Validation of a Physics‐Based Low‐Order Thermo‐Acoustic Model of a Liquid‐Fueled Gas

Validation of a Physics Based Low Order Thermo Acoustic Model of a Liquid Fueled‐‐‐ Gas Turbine‐‐‐ Combustor‐‐‐ and its Application for Predicting‐‐‐ Combustion Driven Oscillations

A dissertation submitted to the Graduate School of the University of Cincinnati

in partial fulfilment of the requirements for the degree of Doctor of Philosophy (Ph.D.)

In the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering and Applied Sciences

Michael Knadler

B.S., Aerospace Engineering, University of Cincinnati, 2011

November 2017

Committee Chair: Dr. Jongguen Lee, Ph.D.

Abstract

This research validates a physics based model for the thermo-acoustic behavior of a liquid-fueled gas turbine combustor as a tool for diagnosing the cause of combustion oscillations. A single nozzle, acoustically tunable gas turbine combustion rig fueled with Jet-A was built capable of operating in the unsteady combustion regime. A parametric study was performed with the experimental rig to determine the operating conditions resulting in thermoacoustic instabilities.

The flame transfer function has been determined for varying fuel injection and flame stabilization arrangements to better understand the feedback loop concerning the heat release and acoustics. The acoustic impedance of the boundaries of the combustion system was experimentally determined. The results were implemented in a COMSOL Multiphysics model as complex impedance boundary conditions at the inlet and exit and a source term to model the flame and heat release. The validity of that model was determined based on an eigenvalue study comparing both the frequency and growth rate of the eigenvalues with the experimentally measured frequencies and pressures of the stable and unstable operating conditions. The model demonstrated that it can accurately predict the instability of the examined operating conditions.

The model also closely predicted the frequency of instability and demonstrated the usefulness of including the experimentally determined acoustic boundary conditions over idealized sound hard boundaries.

Acknowledgments:

I would first like to thank my advisor, Dr. Jongguen Lee, for his year of guidance and patience in leading me through my research.

I would also like to thank my committee members, Dr. Jay Kim, Dr. Kwanwoo Kim, and Dr. Mark Turner for their insight and advice throughout my dissertation research and studies.

Also in need of recognition are my fellow graduate students who help me both in completing my research, Arda Cakmakci and Thomas Caley, and keeping me sane in and out of the lab, Jun Hee Han.

Of course nothing would get done in the lab without Curt Fox who has helped me with most every piece of technical equipment I have needed.

And finally, Kiev, for always reminding me to never work too hard.

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TABLE OF CONTENTS

1. Introduction ...... 1 1.1. The Thermoacoustic Problem ...... 1 1.2. Objectives ...... 1 2. Theory of Thermoacoustic Instability ...... 3 2.1. The Rayleigh Criterion for Thermoacoustic Instability ...... 3 2.2. Mathematical Model for Thermoacoustic Instability ...... 8 2.3. One Dimensional Wave Theory for Acoustic Perturbations ...... 11 3. Previous Approaches to Thermoacoustic Modeling ...... 16 3.1. Computational Fluid Dynamics (CFD) ...... 16 3.2. Thermoacoustic Network Model...... 17 4. Finite Element Modeling for Thermoacoustic Instabilities ...... 26 4.1. Acoustic Wave Coefficients ...... 26 4.1.1. Determining Acoustic Wave Coefficients ...... 26 4.1.2. Acoustic Wave Coefficient Error Analysis ...... 28 4.2. Acoustic Impedance Measurements and Boundary Conditions ...... 29 4.3. Flame Modelling ...... 39 5. Experimental Details ...... 54 5.1. Single Nozzle Acoustically Tunable Gas Turbine Combustion Rig Setup ...... 54 5.1.1. Inline Heater ...... 55 5.1.2. Air Siren ...... 55 5.1.3. Inlet Plenum ...... 57 5.1.4. Fuel Nozzle and Swirler ...... 57 5.1.5. Combustion Chamber and Pressure Vessel ...... 61 5.1.6. Transition Tube ...... 62 5.1.7. Pressure Screws ...... 62 5.2. Dynamic Pressure Sensor Setup ...... 63 5.3. Flame Transfer Function Setup ...... 67 5.4. High Speed Camera Setup ...... 72 5.5. COMSOL Model Development ...... 72 6. Results ...... 78 6.1. Stability Map ...... 78 6.2. Flame Transfer Function Measurement ...... 83

6.3. High Speed Flame Imaging ...... 86 6.3.1. High Speed Color Imaging ...... 87 6.3.2. OH* ICCD Imaging ...... 89 6.4. Time Delay Determination ...... 94 6.5. Acoustic Boundary Condition Measurement ...... 98 6.6. Combustor Temperature Profile ...... 101 6.7. Eigenfrequency Study ...... 102 7. Conclusion ...... 110 8. Bibliography ...... 112 A. APPENDIX A: Flush to Recess Mounting Calibration and MATLAB Code ...... 116 B. APPENDIX B: Flame Transfer Function MATLAB Code ...... 125 C. APPENDIX C: Three-line Pyrometry Theory and Calibration ...... 131 D. APPENDIX D: Impedance Calculation MATLAB Code...... 134

LIST OF FIGURES

Figure 2.1: Thermodynamic interpretation of the Rayleigh criterion. Heat addition in phase with pressure (red) and out of phase with pressure (green) ...... 6 Figure 2.2: Block diagram representation of feedback loop between acoustic fluctuations and heat release ...... 8 Figure 2.3: Acoustic wave propagation in a duct ...... 13 Figure 3.1: (a) Schematic of a model gas turbine combustor with acoustic waves A and B upstream and downstream of the flame (b) Block diagram of thermoacoustic network model ... 18 Figure 3.2: (left) Upstream and downstream acoustic variables with relation to thermoacoustic network element (right) Reimann invariants upstream and downstream with respective directions of flow ...... 19 Figure 3.3: Converging-diverging nozzle ...... 20 Figure 3.4: Combustor with premix duct used to develop transfer matrices presented in Equations 3.2.22 and 3.2.23 ...... 23 Figure 4.1: Flow through an orifice showing vena contracta effect ...... 32 Figure 4.2: Orifice showing vena contracta producing edge vortices ...... 32 Figure 4.3: Effect of bias flow on acoustic impedance as determined by Jing and Sun [29] ...... 34 Figure 4.4: Su's [38] impedance model and its dependence on frequency ...... 34 Figure 4.5: Testud's [39] model for predicting orifice whistling. Negative indicates whistling potential...... 36 Figure 4.6: RANS simulation confirming orifice whistling potential model ...... 36 Figure 4.7: Converging-diverging nozzle with shock used to model choked orifice impedance . 37 Figure 4.8: Magnitude and phase of reflection coefficient comparison between numerical (circle) and analytical (solid) results of Stow [43] ...... 38 Figure 4.9: Impedance comparison between finite difference solver, Euler solver, and analytical results with a compaact nozzle assumption [46] ...... 39 Figure 4.10: Schematic of combustor used to derive T22 flame model ...... 41 Figure 4.11: Comparison of measured (red crosses) and modeled (blue line) values for transfer matrix element T22 for various temperatures [48] ...... 45 Figure 4.12: (left) Absolute value and (right) phase of flame transfer matrix elements comparison for measured (black square), Rankine-Hugoniot approximation (red line), and OH* chemiluminescence measurement (blue diamonds) results ...... 47 Figure 4.13: (left) Effect of β on first resonant frequency of combustor comparing exact solution (solid line) and one term Galerkan approximation (dashed line), (right) effect of time delay on growth rate for varying β [13] ...... 50 Figure 4.14: Effect of using temperature jump on first resonant frequency for single temperature jump (solid line), uniform temperature rise (dashed line), and five smaller jump approximations (-.-) ...... 51 Figure 5.1: Overview of single nozzle, acoustically tunable, gas turbine combustion rig ...... 54 Figure 5.2: Cross sectional view of combustion rig ...... 55 Figure 5.3: Inlet section connected to air modulation siren and siren bypass line ...... 56 Figure 5.4: Air modulation siren ...... 56 Figure 5.5: 3D printed and cross sectional CAD view of swirler used in testing ...... 58 Figure 5.6: Swirler mounted on effusion plate viewed from upstream/inlet side (left) and downstream/combustor side (right). Splash plate not included...... 60

Figure 5.7: Downstream/Combustor side view of swirler mounted with splash plate attached and flush with flared exit of swirler ...... 60 Figure 5.8: Pressure drop vs. main air flow rate for open and closed effusion hole cases ...... 61 Figure 5.9: Piezoelectric pressure transducer seated in wave guide mount for recessed mounting ...... 63 Figure 5.10: Inlet orifice impedance test setup ...... 64 Figure 5.11: Schematic of exit impedance setup ...... 66 Figure 5.12: Exit orifice impedance testing setup ...... 66 Figure 5.13: Microphone mounting on transition tube for exit orifice impedance testing ...... 67 Figure 5.14: Optical arrangement of the three line pyrometer...... 69 Figure 5.15: Time traces of downstream pressure,, three line pyrometer emissions, and OH* chemiluminescence emissions showing in-phase relationship ...... 70 Figure 5.16: (left) Schematic of OH* chemiluminescence setup focusing emissions from the side window of the combustor onto the PMT sensor and (right) actual experimental hardware in place for testing ...... 71 Figure 5.17: COMSOL geometry with elements representing (from left to right) inlet, nozzle, combustor, flame zone, and transition tube ...... 73 Figure 5.18: COMSOL geometry for the 0.0508m-long transition tube case with fine mesh visible ...... 74 Figure 6.1: Acoustic pressure fluctuations in the combustor for the 0.254m-long transition tube setup ...... 79 Figure 6.2: Acoustic pressure fluctuations in the combustor for the 0.508m-long transition tube setup ...... 79 Figure 6.3: Combustor pressure fluctuation spectrum for Case 1 ...... 81 Figure 6.4: Combustor pressure fluctuation spectrum for Case 2 ...... 82 Figure 6.5: Combustor pressure fluctuation spectrum for Case 3 ...... 82 Figure 6.6: Combustor pressure fluctuation spectrum for Case 4 ...... 83 Figure 6.7: Exemplary plots used in determining the gain of the FTF for (top) 1500SLPM case fordced at 400Hz and (bottom) 2000SLPM case forced at 400Hz ...... 84 Figure 6.8: Gain of the FTF for 2000SLPM and 1500SLPM cases ...... 85 Figure 6.9: Phase of the FTF for 2000SLPM and 1500SLPM cases ...... 86 Figure 6.10: Averaged high speed color camera images for (top left) Case 1 (top right) Case 2 and (bottom) Case 4 showing pressure fluctuations in the inlet (P1') and combustor (Pcomb'), frequency of fluctuations ...... 88 Figure 6.11: Averaged ICCD images for (top left) Case 1 (top right) Case 2 and (bottom) Case 4 with locations of peak and centroid intensity ...... 90 Figure 6.12: Average ICCD images of 2000SLPM 20in case under forcing from 200Hz, 400Hz, 600Hz, and 800Hz ...... 91 Figure 6.13: Average ICCD images of 1500SLPM 20in case under forcing from 200Hz, 400Hz, 600Hz, and 800Hz ...... 91 Figure 6.14: Phase averaged ICCD images for the 2000SLPM 20in case under natural instability ...... 93 Figure 6.15: Intensity contours for the phase averaged ICCD images for the 2000SLPM 20in case under natural instability ...... 93 Figure 6.16: Movement of centroid location downstream of nozzle for the 2000SLPM 20in case under 200Hz forcing by the siren...... 94

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Figure 6.17: Video FFT analyzed OH* emission peaks for (top left) 200Hz, (top right) 400Hz, (bottom left) 600Hz, and (bottom right) 800Hz ...... 97 Figure 6.18: Real component to impedance for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases ...... 99 Figure 6.19: Imaginary component to impedance for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases ...... 99 Figure 6.20: Reflection coefficient for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases ...... 100 Figure 6.21: Percent error in acoustic pressure amplitude predictions for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases ...... 100 Figure 6.22: Temperature profiles used in COMSOL model ...... 102 Figure 6.23: Mode shape for Case 1 comparing predicted COMSOL values to experimentally determined acoustic pressure amplitudes in inlet and combustor with error bars indicating one standard deviation ...... 108 Figure 6.24: Mode shape for Case 2 comparing predicted COMSOL values to experimentally determined acoustic pressure amplitudes in inlet and combustor with error bars indicating one standard deviation ...... 109

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LIST OF TABLES

Table 5.1: Swirler Geometry ...... 58 Table 5.2: Measured Effective Area and Projected Mass Flow Rates for Ten SN0.6 Swirlers [54] ...... 59 Table 5.3: Inlet transducer mount locations ...... 65 Table 5.4: Exit orifice impedance test transducer locations ...... 66 Table 5.6: Mesh density impact on eigenfrequency of rig model without flow or source present74 Table 5.7: COMSOL grid impact study results with experimental operating conditions in place for 2000SLPM 0.508m-long transition tube case ...... 75 Table 5.8: Exemplary input parameters for the COMSOL model ...... 76 Table 6.1: Equivalence ratios for stability map operating conditions ...... 78 Table 6.2: Time delay calculations based on known convection distances and calculated velocities ...... 96 Table 6.3: Results of final COMSOL model compared to experimentally determined acoustic pressure fluctuations in the combustor and their respective frequencies ...... 102 Table 6.4: Effect of impedance boundary conditions on converged eigenfrequencies in the COMSOL model ...... 103 Table 6.5: Effect of temperature gradient on converged eigenfrequencies in COMSOL model 104 Table 6.6: Model prediction with additional time delays ...... 105 Table 6.7: Eigenfrequency predictions using visible spectrum emissions in calculating flame zone and time delay ...... 107

1. Introduction

1.1. The Thermoacoustic Problem

In a drive to produce more efficient and lower NOx emission gas turbine engines techniques have been employed in engine design that increase the system’s risk for self-sustained combustion instabilities. These included the use of lean premixed prevaporized (LPP) combustion that along with reducing local hot spots in the combustion chamber, leading to lower

NOx emissions, reduce both flame anchoring and dampening [1]. Several consequences of LPP combustion lead to this increased susceptibility for thermoacoustic oscillations including a lower dilution air supply, leaner equivalence ratio, and a pressure antinode located near the flame [2].

These oscillations, under the right conditions, can cause undesirable pressure fluctuations in the combustor of such amplitude as to significantly increase the wear and tear on engine components leading to unsafe operating conditions and failure. In addition, thermoacoustic instabilities are not just a local phenomenon. Acoustics of the entire combustion system, from inlet boundary conditions through outlet boundary conditions and everything in between through which the acoustic waves propagate, can affect the combustor’s stability [3]. Although local flow characteristics can play an important role, a global view of the system must be taken to properly diagnose thermoacoustic instabilities. This often inhibits the further development of stable low- emission combustors and so a greater understanding of the mechanisms which initiate and sustain the instabilities and various operating conditions is needed.

1.2. Objectives

The objective of this study is to develop and validate a theoretical finite element model capable of predicting the formation of combustion driven pressure oscillations through a detailed understanding of the physical response of the flame to periodic disturbances, the acoustic

1 conditions at the boundaries, and the propagation of acoustic waves in the air supply and combustor. This knowledge is currently limited and the validation of these combustion dynamics models is not found.

2. Theory of Thermoacoustic Instability

2.1. The Rayleigh Criterion for Thermoacoustic Instability

Unstable combustion is the self-sustained combustion oscillations at the resonant frequency of the combustion chamber which are the result of closed loop coupling between unsteady heat release and pressure fluctuations. These flame dynamics have been observed and studies by interested researchers for well over the past two centuries. The first recorded observation was of what became known as the “singing” flame by Higgins in 1777 [4]. Higgins and other researchers discovered that anchoring a flame on a smaller diameter fuel line inside a larger diameter tube would produce an audible level of sound by exciting the fundamental or harmonic modes of the larger tube, similar to the excitation of an organ pipe. Years later a “dancing” flame was observed by LeConte where a flame would pulse in sync to beats of music (if only the flame could learn to act then it would a true triple threat!). Around the same time, Rijke discovered that sound could be generated in a vertical tube, open on both ends, by heating a metal gauze placed exactly at one quarter the distance from the bottom to top of the tube. This setup, known as the

Rijke tube, comes about as a result of acoustic velocity fluctuations in the bottom and top halves of the tube being opposite each other and the heat source being placed at the L/4 location causing the velocity fluctuations to lead (in the bottom half) or lag (in the top half) by 90 o

These phenomena were first described by Lord Rayleigh in 1878 [5]. In his own words the thermoacoustic phenomenon is described as follows: “If heat be periodically communicated to, and abstracted from, a mass of air vibrating in a cylinder bounded by a piston, the effect produced will depend upon the phase of the vibration at which the transfer of heat takes place. If heat be given to the air at the moment of greatest condensation or to be taken from it at the moment of greatest rarefaction, the vibration is encouraged. On the other hand, if heat be given

3 at the moment of greatest rarefaction, or extracted at the moment of greatest condensation, the vibration is discouraged”. In practical term applied to gas turbines, the acoustic waves cause fluctuations in heat release by altering the fuel and/or air mass flow rate. The fluctuations in resulting heat release then amplify the acoustic waves which results in self-sustained instabilities.

When these heat release fluctuations and pressure fluctuations are in phase the system will be naturally unstable and when they are out of phase natural dampening will occur. More specifically, any upstream flow perturbation can perturb the heat release further downstream at the flame. When the heat release fluctuates, then volume expansion of the flame fluctuates as well. As this cycle continues the volume of the expansion fluctuations of the flame will release acoustic waves throughout the combustion chambers at a frequency dependent upon the initial perturbation frequency. The acoustic pressure wave will propagate all along the combustor, eventually reflecting back upstream and causing acoustic velocity fluctuations in the flame.

This theory can be formulated into an inequality integrating the relationship between pressure and heat release fluctuations and the natural dissipation of the system.

2.1.1 ′, ′, > Φ, In this equation, p’ is the acoustic pressure fluctuation, q’ is the heat release fluctuation, Φ is the wave energy dissipation, τ is the period of the fluctuation, and V is the volume of combustor. In other words, if the mechanical energy added to the system through heat release and pressure per cycle is greater than the energy the system is able to dissipate over that same cycle then thermoacoustic instability will occur.

To derive this relationship from first principles, one starts with the acoustic energy density in a one dimensional acoustic field given as such [3].

2.1.2 ̅′ ′ = + 2 2̅ Where the first term on the RHS of Equation 2.1.2 is the kinetic acoustic energy and the second term is the potential acoustic energy. To sustain acoustic waves the system will transfer energy between these two components on a cyclical basis. Then, under the condition of zero mean velocity and no spatial change in mean variables, the momentum and energy conservation equations reduce to Equations 2.1.3 and 2.1.4, respectively.

2.1.3 ′ ′ ̅ + = 0 2.1.4 ′ ′ + ̅ = − 1′ Multiplying Equation 2.1.3 by u’ and 2.1.4 by allows the resulting expressions to be added in ̅ such a way that using Equation 2.1.2 results in the following expression.

2.1.5 ′ − 1 + + ′ = ′′ ̅ Equation 2.1.5 can then be integrated spatially over the length of the combustor, L, and temporally over the period of oscillation, τ.

2.1.6 − 1 Δ ′ = ′, ′, − Δ − Φ, ̅ Where ∆τ is changes in time and ∆L is changes in length. The LHS of Equation 2.1.6 is the change in acoustic energy of the combustor per cross sectional area. The first term on the RHS of the equation is known as the Rayleigh integral and resembles the LHS of Equation 2.1.1. The second term is the acoustic energy flux across the control surface of the field defined as = , and the final term accounts for all the dissipation of energy in the acoustic field. This ′′ energy balance shows that if the pressure and heat release fluctuations are in phase and the gain

5 between them is large enough (represented by the Rayleigh integral) to overcome the flux and dissipative effects (the second and third terms on the RHS) the acoustic energy in the combustor will increase, leading to thermoacoustic instabilities.

The Rayleigh Criterion can also be thought of as a thermodynamics cycle. A volume of gas can be expanded and compressed by being acted upon by an acoustic wave. Since acoustic waves are isentropic the volume will move along an isentrope when plotted on a p-v diagram as shown by the solid line in the figure below.

Figure 2.1: Thermodynamic interpretation of the Rayleigh criterion. Heat addition in phase with pressure (red) and out of phase with pressure (green) When heat is added to the system, the specific volume of the gas will increase. If that heat addition is in phase with the pressure fluctuations, the gas’s state moves clockwise around the cycle labeled 1-2’-3’-4’. This results in a thermoacoustic heat engine providing mechanical energy to the sound wave and leading to combustion instabilities. If the heat addition is out of phase with the pressure fluctuations then the cycle proceeds counterclockwise along the path labeled 1-2”-3”-4”. In this case mechanical energy is extracted from the sound wave and the combustion will remain stable.

The mechanical work from the above thermodynamic cycle can be described by Equation

2.1.7 [3].

2.1.7 ′ ′ ′ = − + = 0 + ′ ~ ′′ Where v’ is the change in specific volume. This change is split into two different parts.

The first, represented as , is the isentropic component and the second, , is the component − ′ due to heat addition. The rate at which this term changes is then proportional to the fluctuations in heat addition. Then, if the final integral over the period of one oscillation is positive, the work done by the engine in positive, energy is provided to the acoustic waves, and thermoacoustic instabilities arise.

Describing this relationship between heat release and velocity fluctuations in a manner more fitting the models that need to be developed and implemented, it can be described as a function, F. The unsteady heat release can be defined as a source term, Q’ and the velocity fluctuations as u’. Since it is also known that the volume expansion of the flame will be proportional to the heat release a proportionality constant k is also introduced resulting in the following equation [6].

2.1.8 Δu = kQ = kFu′ Impedances (defined as the ratio of acoustic pressures to acoustic velocities) of the system affecting the stability are introduced through the use of a geometric transfer function G described in the equation below.

2.1.9 − = + These equations can then be combined into the feedback loop shown in the figure below.

Figure 2.2: Block diagram representation of feedback loop between acoustic fluctuations and heat release 2.2. Mathematical Model for Thermoacoustic Instability

To begin the derivation of the governing equations of the thermoacoustic mathematical model upon which the current study will rest, several assumptions need to be made/reiterated [4]:

• One dimensional flow: specifically in the longitudinal direction of the combustor

• Inviscid flow: dissipation effects of the duct on the acoustic waves will be ignored

• Thermal conductivity to the system’s surroundings is negligible

• Perfect gas (i.e. ) = • Stationary flow

Under these assumptions the conservation equations for mass, momentum, and energy are reduced to Equations 2.2.1, 2.2.2, and 2.2.3, respectively.

2.2.1 + = 0 2.2.2 + + = 0 2.2.3 + = − +

Where e is the specific internal energy and q is the heat release rate per unit volume. Since a perfect gas was assumed, it can be shown that . With that added relation, Equations = 2.1.1 and 2.2.3 can be combined to yield the following.

2.2.4 + + = − 1 For each of the flow variables the instantaneous va lue of each is determined as the sum of its averaged component, identified by the bar above the variable, and its fluctuating component, identified by the ‘ as shown in the following equations.

2.2.5 = ̅ + ′ 2.2.6 = + ′ 2.2.7 = ̅ + ′ 2.2.8 = + ′ The conservation equations can now be treated for both the averaged and fluctuating components. For just the average values Equations 2.2.1, 2.2.2, and 2.2.3 become

2.2.9 = 0 2.2.10 ̅ + = 0 2.2.11 ̅ + ̅ = − 1 For the fluctuating components, Equations 2.2.2 and 2.2.4 are utilized and result in the following relationships when the second order terms are neglected.

2.2.12 ′ ′ ′ ̅ + + ̅ + ′ + = 0 2.2.13 ′ ′ ̅ + + + ̅ + ′ = − 1′ ′ 9

It is also assumed that the flame zone is located at some point, x f, and the heat release fluctuations can then be represented as Equation 2.2.14.

2.2.14 ′, = ′ − Where q’ f is the heat release per unit area and δ is the Dirac delta function. Based on this functional representation it can be concluded that the mean variables are constant through the combustor except for a step change occurring at the flame zone location x f. The effect of the step change on the pressure is negligibly small compared to velocity or density and the spatial gradient of the mean pressure and velocity will be small as well. With these simplifications,

Equation 2.2.12 then becomes

2.2.15 ′ ′ ′ + + ̅ = − 1′ Thus, after partially differentiating Equation 2.2.15 with respect to spatial location and Equation

2.2.13 with respect to time and algebraically manipulating the results, the following relationship can be formed.

2.2.16 ′ ′ ′ ′ ′ − ̅ 1 − + 2 = − 1 + These two equations (2.2.15 and 2.2.16) are the describing functions for an acoustic pressure and velocity field forced by an unsteady heat release source, q’, located within the combustor. If the flow is negligibly small (i.e. and are close to zero), then a further simplification can be made to the above equations.

2.2.17 ′ ′ + ̅ = − 1′ − 2.2.18 ′ ′ ′ − ̅ = − 1 −

When no heat source is present in the system Equation 2.2.18 (the RHS is zero) will become the classic acoustic wave equation, which will be discussed in further detail in Section 2.3: “One

Dimensional Classic Acoustic Wave Theory” where the theory behind the acoustic characterization of passive components is handled. It is important to note here, however, that the solution of the homogenous wave equation is well known and shown in Equations 2.2.19-2.2.21.

2.2.19 ′, = ̅ 2.2.20 = sin + 2.2.21 = sin ̅ + Where is the function representing the spatial component to the solution, is the function representing the temporal component, k is the wave number, and and are phases determined by the boundary conditions. As sine is a periodic function an infinite number of solutions, or eigenvalues, will exist. Each unique will represent a different spatial mode and when combined with the temporal component , will yield a solution, . Each ′, ′, represents a standing wave with a frequency . = ̅ 2.3. One Dimensional Wave Theory for Acoustic Perturbations

Any approach to the thermoacoustic problems necessitates an understanding of the wave propagation throughout the combustor. In essence, one-dimensional wave theory assumes that all variables are constant in any direction perpendicular to the flow. All acoustic properties are considered as plane waves and only propagate in a longitudinal direction to the flow. In order to utilize this approximation the model must assume the walls in the test section are acoustically hard, the frequencies of interest are in a range where the wavelengths are less than the cross sectional area of the duct. In general, this criterion is covered when the following equation holds true [7].

2.3.1 < 0.586 Where f is the frequency (Hz), c is the speed of sound through the medium (m/s), and d is the diameter of the tube (m). For the present study the speed of sound in the inlet was approximately

425m/s and the cross sectional diameter was 0.04445m resulting in a frequency range of up to

5600Hz, well above the 200-800Hz range where measurements were made.

Conditions inside of the duct are defined by the pressure, velocity, and density of the flow. Any fluid motion within the duct must hold to both conservation of mass and Newton’s

Second Law of Motion. So in the case of the acoustic components, the equations describing the fluid can be represented as follows.

2.3.3 ′ 1 ′ ̅ = − 2.3.4 ′ ′ ̅ = − The latter equation being Euler’s Equation where it is assumed there is no mean velocity in the duct, or . = 0 Since acoustic pressure and velocity are the quantities of interest in the above equations the next step is to eliminate either and solve for the other. For example, if the first equation is differentiated with respect to time and the second with respect to x location, the left hand sides will be identical, eliminating the u’ term and p’ can be solved for. Doing so results in the following equation.

2.3.5 ′ 1 ′ = The same can also be done to find the equation desc ribing the acoustic velocity wave propagation, resulting in the equation below.

2.3.6 ′ 1 ′ = The acoustic wave at any point in the combustor can be thought of as the sum of two waves propagating through the combustor: the incident wave, whose amplitude will be denoted as A, and the reflected wave, whose amplitude will be denoted as B, as shown below.

Figure 2.3: Acoustic wave propagation in a duct The wave amplitudes A and B are constant throughout any constant are duct (although each frequency will have its own unique set of wave amplitudes) allowing for the determination of the incident and reflected pressure wave functions along any location, x, in the duct and at any time t through the use of a complex exponential. Since the disturbances of interest can be represented as simple harmonic functions a fluctuation such as p’ could be represented in a form similar to p·exp(i ω). Defining the amplitude of A and B as the magnitude of the wave at the point x=0 (the inlet boundary) the equations for each pressure wave and total acoustic pressure fluctuation in the duct can be represented as follows [8].

2.3.7 , = ∗ 2.3.8 , = ∗ 2.3.9 ′, = , + , = ∗ + ∗ Where ω is the angular frequency, k is the wave number ( ω/c), x is the location along the longitudinal axis of the duct, and M is the Mach number. Acoustic velocity can also be determined in a similar way per the equation below.

2.3.10 ′, 1 , = = ∗ − ∗ ̅ ̅ Since the current study will involve conditions with flow present in the duct the next step is to add a mean velocity ( ) into the foundational equations and re-derive the acoustic pressure > 0 and velocity. Doing so results in the following change to the Euler Equation.

2.3.11 ′ ′ 1 ′ + = − ̅ Decomposing the modified Euler Equation into its components travelling the incident, A, and reflected, B, directions yields the following set of equations.

2.3.12 1 ′ + ′ = − ̅

2.3.13 1 ′ + ′ = − ̅ + - In order to satisfy the above equations, modified wave numbers, k and k , are introduced and defined as such.

2.3.14 = + 2.3.15 = Replacing the k’s in 2.3.7 and 2.3.8 with the incident− and reflected planar wave numbers k + and k-, respectively, and applying the same mathematical scheme used to get equations 2.3.9 and

2.3.10 results in the acoustic pressure and velocity equations with the presence of mean flow as shown below.

2.3.16 ′, = , + , = ∗ + ∗

2.3.17 ′, 1 , = = ∗ − ∗ ̅ ̅

Or alternatively in terms of the flow Mach number (where M=u/c), the following form of equations 2.3.16 and 2.3.17 takes shape.

2.3.18 ′, = , + , = ∗ + ∗

2.3.19 ′, 1 , = = ∗ − ∗ ̅ ̅

3. Previous Approaches to Thermoacoustic Modeling

Various approaches to modeling thermoacoustic instabilities in gas turbine combustors have been tried through the years. Presented below is a brief overview of some of the major areas of study and commentary on their respective strengths and weaknesses.

3.1. Computational Fluid Dynamics (CFD)

CFD approaches to solving the thermoacoustic equations generally take one of three forms:

Reynolds-Averaged Navier-Stokes (RANS), Large Eddy Simulation (LES), or Direct Numerical

Simulation (DNS). Each resolves the turbulent equations of motion in varying degrees of accuracy, simplicity, and efficiency. RANS tends to be the most commonly employed turbulence model and involves decomposing necessary stochastic flow variables in to mean and fluctuation components [9]. This variable split, when inserted back into the governing equations, yields a very similar set of equations but with the addition of the Reynold’s stress tensor, , ′′ accounting for the velocity fluctuations. This additional term has no direct analytical expression that can be solved for and so a RANS model has to be implemented to solve for the time averaged values. The simplest is the eddy viscosity model where the Reynold’s stress tensor is assumed proportional to the strain rate given by Equation 3.1.1 and an eddy viscosity, νT.

3.1.1 1 = + 2 The eddy viscosity requires another model the most common of which are the Spalart-Allmara, k-ϵ, and k-ω models. These models operate under the assumption that νT is isentropic and only depends on the local flow field and must always be calibrated against simple test cases.

Advantages to employing a RANS strategy include computational inexpensiveness and an ability to use a relatively coarser mesh. However, by using this model the range of time scales has to be separated from the turbulent scales as RANS models are only applicable to statistically stationary

16 turbulent flows. RANS also tends to smooth out higher frequency fluctuations, which becomes problematic if the disturbances are not high amplitude.

Another CFD approach that has been utilized is Large Eddy Simulation (LES). In this simulation large energy containing scales are resolved while smaller, dissipative energy scales are modelled. Small energy scales are defined as those not geometry dependent and hence fall under universal rules that can be modeled. The Navier-Stokes Equations are low-pass filtered into resolved and unresolved components on a scale dependent upon the cell size of the model, typically the cubed root of the cell volume. However, this approach also introduces additional terms into the governing equations, here known as sub-grid-scale terms that require additional modelling. Another drawback to LES modeling is that it is much more computationally expensive than the RANS technique.

The third technique occasionally utilized is Direct Numerical Simulation. Here, all temporal and spatial scales are directly solved for without any turbulent modelling as seen in previous approaches. As such, this is by far the most computationally exhausting approach, especially for very large Reynolds numbers that produce very small scale lengths. DNS is typically only ever adopted for simplified geometries and generally only for low Reynold’s number flows.

3.2. Thermoacoustic Network Model

A thermoacoustic network model is the simplest way to solve the thermoacoustic problem. Here, the combustion system is modeled as a series of acoustic two-ports where pressure and velocity are related between each section. Each section is modeled as a set of one dimensional waves propagating upstream and downstream within the network element with interactions between

17 geometry, boundaries, flow field, and sources modeled as well. A general schematic of the thermoacoustic network model can be seen in the figure below.

Zo ZS Z2 Combustor @ T 2, P 2

A(x) M2(x) Inlet duct @ T 1, P 1 B(x)

A(x) Flame M (x) B(x) 1

Swirler (a)

Inlet Exit Inlet duct boundary Swirler Flame Combustor boundary (diffuser) condition condition

(b)

Figure 3.1: (a) Schematic of a model gas turbine combustor with acoustic waves A and B upstream and downstream of the flame (b) Block diagram of thermoacoustic network model The crux of the model comes down to a transfer matrix used to relate the outlet (upstream) pressure and velocity from one acoustic element to the inlet (downstream) pressure and velocity of the next element as shown in Equation 3.1.1.

′ ′ 3.2.1 = ′ ′ Where subscripts u and d represent upstream and downstream conditions, respectively, and T ij indicates the i-th and j-th elements of the transfer matrix. Another useful form of this relationship casts the matrix into a relationship between the downstream and upstream propagating Riemann invariants and . These quantities are related to acoustic pressure and velocity by Equations 3.2.2 and 32.3.

3.2.2 ′ = + 3.2.3 ′ = − Depending on the nature of the acoustics problem being investigated the decomposition of pressure and velocity into the Reimann invariants can be advantageous. Since and have a defined direction of propagation, as seen in the figure below, the influence of particular elements in the thermoacoustic network can be better understood [10].

Figure 3.2: (left) Upstream and downstream acoustic variables with relation to thermoacoustic network element (right) Reimann invariants upstream and downstream with respective directions of flow Physical variables, though, tend to be more useful in combustion related applications due to the influence the velocity modulation has over the equivalence ratio of the flame and therefore the flame dynamics and heat release [11]. Using the Reimann invariants, a transfer matrix relationship can be set up analogous to that shown in Equation 3.2.4.

3.2.4 = While the transfer matrix approach relates the upstream and downstream variables across the element interface, a similar approach utilizing a scattering matrix relates the incoming and outgoing acoustics waves by mixing the upstream and downstream Riemann invariants on either side of the equation.

3.2.5 =

The task then becomes properly setting up the transfer or scattering matrix, which is fluctuation, geometry, and condition dependent. To begin with a simple case, assuming the fluctuations are harmonic disturbances in time (i.e. modeled as e iωt) a duct with length, L, can have the following transfer matrix [2].

cos sin 3.2.6 = 1 sin cos Where Z is the characteristic impedance equal to ρc. Equivalently simple matrices can be derived when using the Reimann invariants for the transfer matrix (Equation 3.2.4) and the scattering matrix (Equation 3.2.5)

3.2.7 0 = 0 3.2.8 0 = The next element considered is a contraction0 element, such as a converging/diverging nozzle as shown in the figure below [3].

Figure 3.3: Converging-diverging nozzle

Here, x u is the upstream location, x d is the downstream location, A u is the cross sectional area at the upstream location, A d is the cross sectional area at the downstream location, and l is the distance between the two locations (i.e. l = x d-xu). The element must also be acoustically

20 compact meaning k l << 1. To begin the derivation from the conservation equations, the first step is to derive the relationship between upstream and downstream acoustic perturbations by integrating the unsteady Bernoulli equation.

3.2.9 + + = 0 2 − 1 Where φ is the velocity potential. Integrating that particular term yields the following relationship.

3.2.10 = ≈ Since the assumption of acoustical compactness was made it was reasonable to approximate u(x) as u cA(x) in the above equation. Another term called the extended length, l ext , is now introduced in Equation 3.2.11 to further simplify the above expression.

3.2.11 = If harmonic time dependence is assumed, then combining Equations 3.2.10 and 3.2.11 yields

3.2.12 = ′ The remaining two terms on the LHS of Equation 3.2.9 are much more straightforward and can simply be evaluated at x u and x d. Doing so and combining the results with Equation 3.2.12, linearizing, and normalizing by the speed of sound, c, gives Equation 3.2.13.

3.2.13 ′ ′ + + = The above derivation accounted for the conservation of momentum between the upstream and downstream locations. The second step is to conserve mass. Here the derivation begins with

Equation 3.2.14.

3.2.14 + = 0 Applying this equation to the aforementioned geometry results in Equation 3.2.15.

3.2.15 + = 0 The assumption is now made that the perturbations are harmonic and the wavelength is longer than the length of the acoustic element (i.e. k l = c/ ω > l) [12]. Density perturbations must also be assumed to be of the form given in Equation 3.2.16.

3.2.16 = + With these additional assumptions, Equation 3.2.15 now becomes

3.2.17 ′ + + ′ = Similar to the extended length introduced in the derivation of the conservation of momentum equation, a reduced length, l red , is now defined to help simplify Equation 3.2.15.

3.2.18 = Combining Equation 3.2.17 and 3.2.18 and dividing by density yields the desired mass conservation equation.

3.2.19 ′ ′ + + ≈ 0 From Equations 3.2.13 and 3.2.19 the transfer matrix equation between the upstream and downstream sections of the element can be determined.

′ ′ 3.2.20 1 − − = − ′ ′

In Equation 3.2.20 α is the upstream to downstream area ratio (i.e. α = A u/A d), ζM is the pressure loss term, and leff is the effective length. The pressure loss term is included to account for loss of acoustic energy and, if large enough, cannot be neglected. The effective length term is a result of combining the extended length term (Equation 3.2.11) with end correction terms, as shown below, that can appear at the upstream and downstream boundaries.

3.2.21 = + , + , Dowling and Stow [13] expanded upon the simple duct model to include an analytically determined transfer matrix for a simple duct of uniform cross sectional area with a flame model

(Equation 3.2.22) as well as a premix duct (Zone labeled 3 in the figure below) with an included combustion zone (3.2.23).

Figure 3.4: Combustor with premix duct used to develop transfer matrices presented in Equations 3.2.22 and 3.2.23 For both of these transfer matrices to hold true the mean flow must be negligible.

3.2.22 1 0 = 0 1 −

1 3.2.23 = 0

The flame model used in Equation 3.2.22 is based on the relationship shown in Equation 3.2.24 where τ is the time delay, Q’ is the rate of heat input per unit area, and β is a nondimensional constant.

3.2.24 ′ = − ′ − − 1 For the case of the premix duct, it is assumed that the duct is short, has a small cross-sectional area, no combustion takes places within it, the flow is incompressible, and the pressure difference from the zones labeled one to two are related to the change in momentum in the duct.

Those assumptions result in the following relationships upon which Equation 3.2.23 is based.

3.2.25 − = = Where A is the cross-sectional area and L is the effective axial length.

Campa and Comporeale [14] continued this work by furthering the analytical transfer matrix model of the premixing duct. Under the assumption of one-dimensional flow they obtained the following linearized conservation equations.

3.2.26 ′ + ′ = 0

3.2.27 ′ ′ + + + ′ = 0 Where l eff is the effective length of the compact element and ζ is the acoustic pressure loss coefficient later laid out by Laera, et al. [15] to be as follows.

3.2.28 2Δ = The transfer matrix for the compact element then becomes Equation 3.2.29.

3.2.29 1 − 1 + − = − + Where α is the upstream to downstream area ratio A u/A d.

Although relatively simple and easily implemented, the above matrices fail to properly model and complex geometry in the combustion system (e.g. a burner) or any sources (e.g. a flame). Instead, these matrices must be experimentally determined. In order to do so, two independent test cases must be considered. Since the desired matrix will have four terms, four independent equations are required and each test case can only provide two equations, one for pressure and one for velocity (or one for f and one for g). A common method for creating these test cases is placing a loudspeaker at either end of an experimental setup and an anechoic termination at the other. Then switching the ends gives the second test condition. Acoustic pressures can be measured on either side of a test piece at both conditions to yield the necessary measurements simply by solving the equation below where a indicates one test state and b indicates the second test state [16].

0 0 ̅ ̅ 0 0 3.2.30 ̅ = 0 0 ̅ ̅ 0 0 ̅ This setup has been utilized through several experiments and verified to provide accurate acoustic data and transfer matrix measurements [2] [10] [16] [17].

4. Finite Element Modeling for Thermoacoustic Instabilities

The present study will utilize a finite element model for predicting thermoacoustic instabilities.

In order to do so the two main drivers of thermoacoustic instabilities, acoustic propagation and heat release (driven by the flame response), must be analytically modeled and implemented in the finite element model. The following is discussion of those two fields and how they will be implemented in the model development.

4.1. Acoustic Wave Coefficients

4.1.1. Determining Acoustic Wave Coefficients

A necessary part of this model is determining the magnitudes of the incident and reflected planar waves in the duct. One common method utilized to accomplish this goal is measuring the acoustic pressure with highly sensitive microphones. A setup of as few as two microphones can be used to determine the incident and reflected components of a plane wave but an overdetermined system of three or more microphones will reduce measurement errors [18] [19].

One implementation of the two-microphone method to determine the acoustic wave coefficients was conducted by Yi and Santavicca [20]. Using the definitions of acoustic pressure and velocity as defined in Equations 2.3.18 and 2.3.19, the complex wave coefficients can be determined from two pressure measurements at unique locations x 1 and x 2 as shown below.

2 2 + − + 4.1.1 1 + 1 − 1 + 1 − = 2 2 − 1 − 1 −

− 4.1.2 1 + 1 + = 2 2 − 1 − 1 −

This method did require a third reference sensor, however, as well as being only two- microphones, subject to several sources of error that will be further discussed in the next section including limited frequency spectrum and sensor location aligning with an acoustic node.

Therefore, the multi-microphone method will be used in the present study.

To begin the derivation of the necessary multi-microphone method equations, start from the simplest two-microphone method case. The relationship between acoustic pressure and wave amplitudes given in Equations 4.1.1 and 4.1.2 can be cast in matrix form as shown in Equation

4.1.3 [21].

4.1.3 ′ = ′ In order for this system of equations to be solvabl e, each resulting equation must be linearly independent. In other words, the determinant of the first matrix must be non-zero. For this condition to hold, Equation 4.1.4 must then be satisfied.

4.1.4 − + ≠ 2 , = 1,2,3 … When additional microphones are added to the array, Equation 4.1.3 simply becomes an over determined system of equations consisting of n equations (n being the number of microphones used) and two unknowns, A and B.

′ 4.1.5 ′ = ⋮ ⋮ ⋮ ′ Equation 4.1.5 can then be solved using a Moore-Penrose pseudo-inverse under the condition that at least two of the rows are linearly independent.

4.1.2. Acoustic Wave Coefficient Error Analysis

Although the two microphone method had been held as the standard for measuring in-duct acoustic properties there are several drawbacks that result in the multi-microphone method being a superior setup [22]. In his work on the multi-microphone method, Jang derived an equation for the singularity condition of the microphone array [23].

4.1.6 2 2 − − = 0 1 − 1 − For the two microphone case, Equation 4.1.6 will reduce down to the half-wavelength restriction, which results in large measurements errors when the spacing between the two microphones (x 12 ) approaches half of the wavelength of a frequency of interest.

4.1.7 2 cos = 1 This restriction, well documented in two-microphone1 − method literature [18] [23] [22], imposes the restriction on spacing according to Equation 4.1.8.

4.1.8 < 1 − It was also noted that the minimum error in the two-microphone method was achieved when the spacing between the sensors satisfied Equation 4.1.9.

4.1.9 = Numerical investigations by Boden yielded the addit2ional information that if the spacing is even further restricted to a region defined by Equation 4.1.10 the total error would lay within an order of magnitude of the minimum achievable error.

4.1.10 0.1 < < 0.8

However, increasing the number of microphones in the array eliminates this singularity condition resulting in far more robust and accurate measurements over a broader range of frequencies.

Therefore, a three microphone setup was utilized for acoustic measurements throughout this study.

Proper spacing for the multi-microphone setup then needed to be determined in order to minimize measurement errors. Jang suggested that using an equidistant spacing between all three sensors would result in the lowest error as this reduces the setup’s singularity factor to its minimum, reducing the setups sensitivity to input errors [23]. Fischer, however, determined that although the inability of the two-microphone method to work at the half wavelength spacing condition is eliminated through the use of three microphones, there is still a high sensitivity to errors around that frequency when all the relative distances between sensors correspond to the half wavelength [19]. Non-uniform spacing between the microphones, though, pushes error sensitivity ranges into frequencies well above those of interest and should be the desired setup.

He also concludes that utilizing any more than three microphones yields such little improvement in accuracy that it would not be worth the additional cost and hassle to include. The current study proceeded along Fischer’s line of reasoning since experimental data collected in the test rig resulted in lower measurement error when irregular spacing was use than for the uniform case.

4.2. Acoustic Impedance Measurements and Boundary Conditions

Early efforts to implement the two-microphone method to calculate acoustic impedance involved the use of auto and cross spectral densities from two fixed microphones and bypassed the determination of the wave coefficients entirely [24]. Transfer matrix methods were also developed based on the cross spectral densities to determine acoustic properties [25]. However, these methods were hindered by errors arising from phase mismatching and cumbersome

29 calibrations required before tests could be run. A single microphone technique for impedance measurements was even developed to try and avoid these drawbacks [26]. Since none of these methods are as accurate or robust as the multi-microphone method, though, the present study uses the wave coefficients as determined by the procedure in the preceding sections to determine acoustic impedance and the reflection coefficient.

Once the wave coefficients in the duct are determined they can be applied to calculating the acoustic impedance at the boundary of the system. Recalling that the specific acoustic impedance, Z, is simply the ratio of acoustic pressure to velocity, the value at a boundary can be determined as given below.

4.2.1 ′ + = = ̅ = + Since A and B are both functions of frequency,′ the− impedance will then too be frequency dependent. The complex form of acoustic impedance involves the resistance, R, and the reactance, X, which are measures of the absorption and dissipation of acoustic vibration energy and the storing and releasing of energy without dissipation, respectively [27]. Since the orifice is being used as a boundary for the model, only the impedance on the inner side is of interest and as such a pair or array of microphones on only one side of the orifice can be utilized [23] [28]. In order to avoid potential errors in the measurements the tests should be conducted in a smooth bore tube with a diameter much smaller than the wavelength of the frequencies of interest. The smooth bore helps to eliminate any effect tube geometry may have on the acoustic waves and isolate the effect of the orifice. It is also helpful to define a reflection coefficient as the ratio of the upstream propagating wave to the downstream wave. In other words, it is a measure of the fraction of energy being reflected by the boundary.

4.2.2 = A reflection coefficient of one means all incoming energy is reflected by the boundary (the ideal sound hard boundary condition). Less than one, the reflection coefficient indicates that energy is being absorbed by the boundary and emitting less than is received. Greater than one means the boundary is adding energy to the system. Since the acoustic measurement tests were taken at different ambient conditions than the combustion tests they will be used to model, it is also useful to normalize the impedance by the density and speed of sound to form a specific acoustic impedance defined below.

4.2.3 + = The final form of the impedance to be utilized in the− present study comes with a correction term added to account for the thickness of the orifice [29].

4.2.4 + = − Where T is the thickness of the orifice plate and − σ is the open area ratio, which is a measure of the area of the orifice to the entire orifice plate. This correction factor was the result correcting analytical results based on thin plate end corrections determined by Howe [30] to produce good agreement with experimental results when utilizing a thick orifice plate. As the ratio of orifice radius to plate thickness is one here, this is considered a thick plate. A further discussion of plate thickness will be conducted below.

Orifices and their interactions with acoustic waves have been well studied through the years. The work has focused on several areas, three of which will be highlighted here: vena contracta effects, edge dissipation, and orifice whistling. The vena contracta effect describes the narrowing of the flow as it travels through an orifice.

Figure 4.1: Flow through an orifice showing vena contracta effect As can be seen in the above figure, the vena contracta effect causes the flow to narrow into a stream of cross sectional area A j smaller than that of the orifice. When this jet breaks down in a turbulent mixing zone before expanding to a uniform flow of cross-sectional area A 2 downstream of the orifice sound production can occur [31]. The resulting source produces the upstream and downstream travelling acoustic waves labelled and , respectively. Edge dissipation effects also affect orifice acoustics and are related to the vena contracta effect. After the flow narrows into a jet, vortices can form which draw power from the flow and dissipate it as kinetic energy [32] [33] [34] [35]. This can also complicate the one-dimensional flow assumption of the acoustic theory used in this study and so care should be taken to locate sensors such a distance downstream of an orifice that its turbulent effects can be mitigated or eliminated.

Figure 4.2: Orifice showing vena contracta producing edge vortices

These acoustic vortices originate without the presence of bias flow as air is drawn from one side of the orifice and thrown along the axis of the other, alternating from side to side at the frequency of the sound generated [33]. In the presence of bias flow the loss of flow energy is explained by the interplay between the acoustic energy and vertical energy, the latter of which is drawn downstream by the flow and dissipated as heat [35]. The sharp edge of the orifice caused a fluctuating vorticity field which accomplished this energy transfer. Zhou and Boden [36] developed an analytical model to study acoustic properties of an orifice based on the modified

Cummings Equation given below [37].

4.2.5 1 + + Δ + Δ + ∙ = 2 Where is the effective orifice thickness, V(t) is the fluctuating acoustic velocity in the orifice, U is the mean bias flow velocity, ∆p(t) is the fluctuating pressure difference over the orifice, ∆P is the steady pressure drop across the orifice, and C c(t) is the discharge coefficient, time dependent due to the vena contracta effect. The model also proposed splitting the discharge coefficient into two parts, C CM and C CA , for the parts dependent upon mean flow and acoustic flow, respectively, further complicating the model. The equation used to calculate acoustic impedance of the orifice is given in Equation 4.2.6.

4.2.6 = + ̅ Their study showed that an analytical model for orifice impedance could work but differences start to emerge as bias flow velocity and frequency increases, especially for the sharp edged orifice cases where vortex shedding becomes an issue.

Jing and Sun [29] study how plate thickness can affect the acoustic impedance especially in the regime where the ratio of plate thickness to orifice radius is at one or above, as it will be in

33 the present study. They concluded that there was a large effect of bias flow on resistance, most notably on the resistance with lesser but still significant variations occurring in resistance.

Figure 4.3: Effect of bias flow on acoustic impedance as determined by Jing and Sun [29] Su et al. [38] ran CFD simulations to compare with the results of Jing and Sun’s model with experimentally determined data. Although all generally agreed on the trends of the resistance and reactance, the CFD model appeared to by the superior choice at predicting the experimental data leading to the need for caution when choosing an analytical model to determine orifice impedance.

Figure 4.4: Su's [38] impedance model and its dependence on frequency

A third area of study involving the interactions of orifices and acoustic waves is orifice whistling. Orifice whistling is generally not taken into account in design stages as it only occurs occasionally but when it does can generate high level noise and vibration causing part fatigue and failure. It is related to self-sustained oscillations, like thermoacoustic instability, due to the instability of the shear flow in an orifice due to either hydrodynamic or acoustic feedback [39]. If hydrodynamic, it is a result of vortices generated in the shear layer reaching an area with a velocity gradient, such as an area expansion. Either way the velocity fluctuations caused by the feedback perturb the vorticity in the shear layers, transferring energy from the flow to the oscillations. The potential of an orifice to whistle can be discussed by first defining an exergy wave [40].

4.2.7 ± ± Π = 1 ± Where M 0 is the steady flow Mach number, p is the pressure, and ± describes the direction of propagation with regards to the flow direction (+ in the direction of flow and – against it). The exergy scattering matrix, S e, can then be defined as

4.2.8 Π Π = Π Π If one of the eigenvalues of the matrix is negative then the orifice is prone to whistling ∗ − because it can amplify acoustic waves. Testud [39] developed an accurate model for predicting an orifice’s whistling potential based on Strouhal number and concluded the greatest potential lies when it is between 0.2 and 0.4.

Figure 4.5: Testud's [39] model for predicting orifice whistling. Negative indicates whistling potential Later experiments by Lacombe and Moussou [41] and RANS simulations by Kierkegaard, et al.

[42] confirmed Testud’s model for orifice whistling potentiality.

Figure 4.6: RANS simulation confirming orifice whistling potential model The impedance of choked inlets, usually modeled as a convergent-divergent nozzle, has also been of interest to several researchers. Stow, et al. [43], based off of a model by Marbel and

Candel [44], studied a choked inlet with a shock in the divergent section.

Figure 4.7: Converging-diverging nozzle with shock used to model choked orifice impedance Due to the presence of a shock no acoustic waves should be able to propagate upstream through the nozzle. Thus, any model of the reflection coefficient for this orifice should agree with that theory. Downstream of the shock, however, there will be an upstream propagating acoustic wave and once reflected, will result in downstream propagating acoustic, entropy, and vorticity waves.

The model for the reflection coefficient is based on the work on the interactions of flow disturbances and shocks by Kuo and Dowling [45] and is reported as Equation 4.2.9.

1 1 ̅ 4.2.9 ̅ / − − / ̅ = − ̅/ /1 1

Where, the terms represent the influence of the acoustic wave, the terms ̅/ ̅/ represent the influence of the vorticity waves, the /1 1 terms represent the influence of the entropy wave, and R is the mean radius. Their numerical results agreed well with the analytical model up to a normalized frequency 0.2.

Figure 4.8: Magnitude and phase of reflection coefficient comparison between numerical (circle) and analytical (solid) results of Stow [43] They argue that their analysis should be suitable to any gas turbine combustor application as the non-dimensional frequency is typically less than 0.1.

Lamarque and Poinsot [46] also developed a numerical model based on Marbel and

Candle’s work, but this study examined the impedance of the orifice instead of just the reflections. They compared the real and imaginary components of the impedance for analytical formulas with a compact nozzle, numerical methods with the linearized Euler Equations and a finite difference solver in Fourier space, and a full space time solver of the Euler Equations.

Their finite difference method tended to agree well with the Euler space-time solver and the asymptotic behavior of the impedance components approach the analytical value as expected.

Peaks in the real component and sign changes in the imaginary component coincide with acoustic modes of the nozzle.

Figure 4.9: Impedance comparison between finite difference solver, Euler solver, and analytical results with a compaact nozzle assumption [46] 4.3. Flame Modelling

Proper modelling of the flame and its contribution to the inhomogeneous wave equation or acoustic network is a necessary step to determining the eigenfrequencies of a system. Several approaches have been undertaken in previous studies including treating the flame as a lumped element in the transfer matrix approach [2] [6] [47] [48]. One of the early approaches to flame modeling in the transfer matrix method treated the flame as an acoustic source term to the burner matrix. This additional term would change the transfer matrix setup as follows.

4.3.1 ′ ′ = + ′ ′ Where p s and u s are the pressure and velocity components of the source term due to the flame.

However, an advantage to treating the flame and burner as one single element (i.e. without the

39 source term matrix) is that the matrix can be determined with a two source method like the acoustic variables were before, with a test state having upstream forcing and a test state having downstream forcing.

If it is assumed that the transfer matrix for the burner element does not change in the presence of a flame, then a standalone flame transfer matrix can also be determined. In other words, a flame transfer matrix, F, can be calculated if a burner/flame, lumped transfer matrix,

BF, can be determined experimentally as stated above, and a burner transfer matrix, B, without the flame present can be determined simply by multiplying the lumped matrix by the inverse burner matrix [48].

4.3.2 = The process is generally simplified even further by noting that the T 22 element, relating upstream and downstream acoustic velocities, is generally constant without the presence of a flame and most affected by combustion occurring [6]. T 11 appeared to be consistent across frequencies, T 12 is close enough to zero to be considered as such, and T 21 is comparatively small towards T 22 that its contribution would be greatly overshadowed.

To determine the analytical model for T 22 three relationships were taken into account.

The first was the jump condition across the flame. A schematic of the combustor and flame front is presented below where 1 indicates upstream of the flame, 2 indicates downstream of the flame, and i indicates the fuel injector location.

Figure 4.10: Schematic of combustor used to derive T22 flame model Due to the assumption of compactness, the flame itself is treated as a discontinuity. The equations for continuity, momentum, energy, and state across the flame are as follows [48].

4.3.3 − = − = ≡ 4.3.4 ‖‖ + ‖‖ = 0

4.3.5 1 + + ‖ ‖ = 2

4.3.6 = 0 Where denotes the jump across the flame, S represents the motion of the discontinuity with ‖∙‖ respect to fixed coordinates, and S f with respect to the incoming fluid. The heat release present in

Equation 4.3.7 is further modeled as proportional to the fuel flow.

4.3.7 = ℎ Where S f is the burning velocity, y f is the mass fraction of the fuel in the incoming fuel/air mixture, and h f is the reaction enthalpy of the fuel. Equations 4.3.3-4.3.7 can then be combined and linearized to form the relationship between the acoustic velocities on either side of the flame.

4.3.8 ′ − ′ − 1 − 1 − 1 = − 1 + − + − ̅ − 1

The second relationship was the equivalence ratio fluctuations. At the fuel injection location the mass fraction of the fuel is determined as

4.3.9 , = + Where m f is the mass flow rate of fuel and m a is the mass flow rate of air. Since y f is dependent upon fuel and air flow rates, a fluctuation in either will result in a fluctuation in y f. For the lean combustion conditions used in the experiment, m a>>m f and so the normalized fuel fluctuations at the injector can be modeled as so.

4.3.10 , ′ , ′, = − − , , , The equivalence ratio fluctuations at the flame front can then be described by using a time lag model from the injection point [49] since the fuel concentration fluctuations at the flame front at a time, t, will simply be the same as the fluctuations at the injection location at a time t-τ.

4.3.11 ′, = ′, − This simple model only holds under the assumptions that fuel injection and combustion each take place at one fixed location and there is no dispersion. For most practical combustors these do not hold up and so a distributed time lag function must be added. This is essentially determining an effective equivalence ratio by average the time delays of each position on the flame surface, S.

4.3.12 1 ′, = ′, − Using a probability density function, ξ(τ) to model the time delay dependence and converting from time to frequency domains, 4.3.12 can be rewritten as

4.3.13 , = ,

Where ^ denotes the Laplace transformed fluctuation quantity. Since the integral in Equation

4.3.13 is the Fourier transform it can then be simplified to

4.3.14 ∗ , = ,ℱ Where is the complex conjugate of the Fourier transform. Then, under the assumption of a ∗ ℱ Gaussian probability distribution for the time delays, the fluctuations at the flame location can finally be modeled as

4.3.15 1 , = , − ̅ − 2 Where is the mean time delay and their standard deviation. ̅ The third relationship necessary to derive is the fluctuation in flame speed. Since turbulent, premixed combustion at a high Reynold’s number is of interest here, it can be assumed that the turbulent flame speed is proportional to the turbulence intensity and can be shown that any periodic acoustic perturbation will result in a periodic perturbation in turbulence intensity

[48]. To begin the derivation, consider a combustor with mean flow but no combustion. Part of the flow energy is converted to turbulent kinetic energy, which eventually dissipates and is converted into heat. This irreversible loss can be represented by the following form of the

Bernoulli equation.

4.3.16 ‖‖ 1 1 1 = − ‖ ‖ − ‖‖ − 2 2 Where is the velocity potential and is the loss term, generally determined experimentally for a given combustor geometry. Thus, the flow energy converted to turbulent kinetic energy can be given by the final term in Equation 4.3.16.

4.3.17 1 1 − = 2 2

Where is the turbulence intensity. Linearizing Equation 4.3.16 and relating the velocities to the mean burner exit velocity, , results in the following relationship. 4.3.18 ̂ − ̂ = − + ̅ Where L b is the measure for the potential drop and is the loss term in terms of the burner exit velocity. As with fuel mass fraction fluctuations, the turbulence intensity fluctuations will flow with the mean velocity from point of initiation to the flame front. A turbulent time lag, τt, is associated with this convection. It is then assumed that the turbulent intensity fluctuations decay at the same rate as mean intensity and the following relationship can be formed relating intensity fluctuations at the point of incidence and flame front.

4.3.19 − − − = = = As stated previously, the turbulent flame speed fluctuations are considered proportional to the turbulence intensity fluctuations and so Equation 4.3.19 can be expanded to include the S f.

4.3.20 − = = ̅ As with the fuel mass fraction fluctuations, a Gaussian distribution of time delays will be utilized giving the final expression for flame speed fluctuations as Equation 4.3.21.

4.3.21 1 = − ̅ − ̅ 2 Finally, with the three relationships derived it is possible to analytically model the transfer matrix term T 22 as follows.

4.3.22 1 1 = = 1 + − 1 − ̅ − − − ̅ − 2 2

Experimental verification of the flame model produced good agreement between experimental data and model predictions [48].

Figure 4.11: Comparison of measured (red crosses) and modeled (blue line) values for transfer matrix element T22 for various temperatures [48] However, there were a few drawbacks to utilizing this method including the time delay being curve fitted from experimental data (model cannot be universally applied to flames without also running experiments to verify it) and the experiment requiring three independent tests states to determine the transfer matrix. The experiments also failed to include anechoic terminations which are required when trying to isolate one acoustic network element.

Alemela, et al. [50] conducted a study comparing several different methods for determining the transfer matrix, including the multi-microphone method, a hybrid method of

Rankine-Hugoniot relationships and a measured flame transfer function (FTF), and a hybrid method of Rankine-Hugoniot relationships and an n-τ-σ FTF model. The multi-microphone method has been discussed previously so instead of rehashing all information it will now just be added that the study used an array of four microphones on each side of the test element and determined the complex wave amplitudes A and B by minimizing the following equation.

4.3.23 || ∙ − + → And the transfer matrix utilized is the one Campa and Comporeale used as presented in Equation

3.2.29.

The next method compared was the Rankine-Hugoniot with measured FTF model.

Treating the flame as a compact element it can be thought of as a discontinuity and the jump conditions across the flame can be derived from the conservation equations [51]. These are the

Rankine-Hugoniot relations and the linearized equations with second and higher order terms neglected are presented in the equations below.

4.3.24 ′ = − − 1 +

4.3.25 = + − 1 − ̅ Where h indicates the hot side (downstream of flame), c indicates cold side (upstream of flame) and β is the ratio of specific impedances, . The next step is to determine how the acoustic perturbations affect the heat release through a flame transfer function. Assuming the heat release is mainly affected by upstream velocity perturbations a frequency dependent FTF can be modeled as so.

/ /̅ 4.3.26 = / / = Here I represents the intensity of OH*-chemiluminescence collected in a photomultiplier tube as this is seen as being proportional to and representative of the heat release of a flame [52].

Equations 4.3.24-4.3.26 can then be combined to obtain transfer matrix for this approach.

4.3.27 − − 1 1 + ̅ = ̅ − − 1 1 + − 1 The third method compared in the study involves the same transfer matrix as Equation

4.3.27, but instead of measuring the FTF an n-τ-σ modeled is utilized based on the work by

Schuermanns [48].

4.3.28 1 = − 1 − − 2 Where n indicates the interaction index (a measure of the intensity of the coupling between velocity and heat release). The results of their study showed good agreement between the two experimental methods (multi-microphone and hybrid measured FTF) across all matrix elements in both amplitude and phase, as well as good agreement with the hybrid n-τ-σ model in all matrix elements except for T 12 .

Figure 4.12: (left) Absolute value and (right) phase of flame transfer matrix elements comparison for measured (black square), Rankine-Hugoniot approximation (red line), and OH* chemiluminescence measurement (blue diamonds) results Dowling and Stow [13] took a different approach to flame modelling. They thought of the flame zone as a single axial plane located at a location x=b. The location would also be related to the incoming air velocity by a time delay, τ, which is the time between the fuel’s injection and its combustion. The heat release at the combustion zone, q’, could then be modeled as shown below.

4.3.29 , = − Where Q’ is the heat release per unit area and is given by

4.3.30 = − − − 1 Where β is a nondimensional number expect to lie within the range of zero to ten. In Equation

4.3.24, is taken to be the velocity fluctuation just upstream of the flame zone even though by the definition of τ it should be the fluctuations at the injection point. The authors do note, however, that since the distance between fuel injection and combustion is typically small compared to perturbation wavelengths and so the phase difference will be negligible. With this flame model all heat addition is assumed in the small region of the flame zone and so the governing equations at all locations xb reduce to the homogeneous wave equation.

Then, integrating across the flame zone from x=b -, just upstream of the flame, to x=b +, just downstream of the flame zone give the pressure and velocity jump conditions.

4.3.31 ′ = 0

4.3.32 ′ − 1 ′ = − ̅ Or, expressing Equation 4.3.31 in another manner

4.3.33 − 1 ′ = ̅̅ Substituting in the heat release model given in Equation 4.3.30 into Equation 4.3.33 results in the following.

4.3.34 , = , − , −

Then, in order to find the eigenfrequencies of the system, only solutions with a time dependence, eiωt, were considered. In the upstream section of the combustor the imposed boundary condition

was imposed resulting in the following equations. ̂0 = 0 4.3.35 ̂ = sin

4.3.36 = cos ̅̅ Similarly, the boundary condition imposed at the end of the combustor results in the = 0 following equations.

4.3.37 ̂ = cos −

4.3.38 = sin − ̅̅ The pressure jump condition can then be expressed as

4.3.39 sin = cos − When the velocity jump condition is reevaluated in a similar manner and divided by Equation

4.3.37, the equation becomes

4.3.40 tan tan − = 1 − e Which can be used to find the resonant frequencies of the system. Dowling and Stow further examined the model by varying β and τ independently to see their effects on the first resonant mode.

Figure 4.13: (left) Effect of βββ on first resonant frequency of combustor comparing exact solution (solid line) and one term Galerkan approximation (dashed line), (right) effect of time delay on growth rate for varying βββ [13]

For a β of zero the roots are simply ω=ωn. If only β is changed and τ stays fixed at zero then only the frequency of oscillation will shift, not the stability. In order to become unstable, there must be a time delay. When , both the frequency of oscillation and the growth rate will be ≠ 0 affected. They further determined that in order for the system to go unstable the time delay had to lay within a range determined by the following equation.

4.3.41 − < ℜ − ℎ cot < 0 Their study also examined the effect of a temperature gradient, taken as a step increase from T 1 to T 2 across the flame zone, on the frequency of the system. To avoid reiterating the entire genesis of the governing equations, note that the derivation follows the same procedure and begin at the jump conditions now given as

4.3.42 ′ = 0

4.3.43 1 ′ − 1 ′ = − ̅ Where this time Equation 4.3.43 can be rewritten as

4.3.44 − 1 ′ = ̅ Now, the equation used to evaluate the resonant frequencies becomes

4.3.45 ̅̅ tan tan − = 1 − e ̅ ̅ Thus, the temperature can be seen to have an impact on the wave numbers upstream and downstream of the flame, affecting acoustic wave propagation, as well as introducing a flame impedance effect described by . The effects of varying temperature jumps, as well as ̅ ̅ compact versus distributed heat releases, can be seen in the figure below.

Figure 4.14: Effect of using temperature jump on first resonant frequency for single temperature jump (solid line), uniform temperature rise (dashed line), and five smaller jump approximations (-.-) Campa and Camporeale [14] continued with this model by integrating the equations into a burner transfer matrix model developed in COMSOL. The heat release fluctuations were added to the governing equations of the internal domain of an acoustic network element. In this setup the inhomogeneous wave equation now becomes

4.3.46 1 1 ̂ − ∇∇̂ = − − − exp ̅̅ ̅ Where λ is –iω. They also note that β and τ need to be determined by experiments or CFD models but chose arbitrary, but through experience admissible, values for each. In order to model such an equation in COMSOL, a source term had to be determined compatible with the software.

The necessary equation to be satisfied in the COMSOL environment is given in Equation 4.3.47.

4.3.47 ̂ ∇ ̂ − = ̅ ̅ ̅ Where Q CM is the COMSOL source term. The ^ accent indicates a complex quantity, which for the fluctuation variables is defined by the equation below, pressure given as an example.

4.3.48 = ℜ̂exp When the above relationship is substituted into the inhomogeneous wave equation it can then be expressed as

4.3.49 ∇ ̂ − 1 ̂ − = − ̅̅ ̅ ̅̅ The relationship between Equations 4.3.47 and 4.3.49 can now clearly be seen and so the source term inputted into the COMSOL model can be expressed as such.

4.3.50 − 1 = − ̅̅ For the present study, a similar approach is used for the source term implemented in

COMSOL. Recalling the definition of the FTF given in Equation 4.3.26, the relationship can be re-expressed in terms of a gain, G, and phase, ϕ, given as and . = | | = / 4.3.51 = () () / = Equation 4.3.50 can then be combined with 4.3.51 to produce an acoustic source in terms of a measured flame transfer function.

− 1 − 1 4.3.52 = − () = ()exp ( ()) ̅̅ ̅̅

By setting the temperature, pressure, and medium conditions within COMSOL density, the speed of sound, ratio of specific heat capacities, and velocities can be calculated in the program.

Magnitude of the gain of the FTF and phase at each frequency were determined experimentally, as will be discussed later, and implemented as an interpolation function.

5. Experimental Details

5.1. Single Nozzle Acoustically Tunable Gas Turbine Combustion Rig Setup

Figure 5.1 shows the combustor test rig consisting of an inline heater to preheat the inlet air, an air modulation device (siren), a siren interface section to allow for air to flow through and bypass the modulation device, an inlet plenum, a fuel nozzle mounted inside a radial-radial swirler body, a combustion chamber optically accessible through quartz glass windows, a pressure vessel to contain the high pressure combustion tests, a variable length transition tube to acoustically tune the combustor, an outlet pressure screw to regulate the combustor pressure, and a water cooled exhaust plenum running to a muffler. Each section of the combustion rig is discussed in further detail below.

Figure 5.1: Overview of single nozzle, acoustically tunable, gas turbine combustion rig

Figure 5.2: Cross sectional view of combustion rig 5.1.1. Inline Heater

The inline air heater used to preheat the air up to the desired inlet temperature was a specialty flanged inline air heater made by Osram-Sylvania (Part #F073377, 72kW). All of the main airflow for the combustion tests was run through the heater and flow measurements were taken upstream of the heater using a Teledyne LS-4F flow sensor (flow capacity of up to 3000SLPM with a full scale accuracy of ±1%) displayed on a Teladyne THCD-100 digital display.

5.1.2. Air Siren

Inlet air was modulated for the flame transfer function tests using an air siren (Figure 5.3 front and Figure 5.4) consisting of a stator, rotor, and DC motor capable of changing the frequency of modulation.

Figure 5.3: Inlet section connected to air modulation siren and siren bypass line

Figure 5.4: Air modulation siren Figure 5.3 shows the motor, a direct current permanent magnet motor controlled by a motor controller (Leescan, Model # 17430.00 F07) connected to the siren. The siren works by connecting the motor to the rotor shaft with a set of gears (motor:rotor gear ratio 4:1) and

56 spinning a rotor at a desired frequency. The rotor, shown in Figure 5.4, is a disc containing a number of holes, in this case ten, that rotate through the flow path inside of the siren. With this setup modulation frequencies of up to 1000Hz are obtainable. The placement of the rotor can be adjusted using a set of screws and springs in the rear of the siren to ensure the rotor would not scrape against the inside of the stator during operation. The experimental rig was also fitted with a siren interface which includes a bypass line to allow for adjusting the level of modulation (i.e. the instantaneous peak amplitude of pressure fluctuations as measured in the inlet by the pressure transducers and LabView program during rig operation). The lines running from the heater through the siren and heater to the bypass were both fitted with globe valves to control flow rate.

When the bypass line in restricted, more air flows through the siren increasing levels of modulation in the inlet and when the bypass line is less restricted the opposite effect occurs and modulation levels are lowered.

5.1.3. Inlet Plenum

A stainless steel inlet plenum connects the siren interface to the combustion chamber. It measures 0.254m long and is square on the inside (0.04445m x 0.04445m). The plenum is fitted with a static pressure port connect to a digital pressure sensor (Ashcroft, Model

#302174SDD2LXFF160#), a temperature port where a type K thermocouple (Omega KMQXL-

062G-6 connected to an Omega display CNi3253) measures the inlet temperature, and several connection ports for the dynamics pressure sensors used to measure acoustic pressure that will be discussed in more detail in Section 3.2.

5.1.4. Fuel Nozzle and Swirler

A single point fuel nozzle (Parker Hannifin, Part#: 6060097E01, Maximum Flow Rate: 0.5GPH water at 100psi, hollow cone, 75 o spray angle) was utilized with a counter rotating radial-radial

57 swirler with a swirl number of 0.6 designed by Yi-Huan Kao at the UC Combustion Research

Lab and 3D printed by the University of Cincinnati Research Institute (UCRI). Specific information on the geometry can be found in the table below. Previous research has been done with respect to the swirler mounting by Kao [53] and Haseman [54] and concluded that the mounting location has an effect on the combustion dynamics characteristics. For the present study swirler was mounted so that the flared end would sit flush with the splash plate at the entrance to the combustor. The fuel nozzle was mounted flush with the end of the mounting piece (gray section in the CAD model in Figure 5.5) to allow the spray to impinge upon the venture passage’s inner surface (teal section in CAD model in Figure 5.5) and allow for better atomization.

Figure 5.5: 3D printed and cross sectional CAD view of swirler used in testing

SN0.6 Swirler Inner Passage Outer Passage Height (m) 0.0020 0.0038 # of Vanes 22 20 Vane Thickness (m) 6.35E-4 6.35E-4 Vane Angle (deg) 52.8 70.2 Flare Angle (deg) 36 Exit Diameter (m) 0.0254 Table 5.1: Swirler Geometry Extensive aerodynamics testing on ten separate 3D printed swirlers of the same design used in this study (referred to as UCRI-2 in cited works) established the replicability of the 3D printed

58 part and adherence to design specifications [54] [55] [56]. The results of the studies are summarized in Table 5.2.

Swirler Projected ṁ at 4% dp, 60°F Projected ṁ at 4% dp, 400°F 2 Number Aeff (in ) % Var (lbm/hr) (lbm/hr) 1 0.1517 6.0 77.74 60.45 2 0.1492 4.4 76.43 59.43 3 0.1371 -4.0 70.27 54.64 4 0.1325 -7.7 67.88 52.78 5 0.1349 -5.7 69.12 53.75 6 0.1385 -3.0 70.99 55.20 7 0.1466 2.7 75.10 58.39 8 0.1393 -2.4 71.39 55.51 9 0.1476 3.4 75.64 58.81 10 0.1490 4.2 76.33 59.35 Table 5.2: Measured Effective Area and Projected Mass Flow Rates for Ten SN0.6 Swirlers [54] Variation percent was based on the deviation from the designed value of 0.13in 2.

It was then necessary to establish the effective area of the swirler used in the present study (separate from any used in the previous studies and characterizations). The rig design also utilizes an effusion plate with ninety-two holes surrounding it that allow air to bypass the nozzle and form a buffer zone between the flame and the quartz window so the flow split through and around the swirler must be determined for accurate equivalence ratio calculations. The effusion plate setup can be seen in Figure 5.6. The flow split determined was between the swirler as a whole (i.e. both inner and outer vaned passages), with fuel nozzle in place to block the center nozzle passage, and the total of the ninety-two, 0.00159m diameter holes surrounding it. When testing and during normal operation of the rig a splash plate flush with the exit flare of the swirler and with a 0.003175m gap on all sides, shown in Figure 5.7, is present.

Figure 5.6: Swirler mounted on effusion plate viewed from upstream/inlet side (left) and downstream/combustor side (right). Splash plate not included.

Figure 5.7: Downstream/Combustor side view of swirler mounted with splash plate attached and flush with flared exit of swirler

The pressure drop was measured for several flow rates with the effusion holes open and blocked and used to calculate the effective area for each configuration using Equation 5.1.1.

5.1.1 = 2∆ This resulted in an effective area of 9.68E-5m 2 for the swirler, in line with values determined from the previous swirler characterizations, and an effective area of 8.45E-5m 2 for the effusion plate. This gives a flow split of 55%/45% through swirler/effusion holes.

Main Air Flow Rate vs. dP 18.00 16.00 14.00 Effusion 12.00 Holes Open 10.00 8.00 Effusion dP (%) 6.00 Holes Blocked 4.00 2.00 0.00 900 1400 1900 2400 m_main (SLPM)

Figure 5.8: Pressure drop vs. main air flow rate for open and closed effusion hole cases 5.1.5. Combustion Chamber and Pressure Vessel

The combustion chamber is made of 9.5mm thick stainless steel and measures 0.0635m x

0.0635m on the inside and 0.254m long. On three sides of the combustor, aligned with the exit plane of the swirler, are optically accessible areas allowing a 0.0508m by 0.09525m field of view. These hold 6.35mm thick quartz glass windows, sealed with graphite gaskets, allowing optical access to the entire flame. The top window can be swapped out for a stainless steel dummy window with 3.175mm diameter stainless steel tubing attached that run out through the top of the pressure vessel to accommodate dynamic pressure measurements in the combustion

61 chamber. The combustor is also fitted with an ignition port on the bottom through which a stainless steel tube carrying hydrogen and housing a spark plug is run in order to ignite the flame.

The entire combustor is housed inside a pressure vessel made of 19mm thick stainless steel with

12.7mm thick quartz glass windows. A separate cooling air line runs room temperature air into the pressure vessel to pressurize the chamber and provide impinging airflow over the windows to reduce thermal stress and prevent them from breaking during testing. This design allows for a pressure difference between the pressure vessel and ambient conditions of up to 500kPa. The pressure differential across the inner vessel (combustion chamber to pressure vessel) is kept to no more than 0.7kPa to both avoid stress on the thinner inner quartz windows and reduce any cross flow between the cooling air in the pressure vessel and main air flow in the combustor.

5.1.6. Transition Tube

The end of the combustion chamber is butted up against a transition tube made from a stainless steel pipe nipple (3” NPT). Tubes 0.254m-, 0.381-, and 0.508m-long were fabricated to allow for acoustic tuning of the combustor. By changing combustor length, the resonant frequency of the combustor will change as well. All transition tubes are fitted with three thermocouple ports thorough which temperature gradient measurements can be obtained.

5.1.7. Pressure Screws

Combustor pressure is set through the use of bored out screws fitted into a water cooled flange at the end of the transition tube. The screws have varying diameters to vary the combustor pressure during the tests and can be swapped out between runs. The screws are designed to choke the exit so no downstream perturbations in the exhaust line can propagate back into the combustor and influence the acoustic measurements.

5.2. Dynamic Pressure Sensor Setup

In order to measure acoustic pressure, an array of dynamic pressure sensors (PCB

Piezoelectronics Model # 112A22) were used. The sensor outputs were run through a PCB

Piezoelectronics signal conditioner (Model # 482C) where the measured signal is amplified before running to the DAQ (data acquisition) board to be acquired by the computer. In order to protect the sensors from damage sue to heat from the preheated inlet air or combustion they were recess-mounted on a wave guide designed according to specifications obtained from PCB

Peizoelectronics.

Pressure Sensor Waveguide Sensor Housing Tube

Figure 5.9: Piezoelectric pressure transducer seated in wave guide mount for recessed mounting The wave guides are made with stainless steel tubing (Outer Diameter = 3.175mm) and those mounted in the inlet section were 89mm long while the ones mounted in the combustor section were comprised of a 152.4mm long tube welded to the dummy plate on top of the combustor butted up against a wave guide with a 50.8mm long tube and connected through a union. In each case the inner most end of the wave guide was flush-mounted with the inner wall of the rig.

Mounting the pressure sensors in such a way can result in attenuation or amplification of the original pressure signal that would be collected using a flush-mounted sensor. To account for

63 this difference each wave guide had to be calibrated against a flush mounted sensor in a calibration tube. The theory and process for this calibration can be found in Appendix A.

Orifice Impedance Test Setup

The acoustic boundary conditions at the inlet and exit of the combustion rig were measured using a multi-microphone setup based on the theory and error analysis described in sections 4.1 and 4.2. A schematic for the inlet orifice impedance setup can be seen in Figure

5.10.

Figure 5.10: Inlet orifice impedance test setup Impedance measurements are taken with an array of three dynamics pressure transducers in the inlet and a speaker downstream forcing the acoustic pressure. A fourth transducer was used as an error check to confirm the reliability of the measured values. Data was taken at frequencies between 200 and 800Hz as well as at main air flow rates of 1500SLPM and 2000SLPM to match the region of instability and flame transfer functions. To isolate the effect of the orifice and maximize the signal level from the sensors, the swirler and nozzle were removed leaving the combustor and inlet completely open. The pressure difference across the orifice was checked to ensure that the flow was choked as it would be during normal combustion testing. The three

64 microphones used in the test were located at positions x 1, x 2, and x 4 while the microphone at x 3 was used as an error check. The exact locations of each can be seen in Table 5.4.

Mount Distance Downstream of Orifice, x (m) 1 0.252 2 0.214 3 0.176 4 0.125 Table 5.3: Inlet transducer mount locations The exit impedance tests were setup in a similar manner to the inlet impedance measurements except the pressure screw was the orifice of interest. A schematic of the test setup can be seen in

Figure 5.11 and photos of the testing setup are Figures 5.12 and 5.13. An array of four microphones was mounted in the 0.508m-long transition tube, this time with mounts labeled two, three, and four being used to calculate the wave coefficients and mount one being used for determining measurement accuracy. For the exit impedance test the loudspeaker was housed inside a speaker housing to keep the pressure difference across the diaphragm of the speaker within operating limits. This is required since the pressure inside of the transition tube is at least twice the ambient pressure to ensure the orifice is choked. Air flowed into the speaker housing and around the speaker, offset from the exit to the housing on screws to allow air to flow around it, through a reducing coupling, and into the 0.508m-long transition tube. For the inlet impedance test, a speaker housing was not necessary since the duct was nominally at atmospheric pressure.

The air exited through outlet holes between the speaker’s diaphragm and the orifice so pressure upstream of the orifice was at least twice as high as ambient pressure while the speaker was located downstream of the orifice. Locations for the microphone locations for the exit impedance test can be seen in Table 5.4.

Mount Distance Upstream of Orifice, x (in) 1 0.438 2 0.387 3 0.260 4 0.108 Table 5.4: Exit orifice impedance test transducer locations

Figure 5.11: Schematic of exit impedance setup

Figure 5.12: Exit orifice impedance testing setup

Figure 5.13: Microphone mounting on transition tube for exit orifice impedance testing Which microphone locations are used to determine the wave coefficients and impedance was decided based on avoiding acoustic pressure nodes at the microphones for the frequencies of interest. At an acoustic pressure node the pressure is at a minimum and such a low value can cause large errors in the calculations. Caution must be taken to avoid such locations along the duct, a very difficult task over the range of frequencies of interest, 200Hz-800Hz. Therefore, the locations of the sensors were selected after calculating the quarter and three-quarter wavelength distances from the source for each frequency and avoiding those near the known frequencies of instability.

5.3. Flame Transfer Function Setup

The flame transfer function defines the relationship between the inlet velocity fluctuations just upstream of the nozzle and the heat release fluctuations in the combustor. The normalized inlet

velocity fluctuations, , upstream of the nozzle was used as the input function for the FTF. The inlet velocity fluctuations can be measured using a two microphone method [57]. This method 67 uses the finite difference approximation given in Equation 5.4.1 to model the acoustic velocity at a point half way between two microphones.

5.4.1 1 − ≈ − Δ Δ Since the inlet to the rig was equipped with multiple microphone ports, four transducers total were mounted when running the tests and each was analyzed in pairs to determine the velocity.

In other words, microphones one, two, three, and four all collected data during the tests and then the calculated velocities using microphones one and two were compared to using one and three were compared to using one and four and so on. In total, six separate pairings of microphones, with four different spacings were available. As was discussed in Section 4.1, microphone spacing does have to be taken into account based around the frequencies of interest. Therefore, the frequency ranges based on Equation 4.1.10 for each pair of microphones were the only points used for each set. The locations for each microphone were the same as in the inlet impedance tests and can be found in Table 5.4. The gain and phase corrections as discussed in Section 5.2 were also applied to these measurements. When spacing was controlled for each frequency had between two and six gain values from the different microphone pairs averaged out to produce the final gain curve implemented in the COMSOL model. The results will be presented in Section

6.2.

The output function for the FTF is the unsteady heat release and was measured using

OH* chemiluminescence. When combustion occurs electronically excited hydroxyl radicals

(OH*) are formed at the flame front. Upon relaxation these molecules emit electromagnetic radiation in a narrow band around 308nm and this is widely accepted as proportional to heat release and has been shown to correlate well with lean burning flames [52]. However, as some of the test conditions run in the present study encroach upon the stoichiometric and richer burning

68 equivalence ratios it has to be established that OH* chemiluminescence is an appropriate measure of heat release here. The flame produced in this study is yellow and very sooty. As such, any emissions collected from the flame will include both OH* indicating heat release and background emissions from the soot. It has been shown through emission spectroscopy and coherent anti-Stokes Raman scattering (CARS) techniques that soot particle temperature is good representation of flame temperature [58]. Therefore, if it can be shown that soot temperature fluctuations are correlated with inlet velocity fluctuations and OH*- chemiluminescence, then emissions from a sooty flame can be used as an indicator of heat release. Research conducted by

Knadler, et al. [59] and Cakmacki, et al. [60] concluded that even under sooty conditions emissions taken with a three-line pyrometer, used to measure soot temperature, were in phase with inlet pressure fluctuations indicating that it is a reliable correlation to combustion dynamics.

The three line pyrometer setup used in the previous studies by Knadler and Cakmacki, can be seen in Figure 5.14. A more complete discussion on three-line pyrometry, including theory and calibration can be found in Appendix C.

Figure 5.14: Optical arrangement of the three line pyrometer

It consists of three separate photo-multiplier tubes (PMTs) arranged behind three different bandpass filters (660nm, 730nm, and 800nm, FWHM = 10nm). A collection lens collects emissions from the entire flame and using a set of two dichroic mirrors, is redirected into the three PMTs. The wavelengths were chosen to exclude molecular emissions, such as OH*.

To ensure soot temperature fluctuations were in phase with pressure fluctuations for the current study, data was collected using both the three line pyrometer and the OH* chemiluminescence setup under naturally occurring instability. The time traces for each of the three lines of the pyrometer, the OH* chemiluminescence measurement, and the pressure transducer located farthest downstream in the inlet (Mount 1) were plotted against each other and can be seen in Figure 5.15.

Time Traces Under Naturally Occurring Instability 1

0.8

0.6

0.4

0.2 Downstream Pressure 800nm 0 730nm 4.1 4.102 4.104 4.106 4.108 4.11 -0.2 660nm

NormalizedAmplitude OH -0.4

-0.6

-0.8

-1 Time (s)

Figure 5.15: Time traces of downstream pressure,, three line pyrometer emissions, and OH* chemiluminescence emissions showing in-phase relationship

Since OH* and three line pyrometer measurements were taken at two separate times (each required optical access to the same window and thus could not be run simultaneously) the time indicated on the graph is arbitrary. However, each is correctly plotted in its relation to the downstream pressure, corrected for phase delay between flush and recess mounting, which was consistent between the two runs. Thus, since both soot temperature and OH* chemiluminescence are fluctuating in phase with the inlet pressure fluctuations, each can be considered a valid indicator of combustion dynamics and OH* chemiluminescence is an appropriate output function for the FTF.

A PMT (Products for Research, PR1402SHCE PMT) was used to gather the OH* emission and was equipped with a narrow bandpass filter (Andover Co., 307FS10-25, center wavelength=308nm, bandwidth=5nm). The chemiluminescence emission from the whole flame was collected by a lens (Uncoated UV Plano-Convex lens, f=75.0, ϕ=1.0”) and directed to the sensor of the PMT. The signal from the PMT is run with a high voltage supply (Pacific Precision

Instruments Model #206) and the signal from it was amplified (Driel Model#70710) before being digitized and recorded. Figure 5.16 shows a schematic of the OH* chemiluminescence measurement setup and a photo of the equipment used.

Figure 5.16: (left) Schematic of OH* chemiluminescence setup focusing emissions from the side window of the combustor onto the PMT sensor and (right) actual experimental hardware in place for testing

All of the FTF data was collected using a custom made LabView program which collected 65536 data points per channel at a rate of 8000Hz. The data was then processed through MATLAB where all corrections were applied.

5.4. High Speed Camera Setup

A color high speed camera (Vision Research Phantom Miro LC310) equipped with a Nikon

Micro-Nikor lens (f=105mm) was used to capture instantaneous, color flame images. The camera is capable of recording up to 10,000 frames per second at a 512x512 pixel resolution and is controlled by the provided Phantom Camera Control (PCC) software. It produces .cine files which can be viewed and converted through the Phantom CineViewer software. Also utilized was a black and white high speed camera (Pi-Max 4 1024i) with a 308nm narrow bandpass filter and a high speed intensifier (Lambert Instruments, HiCATT XXCUD) to capture instantaneous emissions occurring in the OH* band. This camera is capable of recording 1024x1024 pixel resolution images with up to a 32MHz frame rate. High speed videos in this study were collected at a 512x512 pixel resolution at 10,000fps with an exposure time of 50 µs.

5.5. COMSOL Model Development

The finite element model used to predict combustion instabilities was developed in COMOSL

Multiphysics, a commercially available 3D finite element program. This software allows for the development of the CAD model inside the program environment as well as innate coupling of the necessary physics domains (thermodynamics, acoustics, flow, etc.) to produce more accurate results. Equations from various physical domains are solved simultaneously allowing the interplay of variables affected by multiple domains to feedback into each other and be solved iteratively. The basic geometry for the model is shown in Figure 5.17. The model runs from left

72 to right with the origin of the coordinate system being placed at the centroid of the far left of the square inlet. Distances on the axes are measure in meters.

Figure 5.17: COMSOL geometry with elements representing (from left to right) inlet, nozzle, combustor, flame zone, and transition tube The geometry consisted of five primitive elements native to the COMSOL environment: a block for the inlet, a cylinder for the nozzle, a block for the combustor, a block for the flame zone, and a cylinder for the transition tube. Each was modeled to match the actual rig geometry as closely as possible. To further simplify the geometry the nozzle was modeled as a cylindrical element with the same length as the actual nozzle (0.023m or the distance from the fuel injection location at the base of the swirler to the flared exit sitting flush with the entrance to the combustor) and same area as the effective area of the nozzle and effusion holes from the data measured in Figure

5.8). Removing the component entirely would have resulted in an overall shorter combustor and changed the resonant frequency leading to inaccurate results.

A study was conducted on the 0.0598m-long transition tube geometry to see how the mesh density would affect the results. The test was run with no source, flow, or impedance and the resonant frequency of the model was recorded for each of the physics controlled meshing options available in COMSOL. Table 5.6 shows the converged eigenfrequency for each of the

73 built-in mesh densities in COMSOL along with the domain elements, degrees of freedom, and solution time for each.

Mesh Density Domain Elements Deg. of Freedom Time per Iteration (s) Eigenfreq (Hz) Extremely Coarse 179 442 3 474.7 Extra Coarse 477 960 3 473.35 Coarser 577 1238 3 475.45 Coarse 1358 2687 4 475.04 Normal 2922 5160 4 473.73 Fine 6620 10879 6 474.31 Finer 10101 16315 8 474.29 Extra Fine 23047 35460 14 474.24 Extremely Fine 58843 87719 31 474.22 Table 5.5: Mesh density impact on eigenfrequency of rig model without flow or source present Every option from the Fine mesh and below results in an eigenfrequency within 0.019% of the finest mesh, an order of magnitude closer than the normal mesh and above. The Fine mesh, being the fastest computationally and within reasonable potential error of the finest mesh available, was chosen as the mesh density for all work from this point forward. It can be seen applied to the

0.0508m-long transition tube geometry in Figure 5.18.

Figure 5.18: COMSOL geometry for the 0.0508m-long transition tube case with fine mesh visible The study was also replicated after the final results were obtained with the 2000SLPM

0.508m-long transition tube case to see how much the mesh density would affect the results with realistic parameters in place. Those results are summarized in Table 5.7.

Mesh Density Domain Elements Deg. of Freedom Time per Iteration (s) Eigenfreq (Hz) Extremely Coarse 274 1240 5 410.58-6.0952i Extra Coarse 545 2139 5 423.95-4.2933i Coarser 962 3647 6 424.30-3.8772i Coarse 1361 5042 6 428.55-2.8649i Normal 3204 10311 9 428.06-3.3716i Fine 7642 21985 12 429.95-3.0511i Finer 12569 34850 23 430.55-2.9629i Extra Fine 26602 69705 38 430.82-2.9077i Extremely Fine 66152 166443 191 431.27-2.7745i Table 5.6: COMSOL grid impact study results with experimental operating conditions in place for 2000SLPM 0.508m- long transition tube case The fine mesh is shown to still produce accurate results on par with the densest mesh available in

COMSOL. Instability was still predicted accurately with only a 0.3% difference in the fluctuation frequency for a 95% reduction in solution time per solver iteration.

The governing equation for the pressure acoustics module of COMSOL in the frequency domain is the simplified 1D Helmholtz Equation given in Equation 5.6.1.

5.6.1 1 ∇ ∙ − ∇ − − = Where ρ0 is the density of the fluid, p is the pressure, c is the speed of sound, ω is the angular frequency, and q and Q are acoustic source terms, dipole and monopole, respectively. In the case of this model, the dipole source term, q, will be set to 0 and the monopole source term, Q, will be used to model the flame. This is where the FTF must be implemented into COMSOL. The monopole source term used was defined in Section 4.3 Equation 4.3.52 under the discussion of flame modeling techniques and is restated below.

4.3.52 − 1 Q = iω exp Where γ is the ratio of specific heats, is the velocity fluctuation, is the mean heat release, G is the gain of the FTF as a function of frequency, and φ is the phase of the FTF as a function of

75 frequency. The heat release model is set to be emitted over the volume of a block element located in the combustor. The exact location and dimensions of the heat release zone were determined experimentally through the use of high speed camera imaging and are discussed in

Section 6.3.

Non-hard wall boundary conditions can also be added to improve model accuracy. For those boundaries where the impedance will be measured the information will be modeled in

COMSOL according to Equation 5.6.2.

5.6.2 1 − ∙ − ∇ = − Where n is the unit normal vector to the boundary, p t is the total acoustic pressure field, and Z i is the specific acoustic impedance. Many of the variables defining the experimental environment are calculated in COMSOL automatically. However, some of the basics that differentiate between the various operating conditions need to be inputted before COMSOL can apply them to the eigenstudy. Those values that needed to be inputted can be found in Table 5.8. Along with the flame transfer function, inlet and exit impedance, and temperature profile, these measurements are sufficient to produce an accurate model of the operating conditions.

Input Value Input Parameter 1500SLPM Cases 2000SLPM Cases Combustor Pressure 303975 Pa 364770 Pa Air Flow Rate 0.030625 kg/s 0.040833 kg/s Gas Constant 287 J/kg/K Mean Heat Release 43.389 kW Temperature Upstream of Flame 449.8 K Inlet Speed of Sound 425.13 m/s Air Velocity in Inlet 6.58 m/s 7.31 m/s Air Velocity in Combustor 9.06 m/s 9.07 m/s Inlet Density 2.35 kg/m 3 2.83 kg/m 3 Table 5.7: Exemplary input parameters for the COMSOL model

Density and speed of sound through the rest of the combustor will change along with the temperature gradient specified and so are not listed here but are calculated at the necessary points by COMSOL.

6. Results

6.1. Stability Map

A parametric study of the combustor was first run to determine the natural stability regimes for various operating conditions. Main air flow rates were set to 1500, 1750, and

2000SLPM. Due to minimum flow rate constraints on the heater, 1500SLPM was the minimum flow rate allowable without risk of burning out the heating element. The inlet temperature was varied between 422K, 450K, and 478K. Fuel flow rates were set to 8.8E-4, 10.1E-4, 11.3E-4, and 12.6E-4 kg/s and a table of the resultant equivalence ratios at each air and fuel flow rate can be seen below.

ϕ Fuel Flow Rate (pph) 7 8 9 10 Total Air 1500 0.767 0.877 0.987 1.096 Flow Rate 1750 0.658 0.752 0.846 0.940 (SLPM) 2000 0.576 0.658 0.740 0.822

Table 6.1: Equivalence ratios for stability map operating conditions The 0.254m and 0.508m-long transition tubes were used to alter the resonant frequency of the combustor as well. These various operating conditions resulted in combustor gauge pressures ranging from 172-310kPa. The resultant stability map for each transition tube is shown below.

Pressure fluctuations shown are those taken by the dynamic pressure transducer located in the combustor with gain corrections applied.

Combustor Pressure Fluctuations

0.5

0.4

0.3

P' P' (psi) 0.2 8.8E-4 kg/s 0.1 10.1E-4 kg/s 0 11.3E-4 kg/s 11.3E-4 kg/s 12.6E-4 kg/s 8.8E-4 kg/s

Figure 6.1: Acoustic pressure fluctuations in the combustor for the 0.254m-long transition tube setup

Combustor Pressure Fluctuations

0.3 0.25 0.2 0.15

P' P' (psi) 8.8E-4 kg/s 0.1 10.1E-4 kg/s 0.05 0 11.3E-4 kg/s 11.3E-4 kg/s 12.6E-4 kg/s 8.8E-4 kg/s

Figure 6.2: Acoustic pressure fluctuations in the combustor for the 0.508m-long transition tube setup

Frequency ranges for the observed pressure fluctuations in the combustor were approximately

600-700Hz for the 0.254m-long transition tube and 400-500Hz for the 0.508m-long transition tube. The greatest natural instability was observed for the 10” transition tube configuration at a main air flow rate of 2000SLPM, inlet temperature of 450K, and a fuel flow rate of 8.8E-4 kg/s.

This resulted in pressure fluctuations of 3.1kPa at a frequency of 631Hz. A similarly unstable condition arose from the nearly identical operating conditions but with a fuel flow rate of 8pph instead. This resulted in pressure fluctuations of 2.62kPa at a frequency of 645Hz. Comparing the 0.254m and 0.508m transition tubes, the 0.508m generally resulted in lower maximum pressure fluctuations but across the board showed slightly higher fluctuations as a whole. As the purpose of the study is to develop a model capable of predicting natural instabilities, several operating conditions had to be chosen to validate the model against and should involve both stable and unstable regimes. The four operating conditions chosen were

• Case 1: 2000SLPM main air flow, 450K inlet temperature, 10.1 kg/s fuel flow, 0.254m

transition tube (unstable)

• Case 2: 2000SLPM main air flow, 450K inlet temperature, 10.1 kg/s fuel flow, 0.508m

transition tube (unstable)

• Case 3: 1500SLPM main air flow, 450K inlet temperature, 10.1 kg/s fuel flow, 0.254m

transition tube (stable)

• Case 4: 1500SLPM main air flow, 450K inlet temperature, 10.1 kg/s fuel flow, 0.508m

transition tube (stable)

This gives two stable and two unstable conditions utilizing two different air flow rates and two different geometries providing a range of values to test the model against. Each condition will

80 henceforth be referred to by its case number listed above. Example spectra from the combustor pressure plots for the four conditions are shown below.

Figure 6.3: Combustor pressure fluctuation spectrum for Case 1

Figure 6.4: Combustor pressure fluctuation spectrum for Case 2

Figure 6.5: Combustor pressure fluctuation spectrum for Case 3

Figure 6.6: Combustor pressure fluctuation spectrum for Case 4 6.2. Flame Transfer Function Measurement

The next step in model development is to understand the flame’s response to disturbances in the flow. This was accomplished through the procedure laid out in Section 5.4. The main air flow was modulated between frequencies of 200Hz and 800Hz to gather the flame response over frequencies covering all cases of natural instability. Since flame response is dependent on equivalence ratio FTFs for both the 1500SLPM and 2000SLPM cases were determined. The gain for each frequency was determined by calculating the slope of a linear trend line when normalized velocity measurements are plotted against normalized chemiluminescence measurements. Data was collected over a range of modulation intensities to ensure a linear response of the flame.

1500SLPM, 400Hz 9 y = 0.487x 8 7 6 5 4 OH'/OH 3 2 1 0 0 5 10 15 20 V'/V

2000SLPM, 400Hz 25 y = 0.9405x 20

OH'/OH 10

0 0 5 10 15 20 25 30 V'/V

Figure 6.7: Exemplary plots used in determining the gain of the FTF for (top) 1500SLPM case fordced at 400Hz and (bottom) 2000SLPM case forced at 400Hz The gain and phase of each condition can be seen in the plots below. The phase is used to calculate the total time delay between the two microphone measurement location and flame location by Equation 6.2.1.

6.2.1 Δ 1 = − × Δ 360

Since the convection time through the inlet, nozzle, and combustor can be calculated based on velocity and measurement locations, a general idea of the appropriate time delay can be determined and used as a basis for unwrapping the phase measurements to the necessary angle.

Since the phase difference between the two measurements will only be reported in a 360 o interval by the MATLAB program, the unwrapping accounts for the additional full cycles of delay. As will be discussed further in Section 6.4, the flame model inputted into COMSOL needs only the time delay between fuel injection and combustion and so the convection time from velocity measurement location to fuel injection will be subtracted off the values calculated from the phase plots, 16.878ms and 18.994ms for 2000SLPM and 15000SLPM, respectively.

Flame Transfer Function Gain 1.4 1.2 1 0.8 1500SLPM

Gain 0.6 2000SLPM 0.4 1500 Avg 0.2 2000 Avg 0 150 350 550 750 Frequency (Hz)

Figure 6.8: Gain of the FTF for 2000SLPM and 1500SLPM cases

Flame Transer Function Phase 0 0 200 400 600 800 1000 -1000

-2000

-3000 y = -6.0759x 1500SLPM 2000SLPM -4000 Phase Phase Angle (deg)

-5000 y = -6.8379x

-6000 Frequency (Hz)

Figure 6.9: Phase of the FTF for 2000SLPM and 1500SLPM cases As expected the FTF acts as a sort of low pass filter with higher gain at lower frequencies trailing off to lesser response at higher frequencies [61] [62]. This is due to the increased dispersion of the perturbations at higher frequencies, lessening their impact on the flame. As discussed in

Section 5.4, the final gain curve was produced by averaging several data points, each presented in Figure 6.8, from various two microphone pairs controlled for spacing at each frequency. Since fluctuations are now forced instead of natural, the travelling wave should produce similar acoustic pressure amplitudes at each microphone location in the inlet, resulting in similar velocity perturbations calculated with each set of microphones, finally yielding similar gain values for each frequency. The tight grouping of the included pairs at each frequency for both flow rates that can be seen in Figure 6.8 indicates that the measurements are accurate and the spacing was properly controlled for at each frequency.

6.3. High Speed Flame Imaging

To determine the location of the flame zone and time delay used in the finite element model, high speed videos of the flame naturally unstable, naturally stable, and under modulation were

86 taken with both a color high speed camera and an integrated charge coupling device (ICCD) camera with a UV filter attached to the lens to capture chemiluminescence fluctuations.

6.3.1. High Speed Color Imaging

The high speed color camera was set up to capture the natural flame fluctuations under several operating conditions. Both unstable conditions (Case 1 and Case 2) and one stable condition

(Case 4) were captured. The overall averaged image for each condition along with the upstream and combustor pressure fluctuations and locations of peak and average intensities is shown below.

Figure 6.10: Averaged high speed color camera images for (top left) Case 1 (top right) Case 2 and (bottom) Case 4 showing pressure fluctuations in the inlet (P1') and combustor (Pcomb'), frequency of fluctuations

Each video was collected at a frame rate of 10,000fps with a 50 µs exposure and 512x512 pixel resolution. Overall the two unstable flames resemble each other very closely with similar intensity centroid locations, 0.025m and 0.026m downstream of the nozzle, and maximum intensity locations, 0.0190m and 0.0196m. The stable flame, however, is longer and fuller than its leaner counterparts with centroid and maximum intensity locations of 0.030m and 0.0278m, respectively. From this it can be concluded that at least in the visible spectrum the level of modulation in the combustor does not affect the shape or location of the flame, only the equivalence ratio will matter.

6.3.2. OH* ICCD Imaging

Since OH* chemiluminescence is being used as the measure of heat release then the more important images to analyze are those taken with the UV filter and ICCD camera. These images were taken at the same operating conditions as the color high speed camera with the addition of forced modulation conditions at 200Hz, 400Hz, 600Hz, and 800Hz as well for both the

2000SLPM and 1500SLPM flow rates. Averaged images for all of these conditions are presented below.

Figure 6.11: Averaged ICCD images for (top left) Case 1 (top right) Case 2 and (bottom) Case 4 with locations of peak and centroid intensity

Figure 6.12: Average ICCD images of 2000SLPM 20in case under forcing from 200Hz, 400Hz, 600Hz, and 800Hz

Figure 6.13: Average ICCD images of 1500SLPM 20in case under forcing from 200Hz, 400Hz, 600Hz, and 800Hz As with the color high speed images, the two unstable cases resulted in very similar centroid locations. However, when only taking OH* emissions into account, the stable case also had a similar centroid. All three were located 0.023m±0.001m downstream from the nozzle, a less than

5% difference. Even for the cases of forced modulation the centroid location fell within that range for all but one test condition. The locations of peak intensity, though, varied between all

91 three cases measuring in at 0.0140m for Case 1, 0.0128m for Case 2, and 0.0120m for the Case

4, up to a 16% variation between operating conditions. Qualitatively, the OH* filtered images are also more symmetric than the color camera images. Since the OH* chemiluminescence utilizing the PMT setup is capturing the heat release globally and the centroid location was similar between all cases examined, the flame zone location in the COMSOL model was selected to be

0.023m downstream of the nozzle.

The width of the block element used to model the flame zone also needed to be determined. Since intensity centroid location was used to place the flame zone, it was also used to determine the width. Under the forced modulation conditions the images gathered were phase averaged to produce a single average image of the flame at certain phase increments based on the frame rate of the camera and frequency of fluctuations (NB: The phase of each image is arbitrary and not correlated to pressure fluctuations, overall intensity fluctuations, or the phase from any other condition).

Figure 6.14: Phase averaged ICCD images for the 2000SLPM 20in case under natural instability

Figure 6.15: Intensity contours for the phase averaged ICCD images for the 2000SLPM 20in case under natural instability

This not only shows the flame shape dynamics through one cycle of fluctuation but the centroid location from each image can be extracted to show its movement as well. Although different for each condition run, the most extreme centroid movement was found in the 200Hz, 2000SLPM,

20” case where the centroid moved approximately 0.005m through the cycle. Thus, in order to cover all potential locations of the centroid, the flame zone in the model was given a dimension of 0.005m in the axial direction.

200Hz Centroid Movement 0.026 0.025 0.024 0.023 0.022 200Hz

nozzle(m) 0.021 200Hz Avg 0.02 0.019

CentroidLocation downstreamof 0 100 200 300 400 Phase Angle (deg)

Figure 6.16: Movement of centroid location downstream of nozzle for the 2000SLPM 20in case under 200Hz forcing by the siren 6.4. Time Delay Determination

One of the most important aspects to both the flame model and thermoacoustic instabilities in general is the time delay between the velocity fluctuations and heat release fluctuations. As mentioned in Section 2.1, in order for natural instability to even occur the pressure fluctuations and heat release fluctuations must be in phase with each other. As discussed in Section 6.2 the time delay given by the FTF describes the total convection time from two microphone measurement location of the velocity fluctuations to the heat release fluctuations. Therefore, the total convection time should closely match the measured values of 16.878ms and 18.994ms for

94 the 2000SLPM and 1500SLPM cases, respectively. This essentially requires the calculation of flow time through three sections of the combustion rig: from two microphone location to the fuel injection location ( τin ), fuel injections through the end of the nozzle (τnoz ), and from the end of the nozzle to the heat release zone (τcomb ). The sum of all of these partial time delays should approximate to the total time delay, τtot , as determined by the FTF. As mentioned in the previous section, the heat release zone is taken to be 0.023m into the combustor so that value will be used for the calculations.

6.4.1 = + + Since temperature, pressure, and inlet geometry are all known the flow velocity and distance traveled are easily available and the convection time delay for that component can be calculated rather simply. For the nozzle component approximations were made based on the known flow split and effective area were used when calculating velocity but the exact distance from fuel injector to nozzle exit is known and utilized. Finally, for the combustor component distance traveled is again known but the only other known is combustor pressure. Therefore an approximation using an extrapolated temperature profile from downstream measurement locations and assumption of the entire flow having reconverged when travelling through the combustor cross-section was used to calculate combustor velocity. The flow velocities, distances, and calculated time delays for each section are listed in the table below for the 2000SLPM and

1500SLPM conditions.

2000SLPM Component Distance (m) Velocity (m/s) Time Delay (ms) Inlet 0.099 7.31 13.522 Nozzle 0.023 30.4 0.756 Combustor 0.023 9.07 2.536 Total 16.814 1500SLPM Component Distance (m) Velocity (m/s) Time Delay (ms) Inlet 0.099 6.58 15.024 Nozzle 0.023 22.8 1.009 Combustor 0.023 9.06 2.539 Total 18.572 Table 6.2: Time delay calculations based on known convection distances and calculated velocities The theoretical and measured values for the time delay agree well with the 200SLPM case only having an error 0.38% and the 1500SLPM case having an error of 2.22%. As the model only requires the time delay from nozzle and combustor components then the final determination comes down to a choice between using the sum of the two calculated components ( τnoz +τcomb ) or taking the FTF time delay and subtracting off the inlet component. Since the FTF was confidently measured several times and the inlet component is the most accurate from the theoretical model based having made fewer assumptions about its state variables when calculating that velocity component the latter of the two choices was selected. Thus, the final time delays used in the COMSOL model are 16.878ms – 13.522ms = 3.356ms for the

2000SLPM cases and 18.994ms – 15.024ms = 3.970ms for the 1500SLPM cases.

As an additional check to the time delay calculation in the combustor, the ICCD camera high speed videos taken with the 308nm filter were analyzed using a video FFT program. A high speed video is inputted into the program along with camera frame rate and the output shows pixel intensity fluctuations above and below the overall mean value at each pixel location through one cycle of oscillation at the frequency of interest. ICCD videos were taken and analyzed in this manner for forcing frequencies of 200Hz, 400Hz, 600Hz, and 800Hz. Since the

96 time delay in the combustor was calculated to be 2.536ms and 2.539ms for the 2000SLPM and

1500SLPM cases, respectively then by Equation 6.2.1 the slope of a theoretical phase plot should be approximately 0.913. This would indicate the between the 200Hz and 800Hz cases there should be a phase difference of 657 o, or approximately two additional cycles present in the combustor. In the processed videos this would manifest as two additional areas of local maximum intensity in the 800Hz case compared to the 200Hz case. Peak emissions for each cycle for the 2000SLPM case are presented in Figure 6.17.

Figure 6.17: Video FFT analyzed OH* emission peaks for (top left) 200Hz, (top right) 400Hz, (bottom left) 600Hz, and (bottom right) 800Hz

In the 200Hz forcing case presented, one red area indicating a local cycle peak emission intensity can be seen at the x=50 axial location (axes in Figure 6.17 indicate pixel location after condensing the 512x512 pixel video down by a factor of three in each dimension). In the 800Hz case, three axial locations, x=15, x=50, and x=100, all have an emission peak, indicated by the above mean intensity yellow/orange/red shading. This confirms the time delay associated only with the combustor should be approximately 2.536ms and the assumptions made in calculating that value were justified.

6.5. Acoustic Boundary Condition Measurement

Inlet and exit boundary conditions were measured using the method described in Section 5.3. and the code utilizing that theory and used to produce the data seen in this section is found in

Appendix D. The plots for each boundary condition are split into the real (resistance) and imaginary (reactance) components. Data was taken at frequencies between 200Hz and 800Hz.

Since the impedance measurements were taken at room temperature and the combustion tests are run at a much higher temperature a temperature effect had to be taken into account. In order to do so instead of plotting the impedance values against frequency they were plotted against wave number. Since wave number normalizes the frequency by speed of sound and speed of sound is dependent upon temperature the impedance value used by COMSOL will be appropriately shifted. The values were imported into COMSOL as an interpolation function that could linearly interpolate between measured values to obtain the proper boundary condition. It is also helpful to examine the boundary conditions as reflection coefficients, as discussed in Section 4.2, to see how the impedance compares to the theoretical sound hard boundary condition. Since the reflection coefficient is a ratio of the magnitude of the reflected wave to incident wave the sound

98 hard boundary would have a value of one. Above one the boundary is adding energy to the system and below one it is extracting it.

Real Impedance 70 60 50 40 30 1500 Inlet 20 2000 Inlet 10 1500 Exit

Real Impedance 0 2000 Exit -10 0.5 1 1.5 2 2.5 -20 -30 Wave Number (f/c)

Figure 6.18: Real component to impedance for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases

Imaginary Impedance 60 40 20 0 1500 Inlet 0.5 1 1.5 2 2.5 -20 2000 Inlet -40 1500 Exit -60 2000 Exit Imaginary Impedance -80 -100 Wave Number (f/c)

Figure 6.19: Imaginary component to impedance for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases

Reflection Coefficient 1.2

0.8 1500 Inlet 0.6 2000 Inlet 0.4 1500 Exit

Reflection Coefficient 0.2 2000 Exit 0 0.5 1 1.5 2 2.5 Wave Number (f/c)

Figure 6.20: Reflection coefficient for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases Generally the reflection coefficient stays at or below one as expected. Slight deviations from this trend tend to occur around wave numbers where the error between measured and predicted pressures was high.

Predicted Pressure % Error 200

150

100 1500 Inlet 50 2000 Inlet 0 1500 Exit 0.5 1 1.5 2 2.5 -50 2000 Exit

PredictedPressure Error % -100

-150 Wave Number (f/c)

Figure 6.21: Percent error in acoustic pressure amplitude predictions for upstream and downstream orifices for both 1500SLPM and 2000SLPM cases These also coincide with the conditions that place a pressure node near one of the three microphone measurement locations. Since a wide range of frequencies was being examined it

100 was impossible to choose microphone locations that completely eliminated the possibility of node interference but the microphone setup was selected to minimize the error at the frequencies known to be of interest from the stability mapping.

6.6. Combustor Temperature Profile

Since temperature has a great effect on speed of sound and therefore frequency, an accurate temperature profile will improve the reliability of any model. Typically the temperature is considered a jump condition across the flame zone with a constant upstream and different but still constant downstream temperature. However, by measuring the temperature at various locations in the transition tube, it has been shown that this is not necessarily the case. For the

0.254m-long transition tube setup two thermocouple ports were available located at 0.319m and

0.370m downstream of the nozzle. For the 0.508m-long transition tube setup three thermocouple ports were available located at 0.319m, 0.493m, and 0.649m downstream from the nozzle. Data between each measurement point was linearly interpolated and extrapolated from the measurement points upstream to the flame zone and downstream to the exit. The inlet to the nozzle temperature was held constant at 450K. The final temperature profile for each of the cases can be seen below.

101

COMSOL Model Temperature Profile 1400 1200 F)

o 1000 800 Case1 600 Case 2 400 Case 3 Temperature( 200 Case 4 0 0 0.2 0.4 0.6 0.8 1 Downstream from Inlet (m)

Figure 6.22: Temperature profiles used in COMSOL model 6.7. Eigenfrequency Study

With all the pieces now in place the COMSOL model can be run through an eigenfrequency study to see where the predicted frequencies lie and analyze their stability. The results of the study for the COMSOL model including all appropriate boundary conditions and temperature refinements and how they compare to the experimentally determined values are presented in the table below.

Eigenfrequency Experimental Case COMSOL P' (kPa) Frequency (Hz) 1 648.75-5.2283i 2.62 645 2 429.92-3.0511i 0.979 422 3 667.6+10.342i 0.0417 675 4 468.13+6.2531i 0.076 452 Table 6.3: Results of final COMSOL model compared to experimentally determined acoustic pressure fluctuations in the combustor and their respective frequencies The model correctly predicts instability for the two 200SLPM cases and stability for the two

1500SLPM cases. With regards to frequency Case 1 had an error of 0.578%, Case 2 had an error of 1.84%, Case 3 had an error of 1.11% and Case 4 had an error of 3.45%.

102

Taking these as the hallmark values, the refinements to the model can essentially be

“turned off” to show the importance of the proper boundary conditions and temperature profile in turn. The comparison between the eigenfrequencies for the measured boundary condition models versus the sound hard boundary condition model is shown in the table below.

Eigenfrequency Experimental Case Boundary Conditions COMSOL P' (kPa) Freq (Hz) 1 Impedance 648.75-5.2283i 2.62 645 1 Sound Hard 649.03-6.9521i

2 Impedance 429.92-3.0511i 0.979 422 2 Sound Hard 435.44-2.8887i

3 Impedance 667.6+10.342i 0.0417 675 3 Sound Hard 674.91+5.3201i

4 Impedance 468.13+6.2531i 0.076 452 4 Sound Hard 473.04+5.289i Table 6.4: Effect of impedance boundary conditions on converged eigenfrequencies in the COMSOL model With the exception of Case 3, all tested conditions are closer to the experimental frequency when the impedance boundary condition is included then with the sound hard boundary. Model to experimental errors improved from 0.621% to 0.578% for Case 1, 3.09% to 1.84% for the Case

2, and 4.45% to 3.45% for Case 4. Although Case 3 does appear to converge more accurately with the sound hard boundary condition it should be noted that it is a stable case and lacks a clearly defined acoustic pressure peak as shown in the stability map section. Each of these experimental values was the result of averaging several data files and while the unstable conditions were clearly and consistently unstable in both frequency and magnitude, the stable conditions had a constantly low pressure level over a range of frequencies approximately ±25Hz from the reported average value. Considering the sound hard boundary more accurate when both

103 correctly predict stability and closely approximate the range of very low level response frequencies could be disingenuous and all examined data points should be considered.

The differences between the constant temperature and temperature gradient results can also be examined. For the constant temperature case the downstream temperature was set to

1088K for all cases.

Eigenfrequency Experimental Temperature Frequency Case COMSOL P' (kPa) Profile (Hz) 1 Gradient 648.75-5.2283i 2.62 645 1 Constant 637.79-5.5397i

2 Gradient 429.92-3.0511i 0.979 422 2 Constant 423.03-5.6986i

3 Gradient 667.6+10.342i 0.0417 675 3 Constant 653.51+7.2008i

4 Gradient 468.13+6.2531i 0.076 452 4 Constant 464.73+7.8268i Table 6.5: Effect of temperature gradient on converged eigenfrequencies in COMSOL model Here, errors in predicting the frequency for the constant temperature profile cases are 1.13% for

Case 1, 0.243% for Case 2, 3.29% for Case 3, and 2.74% for Case 4. For Cases 3 and 4 the predictions are improved using the temperature gradient while for Cases 1 and 2 the constant temperature profile yields the more accurate results. Since the temperature is based on only two or three measurement locations in the transition tube it could be that the longer tube will result in a much more inaccurate temperature profile when extrapolated over the length of the combustor resulting in less accurate predictions. This should not discredit the need for replacing a simple temperature jump model with a more realistic gradient model but rather highlight the care that must be taken when measuring the profile. It is possible that measuring the temperature in the

104 flame zone and at the exit could refine the model even further and produce improved eigenfrequency study results.

In order the check the validity of the time delay calculations the same procedure laid out in Section 6.4 was carried out again, but with additional unwrapping performed on the phase plot. Since the time delay at each frequency can only be calculated in a 180 o interval, several orders of unwrapping had to be performed on each delay to properly align it on the phase plot, each representing an additional cycle of delay between the velocity and heat release measurements. From the data given in Figure 6.9, one and two additional unwrappings were performed resulting in four new time delays (two each for each additional unwrapping). The resulting time delays were 5.110ms and 6.864ms for the 2000SLPM cases and 5.725ms and

7.479ms for the 1500SLPM cases. The model was then run again at each time delay to examine its effect on the predicted eigenfrequencies and the results are presented in Table 6.6.

Eigenfrequency COMSOL 2000SLPM 5.11 6.864 Experimental τττ=3.356ms 1500 SLPM Case Boundary Conditions 5.725 7.479 P' (kPa) Freq (Hz) τττ=3.97ms 1 Impedance 637.79-5.5397i 643.11-5.4115i 638.61-3.3521i 2.62 645 1 Sound Hard Boundary 638.97-6.8767i 643.94-6.8761i 640.04-4.5134i

2 Impedance 423.03-5.6986i 442.23-7.7567i 449.09-3.3157i 0.979 422 2 Sound Hard Boundary 428.29-5.6379i 445.71-7.3692i 452.47-4.2899i

3 Impedance 653.51+7.2008i 675.38+11.947i 675.84+1.4778i 0.0417 674 3 Sound Hard Boundary 659.62+4.1882i 682.64+3.4281i 681.49-2.754i

4 Impedance 464.73+7.8268i 457.71+6.072i 458.01-1.7029i 0.076 452 4 Sound Hard Boundary 469.69+6.4071i 462.91+6.3986i 461.97-1.6107i Table 6.6: Model prediction with additional time delays

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Through the addition of one unwrapping, the model maintains its ability to correctly predict the stability for each case examined. However, when two additional unwrappings are performed the model fails to correctly predict stability for Case 4. Also important to note, for Case 3 with two additional unwrappings, the boundary conditions become important to stability prediction. The inclusion of the impedance boundary condition is necessary for the model to predict stability, whereas the sound hard boundary condition will incorrectly predict instability. In all previous cases the model predicted eigenfrequencies either stable or unstable enough that boundary conditions did not matter. However, this does highlight the fact that proper boundary condition modelling can be necessary under certain operating conditions and care must be taken to include them if the model is to be extrapolated to further research.

One last set of runs were completed on the FEM model to validate the assumption of flame location. The emissions from the visible spectrum were analyzed to determine the flame zone location and time delay. The color high speed camera videos were analyzed using the video

FFT software as well to create an average fluctuating flame image for Case 2 and Case 4. The location for the flame zone was set to be the location of maximum intensity fluctuations. Since the visible spectrum extends much farther axially down the combustor the flame zone moved significantly downstream. The 2000SLPM flames and 1500SLPM flames also differed significantly as compared to the OH* band emissions. The flame zone was moved to be centered around 0.030m for Cases 1 and 2 and 0.0459m for Cases 3 and 4. Since the FTF was only taken for the OH* emissions the time delay could only be calculated theoretically as laid out in Table

6.2. The same procedure was taken but the combustor time delay component was calculated with a different axial location. Then only the nozzle and combustor time delays were used in the model. The resulting time delays for the visible emissions were 4.064ms for the 2000SLPM

106 cases and 6.075ms for the 1500SLPM cases. The results for the visible study are presented in

Table 6.7.

Eigenfrequency Experimental Case Boundary Conditions COMSOL P' (kPa) Freq (Hz) 1 Impedance 647.4-0.22176i 2.62 645 1 Sound Hard 648.06-2.3024i

2 Impedance 433.41+1.0983i 0.979 422 2 Sound Hard 438.05-0.1608i

3 Impedance 668.76+5.8128i 0.0417 675 3 Sound Hard 675.25+1.8513i

4 Impedance 457.68-1.1904i 0.076 452 4 Sound Hard 461.97-1.9088i Table 6.7: Eigenfrequency predictions using visible spectrum emissions in calculating flame zone and time delay Instability is still correctly predicted for Case 1 and the frequency of fluctuations decreases slightly. For Case 2, instability is now only correctly predicted for the sound hard boundary case and fluctuation frequencies shifted away from the measured value of 422Hz. Case 3 remained mostly unchanged with stability predictions still accurate and frequency shifts of less approximately 1Hz for both boundary conditions. Case 4, however, is now entirely shifted into the unstable regime resulting in an incorrect prediction. Since visible emissions are in phase with

OH* chemiluminescence emissions it should theoretically be possible to construct a model based around that emission band. However, since the FTF utilized in this model is still based on OH* emissions and the measured phase from the FTF used to determine the time delay in the previous iterations of the model, it is understandable that trying a hybrid approach with the visible spectrum and assumptions made with regards to the calculated time delay in the combustor and nozzle could result in less accurate eigenfrequency predictions. Further work to refine the visible

107 spectrum emissions profile could yield more accurate results but is beyond the scope of this thesis.

A major benefit to the FEM code is the ability to extract pressure mode traces along the combustor as another check against the experimental results. Acoustic pressure contours were extracted from the beginning of the inlet section through the nozzle element and to the end of the combustor. This was then plotted against the measured acoustic pressure at the four locations in the inlet section and the one in the combustor section. It does need to be noted though that the results of the COMSOL model and experimental model are plotted on two different scales.

Eigenmode modelling cannot be used as a measure of pressure amplitude since by nature it uses a linearity scheme that converges to arbitrary values on the order of unity. It is useful for identifying mode shapes within a model but cannot be used to predict acoustic pressure amplitude in the real world or compare relative amplitudes across models. However, good agreement between the shape of the eigenmode and scaled experimental amplitudes will indicate an accurate eigenfrequency prediction for the model. The acoustic pressure modes for the unstable cases are presented in Figures 6.23 and 6.24.

Case 1 3 6 2 4 1 2 0 0 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -2 (kPa) -2 -4 -3 -6 Distance Downstream (m) Acoustic PressureCOMSOL

COMSOL Experimental Acoustic PressureExperimental

Figure 6.23: Mode shape for Case 1 comparing predicted COMSOL values to experimentally determined acoustic pressure amplitudes in inlet and combustor with error bars indicating one standard deviation

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Case 2 2.5 2.5

1.5 1.5

0.5 0.5

-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.5

-1.5 -1.5 (kPa)

-2.5 -2.5 Distance Downstream (m) AcousticPressure COMSOL

COMSOL Experimental Acoustic PressureExperimental

Figure 6.24: Mode shape for Case 2 comparing predicted COMSOL values to experimentally determined acoustic pressure amplitudes in inlet and combustor with error bars indicating one standard deviation Through the inlet the agreement between the predicted and measured mode shape is good. Four of the five experimental pressures fall on or near the predicted pressure line and generally within the standard deviation of each value. However, the model tends to over predict the relative pressure in the combustor. Case 2 falls well below the predicted mode shape and Case 1 barely reaches to the prediction within one standard deviation. As the nozzle was modeled as a simple cylindrical element within COMSOL it is possible that the actual nozzle in the experimental rig exhibits a greater effect on the acoustics of the system. No acoustic characterization of the nozzle was carried out and it is quite possible that doing so would alter the relative pressure in each section and bring the prediction more in line with the experimental data. This, however, is a topic for further research.

109

7. Conclusion

A finite element model was created in COMSOL to predict the resonant frequency and stability of several configurations of a single nozzle, acoustically tunable gas turbine combustion rig. The rig was modeled as a series of FEM elements with internal state variables either set from experimentally determined values or governed by one dimensional wave theory. The acoustic boundary conditions for the inlet and exit were modeled by acoustic impedance experimentally determined through multi-microphone method testing in the rig. The effects of both frequency and flow rate were taken into account when modeling the impedance. An appropriate flame model was established based on an experimentally determined flame transfer function. Care was taken to appropriately model the gain, phase, and flame location through the use of two- microphone method velocity measurements, OH* chemiluminescence, and both color and OH* filtered ICCD high speed imagery. The time delay, based on phase of the FTF, was even more closely examined as it is recognized as one of the most important factors in determining the stability of the system and so FTF phase, OH* imaging, and theoretical convection time were all checked against each other to ensure accuracy. The temperature profile in the transition tube was measured in order to improve model accuracy as well. The model proved its ability to accurately predict both frequency and stability through an eigenvalue study with final refinements producing results accurate to within 2% of the measured frequency for unstable cases and 4% for stable cases. It also correctly predicted stability or instability for all cases. Proper modeling of acoustic impedance and temperature profile were shown to improve model accuracy in most cases examined. Eigenmode pressure shapes closely matched those determined experimentally through the inlet, but over predicted the acoustic pressure in the combustor. Future work based on this study that could be used to improve model accuracy include determining a more exact

110 temperature profile for the entire length of the combustor and acoustically characterizing the burner element to see its effect on the pressure in the combustor section.

111

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A. APPENDIX A: Flush to Recess Mounting Calibration and MATLAB Code

Shown in Figure A.1 is the calibration tube used for flush to recess mounted corrections. This tube was a 0.293m long aluminum tube with an inner diameter of 0.0483m.

Figure A.1: Cross section view of calibration tube A loudspeaker (BMS Model# 459ND-MID) would be mounted to one end of the tube and two pressure sensors, one flush mounted and one recess mounted, would be attached side by side to the other end. For the transducer mounted on the optical section of the combustor, however, since part of the wave guide was welded to the dummy stainless steel window on the rig, it was not possible to utilize the calibration tube. Instead, the exhaust section and choked exit of the rig was removed and a speaker was mounted in its place. A plug was mounted onto the dummy window allowing for a sensor to be flush mounted on the combustor wall and calibration was carried out in that manner. This calibration procedure produces the response of recess-mounted

116 sensors with respect to the flush-mounted sensors in the form of a transfer function. To calculate the gain and phase the ratio of the cross correlation of the flush- and recess-mounted signals to the auto correlation of the flush-mounted signal was examined at each forcing frequency with the phase delay being extracted from the lag of the two input signals as shown in Equation A.1.

A.1 ⋆ exp = ⋆ Where G represents the flush to recess mounted gain (i.e. multiply the flush-mounted signal by the gain to obtain the recess mounted signal), represents the phase lag, is the forcing frequency, is the flush-mounted time series, is the recess-mounted time series, and ⋆ represents the cross correlation function. The gain and phase of the transfer function represent the amplification or attenuation of the pressure signal’s magnitude as well as the phase shift, or time delay, experienced by the signal from the assumed 1D travelling wave in the inlet or combustor to the sensor itself. Correcting for the change in signal is necessary for accurate calculations of both the magnitude of the inlet fluctuations but also the overall time delay between the two-microphone method measurement location and flame zone. Calibrations were performed for various wave guides whose characteristics are summarized in Table A.1. Mount numbering is the same as given in Tables 5.3 and 5.4 and can be used to match each mount with its location for all combustion and impedance tests. Calibration curves for each mount are shown in Figures A.2-A.11.

Mount Wave Guide Length (mm) Wave Guide Inner Diameter (mm) 1 89 0.91 2 89 0.91 3 89 1.83 4 89 1.83 Combustor 152.4+50.8 0.91 Table A.1: Recess-mounted wave guide dimensions

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Mount 1 Gain 4 3.5 3 2.5 2 Gain 1.5 1 0.5 0 200 300 400 500 600 700 800 900 Frequency, Hz

Figure A.2: Gain calibration curve for transducer mount 1

Mount 1 Phase Difference 0 -20 200 300 400 500 600 700 800 900 -40 -60 -80 -100 -120 -140 Phase Angle, Degrees Phase Angle, -160 -180 Frequency, Hz

Figure A.3: Phase calibration curve for transducer mount 1

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Mount 2 Gain 4 3.5 3 2.5 2 Gain 1.5 1 0.5 0 200 300 400 500 600 700 800 900 Frequency, Hz

Figure A.4: Gain calibration curve for transducer mount 2

Mount 2 Phase Difference 0 -20 200 300 400 500 600 700 800 900 -40 -60 -80 -100 -120 -140 Phase Angle, Degrees Phase Angle, -160 -180 Frequency, Hz

Figure A.5: Phase calibration curve for transducer mount 2

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Mount 3 Gain 10 9 8 7 6 5 Gain 4 3 2 1 0 200 300 400 500 600 700 800 900 Frequency, Hz

Figure A.6: Gain calibration curve for transducer mount 3

Mount 3 Phase Difference 0 -10 200 300 400 500 600 700 800 900 -20 -30 -40 -50 -60 -70 Phase Angle, Degrees Phase Angle, -80 -90 Frequency, Hz

Figure A.7: Phase calibration curve for transducer mount 3

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Mount 4 Gain 9 8 7 6 5

Gain 4 3 2 1 0 200 300 400 500 600 700 800 900 Frequency, Hz

Figure A.8: Gain calibration curve for transducer mount 4

Mount 4 Phase Difference 0 -20 200 300 400 500 600 700 800 900 -40 -60 -80 -100 -120

Phase Angle, Degrees Phase Angle, -140 -160 Frequency, Hz

Figure A.9: Phase calibration curve for transducer mount 4

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Combustor Mount Gain 6

3 Gain 2

0 200 400 600 800 Frequency, Hz

Figure A.10: Gain calibration curve for transducer mounted in combustor

Combustor Mount Phase 0 -20 200 400 600 800 -40 -60 -80 -100 -120 -140 Phase Angle, Degrees PhaseAngle, -160 -180 Frequency, Hz

Figure A.11: Phase calibration curve for transducer mounted in combustor Previous studies performed by Lee [63] support the validity of the measured calibrations. Narrow bored pressure tubes produced spikes in transducer responses on the order seen in the present study. Shorter tube lengths also were shown to produce greater signal amplification. Although

122 the shortest tube lengths in Lee’s study were 320mm long, the trends produced do indicate amplification on the level seen in this calibration had Lee’s study continued to decrease tube length to the 76.2mm-203.2mm used here. Differences in wave guide tube’s inner diameter is the main reason behind the different trends that can be seen in mounts one, two, combustor and mounts three and four. The measured signal in the recess-mounted setup is generally amplified with slight amplification at lower frequencies and increasing amplification as frequency increases. This is likely due to the cavity in the top of the mount, where the sensor is housed, acting as a Helmholtz resonator around the frequencies of amplification.

These gain and phase calibrations were applied to the measured data by creating an interpolation function in MATLAB that could be divided into the FFT spectrum of each pressure signal to correct for the amplitude at each frequency as shown in Equation 5.2.2.

A.2 = Phase corrections were implemented by determining the proper phase angle for correction and multiplying the FFT of the time series by the complex term , where is the cos+∙sin phase shift..

%Flush to recess calibration method using cross and auto correlations %Input: raw calibration data %Output: gain and phase data at each frequency clc; clear all ; close all ; format long gg = 1; %% Raw Data for fstr = { '200' '250' '300' '350' '400' '450' '500' '550' '600' '650' '700' '750' '800' } %list of frequencies run AfterStr = 'Hz.lvm' ; %file name ending

%create filename freq = fstr{fi} LVMName1 = strcat(freq,AfterStr)

%load in raw data

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[Raw1(:,1), Raw1(:,2), Raw1(:,3)] = textread(LVMName1, '%f %f %f' ); f = sscanf(LVMName1, '%d%s' ); f=f(1)

%create time series from raw data start = 5001; ende = start + 4999; time = Raw1(start:ende,1); %list of times P1r1 = Raw1(start:ende,2); P1m1 = P1r1.*1000./.01505./100; %channel 1 converted to pascals P2r1 = Raw1(start:ende,3); P2m1 = P2r1.*1000./.01517./100; %channel 2 converted to pascals flushsig = P1m1; %flush mounted signal recesssig = P2m1; %recess mounted signal

%% gain and phase flush = xcorr(flushsig); [Amp_flush, Pos_flush] = max(flush); Crosscor = xcorr(flushsig, recesssig); [Amp_recess, Pos_recess] = max(Crosscor); Gain = Amp_recess/Amp_flush; % gain using cross correlation lag = Pos_recess - Pos_flush; %phase using cross correlation Phaseflush = atand((imag(X1(f)))/(real(X1(f)))); Phaserecess = atand((imag(Y1(f)))/(real(Y1(f))));

sheet = num2str(f); filename = 'SPL_And_FFT' ; x1range=strcat( 'N' ,num2str(ji)); outs = [f Gain lag Phaseflush Phaserecess]; xlswrite(filename,outs, sheet, x1range); %output file with gain and phase clear LVMName1 Raw1 gg = gg+1; end

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B. APPENDIX B: Flame Transfer Function MATLAB Code

%Flame transfer function calculation code utilizing two microphone method %Input: raw signals from upstream microphones and OH* data in combustor %Output: velocity and chemiluminescence fluctuations and frequencies at %which they occur close all ; clear all ; clc;

file_tot = 53; %Total number of files to be processed

%Presize matrices for speed of code outdat = zeros(file_tot*8,13); phasedat = zeros(file_tot,4); coh_list = zeros(file_tot,1); phase_list = zeros(file_tot,1); p_in = []; %inlet pressures for each data file (psig)

T_in = []; %inlet temperatures for each data file (K)

Q_main = []; %main air flow rates for each data file (SLPM) ff = []; %forcing frequencies for each data file (Hz) fs = 8000; %sampling frequency delf = fs/8192; %delta F for FFT fft_freqs = fs/2*linspace(0,1,(8192*8)/2+1); %list of FFT frequencies fid2 = fopen( '' ); %file name for OH* background emissions oh_bckgrnd_all = textscan(fid2, '%f %f %f %f %f %f' ,'headerlines' ,172); oh_bckgrnd = cell2mat(oh_bckgrnd_all(6)); fclose(fid2);

%flush to recess mounting gains and phases for each transducer mount f_to_r_freqs = [200 250 300 350 400 450 500 550 600 650 700 750 800]; f_to_r_gain_2 = [1.161757795 1.28038595 1.467389943 1.741077262 2.155104508 2.73323872 3.440107678 3.674331661 3.001892058 2.246364267 1.728092093 1.393935907 1.169820044]; %Divide recessed mounted pressures by gain to get flush mounted pressures f_to_r_phase_2 = [-5.499090583 -7.854169753 -11.04875449 -15.99959371 - 23.90605293 -37.53316935 -60.01341835 -91.51570463 -119.4647774 -136.4860398 -146.2572583 -152.2938182 -156.5240497]; %Subtract from recessed phase to get flush phase f_to_r_gain_1 = [1.141723773 1.272485506 1.433275788 1.670172975 2.019384695 2.489615771 3.275503156 3.747986209 3.328343167 2.540547707 1.938151597 1.542361011 1.278314849]; f_to_r_phase_1 = [-4.668680494 -6.025404317 -10.26351278 -14.7897275 - 21.4084319 -32.92153865 -50.90766032 -80.08697101 -110.4171826 -130.778738 - 142.6601498 -149.7999748 -154.5620365];

125 f_to_r_gain_3 = [1.117180617 1.174216579 1.23973795 1.333935139 1.456865594 1.621732712 1.830935242 2.14142585 2.587226301 3.308943029 4.566602769 6.789738121 8.89230239]; f_to_r_phase_3 = [-0.581707804 -0.909085115 -1.185390529 -1.599350895 - 2.532839103 -3.321206638 -4.971704273 -6.469591912 -8.817428565 -13.05969723 -20.92035797 -38.4842636 -76.9377965]; f_to_r_gain_4 = [1.069406356 1.125811533 1.222177538 1.331670374 1.477144455 1.660602996 1.950875691 2.462045071 3.306468657 4.711574072 7.35251586 8.490531198 5.665041983]; f_to_r_phase_4 = [-0.907840567 -1.182436783 -1.499370926 -2.565351251 - 3.175166504 -4.539385323 -6.611771895 -9.883454865 -15.05881949 -27.08040915 -56.88447337 -108.803672 -141.187662];

%create interpolation curves for each correction f_to_r_gain_1_full = interp1(f_to_r_freqs,f_to_r_gain_1,fft_freqs, 'spline' ,1); f_to_r_gain_1_full_ds = padarray(f_to_r_gain_1_full,[0 length(f_to_r_gain_1_full)], 'symmetric' ,'post' ); f_to_r_gain_1_full_ds(length(f_to_r_gain_1_full_ds)/2+1) = []; f_to_r_gain_1_full_ds(length(f_to_r_gain_1_full_ds)) = []; f_to_r_gain_1_full_ds = transpose(f_to_r_gain_1_full_ds); f_to_r_gain_2_full = interp1(f_to_r_freqs,f_to_r_gain_2,fft_freqs, 'spline' ,1); f_to_r_gain_2_full_ds = padarray(f_to_r_gain_2_full,[0 length(f_to_r_gain_2_full)], 'symmetric' ,'post' ); f_to_r_gain_2_full_ds(length(f_to_r_gain_2_full_ds)/2+1) = []; f_to_r_gain_2_full_ds(length(f_to_r_gain_2_full_ds)) = []; f_to_r_gain_2_full_ds = transpose(f_to_r_gain_2_full_ds); f_to_r_gain_3_full = interp1(f_to_r_freqs,f_to_r_gain_3,fft_freqs, 'spline' ,1); f_to_r_gain_3_full_ds = padarray(f_to_r_gain_3_full,[0 length(f_to_r_gain_3_full)], 'symmetric' ,'post' ); f_to_r_gain_3_full_ds(length(f_to_r_gain_3_full_ds)/2+1) = []; f_to_r_gain_3_full_ds(length(f_to_r_gain_3_full_ds)) = []; f_to_r_gain_3_full_ds = transpose(f_to_r_gain_3_full_ds); f_to_r_gain_4_full = interp1(f_to_r_freqs,f_to_r_gain_4,fft_freqs, 'spline' ,1); f_to_r_gain_4_full_ds = padarray(f_to_r_gain_4_full,[0 length(f_to_r_gain_4_full)], 'symmetric' ,'post' ); f_to_r_gain_4_full_ds(length(f_to_r_gain_4_full_ds)/2+1) = []; f_to_r_gain_4_full_ds(length(f_to_r_gain_4_full_ds)) = []; f_to_r_gain_4_full_ds = transpose(f_to_r_gain_4_full_ds); for ii=1:file_tot

file_name = strcat( '' ,sprintf( '%03d' ,ii), '.dat' ); %create filename

%extract raw data from file fid = fopen(file_name); data = textscan(fid, '%f %f %f %f %f %f' ,'headerlines' ,172); fclose(fid);

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flush_rec_cor_index = find(f_to_r_freqs==ff(ii)); %Find the index of the frequency to pick the proper gain and phase correction

%extract time and pressure signals and correct for gain in signal %conditioner time = cell2mat(data(1)); ds = cell2mat(data(2))./10; %downstream data us = cell2mat(data(4))./10; %upstream data

dx = 0.127; %distance between mics in m

pressure = p_in(ii); %pressure for file currently running temperature = T_in(ii); %temperature for file currently running Qa = Q_main(ii); %main air flow rate for file currently running

temperatureC = (temperature-32)*(5/9);

rho = ((pressure + 14.7)*6894.75729)/(287*(temperatureC+273.15)); Qa_actual = (Qa*(0.001/60))*(14.7/(pressure+14.7))*((temperatureC+273.15)/293.15); %Calculate actual volumetric flow rate from SLPM v_mean = Qa_actual/0.001976;

%filter input signal between desired frequencies lf = 150; hf = 1000; [c2,c1] = butter(3,[lf/(fs/2) hf/(fs/2)], 'bandpass' );

us = filtfilt(c2,c1,us); us_fft_full = fft(us)./f_to_r_gain_3_full_ds; us = ifft(us_fft_full); us(numel(us)+100) = 0;

ds = filtfilt(c2,c1,ds); ds_fft_full = fft(ds)./f_to_r_gain_1_full_ds; ds = ifft(ds_fft_full); ds(numel(ds)+100) = 0;

outdat_temp = zeros(8,9); outdat_temp(:,1) = ff(ii); phasedat_temp = zeros(8,4); coh_list_temp = zeros(8,1); phase_list_temp = zeros(8,1);

%break 8s file into 1s pieces for n=0:7

%correct for flush to recess gain and shift by necessary time phase_cor_ds = f_to_r_phase_1(flush_rec_cor_index); phase_cor_us = f_to_r_phase_2(flush_rec_cor_index);

time_delay_ds = phase_cor_ds/360/ff(ii); time_delay_us = phase_cor_us/360/ff(ii);

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ind_ds = find(time>=time_delay_ds,1); ind_us = find(time>=time_delay_us,1);

%take 1s of raw data to process at a time time = time(1:8192); us2 = us(1+8192*n+ind_us:8192*n+8192+ind_us); ds2 = ds(1+8192*n+ind_ds:8192*n+8192+ind_ds); press_mean = mean(ds2)+p_in(ii)+14.7;

if length(data)>3 oh_unfilt = cell2mat(data(6)); end

%filter OH* data by subtracting background emissions off oh_unfilt2 = oh_unfilt(1+8192*n:8192*n+8192); oh_filt = oh_unfilt2 - mean(oh_bckgrnd); oh_mean = mean(oh_filt);

%calculate two microphone method velocity fluctuations [x,l] = xcorr(us2,ds2);

[vtmm,~] = TMM(us2,ds2,dx,rho,fs); vtmm_SSPSD = SSPSD(vtmm,fs); oh_SSPSD = SSPSD(oh_filt,fs); us_SSPSD = SSPSD(us2,fs); ds_SSPSD = SSPSD(ds2,fs); freqs = fs/2*linspace(0,1,(8192)/2+1); pp = sqrt(us_SSPSD)*delf; ip = sqrt(oh_SSPSD)*delf; press_p = sqrt(ds_SSPSD)*delf;

[xc,lags] = xcorr(pp,ip); xc2 = CrossCorrelation(pp,ip,fs);

%fluctuation spectra for velocity, OH*, and pressure v_mod_spectrum = sqrt(abs(vtmm_SSPSD*delf))/v_mean; oh_mod_spectrum = sqrt(abs(oh_SSPSD*delf))/oh_mean; p_mod_spectrum = sqrt(abs(ds_SSPSD*delf))/press_mean; figure(1) plot(freqs,v_mod_spectrum); xlabel( 'Frequency (Hz)' ); ylabel( 'V\prime_{rms}/V_{mean}' ); title( 'Velocity Modulation Spectrum' ); grid;

figure(2) plot(freqs,oh_mod_spectrum);

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xlabel( 'Frequency (Hz)' ); ylabel( 'OH\prime_{rms}/OH_{mean}' ); title( 'OH Modulation Sprectrum' ); grid;

figure(3) plot(freqs,p_mod_spectrum); xlabel( 'Frequency (Hz)' ); ylabel( 'P\prime_{rms}/P_{mean}' ); title( 'Pressure Modulation Sprectrum' ); grid;

%extract peak data from spectra freqpeak = find(freqs>=ff(ii)); freqs_short = freqpeak-50:freqpeak+50; [a,b] = max(v_mod_spectrum(freqs_short)); [c,d] = max(oh_mod_spectrum(freqs_short)); [p_max,p_freq] = max(p_mod_spectrum(freqs_short));

disp([ 'Maximum Velocity Modulation Intensity = ' ,num2str(a*100), '% ','at ' ,num2str(freqs(freqs_short(b))), 'Hz' ]); disp([ 'Maximum OH Modulation Intensity = ' ,num2str(c*100), '% ' ,'at ',num2str(freqs(freqs_short(d))), 'Hz' ]); disp([ 'Maximum Pressure Modulation Intensity = ',num2str(p_max*100), '% ' ,'at ' ,num2str(freqs(freqs_short(p_freq))), 'Hz' ]);

gain = (c/a); us_phase = angle(fft(us2)*(1/fs)); ds_phase = angle(fft(ds2)*(1/fs)); vtmm_phase = angle(fft(vtmm)*(1/fs)); oh_phase = angle(fft(oh_filt)*(1/fs));

vtmm_phase_ff = (180/pi)*unwrap(vtmm_phase(fft_freqs==freqs(freqs_short(b)))); % - f_to_r_phase(flush_rec_cor_index); oh_phase_ff = (180/pi)*unwrap(oh_phase(fft_freqs==freqs(freqs_short(b))));

vtmm_ifft = ifft(vtmm_SSPSD, 'symmetric' ); us_ifft = ifft(us_SSPSD, 'symmetric' ); ds_ifft = ifft(ds_SSPSD, 'symmetric' );

for i=1:length(vtmm_ifft) if vtmm_ifft(i)>=0 vtmm_td(i) = sqrt(vtmm_ifft(i)); else vtmm_td(i) = -sqrt(-vtmm_ifft(i)); end end

phase = linspace(0,2*pi,length(vtmm_td));

oh_mean_list(n+1,1) = oh_mean;

%create output data matrix

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outdat_temp(n+1,2) = a*100; outdat_temp(n+1,3) = c*100; outdat_temp(n+1,4) = freqs(freqs_short(b)); outdat_temp(n+1,5) = freqs(freqs_short(d)); outdat_temp(n+1,6) = gain; outdat_temp(n+1,7) = vtmm_phase_ff-oh_phase_ff; outdat_temp(n+1,8) = p_max*100; outdat_temp(n+1,9) = v_mean; outdat_temp(n+1,10) = oh_mean; outdat_temp(n+1,11) = a*v_mean; outdat_temp(n+1,12) = c*oh_mean;

phasedat_temp(n+1,1) = vtmm_phase_ff; phasedat_temp(n+1,4) = oh_phase_ff;

%calculate coherence to check reliability of data [cxy,f] = mscohere(oh_filt,ds2,fs/2,0,8192,fs); cohfnd = find(f==freqs(freqs_short(d))); coh_list_temp(n+1,1) = cxy(cohfnd); [gxy,f2] = cpsd(oh_filt,ds2,fs/2,0,8192,fs); phsxy = atan2(-imag(gxy),real(gxy))*180/pi; phase_list_temp(n+1,1) = phsxy(cohfnd);

outdat_temp(n+1,13) = cxy(cohfnd);

end

outdat((ii-1)*8+1:ii*8,1:13) = outdat_temp; end

%output data outdat

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C. APPENDIX C: Three-line Pyrometry Theory and Calibration

In order to determine temperature from an emission spectrum, Planck’s law of radiation is used for spectral energy density determination as seen in Equation C.1.

C.1 , = − 1 Soot temperature is measured with three-line pyrome try utilizing the ratios of emission intensities from three separate wavelengths. Equation C.2 shows the ratio of spectral intensities at two different wavelengths, λ1 and λ2, incorporating the variation in emissivity.

C.2 = Where C 1 and C 2 are radiation constants equal to and 1.191 × 10 ∙ 1.4384 × 10 ∙ , respectively, T is the apparent temperature in Kelvin, is the emissivity as a function of wavelength, and S is the spectral intensity sensitivity of the measurement system. Measuring a voltage level for each wavelength, , the temperature in Equation C.2 can be solved for and results in Equation C.3.

1 1 − C.3 = ln + ln + ln + ln The spectral sensitivity of the optical system for each wavelength must be calibrated to a known irradiance of a blackbody. This requires a light source with known spectral irradiance where signal ratios can be determined and compared to accepted values. Both a uniform source sphere

(Labsphere, LR-4-M, maximum radiance 2900K) and a blackbody furnace (Williamson Model

#550, radiance range 773K-1773K) were utilized for calibration. The signal ratio for each

131 detector is related to the spectral intensity ratio of the known temperature at the blackbody source through Equation C.4.

C.4 , = = , Where is the spectral sensitivity constant and is the monochromatic spectral radiance for , , a blackbody where i=1,2, j=2,3, and i ≠j.

The three-line pyrometer contained three bandpass filters (Edmund Optics,

FWHM=10nm) centered around 660nm, 730nm, and 800nm. Light is collected through a lens and reflected to or passed through two dichroic mirrors, splitting the collected light into three parts, one for each bandpass filter. Each wavelength is collected using its own PMT. Figure 5.14 is reproduced here and shows the optical setup for the three-line pyrometer.

Figure C.1: Three-line pyrometer optical setup Since spectral intensity is lower at shorter wavelengths, the dichroic mirrors are positioned so that more of the collected emission is focused on the 600nm an 730nm PMTs than the 800nm

PMT. The pyrometer was positioned so that it was level and the collection lens was

132 perpendicular to the radiation source both in calibration and testing. Each PMT was individually aligned in the pyrometer to ensure accurate collection at each wavelength. Global emissions form radiation source or flame were used and alignment procedures ensured the entire source image would appear within the PMT sensor area and centered on it. Each PMT and amplifier voltage was then set according to Table C.1.

λ (nm) PMT Voltage Amplifier Voltage 660 0.524 12 730 0.390 12 800 0.435 12 Table C.1: Three-line pyrometer voltage settings

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D. APPENDIX D: Impedance Calculation MATLAB Code clc; clear all ; close all ; format long for flow = { '0' '300' '600' '900' '1200' '1500' '1750' '2000' } %List of all flow rates run in SLPM from filenames

fi=1; gg = 1;

freq_row = 1;

for fstr = { '200' '220' '240' '260' '280' '300' '320' '340' '360' '380' '400' '420' '440' '460' '480' '500' '520' '540' '560' '580' '600' '620' '640' '660' '680' '700' '720' '740' '760' '780' '800' } %List of all frequencies run from filenames

BeforeStr = strcat(num2str(cell2mat(flow)),'SLPM_' ); %Create string for first part of filename AfterStr = 'Hz_1V.lvm' ; %Create string for last part of filename freq = fstr{fi}; %Create string of frequency for filename LVMName1 = strcat(BeforeStr,freq,AfterStr); %Create filename to read data

[Raw1(:,1), Raw1(:,2), Raw1(:,3), Raw1(:,4), Raw1(:,5), Raw1(:,6)] = textread(LVMName1, '%f %f %f %f %f %f' ); %Read in raw data

f = str2double(cell2mat(fstr)); %Frequency to be used for gain and phase calculations

mdot = str2double(cell2mat(flow))*0.001*1.1839/60; %Calculate mass flow rate in kg/s

% Gain for 200-550 Hz if f <=550 gain1 = (-4.748321*10^-12)*f^5 + (8.379082*10^-9)*f^4 - (5.70765*10^-6)*f^3 + (1.892383*10^-3)*f^2 - .3030574*f + 19.83535; gain2 = (-4.467431*10^-12)*f^5 + (7.611275*10^-9)*f^4 - (5.024694*10^-6)*f^3 + (1.62655*10^-3)*f^2 - .255893*f + 16.72897; gain3 = (4.1828*10^-11)*f^4 - (4.5705*10^-8)*f^3 + (2.259*10^- 5)*f^2 -(4.0873*10^-3)*f + 1.331; gain4 = (2.1763*10^-15)*f^6 - (4.4788*10^-12)*f^5 + (3.9129*10^- 9)*f^4 -(1.84*10^-6)*f^3 + (4.915*10^-4)*f^2 - (6.8945*10^-2)*f + 4.9516;

else % Gain for 550 - 800 Hz gain1 = (-2.0947*10^-9)*f^4 + (5.8456*10^-6)*f^3 - (6.0586*10^- 3)*f^2 + 2.7529*f - 458.53; gain2 = (-1.3147*10^-9)*f^4 + (3.6095*10^-6)*f^3 - (3.6647*10^- 3)*f^2 + 1.6201*f - 259; gain3 = (-4.486911*10^-9)*f^4 + (1.206327*10^-5)*f^3 - (1.198383*10^-2)*f^2 + 5.236801*f - 849.4418; gain4 = (-1.045244*10^-8)*f^4 + (2.593736*10^-5)*f^3 - (2.391016*10^-2)*f^2 + 9.732446*f - 1476.408; end

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% Phase Shifts phase1 = (-2.121776*10^-11)*f^5 + (5.867053*10^-8)*f^4 - (5.984628*10^-5)*f^3 + (2.753077*10^-2)*f^2 - (5.830040)*f + 453.4352; phase2 = (-3.052236*10^-11)*f^5 + (8.040262*10^-8)*f^4 - (7.873759*10^-5)*f^3 + (3.510884*10^-2)*f^2 - (7.243043)*f + 551.1548; phase3 = (-1.318415*10^-11)*f^5 + (2.839460*10^-8)*f^4 - (2.368183*10^-5)*f^3 + (9.454303*10^-3)*f^2 - (1.803717)*f + 130.4968; phase4 = (2.34315*10^-11)*f^5 - (6.03362*10^-8)*f^4 + (5.73361*10^- 5)*f^3 - (2.54019*10^-2)*f^2 + (5.24757)*f - 405.439;

phase1 = -1*phase1*pi()/180; phase2 = -1*phase2*pi()/180; phase3 = -1*phase3*pi()/180; phase4 = -1*phase4*pi()/180;

% Call out pressure transducers and convert to Pascals for each second (5 seconds total)

row_num = 1;

for start = [1 5001 10001 15001 20001]

ende = start + 4999; time = Raw1(start:ende,1); P1r1 = Raw1(start:ende,2); P1m1 = P1r1.*1000./.01526./gain1./10; %pascals P2r1 = Raw1(start:ende,3); P2m1 = P2r1.*1000./.01517./gain2./10; %pascals P3r1 = Raw1(start:ende,4); P3m1 = P3r1.*1000./.01443./gain3./10; %pascals P4r1 = Raw1(start:ende,5); P4m1 = P4r1.*1000./.01505./gain4./10; %pascals

% FFT for each set of 5000 data points (one second of measurement) XPre = fft(2/5000.*P1m1); XPre(1) = []; YPre = fft(2/5000.*P2m1); YPre(1) = []; ZPre = fft(2/5000.*P3m1); ZPre(1) = []; UPre = fft(2/5000.*P4m1); UPre(1) = [];

%Phase shift each time series shift1 = complex(cos(phase1),sin(phase1)); shift2 = complex(cos(phase2),sin(phase2)); shift3 = complex(cos(phase3),sin(phase3)); shift4 = complex(cos(phase4),sin(phase4));

X1 = XPre.*shift1; Y1 = YPre.*shift2; Z1 = ZPre.*shift3; U1 = UPre.*shift4;

% Various constants. Locations are measured from the upstream orifice

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T = 298.15; Area = .038176^2; rho = 1.1839; c = sqrt(1.4*287*T); M = mdot/(rho*Area*c); i = 1i; w = 2*pi()*f;

x1 = -.252412; x2 = -.214312; x3 = -.176212; x4 = -.123825; %meters

pref = .00002; %Pa, reference pressure for sound pressure calculations kp = (w/c)*(1/(1+M)); km = (w/c)*(1/(1-M)); k = w/c; % KP used for waves propogating with biased flow, KM used for waves travelling % against biased flow. Mach number calculated from measured mass flow rate

% Using Least Squares with 3 transducers P11 = X1(f); P21 = Y1(f); P31 = Z1(f); P41 = U1(f);

%% Calculate wave coefficients using transducers 1, 2, 3 ExpTerms123 = [exp(-1i*kp*x1) exp(1i*km*x1); exp(-1i*kp*x2) exp(1i*km*x2); exp(-1i*kp*x3) exp(1i*km*x3)];

Pressures123 = [P11; P21; P31];

x11_123 = pinv(ExpTerms123)*Pressures123;

A123 = x11_123(1); B123 = x11_123(2);

% Impedance calculations SWR123 = (abs(A123)+abs(B123))/(abs(A123)-abs(B123)); % Standing wave ratio R_S123 = (SWR123-1)/(SWR123+1); % Reflection coefficient

ZR123 = real((A123+B123)/(A123-B123)); % Real of impedance ZI123 = imag((A123+B123)/(A123-B123))-km*.003175/.01603; %Imaginary of impedance Zabs123 = abs(ZR123+ZI123*i); %ABS of impedance

% Pressure Errors and SPL Calculations

P1_123 = A123*exp(-i*km*x1)+B123*exp(i*kp*x1); P1magmeas_123 = sqrt((real(X1(f)))^2+(imag(X1(f)))^2); %measure value P1magtheor_123 = sqrt((real(P1_123))^2+(imag(P1_123))^2); %theoretical value calculate from A & B P1error_123 = 100*(P1magtheor_123-P1magmeas_123)/P1magmeas_123; P1SPL_123 = 20*log10(P1magmeas_123/pref);

P2_123 = A123*exp(-i*km*x2)+B123*exp(i*kp*x2); P2magmeas_123 = sqrt((real(Y1(f)))^2+(imag(Y1(f)))^2); P2magtheor_123 = sqrt((real(P2_123))^2+(imag(P2_123))^2); P2error_123 = 100*(P2magtheor_123-P2magmeas_123)/P2magmeas_123; P2SPL_123 = 20*log10(P2magmeas_123/pref);

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P3_123 = A123*exp(-i*km*x3)+B123*exp(i*kp*x3); P3magmeas_123 = sqrt((real(Z1(f)))^2+(imag(Z1(f)))^2); P3magtheor_123 = sqrt((real(P3_123))^2+(imag(P3_123))^2); P3error_123 = 100*(P3magtheor_123-P3magmeas_123)/P3magmeas_123; P3SPL_123 = 20*log10(P3magmeas_123/pref);

P4_123 = A123*exp(-i*km*x4)+B123*exp(i*kp*x4); P4magmeas_123 = sqrt((real(U1(f)))^2+(imag(U1(f)))^2); P4magtheor_123 = sqrt((real(P4_123))^2+(imag(P4_123))^2); P4error_123 = 100*(P4magtheor_123-P4magmeas_123)/P4magmeas_123; P4SPL_123 = 20*log10(P4magmeas_123/pref);

%% Calculate wave coefficients using transducers 1, 2, 4 ExpTerms124 = [exp(-1i*kp*x1) exp(1i*km*x1); exp(-1i*kp*x2) exp(1i*km*x2); exp(-1i*kp*x4) exp(1i*km*x4)];

Pressures124 = [P11; P21; P41];

x11_124 = pinv(ExpTerms124)*Pressures124;

A124 = x11_124(1); B124 = x11_124(2);

% Impedance calculations SWR124 = (abs(A124)+abs(B124))/(abs(A124)-abs(B124)); % Standing wave ratio R_S124 = (SWR124-1)/(SWR124+1); % Reflection coefficient

ZR124 = real((A124+B124)/(A124-B124)); % Real of impedance ZI124 = imag((A124+B124)/(A124-B124))-km*.003175/.01603; %Imaginary of impedance Zabs124 = abs(ZR124+ZI124*i); %ABS of impedance

% Pressure Errors and SPL Calculations

P1_124 = A124*exp(-i*km*x1)+B124*exp(i*kp*x1); P1magmeas_124 = sqrt((real(X1(f)))^2+(imag(X1(f)))^2); %measure value P1magtheor_124 = sqrt((real(P1_124))^2+(imag(P1_124))^2); %theoretical value calculate from A & B P1error_124 = 100*(P1magtheor_124-P1magmeas_124)/P1magmeas_124; P1SPL_124 = 20*log10(P1magmeas_124/pref);

P2_124 = A124*exp(-i*km*x2)+B124*exp(i*kp*x2); P2magmeas_124 = sqrt((real(Y1(f)))^2+(imag(Y1(f)))^2); P2magtheor_124 = sqrt((real(P2_124))^2+(imag(P2_124))^2); P2error_124 = 100*(P2magtheor_124-P2magmeas_124)/P2magmeas_124; P2SPL_124 = 20*log10(P2magmeas_124/pref);

P3_124 = A124*exp(-i*km*x3)+B124*exp(i*kp*x3); P3magmeas_124 = sqrt((real(Z1(f)))^2+(imag(Z1(f)))^2); P3magtheor_124 = sqrt((real(P3_124))^2+(imag(P3_124))^2); P3error_124 = 100*(P3magtheor_124-P3magmeas_124)/P3magmeas_124;

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P3SPL_124 = 20*log10(P3magmeas_124/pref);

P4_124 = A124*exp(-i*km*x4)+B124*exp(i*kp*x4); P4magmeas_124 = sqrt((real(U1(f)))^2+(imag(U1(f)))^2); P4magtheor_124 = sqrt((real(P4_124))^2+(imag(P4_124))^2); P4error_124 = 100*(P4magtheor_124-P4magmeas_124)/P4magmeas_124; P4SPL_124 = 20*log10(P4magmeas_124/pref);

%% Calculate wave coefficients using transducers 1, 3, 4 ExpTerms134 = [exp(-1i*kp*x1) exp(1i*km*x1); exp(-1i*kp*x3) exp(1i*km*x3); exp(-1i*kp*x4) exp(1i*km*x4)];

Pressures134 = [P11; P31; P41];

x11_134 = pinv(ExpTerms134)*Pressures134;

A134 = x11_134(1); B134 = x11_134(2);

% Impedance calculations SWR134 = (abs(A134)+abs(B134))/(abs(A134)-abs(B134)); % Standing wave ratio R_S134 = (SWR134-1)/(SWR134+1); % Reflection coefficient

ZR134 = real((A134+B134)/(A134-B134)); % Real of impedance ZI134 = imag((A134+B134)/(A134-B134))-km*.003175/.01603; %Imaginary of impedance Zabs134 = abs(ZR134+ZI134*i); %ABS of impedance

% Pressure Errors and SPL Calculations

P1_134 = A134*exp(-i*km*x1)+B134*exp(i*kp*x1); P1magmeas_134 = sqrt((real(X1(f)))^2+(imag(X1(f)))^2); %measure value P1magtheor_134 = sqrt((real(P1_134))^2+(imag(P1_134))^2); %theoretical value calculate from A & B P1error_134 = 100*(P1magtheor_134-P1magmeas_134)/P1magmeas_134; P1SPL_134 = 20*log10(P1magmeas_134/pref);

P2_134 = A134*exp(-i*km*x2)+B134*exp(i*kp*x2); P2magmeas_134 = sqrt((real(Y1(f)))^2+(imag(Y1(f)))^2); P2magtheor_134 = sqrt((real(P2_134))^2+(imag(P2_134))^2); P2error_134 = 100*(P2magtheor_134-P2magmeas_134)/P2magmeas_134; P2SPL_134 = 20*log10(P2magmeas_134/pref);

P3_134 = A134*exp(-i*km*x3)+B134*exp(i*kp*x3); P3magmeas_134 = sqrt((real(Z1(f)))^2+(imag(Z1(f)))^2); P3magtheor_134 = sqrt((real(P3_134))^2+(imag(P3_134))^2); P3error_134 = 100*(P3magtheor_134-P3magmeas_134)/P3magmeas_134; P3SPL_134 = 20*log10(P3magmeas_134/pref);

P4_134 = A134*exp(-i*km*x4)+B134*exp(i*kp*x4); P4magmeas_134 = sqrt((real(U1(f)))^2+(imag(U1(f)))^2);

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P4magtheor_134 = sqrt((real(P4_134))^2+(imag(P4_134))^2); P4error_134 = 100*(P4magtheor_134-P4magmeas_134)/P4magmeas_134; P4SPL_134 = 20*log10(P4magmeas_134/pref);

%% Calculate wave coefficients using transducers 2, 3, 4 ExpTerms234 = [exp(-1i*kp*x2) exp(1i*km*x2); exp(-1i*kp*x3) exp(1i*km*x3); exp(-1i*kp*x4) exp(1i*km*x4)];

Pressures234 = [P21; P31; P41];

x11_234 = pinv(ExpTerms234)*Pressures234;

A234 = x11_234(1); B234 = x11_234(2);

% Impedance calculations SWR234 = (abs(A234)+abs(B234))/(abs(A234)-abs(B234)); % Standing wave ratio R_S234 = (SWR234-1)/(SWR234+1); % Reflection coefficient

ZR234 = real((A234+B234)/(A234-B234)); % Real of impedance ZI234 = imag((A234+B234)/(A234-B234))-km*.003175/.01603; %Imaginary of impedance Zabs234 = abs(ZR234+ZI234*i); %ABS of impedance

% Pressure Errors and SPL Calculations

P1_234 = A234*exp(-i*km*x1)+B234*exp(i*kp*x1); P1magmeas_234 = sqrt((real(X1(f)))^2+(imag(X1(f)))^2); %measure value P1magtheor_234 = sqrt((real(P1_234))^2+(imag(P1_234))^2); %theoretical value calculate from A & B P1error_234 = 100*(P1magtheor_234-P1magmeas_234)/P1magmeas_234; P1SPL_234 = 20*log10(P1magmeas_234/pref);

P2_234 = A234*exp(-i*km*x2)+B234*exp(i*kp*x2); P2magmeas_234 = sqrt((real(Y1(f)))^2+(imag(Y1(f)))^2); P2magtheor_234 = sqrt((real(P2_234))^2+(imag(P2_234))^2); P2error_234 = 100*(P2magtheor_234-P2magmeas_234)/P2magmeas_234; P2SPL_234 = 20*log10(P2magmeas_234/pref);

P3_234 = A234*exp(-i*km*x3)+B234*exp(i*kp*x3); P3magmeas_234 = sqrt((real(Z1(f)))^2+(imag(Z1(f)))^2); P3magtheor_234 = sqrt((real(P3_234))^2+(imag(P3_234))^2); P3error_234 = 100*(P3magtheor_234-P3magmeas_234)/P3magmeas_234; P3SPL_234 = 20*log10(P3magmeas_234/pref);

P4_234 = A234*exp(-i*km*x4)+B234*exp(i*kp*x4); P4magmeas_234 = sqrt((real(U1(f)))^2+(imag(U1(f)))^2); P4magtheor_234 = sqrt((real(P4_234))^2+(imag(P4_234))^2); P4error_234 = 100*(P4magtheor_234-P4magmeas_234)/P4magmeas_234; P4SPL_234 = 20*log10(P4magmeas_234/pref);

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%% Store outputs for each second of data outs(row_num,:) = [R_S123 ZR123 ZI123 Zabs123 P1error_123 P2error_123 P3error_123 P4error_123 R_S124 ZR124 ZI124 Zabs124 P1error_124 P2error_124 P3error_124 P4error_124 R_S134 ZR134 ZI134 Zabs134 P1error_134 P2error_134 P3error_134 P4error_134 R_S234 ZR234 ZI234 Zabs234 P1error_234 P2error_234 P3error_234 P4error_234];

row_num = row_num+1;

end

%Create matrix of average values for each frequency to output all_outs(freq_row,:) = [f mean(outs)]; freq_row = freq_row+1;

clear LVMName1 Raw1 %Clear filename and raw data variables for next iteration

gg = gg+1;

end

%Write data for all frequencies to file for each flow rate filename = strcat( 'ImpedanceLSM_' ,num2str(cell2mat(flow)), 'SLPM_1V.xlsx' ); cn = 0; ji = 2; sheet = num2str(f); xlswrite(filename, all_outs); end

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