions may be drawn. The signals from the thermocouple and pressure transducers and the physical processes causing them were limited to lower frequencies. At higher frequency, noise due to electro74
Figure 3.33: Corrected data acquisition system frequency response for the same sinusoidal input signal with all channels referenced to channel 0 magnetic interference was greater than the signals. Separate shielded audio cables were used for both the thermocouple and pressure transducer signals. Grounding of the cable shield at the instrument end did not have any noticeable eect on the noise in the measurements. The signal conditioning box was electrically isolated the from the combustor rigs steel support structure. This resulted in a large reduction in noise. The wires used to connect the thermocouples to the signal conditioning board were twisted together and kept to a minimum length of approximately 200 mm. The water pump, compressor and exhaust fan did not contribute to noise in the experiment. However the uorescent lights, computer monitor and nearby DC voltage supplies had a small eect on the noise levels but were essential for the experiment. The spectra for pressure transducers, M1 and M2 is compared to the
Figure 3.34: Spectra of M1 for the open and choked exit tests compared to the noise oor noise oor in Figures 3.34 and 3.35 for the open and choked exit data. The spectra for signals TC2 and TC4, are compared to the noise oor in Figures 3.36 and 3.37 for the open and choked exit data.
An experimental premixed combustor rig, designed and built by Hield [4], was used for the experiments. Fine wire thermocouples were constructed by a welding technique described by Hart and Elkin [41]. The time constant of the thermocouples was estimated using frequency domain techniques of Strahle [2] and Tagawa [3]. The experimentally determined time constants were found to agree reasonably with theoretical models of the thermocouple system. Pressure transducers were used to measure the pressure uctuations inside the combustor. Both the thermocouple and pressure transducer signals
Figure 3.35: Spectra of M2 for the open and choked exit tests compared to the noise oor
Figure 3.36: Spectra of T2 for the open and choked exit tests compared to the noise oor 77
Figure 3.37: Spectra of T4 for the open and choked exit tests compared to the noise oor were amplied using instrument ampliers to give a substantial improvement in signal to noise ratio. The thermocouple and pressure transducers used for the experiments have been shown to provide reliable measurements of the actual conditions inside the combustor rig.
Chapter 4 Results
This chapter is in three sections. The rst section deals with the existence and frequency of traveling entropy waves in the combustor. The second section investigates the degree of spatial uniformity of the temperature uctuations at the exit of the combustor. In the nal section, results are presented for the response of the exit to incident pressure and entropy disturbances obtained using the so-called two microphone and one thermocouple method developed by Hield [4].
Evidence of traveling entropy disturbances
The importance of traveling entropy waves, relative to acoustic uctuations, on the measured temperature uctuations was investigated rst. A pressure signal was used to estimate the isentropic component of the temperature uctuation, which from equation 2.18 is Tisentropic ( 1) p = . p T (4.1)
The remaining entropy component of the measured temperature uctuation is assumed to travel with the ow. The eect of turbulent mixing in an
Figure 4.1: Locations of thermocouples and pressure transducers unsteady ow may make these traveling entropy waves unrecognizable from one point to the next. Non-dimensional temperature and pressure spectra Non-dimensional spectra for the compensated thermocouple and pressure transducer signals are presented here for the open and choked exit tests. The locations and numbering of the thermocouples and pressure transducers are shown in Figure 4.1. The ends of the thermocouples were positioned in the ow at approximately 10 mm from the inside wall of the pipe. The auto-spectral densities of the pressure transducers signals for the open and choked exit are shown in Figures 4.2 and 4.3. In the open exit case, three sharp peaks can be seen in the spectra at frequencies of approximately 150, 350 and 500 Hz. In the choked exit case, the peaks are rounded and the largest peak occurred at approximately 20 Hz. Figures 4.4 and 4.5 show the auto spectral densities of three out of the six thermocouples (the other three followed T1 closely) for the open and choked exits respectively. Large peaks in temperature spectra can be seen 80
Figure 4.2: Pressure spectra (open exit) for M = 0.034, = 1.0. The location of M1 and M2 is shown in Figure 4.1
Figure 4.3: Pressure spectra (choked exit) for M = 0.1, = 1.0. The location of M1 and M2 is shown in Figure 4.1 81
Figure 4.4: Temperature spectra compared to the estimated isentropic temperature spectra estimated from pressure signal (open exit). The location of the thermocouples and pressure transducers is shown in Figure 4.1 corresponding to the peaks in the pressure spectra in the open exit case. The peaks seen in the choked exit case are less well dened. Isentropic temperature spectra The importance of isentropic temperature uctuations was investigated by comparing the amplitude and phase of the measured temperature to the isentropic temperature uctuations obtained by scaling the pressure transducer signal. This scaling was done using a mean density and temperature determined from the mean temperature and pressure measured in the rig. The relative phase and amplitude ratio of the downstream compensated temperature uctuation (T4) to the downstream pressure uctuation (M1) was determined using the ratio of the spectral densities of the signals. These phase and amplitude ratios are shown for the open and choked exit cases in
Figure 4.5: Temperature spectra compared to the estimated isentropic temperature spectra estimated from pressure signal (choked exit). The location of the thermocouples and pressure transducers is shown in Figure 4.1
Figure 4.6: Amplitude and phase relationship of T to
(open exit) of
the downstream measurement station shown in Figure 4.1 Figures 4.6 and 4.7 respectively. It is clear that the isentropic temperature uctuations were signicantly smaller than the measured temperature uctuations and there is no systematic phase relationship either. This is strong evidence that entropy disturbances are signicant in these experiments. The cross-correlation function The cross-correlation of signal x and signal y is 1 Rxy ( ) = N
xk (t1 )yk (t1 + ) ,
k=1
where each signal is of length N data points. The normalized cross-correlation is Rxy ( ) = Rxy ( ) . Rxx (0)Ryy (0) (4.3)
Thus, the normalized cross-correlation is unity at a time lag of zero if signals x and y are identical. 84
(choked exit) of
the downstream measurement station shown in Figure 4.1 The normalized cross-correlation of thermocouple (TC4) and pressure transducer (M1) signals in the open and choked exit cases are shown in Figures 4.8 and 4.9. For the open exit, there is no signicant cross-correlation between the pressure and temperature uctuations. This is consistent with the view that entropic disturbances are convecting out of the rig without forcing the duct acoustics, as expected of an open exit. The choked exit case showed some correlation between the downstream temperature and pressure signals. The pressure uctuation is not in phase with the temperature uctuation, perhaps indicating that the temperature uctuation is not isentropic. Note that the cross-correlation results presented here are for uncompensated thermocouple signals. However this will not impact the analysis as the dominant frequency observed here is well below the estimated cuto frequency of the thermocouples.
Figure 4.8: Cross-correlation of downstream pressure transducer (M1) with downstream thermocouple (TC4) (open exit)
Figure 4.9: Cross-correlation of downstream pressure transducer (M1) with downstream thermocouple (TC4) (choked exit) 86
Cross-correlation results for axially separated thermocouples The cross-correlation of temperature uctuations measured by two thermocouples separated axially in the combustor pipe, downstream of the ame, were also used to demonstrate the occurrence of traveling entropy waves in the combustor. The mean of each data sample was subtracted before computing the correlations. The cross-correlations of various signals were computed using 20 seconds of data from each test. The cross-correlation of the upstream (TC2) and downstream (TC3) thermocouple was computed. The time lag associated with the transport delay of a downstream traveling random temperature disturbance may be seen as a peak in the cross-correlation function (Figure 4.10). The time lag corresponding to the peaks may be compared with the expected convective time lag of 3 milliseconds, which was estimated based on a convective velocity. This convective velocity was based on the mean density and mass ow rate, calculated using the combustor pressure and mean temperature of the ow which was based on the upstream thermocouple (TC2) and one of the downstream thermocouples (TC5). The cross-correlation method gave reasonable agreement with the expected time delay of 3 ms in the open exit case. In the case of the choked exit the position of the peak in the crosscorrelation indicated that the downstream thermocouple signal was actually leading the upstream signal. This is unexpected and may indicate that the ame is burning back against the direction of ow, although this is speculative. The thermocouple signals from the choked exit test were dominated by the low frequency temperature drift due to the gradual heating of the combustor during the data collection period. Higher frequency temperature uctuations indicative of downstream traveling temperature disturbances were relatively insignicant in comparison to this overall temperature drift and were smaller than those observed in the open exit test.
Figure 4.10: Cross-correlation of the upstream (TC2) and downstream (TC3) thermocouple signals for the open and choked exit tests Frequency of the traveling entropy waves The occurrence of traveling entropic disturbances was demonstrated in the previous section for the case of the open exit using the cross-correlation of two thermocouple signals. However the cross-correlation results did not indicate the frequency of the convected disturbances explicitly. The same information may be presented in terms of the cross-spectral density functions to show the frequency range at which the upstream and downstream thermocouple signals are correlated. Figures 4.11 and 4.12 show the amplitude and phase of the uctuation measured by the upstream thermocouple (TC2) relative to the downstream thermocouple (TC4) for the open and choked exit cases respectively. It is clear that the correlation is limited to low frequency uctuations of up to approximately 40 Hz. The wavelength corresponding to a convected disturbance at 40 Hz is approximately 2 m or 10 times the distance between the two thermocouples and twice the rig length. This result 88
Figure 4.11: Amplitude and phase of TC2 relative to TC4 (open exit) supports the ndings of Muthukrishnan and Strahle [28] that the frequency corresponding to the inverse of the particle residence time in the combustor is important in characterizing the direct combustion related phenomena and entropy disturbances.
Uniformity of the temperature disturbances at the exit
The uniformity of the temperature disturbances at the exit was investigated by analysis of the signals from pairs of thermocouples at the exit. Thermocouple TC4 (located at the side of the pipe) was set as the reference thermocouple. Comparison was made between the signal from the reference thermocouple with three dierent thermocouples at varying circumferential and radial locations, but the same axial location. The closest thermocouple (TC5) was located within four millimeters of the reference. The next nearest 89
Figure 4.12: Amplitude and phase of TC2 relative to TC4 (choked exit) (TC1) was located at the opposite side of the pipe and the third (TC3) was located at the bottom of the pipe. Cross-correlation of downstream thermocouple pairs The cross-correlation results are presented here in Figures 4.13 to 4.18 for the thermocouple combinations described above for the open and choked exit tests. Note that in some of the gures the value of the cross-correlation function at the largest time delays shown is not zero because the thermocouple signals are highly correlated for low frequency temperature uctuations. This correlation of low frequency temperature drift was due to the gradual heating of the combustor during the interval in which the data was taken. Nonetheless, cross-correlation is strong in many cases, at small time delays.
Figure 4.13: Cross-correlation of thermocouples TC4 and TC5 (open exit)
Figure 4.14: Cross-correlation of thermocouples TC4 and TC5 (choked exit)
Cross-spectral analysis of downstream thermocouple pairs The same information contained in the cross-correlations of the previous section is presented here using the cross-spectral density functions in order to show the frequency range in which the temperature uctuations are correlated. Figures 4.19 and 4.20 show the amplitude and phase of TC5 relative to TC4 for the open and choked exit cases. The amplitude ratio is close to unity and the signals are in phase over the frequency range shown. Such results are expected since, as Figure 4.1 shows, these thermocouples are very close to each other. Figures 4.21 and 4.22 show the amplitude and phase of TC1 (the thermocouple at the opposite side of the pipe) relative to TC4 for the open and choked exit cases. Above a frequency of roughly 50 Hz the two thermocouples are uncorrelated. Figures 4.23 and 4.24 show a similar result for TC3, the thermocouple located at the bottom of the pipe. The temperature uctuations are therefore uniform across the duct exit up until a frequency of approximately 50 Hz which corresponds to a convective wavelength of approximately 1.5 m. The eect of isentropic temperature uctuations in this frequency range have been shown to be of negligible amplitude compared to the measured temperature uctuations and therefore may be considered largely entropic in nature. However the eects of turbulent mixing appear signicant and only entropy disturbances of longer wavelength than the length of the pipe have been shown to result in correlated thermocouple signals at the pipe exit.
Figure 4.19: Amplitude and phase of TC5 relative to TC4 (open exit)
Figure 4.20: Amplitude and phase of TC5 relative to TC4 (choked exit)
Results for the two microphone one thermocouple method
The response of the open and choked exit was measured using the 2M1T method explained in Chapter 2. The working frequency range of the two microphone method is considered by the ASTM [48] to be limited to frequencies above which the microphone spacing is equal to one percent of the acoustic wavelength. The same working frequency lower limit will also be assumed for the 2M1T method, which in this case was 40 Hz for the microphone spacing of 208 mm and speed of sound of 805 m/s. Below 40 Hz the method is unable to reliably separate the upstream and downstream traveling pressure waves. In this case the upper frequency range of the method was limited by the signal to noise ratio of the measurements. Results for the pressure reection coecient were plotted for frequencies up to 2000 Hz. The results for the pressure response to entropy disturbances were only plotted for frequencies up to 200 Hz. The relatively poor signal to noise ratio of the temperature measurements compared to the pressure measurements at higher frequencies was a result of the thermocouples relatively slow frequency response.
Response of the open exit to pressure and entropy disturbances
The mean value of the working section temperature during the open exit test was taken to be 1786 K, calculated as the average of the upstream (TC2) and downstream (TC5) thermocouple temperatures. The equivalence ratio was 1.1 and the upstream Mach number was 0.03. The Mach number relevant to the analysis is the downstream working section Mach number of 0.08. Thus the ow was assumed to behave as a low Mach number ow and the results were compared to the theory for reection of sound from an open end with
Figure 4.25: Experimental (.) open exit response to a pressure disturbance compared to theory (-) zero mean ow. Figure 4.25 shows the experimentally determined reection coecient for the open exit compared to the low frequency approximation for the reection coecient of an open end with zero mean ow given in equation 2.17. Agreement between the measurements and theory is good. The experimentally determined open exit response to an incident entropy disturbance is zero as shown in Figure 4.26. This is expected since according to the classical linear theory, the entropy and acoustic elds are uncoupled [4].
Response of a choked exit to a pressure disturbance
The mean value of the working section temperature during the test was 1327 K, calculated as the average of the upstream (TC2) and downstream (TC5) thermocouple temperatures. As the mean data were not recorded for 99
Figure 4.26: Experimental (.) open exit response to an entropy disturbance (average TC1, TC3, TC4) the choked exit test, the upstream and working section absolute pressures were assumed to be 700 kPa and 200 kPa respectively, in keeping with the results at the same operating condition obtained by Hield [4]. Using equation 2.31, Marble and Candels [9] theoretical relationship for a compact choked nozzle gives a pressure reection coecient (Rp ) equal to 0.96 for a Mach number of 0.1. The experimental and theoretical response of the choked exit to a pressure disturbance is shown in Figure 4.27. The linear theory and the experimental response show good agreement over the working frequency range of the method.
Response of the choked exit to an entropy disturbance
As discussed earlier, the convection of entropy disturbances generated by the ame should lead to a strongly non-uniform temperature eld at the nozzle 100
Figure 4.27: Experimental (.) choked exit response to a pressure disturbance compared to theory (-) entrance. As a result, it is unlikely that a single thermocouple measurement will accurately represent the choked exit response. It was shown in section 4.2, that the maximum frequency at which the temperature signals were spatially correlated at the exit was less than the lower limit of the working frequency of the 2M1T method. The response was therefore computed using each individual thermocouple as well as the average of various combinations of the thermocouples. Figures 4.28 and 4.29 show typical results for individual thermocouples positioned at the bottom (TC3) and side (TC4) of the pipe respectively. Figures 4.30 and 4.31 show results for the average of three and ve thermocouples respectively. The theoretical entropy reection coecient is -0.07 based on a Mach number of 0.1. In the working frequency range of the method, there is varying agreement with this theoretical result, as well as some scatter in the spectral data. The causes of this scatter are discussed in some detail by
Figure 4.28: Experimental choked exit response to an entropy disturbance (TC3 only) compared to theory (-)
Figure 4.29: Experimental choked exit response to an entropy disturbance (TC4 only) compared to theory (-) 102
Figure 4.30: Experimental choked exit response to an entropy disturbance (TC1, TC3, TC4) compared to theory (-)
Figure 4.31: Experimental choked exit response to an entropy disturbance (TC0, TC1, TC3, TC4, TC5) compared to theory (-) 103
Hield and Brear [49], and are due to both the sampling duration and magnitude of the measured entropy disturbances, and could not be improved upon in the present set of experiments. Nonetheless, Figure 4.28 shows that the acoustic response of the choked exit to entropic forcing can exhibit both an amplitude and phase that is reasonably similar to Marble and Candels [9] analytic boundary condition. Figures 4.30 and 4.31 exhibit less agreement in amplitude with theory, and Figure 4.29 exhibits poor agreement. Given the results from earlier in this chapter, this observed dependence of the results in Figures 4.28 to 4.31 on the spatial location of the thermocouple measurement is not a surprising result. Specically, the cross-spectral analysis discussed in Section 4.2 and shown in Figures 4.21 to 4.24 showed that the temperature eld immediately upstream of the combustor exit was only uniform for frequencies corresponding to convective wavelengths longer than the combustor itself. Further, these frequencies were below the working range of the 2M1T method. Thus, at higher frequencies (i.e. convective wavelengths shorter than the combustor), the entropy or temperature eld at the combustor exit is non-uniform across the duct diameter, thus resulting in the varying transfer functions in Figures 4.28 to 4.31. This argument can be extended further. As entropy disturbances become less spatially correlated at higher frequency, an instantaneous entropy disturbance obtained by averaging across the duct cross section should have zero amplitude. Thus, in the higher frequency limit, the choked nozzle sees no equivalent 1D entropy disturbance approaching, and so does not produce a reected acoustic wave. As a result, as gures 4.28 to 4.31 all show, the amplitude of the entropy reection coecient tends to zero at higher frequencies.
Coherence analysis of the data
2 The ordinary coherence function 12 for a two input single output system
was taken from Bendat and Piersol [46]. For the incident A and reected B
2 pressure disturbance, AB is 2 AB =
|Rp SAA + Rs SAS |2 . SAA SBB
For the entropy disturbance and the resulting upstream traveling pressure |Rp SSA + Rs SSS |2 = . (4.5) SSS SBB where Rp , Rs , SAA , SBB , SAS and SSA have been dened previously in Sec2 SB 2 disturbance, SB is
tions 2.6.1 and 2.6.2. A value of unity for the ordinary coherence function means that a linear relationship exists between that input alone and the output. Therefore, for a presumed multiple input single output (MISO) system, an ordinary coherence function value of unity between one of the inputs and the output indicates that all other inputs are redundant [46]. This appears to be the case for response of the open exit to the pressure
2 disturbance input (AB = 1) shown in Figure 4.32. This is expected since in
theory the open exit is a SISO system between the downstream and upstream traveling pressure waves. An ordinary coherence function value of zero indicates there is no linear relationship between the input and the output, as is
2 the case for the response of the open exit to an entropy disturbance (SB = 0)
as shown in Figure 4.32. This also appears to be the case for the choked exit (Figure 4.33) in the working frequency range of the 2M1T method (above 40 Hz). In the choked exit case the small value of the ordinary coherence
2 function SB , suggests that the contribution of downstream traveling entropy
disturbances to the upstream pressure disturbances was insignicant in the presence of a much larger contribution due to downstream traveling pressure disturbances. This is expected to result from the relative amplitude of the theoretical pressure to entropy response coecients. 105
Figure 4.32: Ordinary coherence functions for the open exit
Figure 4.33: Ordinary coherence functions for the choked exit
In the choked case a 1 to 2 kPa amplitude pressure oscillation was observed (Figure 4.34). This amplitude is smaller than the 20 kPa amplitude oscillation observed by Hield [11]. It would appear that the operating conditions are dierent and in this case the combustor may also generate smaller entropy disturbances than the earlier experiment. In the absence of large entropy disturbances the combustor is more likely to behave as a SISO system.
Figure 4.34: Raw downstream pressure signal versus time for the choked exit
This chapter presented and analysed experimental results from an investigation of entropy transport and the acoustics in a premixed, turbulent combustor. Cross-correlation results for various thermocouple combinations were rst presented. In the open exit case, these results showed the existence of traveling entropy waves moving downstream at the convective velocity of the ow. Analysis of the relative amplitude and phase of the cross-spectral 107
density of the upstream and downstream thermocouple signal data showed that the correlation of the entropy disturbances was limited to convective wavelengths signicantly longer than the combustor wavelength. The thermocouple data was also used to investigate the spatial uniformity of the measured temperature disturbances at the combustor exit. Again the agreement between thermocouple pairs was limited to low frequency disturbances. Results for the open exit response to pressure disturbances showed good agreement with theory for an open end with low Mach number ow. There was no observed open exit response to entropy disturbances as expected. A coherence analysis of the data also indicated that the open exit acts as a SISO system between the downstream and upstream traveling pressure waves. In the case of the choked exit, the experimental results for the theoretical pressure reection coecient amplitude and phase were consistent with the choked nozzle theory over the working frequency range of the method. The coherence analysis of the choked exit results conrmed that the contribution of entropy disturbances to the upstream traveling pressure wave was insignicant in the presence of a large amplitude downstream traveling pressure disturbances.
1. The response of an open and choked combustor exit to pressure and entropy disturbances was measured. The experiments of Hield and Brear [11] were repeated using a dierent approach giving better results and providing insight into the limitations of the method. Improvements in the signal to noise ratio of the instrumentation was achieved by the use of signal conditioning for the thermocouple and pressure transducer signals. Multiple thermocouples were used to measure the entropy disturbances at the combustor exit. The results for the open exit case agreed reasonably with the linear theory at both small and large amplitudes of forcing. The measurement of smaller amplitude forcing was not possible with the choked exit. The choked exit entropy response was weaker in some cases than that predicted by Marble and Candels [9] linear theory in the working frequency range of the method. In both open and choked exit cases the coherence results did not indicate that there was a signicant linear relationship between the entropy disturbance input and the upstream traveling pressure wave output. This was expected for the open exit system, which may be assumed to behave as a single input single output system without further consideration of the
entropy disturbance. In the case of the choked exit system this was expected since the entropy disturbances measured at the exit were shown to be poorly correlated in space at the higher frequencies in which the 2M1T method was valid (above 40 Hz). 2. The temperature prole at the combustor exit was shown to be uniform at low frequency Using a number of thermocouples arranged at dierent radial locations around the duct gave an indication of the axisymmetry of the temperature prole. It was found that the spatial agreement between pairs of thermocouples was limited to low frequency temperature disturbances, where the wavelengths of the correlated disturbances were greater than the length of the combustor itself. The practical signicance of this result is that at higher frequencies, entropy disturbances are not likely to cause a problem in a choked combustor because they are not spatially correlated. If this is the case, the combustor may be assumed to behave as a SISO system considering only the reection of pressure disturbances. 3. The thermocouple cut-o frequency was estimated using spectral techniques under various convective heat transfer conditions Established spectral techniques for in-situ determination of the time constant of a rst order system were tested using simulated data and the actual insitu thermocouple data. The limits of the eectiveness of the methods in the presence of noise and reduced input bandwidth were shown. The measured thermocouple time constants agreed with simple heat transfer models. The measured time constant results were used in the nal analysis to compensate for the eects of the systems thermal inertia, as this was shown to have a signicant eect on the results.
Recommendations for further Work
Achieving a stable choked operating condition with entropy
disturbances As the premixed experimental combustor exhibits loud thermoacoustic instability over its entire range of ammability limits, it was not possible to achieve an operating condition without large pressure disturbances using the original ame holder. Some alternative ame holder arrangement may be required to achieve a quiet choked operating condition with which entropy disturbances may be studied. An alternative arrangement may be to use an external unsteady source of heat. 2. Improved dynamic temperature measurement techniques Even with thermocouple compensation, the analysis was limited by the bandwidth of the thermocouples as determined by the signal to noise ratio. Since time constant compensation is necessary for all but very low frequency temperature disturbances it is possible that less expensive and more sensitive type-K thermocouples of larger diameter may be substituted for the type-R thermocouples. Larger and more practical type-K thermocouples may actually give a better signal to noise ratio since their thermocouple voltage is six times that of the type-R thermocouple. Another interesting possibility would be the substitution of the ne wire thermocouples with an acoustic thermometry method
in which the temperature is obtained by measuring
the speed of sound across the duct or alternatively an optical technique. The advantage of these methods would be the ability to obtain an instantaneous spatial average of the temperature across the duct.
See for example reference [50]
3. Investigation of the eect of the nozzle geometry on the response The acoustic and entropic response of exit nozzles of dierent geometries could be compared using the existing setup. It is speculated that using a more gradual converging diverging nozzle may result in more linear behavior. 4. Separation of entropy and isentropic temperature disturbances Entropy waves may be separated from acoustic temperature uctuations by comparing a thermocouple exposed to the ow to a thermocouple inside the pipe but not exposed to the ow. At frequencies suciently low to be below the cut o frequency of both thermocouples the arrangement could be used to obtain the relative size of traveling entropy waves by subtraction of the isentropic signal from the signal of the thermocouple in the ow. At higher frequencies the thermocouple exposed to the ow would be expected to respond faster than the thermocouple out of the ow and therefore the time constant of each thermocouple must be compensated for separately. 5. The application of frequency compensation techniques to other measurement systems The Strahle technique was shown to be an eective method to determine the frequency response of an unknown system to an unknown input function. Thermocouples are only one example of a wide set of physical systems which approximate rst order behavior. In fact any system in which the rate of change of a quantity depends on the quantity itself can be modeled as a rst order system. Therefore the techniques used to estimate the frequency response of a thermocouple without knowledge of the input function are equally valid for any measurement device which approximates a rst order system. Another example of such a device is the hot wire used for instantaneous ow velocity measurement. As a typical hot wire diameter is 2.5 m, the possi-
bility of frequency compensation would allow the use of a thicker wire and hence more robust instrument. 6. The measurement of ow rate using the cross-correlation of two thermocouple signals The cross-correlation of two thermocouples separated axially in a turbulent ow downstream of a ame was shown to give a reliable indication of the convective time lag. The technique may prove useful as a relatively unobtrusive technique for ow measurement compared to methods such as the use of orice plates or sonic nozzles. The limitations of the method for dierent ow conditions and thermocouple response requires further investigation.
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Appendix A Additional Data and Results
A.1 Commissioning data
Date: 25/01/08 Time: 3.20pm Ambient Temperature: 24 deg C (outside test cell) 22 deg C (inside test cell) Atmospheric pressure 101.49 kPa (Melbourne) www.bom.gov.au
Ch 0 1 2 3 4 5 6 7
Table A.1: Scale factors for mean measurements Variable Unit Instrument Slope Oset P (upstream) P (work sect) T (upstream) T (work sect) P (fuel) T (fuel) Q (fuel) T (cold junct) kPa kPa K K kPa K m /s K
PTX1400 PTX1400 TC (type-T) TC (type-T) PTX1400 TC (type-T) D10A3255 LM335
253312.5 253312.5 50 50 253312.5 35 0.00011765 100
-253312.5 -253312.5 223.15 223.15 -253312.5 138.15 -7.1568E-5 -2
Table A.2: Zero check for mean measurements Variable Unit Measured (V) Calculated Actual P (upstream) P (work sect) T (upstream) T (work sect) P (fuel) T (fuel) Q (fuel) T (cold junct) kPa kPa K K kPa K m3 /s K 1.394 1.392 1.353 1.393 4.83 0.965 2.967 99.8 99.3 290.8 99.6 307.2 0.00004 294.7 101.5 101.5 296 296 101.5 297 0 295
Table A.3: Span check for mean measurements Variable Unit Measured (V) Calculated Actual P (upstream) P (work sect) T (upstream) T (work sect) P (fuel) T (fuel) Q (fuel) T (cold junct) kPa kPa K K kPa K m3 /s K 2.963 2.940 1.874 2.969 497.3 491.4 316.9 498.8 495.5 489.5 328 501.5 -
Figure A.1: R-type Thermocouple voltage versus temperature
Measured mean data
Date: 28/01/08 Mean open exit temperature using TC2 and TC5 T2 = 1162 K Overall open exit mean temperature 1600 K Calculated = 1.0 , M = 0.034, u2 = 65 m/s , c2 = 805 m/s Choked exit mean ow data not available. Pu s assumed to be 700 kPa and Pw s assumed to be 200 kPa. Mean choked exit temperature using TC2 and TC5 T2 = 1242 K Overall mean choked exit temperature = 1328 K The data used for the premixed combustor open and choked exit tests were taken while the rig was operating at near stoichiometric equivalence ratio. These resulted in higher temperatures and a far greater acoustic noise production than the partially premixed tests. Compared to the partially premixed combustor results there was also a greater uncertainty associated with the position of the ame in the combustor. Gradual deviations of the 123
Table A.4: Open exit test mean values Unit Measured kPa kPa K K kPa K m3 /s K 2.2844e+005 1.0162e+005 291.1137 7.9692e+005 307.0592 1.2460e-004 297.0733
P (upstream) P (work sect) T (upstream) T (work sect) P (fuel) T (fuel) Q (fuel) T (cold junct)
Table A.5: Open exit mean temperatures (20 s of data) TC 0 1 2 1880 3 4 5 1340 1758 1690 1497 1438
Table A.6: Choked exit mean temperatures (20 s of data) TC 0 1 2 1142 3 4 5 1339 1463 1500 1148 1376
Figure A.2: Amplied upstream (TC2) and downstream (TC3) thermocouple signals (open exit) mean temperature were observed throughout the test especially in the choked exit case. Figures A.2 and A.3 show the raw temperature signals from the upstream and downstream thermocouples for the open and choked exit respectively.
Figure A.3: Amplied upstream (TC2) and downstream (TC3) thermocouple signals (choked exit)
Appendix B MATLAB and SIMULINK les
B.1 Import pressure and thermocouple data
Filename: ImportBinaryAllCh.m % Import data for only one channel % mode = r (read only) % machine format = b (IEEE floating point with big endian byte order) fid = fopen([Data_31],r,b); % fid = fopen([DuringWarmUp],r,b); nCoeff = fread(fid,1,double); % header includes 1 double for number of coefficients
nChannels = fread(fid,1,double); scale = fread(fid,[nCoeff nChannels],double); scale=scale; Data = fread(fid,[nChannels inf],int16); %Data= fread(fid,[nChannels 800000],int16); fclose(fid); Fs=scale(1,1) 127
offset=scale(:,2); scaleMult=scale(:,3); for nCount=1:length(Data); Data(:,nCount)=scale(:,3).*Data(:,nCount) + scale(:,2); end time=0:(1/Fs):190;
T0=Data(1,:); T1=Data(2,:); T2=Data(3,:); T3=Data(4,:); T4=Data(5,:); T5=Data(6,:); M1=Data(8,:); M2=Data(7,:);
Import Mean data
Filename: ImportMeanValuesBinary.m
\begin{verbatim} % Import the mean values of mean data % mode = r (read only) % machine format = b (IEEE floating point, big endian byte order) fid = fopen([Mean_30],r,b); nCoeff = fread(fid,1,double);
% header incl 1 double for number of coeffs
nChannels = fread(fid,1,double); scale = fread(fid,[nCoeff nChannels],double); scale=scale; Fsm=scale(1,1); Data = fread(fid,[nChannels inf],int16); %raw data int16 fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % First Apply Scale to change from (16 bit)counts to Volts % % offset=scale(:,2); scaleMult=scale(:,3); for nCount=1:length(Data); Data(:,nCount)=scale(:,3).*Data(:,nCount) + scale(:,2); end
Ch(1)=mean(Data(1,:)); Ch(2)=mean(Data(2,:)); Ch(3)=mean(Data(3,:)); Ch(4)=mean(Data(4,:)); Ch(5)=mean(Data(5,:)); Ch(6)=mean(Data(6,:)); Ch(7)=mean(Data(7,:)); Ch(8)=mean(Data(8,:))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Next apply scale factors to change from volts to SI units
%slope slopes=[253312.5 253312.5 50 50 253312.5 35 0.00011765 100];
%offet offsets=[-253312.5 -253312.5 223.15 223.15 -253312.5 138.15 -7.1568E-5 -2];
variable=Ch.*slopes + offsets;
%units are all Pa, K, m3/s
P_us=variable(1) P_ws=variable(2) T_us=variable(3) T_ws=variable(4) %not connected P_f=variable(5) T_f=variable(6) Q_f=variable(7) T_cj=variable(8)
Import R-type thermocouple lookup table
Filename: ImportRTypeTCdata Interpolates the temperature from the R type thermocouple data provided by NIST .Thermoelectric voltages for temperature between 0 and 1759 deg C in 1 deg C increments. http://srdata.nist.gov/its90/download/download.html fid = fopen(type_r.tab); C= textscan(fid,%f %f %f %f %f %f %f %f %f %f %f %f,headerlines, 17); fclose(fid); 130
% C is a cell array to store the lookup table A =zeros(176,10); A(:,1)=C{2}; A(:,2)=C{3}; A(:,3)=C{4}; A(:,4)=C{5}; A(:,5)=C{6}; A(:,6)=C{7}; A(:,7)=C{8}; A(:,8)=C{9}; A(:,9)=C{10}; A(:,10)=C{11};
TC_mv=zeros(1,1760); entry=1;
for row=1:176; TC_mv(entry:entry+9)= A(row,:); entry=entry+10; end
TdegC=0:1759;
%Examples %Calc T by T=interp1(TC_mv,TdegC,0.1) %0.1 mv %Calc Thermoelectric voltage(mV)by mV=interp1(TdegC,TC_mv,15)%15degC
Calculate variables
Filename: CalcVariables.m Calculates %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % CALCULATION OF OTHER VARIABLES
% Constants gamma=1.4; R=287;
%Temp of Cold Junction in deg C T_cjdC=T_cj-273
%The thermoelectric voltage at the cold junction mV_cj=interp1(TdegC,TC_mv,T_cjdC)
%The mean of the means of the downstream and upstream thermcouples Tm=[mean(T2) mean(T5)]
%T_2mean=mean(Tm) T_2mean=mean(Tm)
%divide by the gain of the sig cond amplifier G=100 and mult % by 1000 to get mV then add the voltage (mV) at the cold junction
mV_tc=T_2mean*10 + mV_cj
T_2dC=interp1(TC_mv,TdegC,mV_tc) 132
T_2=T_2dC+273
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % mass flow rate of air (using 15 hole choked inlet plate) mdot_air=1.9234E-6*P_us/sqrt(T_us) + 0.0010167
%air density from P=rho*R*T rho1=P_ws/(287*T_us) rho2=P_ws/(287*T_2)
%PipeArea PipeArea=pi*(0.049^2)/4;
%DS VELOCITY v1=mdot_air/(PipeArea*rho1) v2=mdot_air/(PipeArea*rho2)
% speeds of sound c1=sqrt(gamma*R*T_us) c2=sqrt(gamma*R*T_2)
% Mach numbers Mach_us=v1/c1 Mach_ds=v2/c2 Mach_Pete=0.014027*P_us/(P_ws)*sqrt(T_ws/T_us)+0.0012277
%mass flow rate of fuel mdot_f=(P_f*Q_f)/(188.6*T_f)
% Equivalence ratio for premixed flame Phi=mdot_f/(0.064*(mdot_air-mdot_f))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % linearised gradient for small thermocouple signals (V/K) at DAQ T_scale=0.1*(interp1(TdegC,TC_mv,ceil(T_2dC)) -interp1(TdegC,TC_mv,floor(T_2dC)))
%Convert to units T0k=(((T0+mV_cj*0.1)/T_scale)+273);%/T_2; T1k=(((T1+mV_cj*0.1)/T_scale)+273);%/T_2; T2k=(((T2+mV_cj*0.1)/T_scale)+273);%/T_2; T3k=(((T3+mV_cj*0.1)/T_scale)+273);%/T_2; T4k=(((T4+mV_cj*0.1)/T_scale)+273);%/T_2; T5k=(((T5+mV_cj*0.1)/T_scale)+273);%/T_2; T6k=(((T6+mV_cj*0.1)/T_scale)+273);%/T_2; M1p=(M1*(100000/5.8));%/P_ws; M2p=(M2*(100000/5.8));%/P_ws;
Compute Spectra
Filename: ComputeSpectraPhaseCorrect.m %Computes the spectra for the mic and TC signals nfft=32768
startsample=1 stopsample=65*Fs
M1=M1p(startsample:stopsample); 134
M2=M2p(startsample:stopsample); T=T4k(startsample:stopsample);
Nsegments=floor(length(M1)/nfft); overlap=0.5; Nsegments=Nsegments*2 - 1 %for overlap 0.5 %Nsegments=8
% hamming window window=zeros(1,nfft); index=1:nfft; %window=ones(1,nfft); window(index)=0.54-0.46*cos(2*pi*index/nfft); U=(1/nfft)*sum(window.*window)
f = Fs*(0:(nfft-1))/nfft; % frequency is in Hz omega = (Fs*2*pi*(0:(nfft-1))/nfft); % omega is in rad/s
% DAQ correction factors H0=zeros(nfft,1); H1=zeros(nfft,1); H2=zeros(nfft,1); H3=zeros(nfft,1); H4=zeros(nfft,1); H5=zeros(nfft,1); H6=zeros(nfft,1); % M2 H7=zeros(nfft,1); % M1
tau=-0.00000625;
for index=1:nfft; H0(index,:)=exp(i*omega(index,:)*0*tau); H1(index,:)=exp(i*omega(index,:)*1*tau); H2(index,:)=exp(i*omega(index,:)*2*tau); H3(index,:)=exp(i*omega(index,:)*3*tau); H4(index,:)=exp(i*omega(index,:)*4*tau); H5(index,:)=exp(i*omega(index,:)*5*tau); H6(index,:)=exp(i*omega(index,:)*6*tau); H7(index,:)=exp(i*omega(index,:)*7*tau); end
A =zeros(nfft,Nsegments); B =zeros(nfft,Nsegments); D =zeros(nfft,Nsegments); entry=1 for n=1:Nsegments; A(:,n)=M1(entry:entry+nfft-1).*window; B(:,n)=M2(entry:entry+nfft-1).*window; D(:,n)=T(entry:entry+nfft-1).*window; entry=entry+nfft*overlap; end
M1hat =zeros(nfft,Nsegments); M2hat =zeros(nfft,Nsegments); T1hat =zeros(nfft,Nsegments); S11=zeros(nfft,Nsegments); S22=zeros(nfft,Nsegments); S12=zeros(nfft,Nsegments); S13=zeros(nfft,Nsegments);
S23=zeros(nfft,Nsegments); S33=zeros(nfft,Nsegments); S11A=zeros(nfft,Nsegments); S22A=zeros(nfft,Nsegments); S12A=zeros(nfft,Nsegments); S13A=zeros(nfft,Nsegments); S23A=zeros(nfft,Nsegments); S33A=zeros(nfft,Nsegments); for n=1:Nsegments;
%Apply the DAQ correction
M1hat(:,n)=H7.*fft(A(:,n), nfft)/nfft; M2hat(:,n)=H6.*fft(B(:,n), nfft)/nfft; T1hat(:,n)=H4.*fft(D(:,n), nfft)/nfft;
S11A(:,n)=M1hat(:,n).*conj(M1hat(:,n))*nfft*2/(U*Fs); S12A(:,n)=M2hat(:,n).*conj(M1hat(:,n))*nfft*2/(U*Fs); S22A(:,n)=M2hat(:,n).*conj(M2hat(:,n))*nfft*2/(U*Fs); S13A(:,n)=T1hat(:,n).*conj(M1hat(:,n))*nfft*2/(U*Fs); S23A(:,n)=T1hat(:,n).*conj(M2hat(:,n))*nfft*2/(U*Fs); S33A(:,n)=T1hat(:,n).*conj(T1hat(:,n))*nfft*2/(U*Fs); end S11=mean(S11A,2); S12=mean(S12A,2); S22=mean(S22A,2); S13=mean(S13A,2); S23=mean(S23A,2); S33=mean(S33A,2);
% For case of 8 segments, 0.5 overlap, data series exactly % 4.5 times the nfft length % the above code is identicle to the MATLAB cpsd function % [S12,F] = cpsd(M2,M1,[],[],nfft,Fs); % cpsd funct requires MATLAB signals processing toolbox
Apply two microphone method
Filename: TwoMicMethodExp.m % 2 microphone method (refer to Seybert and Ross 1977
% % % % % % % % x2
x=0
M1|<----------------->|
M2 |<------------------------->| ..................................................... . . . u-----> A ---------> <-----------B . . . Z(f)
%Constants rho=rho_2 c=c2 u=v2 M=v2/c2 x1=0.085 x2=0.293 % (m) dist from nozzle to pressure tap 1 % density of the air % speed of sound % speed of the flow
freq=Fs*(0:nfft-1)/nfft; omega = freq*2*pi; % omega is in rad/s
for q=1:nfft; k=omega(q)/c; ki=k/(1+M); kr=k/(1-M); V12=[S11(q);S12(q);S21(q);S22(q)];
a11=1; a12=exp(i*ki*x1+i*kr*x1); a13=exp(-i*ki*x1-i*kr*x1); a14=1;
a21=exp(i*ki*x1-i*ki*x2); a22=exp(i*ki*x1+i*kr*x2); a23=exp(-i*kr*x1-i*ki*x2); a24=exp(-i*kr*x1+i*kr*x2);
matrix=[a11,a12,a13,a14;a21,a22,a23,a24; a31,a32,a33,a34;a41,a42,a43,a44];
Minv=inv(matrix); VAB=Minv*V12; SAA(q)=VAB(1,:); SAB(q)=VAB(2,:); end
figure(1); subplot(2,1,1); hold on; plot(freq,abs(Rp),linetype) axis([0 2000 0 4]) xlabel(Frequency (Hz)); ylabel(Amplitude(Rp)); subplot(2,1,2); hold on; plot(freq,angle(Rp)*180/pi,linetype) axis([0 2000 -200 200]) xlabel(Frequency (Hz)); ylabel(Phase(Rp));
Normalized cross-correlation
Actually the result is same as the covariance because the mean is removed before the cross-correlation is taken Filename: CrossCorr.m % Requires MATLAB signals processing toolbox
x=x-mean(x); y=y-mean(y);
[c,lags]=xcorr(x,y,10000,coeff); %coeff specifies normalised 140
[C,I]=max(c) % gives the maxima and the index
stem(c, b) figure(2) hold on plot(x) plot(y,g)
Plotting the spectrogram
The auto spectral density array must be computed rst. Filename: Spectrogram.m %plot spectrogram for 10 seconds of data %nfft must be 4096 %Fs must be 20 000 Hz %overlap is 0.5; %Nsegments=Nsegments*2 - 1
% Make Spectrogram ImageS22=-20*log10(S22A); figure(1) xlabel(Time(s)) ylabel(Frequency (Hz));
%hold on %image(ImageS22(3276:4096,:));%for 4096 0 to 4000 kHz image(ImageS22(3686:4096,:));%for 4096 0 to 2000 kHz %image(ImageS22(29492:32768,:));% for 32768 to 2000 kHz 141
colormap(gray(256)); axis(off)
figure(2) xlabel(Time(s)) ylabel(Frequency (Hz)) axis([0 10 0 2000]);
Strahle method
R=S11./S12; % divide entry by entry Rr=real(R); Ri=imag(R);
B.10
Thermocouple compensation
for index=1:nfft; F1(index,:)=(((i*Fs*2*pi*index)/nfft)+ (70*2*pi))/((i*Fs*2*pi*index)/nfft); F2=F1; F3(index,:)=(((i*Fs*2*pi*index)/nfft)+ (40*2*pi))/(2*40*2*pi); end
%The corrections are applied to the FFT of the signals M1hat(:,n)=H7.*F1.*fft(A(:,n), nfft)/nfft; M2hat(:,n)=H6.*F2.*fft(B(:,n), nfft)/nfft; T1hat(:,n)=H3.*F3.*fft(D(:,n), nfft)/nfft;
Figure B.1: Thermocouple simulation
Figure B.2: Two microphone method simulation
Thermocouple simulation
Variable step solver ode45 (Dormand-Prince) was used.
Two microphone method simulation
Fixed step solver ode3 (Bogacki-Shanpine) was used with step size 5e-5 s.
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